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Full text of "Complexity, Emergent Systems, Life and Complex Biological Systems: Complex Systems Theory and Biodynamics"

Complexity and Dynamics 

Complexity Theories, Dynamical Systems 
and Applications to Biology and Sociology 



PDF generated using the open source mwlib toolkit. See http://code.pediapress.com/ for more information. 
PDF generated at: Sat, 10 Apr 2010 14:47:47 UTC 



Contents 

Articles 

Copyright® 20 10 by I.C. Baianu 
Simplicity and Complexity 



2 



Simplicity 2 

Divine simplicity 6 

Occam's razor 9 

Complexity 24 

Nonlinear system 31 

Kolmogorov complexity 37 

Godel's incompleteness theorems 44 

Tarski's undefinability theorem 58 

Model of hierarchical complexity 6 1 

Complexity theory 70 

Complex adaptive system 7 1 

System Theories and Dynamics 77 

System 77 

Causal loop diagram 82 

Phase space 84 

Negative feedback 86 

Information flow diagram 89 

System theory 90 

Systems thinking 101 

System dynamics 106 

Dynamics 113 

Mathematical Biology, Complex Systems Biology 115 

Mathematical biology 115 

Dynamical systems theory 126 

Living systems 131 

Complex Systems Biology (CSB) 132 

Network theory 138 

Cybernetics 141 

Control theory 149 



Genomics 158 

Interactomics 161 

Chaotic Dynamics i 64 

Butterfly effect 164 

Chaos theory 168 

Lorentz attractor 181 

Rossler attractor 185 

List of chaotic maps 194 

Other Applications 197 

Social network 197 

Sociology and complexity science 207 

Sociocybernetics 212 

Systems engineering 214 

Sociobiology 224 

Theoretical biology 230 

Theoretical genetics 240 

Theoretical ecology 250 

Population dynamics 25 1 

Ecology 253 

Systems ecology 287 

Ecological genetics 291 

Molecular evolution 292 

Evolutionary history of life 295 

Modern evolutionary synthesis 325 

Population genetics 334 

Gene flow 344 

Speciation 347 

Natural selection 355 

The Genetical Theory of Natural Selection 369 

Phylogenetics 37 1 

Human evolution 375 

Systems psychology 391 

Systems engineering 396 

Sociotechnical systems theory 406 

Ontology 412 



Notable Complexity Theoreticians 419 

William Ross Ashby 419 

Ludwig von Bertalanffy 422 

Robert Rosen 427 

Claude Shannon 433 

Richard E. Bellman 442 

Brian Goodwin 445 

John von Neumann 447 

Ilya Prigogine 459 

Gregory Bateson 463 

Otto Rossler 468 

References 

Article Sources and Contributors 470 

Image Sources, Licenses and Contributors 479 

Article Licenses 

License 483 



Copyright® 20 10 by I.C. Baianu 



Simplicity and Complexity 



Simplicity 



Simplicity is a more qualitative word connected to simple. It is a property, condition, or quality which things can be 
judged to have. It usually relates to the burden which a thing puts on someone trying to explain or understand it. 
Something which is easy to understand or explain is simple, in contrast to something complicated. In some uses, 
simplicity can be used to imply beauty, purity or clarity. Simplicity may also be used in a negative connotation to 
denote a deficit or insufficiency of nuance or complexity of a thing, relative to what is supposed to be required. 

The concept of simplicity has been related to truth in the field of epistemology. According to Occam's razor, all other 
things being equal, the simplest theory is the most likely to be true. In the context of human lifestyle, simplicity can 
denote freedom from hardship, effort or confusion. Specifically, it can refer to a simple living lifestyle. 

Simplicity is a theme in the Christian religion. According to St. Thomas Aquinas, God is infinitely simple. The 
Roman Catholic and Anglican religious orders of Franciscans also strive after simplicity. Members of the Religious 
Society of Friends (Quakers) practice the Testimony of Simplicity, which is the simplifying of one's life in order to 
focus on things that are most important and disregard or avoid things that are least important. 

In MCS cognition theory, simplicity is the property of a domain which requires very little information to be 
exhaustively described. The opposite of simplicity is complexity. 

Simplicity in the philosophy of science 

Simplicity is a meta-scientific criterion by which to evaluate competing theories. See also Occam's Razor and 
references. The similar concept of Parsimony is also used in philosophy of science, that is the explanation of a 
phenomenon which is the least involved is held to have superior value to a more involved one. 

Simplicity in philosophy 

The definition provided by Stanford Encyclopedia of Philosophy is that "Other things being equal simpler theories 
are better." 

There is a widespread philosophical presumption that simplicity is a theoretical virtue. This presumption that simpler 
theories are preferable appears in many guises. Often it remains implicit; sometimes it is invoked as a primitive, 
self-evident proposition; other times it is elevated to the status of a 'Principle' and labeled as such (for example, the 
'Principle of Parsimony'). However, it is perhaps best known by the name 'Occam's (or Ockham's) Razor.' Simplicity 
principles have been proposed in various forms by theologians, philosophers, and scientists, from ancient through 
medieval to modern times. Thus Aristotle writes in his Posterior Analytics, 

• We may assume the superiority ceteris paribus of the demonstration which derives from fewer postulates or 
hypotheses. [Aristotle, Posterior Analytics, transl. McKeon, [1963, p. 150].] 

Moving to the medieval period, Aquinas writes 

• If a thing can be done adequately by means of one, it is superfluous to do it by means of several; for we observe 
that nature does not employ two instruments where one suffices (Aquinas 1945, p. 129). 

Kant — in the Critique of Pure Reason — supports the maxim that "rudiments or principles must not be unnecessarily 
multiplied (entia praeter necessitatem non esse multiplicanda)" and argues that this is a regulative idea of pure reason 
which underlies scientists' theorizing about nature (Kant 1950, pp. 538—9). Both Galileo and Newton accepted 



Simplicity 



versions of Occam's Razor. Indeed Newton includes a principle of parsimony as one of his three 'Rules of Reasoning 
in Philosophy' at the beginning of Book III of Principia Mathematica. 

• Rule I: We are to admit no more causes of natural things than such as are both true and sufficient to explain their 
appearances. 

Newton goes on to remark that "Nature is pleased with simplicity, and affects not the pomp of superfluous causes" 
(Newton 1972, p. 398). Galileo, in the course of making a detailed comparison of the Ptolemaic and Copernican 
models of the solar system, maintains that "Nature does not multiply things unnecessarily; that she makes use of the 
easiest and simplest means for producing her effects; that she does nothing in vain, and the like" (Galileo 1962, p. 
397). Nor are scientific advocates of simplicity principles restricted to the ranks of physicists and astronomers. Here 
is the chemist Lavoisier writing in the late 18th Century 

• If all of chemistry can be explained in a satisfactory manner without the help of phlogiston, that is enough to 
render it infinitely likely that the principle does not exist, that it is a hypothetical substance, a gratuitous 
supposition. It is, after all, a principle of logic not to multiply entities unnecessarily (Lavoisier 1862, pp. 623—4). 

Compare this to the following passage from Einstein, writing 150 years later. 

• The grand aim of all science. . .is to cover the greatest possible number of empirical facts by logical deductions 
from the smallest possible number of hypotheses or axioms (Einstein, quoted in Nash 1963, p. 173). 

Editors of a recent volume on simplicity sent out surveys to 25 recent Nobel laureates in economics. Almost all 
replied that simplicity played a role in their research, and that simplicity is a desirable feature of economic theories 
(Zellner et al. 2001, p.2). 

Within philosophy, Occam's Razor (OR) is often wielded against metaphysical theories which involve allegedly 
superfluous ontological apparatus. Thus materialists about the mind may use OR against dualism, on the grounds 
that dualism postulates an extra ontological category for mental phenomena. Similarly, nominalists about abstract 
objects may use OR against their platonist opponents, taking them to task for committing to an uncountably vast 
realm of abstract mathematical entities. The aim of appeals to simplicity in such contexts seem to be more about 
shifting the burden of proof, and less about refuting the less simple theory outright. 

The philosophical issues surrounding the notion of simplicity are numerous and somewhat tangled. The topic has 
been studied in piecemeal fashion by scientists, philosophers, and statisticians. The apparent familiarity of the notion 
of simplicity means that it is often left unanalyzed, while its vagueness and multiplicity of meanings contributes to 
the challenge of pinning the notion down precisely. [Compare Poincare's remark that "simplicity is a vague notion" 
and "everyone calls simple what he finds easy to understand, according to his habits." (quoted in Gauch [2003, p. 
275]).] A distinction is often made between two fundamentally distinct senses of simplicity: syntactic simplicity 
(roughly, the number and complexity of hypotheses), and ontological simplicity (roughly, the number and 
complexity of things postulated). [N.B. some philosophers use the term 'semantic simplicity' for this second 
category, e.g. Sober [2001, p. 14].] These two facets of simplicity are often referred to as elegance and parsimony 
respectively. For the purposes of the present overview we shall follow this usage and reserve 'parsimony' specifically 
for simplicity in the ontological sense. However, the terms 'parsimony' and 'simplicity' are used virtually 
interchangeably in much of the philosophical literature. 

Philosophical interest in these two notions of simplicity may be organized around answers to three basic questions; 
(i) How is simplicity to be defined? [Definition] (ii) What is the role of simplicity principles in different areas of 
inquiry? [Usage] (iii) Is there a rational justification for such simplicity principles? [Justification] 

Answering the definitional question, (i), is more straightforward for parsimony than for elegance. Conversely, more 
progress on the issue, (iii), of rational justification has been made for elegance than for parsimony. The above 
questions can be raised for simplicity principles both within philosophy itself and in application to other areas of 
theorizing, especially empirical science. 



Simplicity 



With respect to question (ii), there is an important distinction to be made between two sorts of simplicity principle. 
Occam's Razor may be formulated as an epistemic principle: if theory T is simpler than theory T*, then it is rational 
(other things being equal) to believe T rather than T*. Or it may be formulated as a methodological principle: if T is 
simpler than T* then it is rational to adopt T as one's working theory for scientific purposes. These two conceptions 
of Occam's Razor require different sorts of justification in answer to question (iii). 

In analyzing simplicity, it can be difficult to keep its two facets — elegance and parsimony — apart. Principles such as 
Occam's Razor are frequently stated in a way which is ambiguous between the two notions, for example, "Don't 
multiply postulations beyond necessity." Here it is unclear whether 'postulation' refers to the entities being 
postulated, or the hypotheses which are doing the postulating, or both. The first reading corresponds to parsimony, 
the second to elegance. Examples of both sorts of simplicity principle can be found in the quotations given earlier in 
this section. 

While these two facets of simplicity are frequently conflated, it is important to treat them as distinct. One reason for 
doing so is that considerations of parsimony and of elegance typically pull in different directions. Postulating extra 
entities may allow a theory to be formulated more simply, while reducing the ontology of a theory may only be 
possible at the price of making it syntactically more complex. For example the postulation of Neptune, at the time 
not directly observable, allowed the perturbations in the orbits of other observed planets to be explained without 
complicating the laws of celestial mechanics. There is typically a trade-off between ontology and ideology — to use 
the terminology favored by Quine — in which contraction in one domain requires expansion in the other. This points 
to another way of characterizing the elegance/parsimony distinction, in terms of simplicity of theory versus 
simplicity of world respectively. [4] Sober [2001] argues that both these facets of simplicity can be interpreted in 
terms of minimization. In the (atypical) case of theoretically idle entities, both forms of minimization pull in the 
same direction; postulating the existence of such entities makes both our theories (of the world) and the world (as 
represented by our theories) less simple than they might be. 

Quotes 

• "Things should be made as simple as possible, but no simpler." — Albert Einstein (1879—1955) 

• "You can always recognize truth by its beauty and simplicity." — Richard Feynman (1918—1988) 

• "Our lives are frittered away by detail; simplify, simplify." — Henry David Thoreau (1817—1862) 

• "Simplicity divides into tools, which are used by Beorma as Royal Highness." — Duke of Beorma 

• "Simplicity is the ultimate sophistication." — Leonardo da Vinci (1452—1519) 

• "If you can't describe it simply, you can't use it simply." — Anon 

• "Simplicity means the achievement of maximum effect with minimum means." — Koichi Kawana, architect of 
botanical gardens 

• "Perfection is achieved, not when there is nothing more to add, but when there is nothing left to take 
away." — Antoine de Saint Exupery 

• "Simplicity is the direct result of profound thought." — Anon 



Simplicity 



See also 

Complexity 
KISS principle 
Occam's razor 
Simplicity theory 
Testimony of Simplicity 
Voluntary simplicity 
Worse is better 

References 

• Craig, E. Ed. (1998) Routledge Encyclopedia of Philosophy. London, Routledge. simplicity (in Scientific Theory) 
p.780-783 

• Dancy, J. and Ernest Sosa, Ed. (1999) A Companion to Epistemology. Maiden, Massachusetts, Blackwell 

Publishers Inc. simplicity p. 477—479. 

Hi 

• Dowe, D. L., S. Gardner & G. Oppy (2007), "Bayes not Bust! Why Simplicity is no Problem for Bayesians , 

Brit. J. Phil. Sci. [3] , Vol. 58, Dec. 2007, 46pp. [Among other things, this paper compares MML with AIC] 

• Edwards, P., Ed. (1967). The Encyclopedia of Philosophy. New York, The Macmillan Company, simplicity 
p.445-448. 

• Kim, J. a. E. S., Ed. (2000). A Companion to Metaphysics. Oxford, Blackwell Publishers, simplicity, parsimony 
p.46 1-462. 

• Maeda, J., (2006) Laws of Simplicity, MIT Press 

• Newton-Smith, W. H., Ed. (2001). A Companion to the Philosophy of Science. Maiden, Massachusetts, 
Blackwell Publishers Ltd. simplicity p.433— 441. 

• Richmond, Samuel A.(1996)"A Simplification of the Theory of Simplicity", Synthese 107 373-393. 

• Scott, Brian(1996) "Technical Notes on a Theory of Simplicity", Synthese 109 281-289. 

• Sarkar, S. Ed. (2002). The Philosophy of Science — An Encyclopedia. London, Routledge. simplicity 

• Wilson, R. A. a. K, Frank C, (1999). The MIT Encyclopedia of the Cognitive Sciences. Cambridge, 
Massachusetts, The MIT Press, parsimony and simplicity p. 627— 629. 

External links 

Mi 
Stanford Encyclopedia of Philosophy entry 

Stanford SIMPLIcity image retrieval system, 1999. 

Extensive bibliography for simplicity in the philosophy of science 

Franciscan rules of simplicity 

ro] 

Complexity vs. Simplicity 

Beyond Simplicity: Tough Issues For A New Era by Albert J. Fritsch, SJ, PhD 

"On Simple Theories Of A Complex World [10] " by W. V. O. Quine 

Art of Simplicity teachings and writings by Claude R. Sheffield 

r 121 

The Simplicity Cycle is a graphical exploration of the relationship between complexity, goodness and time. 
The 30 Most Satisfying Simple Pleasures Life Has to Offer is a great list of simple pleasures. 



Simplicity 



References 



[I] http://plato.stanford.edu/entries/simplicity 

[2] http://bjps.oxfordjournals.org/cgi/content/abstract/axm033vl 

[3] http://bjps.oxfordjournals.org 

[4] http://plato.stanford.edu/entries/simplicity/ 

[5] http://www-db.stanford.edu/IMAGE/ 

[6] http://occamssword.com/CORE%20READINGS%20IN%20THE%20PHILOSOPHY%200F%20SCIENCE.htm 

[7] http://www.francis.or.kr/ReligiousLife/aimsE.htm/ 

[8] http://samvak.tripod.com/complex.html 

[9] http : // w w w . earthhealing. info/beyond, pdf 

[10] http://sveinbjorn.org/simple_theories_of_a_complex_world 

[II] http://www.mywilliespress.com 

[12] http://www.changethis.com/22.SimplicityCycle 

[13] http://www.marcandangel.com/2008/03/23/the-30-most-satisfying-simple-pleasures-life-has-to-offer/ 



Divine simplicity 



In theology, the doctrine of divine simplicity says that God is without parts. The general idea of divine simplicity 
can be stated in this way: the being of God is identical to the attributes of God. In other words, such characteristics as 
omnipresence, goodness, truth, eternity, etc. are identical to his being, not qualities that make up his being. 

In Christian thought 

In Christian thought, God as a simple being is not divisible; God is simple, not composite, not made up of thing upon 
thing. In other words, the characteristics of God are not parts of God that together make God what he is. Because 
God is simple, his properties are identical with himself, and therefore God does not have goodness, but simply is 
goodness. In Christianity, divine simplicity does not deny that the attributes of God are distinguishable; so that it is 
not a contradiction of the doctrine to say, for example, that God is both just and merciful. In light of this idea, 
Thomas Aquinas for whose system of thought the idea of divine simplicity is important, wrote in Summa Theologiae 
that because God is infinitely simple, he can only appear to the finite mind as though he were infinitely complex. 

When theology follows this doctrine, various modes of simplicity are distinguished by subtraction of various kinds 
of composition from the meaning of terms used to describe God. Thus, in quantitative or spatial terms, God is simple 
as opposed to being made up of pieces: he is present in his entirety everywhere that he is present, if he is present 
anywhere. In terms of essences, God is simple as opposed to being made up of form and matter, or body and soul, or 
mind and act, and so on: if distinctions are made when speaking of God's attributes, they are distinctions of the 
"modes" of God's being, rather than real or essential divisions. And so, in terms of subjects and accidents, as in the 
phrase "goodness of God", divine simplicity allows that there is a conceptual distinction between the person of God 
and the personal attribute of goodness, but the doctrine disallows that God's identity or "character" is dependent upon 
goodness, and at the same time the doctrine dictates that it is impossible to consider the goodness in which God 
participates separately from the goodness which God is in Himself. 

Furthermore, it follows from this doctrine that God's attributes can only be spoken of by analogy — since it is not true 
of any created thing that its properties are its being. Consequently, when Christian Scripture is interpreted according 
to the guide of divine simplicity, when it is said that God is good for example, it is nearer to the actual case that the 
Scriptures speak of a likeness to goodness, in man and in human speech, since God's essence is inexpressible; this 
likeness is nevertheless truly comparable to God who is simply goodness, because man is constructed and composed 
by God "in the image and likeness of God". The doctrine assists then for the interpretation of the Scriptures without 
paradox, when it is said for example that the creation is "very good", and also that "none is good but God 
alone" — since only God is good in himself, while nevertheless man is created in the likeness of goodness (and the 
likeness is necessarily imperfect in man, unless that man is also God). This doctrine also helps keep trinitarianism 



Divine simplicity 

from drifting or morphing into tri theism, which is the belief in three different gods: the persons of God are not parts 
or essential differences, but are rather the way in which the one God exists personally. 

The doctrine has been criticized by some Christian theologians, including Alvin Plantinga, who in his essay Does 
God Have a Nature? calls it "a dark saying indeed." Plantinga's criticism is based on his interpretation of 
Aquinas's discussion of it, from which he concludes that if God is identical with his properties, then God himself is a 
property; and a property is not a Person: and therefore, divine simplicity does not describe the Christian God, 
according to Plantinga. K. Scott Oliphint in turn criticizes Plantinga for overlooking the better expressions of divine 
simplicity, saying that his argument is "admirable" as a critique of the impersonalism of speculative philosophy, but 
"not so valuable" as a criticism of the Christian formulation based on verbal revelation. 

John Cobb and David Ray Griffin argue against this idea of divine simplicity. They take a look at the Perfect Being 
Theology, where God is defined as being impassible. Therefore, if God is unaffected by human actions, then God is 
not sympathetic. Therefore God would not be loving. However, God is considered to be loving, which causes God to 
not be a simple being. 

In Jewish thought 

In Jewish philosophy and in Jewish mysticism Divine Simplicity is addressed via discussion of the attributes 
(mKT'D) of God, particularly by Jewish philosophers within the Muslim sphere of influence such as Saadia Gaon, 
Bahya ibn Paquda, Yehuda Halevi, and Maimonides, as well by Raabad III in Provence. 

Some identify Divine simplicity as a corollary of Divine Creation: "In the beginning God created the heaven and the 
earth" (Genesis 1:1). God, as creator is by definition separate from the universe and thus free of any property (and 
hence an absolute unity); see Negative theology. 

For others, conversely, the axiom of Divine Unity (see Shema Yisrael) informs the understanding of Divine 
Simplicity. Bahya ibn Paquda {Duties of the Heart 1:8 ) points out that God's Oneness is "true oneness" (PINm 
nNOri) as opposed to merely "circumstantial oneness" (ilKm riDpT"). He develops this idea to show that an entity 
which is truly one must be free of properties and thus indescribable - and unlike anything else. (Additionally such an 
entity would be absolutely unsubject to change, as well as utterly independent and the root of everything.) [4] 
The implication - of either approach - is so strong that the two concepts are often presented as synonymous: "God is 
not two or more entities, but a single entity of a oneness even more single and unique than any single thing in 
creation... He cannot be sub-divided into different parts — therefore, it is impossible for Him to be anything other 
than one. It is a positive commandment to know this, for it is written (Deuteronomy 6:4) '...the Lord is our God, the 
Lord is one'." (Maimonides, Mishneh Torah, Mada 1:7 .) 

Despite its apparent simplicity, this concept is recognised as raising many difficulties. In particular, insofar as God's 
simplicity does not allow for any structure — even conceptually — Divine simplicity appears to entail the following 
dichotomy. 

• On the one hand, God is absolutely simple, containing no element of form or structure, as above. 

• On the other hand, it is understood that His essence contains every possible element of perfection: "The First 
Foundation is to believe in the existence of the Creator, blessed be He. This means that there exists a Being that is 
perfect (complete) in all ways and He is the cause of all else that exists." (Maimonides 13 principles of faith, First 
Principle ). 

The resultant paradox is famously articulated by Moshe Chaim Luzzatto (Derekh Hashem 1:1:5 ), describing the 
dichotomy as arising out of our inability to comprehend the idea of absolute unity: 



Divine simplicity 

God's existence is absolutely simple, without combinations or additions of any kind. All perfections are found in Him in a perfectly simple 
manner. However, God does not entail separate domains — even though in truth there exist in God qualities which, within us, are separate. . . 
Indeed the true nature of His essence is that it is a single attribute, (yet) one that intrinsically encompasses everything that could be considered 
perfection. All perfection therefore exists in God, not as something added on to His existence, but as an integral part of His intrinsic identity. . . 
This is a concept that is very far from our ability to grasp and imagine. . . 

The Kabbalists address this paradox by explaining that "God created a spiritual dimension... [through which He] 
interacts with the Universe... It is this dimension which makes it possible for us to speak of God's multifaceted 
relationship to the universe without violating the basic principle of His unity and simplicity" (Aryeh Kaplan, 
Innerspace). The Kabbalistic approach is explained in various Chassidic writings; see for example, Shaar Hayichud, 
below, for a detailed discussion. 

See also: Tzimtzum; Negative theology; Jewish principles of faith; Free will In Jewish thought; Kuzari 

See also 

• Tawhid (the Islamic concept of divine unity) 

External links and references 

• General 

T81 

• Divine Simplicity , Stanford Encyclopedia of Philosophy 

• God and Other Necessary Beings , Stanford Encyclopedia of Philosophy 

• Making Sense of Divine Simplicity (PDF), Jeffrey E. Brower, Purdue University 

• Christian material 

• On Three Problems of Divine Simplicity , Alexander R. Pruss, Georgetown University 

• St. Thomas Aquinas: The Doctrine of Divine Simplicity , Michael Sudduth, Analytic Philosophy of 
Religion 

• Jewish material 

• "Paradoxes", in "The Aryeh Kaplan Reader", Aryeh Kaplan, Artscroll 1983, ISBN 0-89906-174-5 

• "Innerspace", Aryeh Kaplan, Moznaim Pub. Corp. 1990, ISBN 0-9401 18-56-4 

• Understanding God [13] , Ch2. in "The Handbook of Jewish Thought", Aryeh Kaplan, Moznaim 1979, ISBN 

0-940118-49-1 

1141 

• Shaar HaYichud - The Gate of Unity , Dovber Schneuri - A detailed explanation of the paradox of divine 

simplicity. 

• Chovot ha-Levavot 1:8 , Bahya ibn Paquda - Online class , Yaakov Feldman 

References 

[I] Plantinga, Alvin. "Does God Have a Nature?" in Plantinga, Alvin, and James F. Sennett. 1998. The analytic theist: an Alvin Plantinga reader. 
Grand Rapids, Mich: W.B. Eerdmans Pub. Co., 228. ISBN 0802842291 ISBN 9780802842299 

[2] Plantinga, cited in Oliphint, K. Scott. 2006. Reasons [for faith]: philosophy in the service of theology. Phillipsburg, N.J.: P&R Pub. ISBN 

0875526454 ISBN 9780875526454 
[3] http://www.daat.ac.il/daat/mahshevt/hovot/la-2.htm 
[4] http://www.torah.org/learning/spiritual-excellence/classes/doh-l-8.html 
[5] http://www.panix.com/~jjbaker/MadaYHT.html 
[6] http://members.aol.com/LazerA/13yesodos.html 
[7] http://www .daat. ac. il/daat/mahshevt/mekorot/ 1 a-2. htm 
[8] http://plato.stanford.edu/entries/divine-simplicity/ 
[9] http://www.seop.leeds.ac.uk/entries/god-necessary-being/ 
[10] http://web.ics.purdue.edu/~brower/Papers/Making%20Sense%20of%20Divine%20Simplicity.pdf 

[II] http://www.georgetown.edu/faculty/ap85/papers/On3ProblemsOfDivineSimplicity.html 



Divine simplicity 

[12] http://www.homestead.com/philofreligion/files/Thomas3.html 
[13] http://www.aish.com/literacy/concepts/Understanding_God.asp 
[14] http://www.truekabbalah.com/ShaarHaYichud.php 



Occam's razor 



Occam's razor (or Ockham's razor ), is the meta-theoretical principle that "entities must not be multiplied 
beyond necessity" (entia non sunt multiplicanda praeter necessitatem) and the conclusion thereof, that the simplest 
solution is usually the correct one. 

The principle is attributed to 14th-century English logician, theologian and Franciscan friar, William of Ockham. 
Occam's razor may be alternatively phrased as pluralitas non est ponenda sine necessitate ("plurality should not be 

[2] 

posited without necessity") . The principle is often expressed in Latin as the lex parsimoniae (translating to the 
law of parsimony, law of economy or law of succinctness). When competing hypotheses are equal in other 
respects, the principle recommends selection of the hypothesis that introduces the fewest assumptions and postulates 
the fewest entities while still sufficiently answering the question. It is in this sense that Occam's razor is usually 
understood. To quote Isaac Newton, "We are to admit no more causes of natural things than such as are both true and 
sufficient to explain their appearances. Therefore, to the same natural effects we must, so far as possible, assign the 

..[3] 

same causes. 

In science, Occam's razor is used as a heuristic (rule of thumb) to guide scientists in the development of theoretical 
models rather than as an arbiter between published models. In the scientific method, Occam's razor is not 

considered an irrefutable principle of logic, and certainly not a scientific result. 



History 



William Seach (c. 1285—1349) is remembered as an influential nominalist but his popular fame as a great 
logician rests chiefly on the maxim attributed to him and known as Occam's razor: Entia non sunt 
multiplicanda praeter necessitatem or "Entities should not be multiplied unnecessarily." The term razor refers 
to the act of shaving away unnecessary assumptions to get to the simplest explanation. No doubt this maxim 
represents correctly the general tendency of his philosophy, but it has not so far been found in any of his 
writings. His nearest pronouncement seems to be Numquam ponenda est pluralitas sine necessitate [Plurality 
must never be posited without necessity], which occurs in his theological work on the Sentences of Peter 
Lombard (Quaestiones et decisiones in quattuor libros Sententiarum Petri Lombardi (ed. Lugd., 1495), i, dist. 
27, qu. 2, K). In his Summa Totius Logicae, i. 12, Ockham cites the principle of economy, Frustra fit per plura 
quod potest fieri per pauciora [It is futile to do with more things that which can be done with fewer]. 

— Thorburn, 1918 [10] , pp. 352-3; Kneale and Kneale, 1962, p. 243. [11] 



The origins of what has come to be known as Occam's razor are 

traceable to the works of earlier philosophers such as Alhazen ""N i /*■ "\ rf»tVT/% 3^ QOnetfl 




(965-1039), [12] Maimonides (1138-1204), John Duns Scotus ' X?^ ty plurfllUfl 

mo« ,^ ti, a • i mc i->™ a a ■ ♦ n non eft ponmla line naeffitatc ? non 

(1265-1308), Thomas Aquinas (c. 1225-1274), and even Aristotle .ncceiTtUBquarCOCbcatponitpueDi' 

(384-322 BC) (Charlesworth 1956). The term "Ockham's razor" ICIVCUrHmc'ulur.wmotUilliinwlt.na? 

first appeared in 1 852 in the works of Sir William Hamilton, 9th Part of a page from Duns Scotus . book 0rdinatio . 

Baronet (1788—1856), centuries after Ockham's death. Ockham did Pluralitas non est ponenda sine necessitate, i.e. 

not invent this "razor," so its association with him may be due to "Plurality is not to be posited without necessity" 
the frequency and effectiveness with which he used it (Ariew 

1976). Though Ockham stated the principle in various ways, the most popular version was written not by him, but by 
John Ponce of Cork in 1639 (Meyer 1957). 



Occam's razor 10 

The version of the Razor most often found in Ockham's work is Numquam ponenda est pluralitas sine necessitate, 
"Plurality ought never be posited without necessity". 

Justifications 

Aesthetic and practical considerations 

Prior to the 20th century, it was a commonly-held belief that nature itself was simple and that simpler hypotheses 
about nature were thus more likely to be true. This notion was deeply rooted in the aesthetic value simplicity holds 
for human thought and the justifications presented for it often drew from theology. Thomas Aquinas made this 
argument in the 13th century, writing, "If a thing can be done adequately by means of one, it is superfluous to do it 

ri3i 

by means of several; for we observe that nature does not employ two instruments where one suffices." 

The common form of the razor, used to distinguish between equally explanatory hypotheses, can be supported by 
appeals to the practical value of simplicity. Hypotheses exist to give accurate explanations of phenomena, and 
simplicity is a valuable aspect of an explanation because it makes the explanation easier to understand and work 
with. Thus, if two hypotheses are equally accurate and neither appears more probable than the other, the simple one 
is to be preferred over the complicated one, because simplicity is practical. 

Beginning in the 20th century, epistemological justifications based on induction, logic, pragmatism, and probability 
theory have become more popular among philosophers. 

Empirical justification 

One way a theory or a principle could be justified is empirically; that is to say, if simpler theories were to have a 
better record of turning out to be correct than more complex ones, that would corroborate Occam's razor. However, 
Occam's razor is not a theory in the classic sense of being a model that explains physical observations, relying on 
induction; rather, it is a heuristic maxim for choosing among such theories and underlies induction. Justifying such a 
guideline against some hypothetical alternative thus fails on account of invoking circular logic. 

There are many different ways of making inductive inferences from past data concerning the success of different 
theories throughout the history of science, and inferring that "simpler theories are, other things being equal, generally 
better than more complex ones" is just one way of many — which only seems more plausible to us because we are 
already assuming the razor to be true (see e.g. Swinburne 1997 and Williams, Gareth T, 2008). This, however, does 
not exclude legitimate attempts at a deductive justification of the razor (and indeed these are inherent to many of its 
modern derivatives). Failing even that, the razor may be accepted a priori on pragmatist grounds. 

One should note the related concept of overfitting, where excessively complex models are affected by statistical 
noise, whereas simpler models may capture the underlying structure better and may thus have better predictive 
performance. It is, however, often difficult to deduce which part of the data is noise (cf. model selection, test set, 
minimum description length, Bayesian inference, etc.). 



Occam's razor 1 1 

Karl Popper 

Karl Popper argues that a preference for simple theories need not appeal to practical or aesthetic considerations. Our 
preference for simplicity may be justified by his falsifiability criterion: We prefer simpler theories to more complex 
ones "because their empirical content is greater; and because they are better testable" (Popper 1992). In other words, 
a simple theory applies to more cases than a more complex one, and is thus more easily falsifiable. 

Elliott Sober 

The philosopher of science Elliott Sober once argued along the same lines as Popper, tying simplicity with 
"informativeness": The simplest theory is the more informative one, in the sense that less information is required in 
order to answer one's questions (Sober 1975). He has since rejected this account of simplicity, purportedly because it 
fails to provide an epistemic justification for simplicity. He now expresses views to the effect that simplicity 
considerations (and considerations of parsimony in particular) do not count unless they reflect something more 
fundamental. Philosophers, he suggests, may have made the error of hypostatizing simplicity (i.e. endowed it with a 
sui generis existence), when it has meaning only when embedded in a specific context (Sober 1992). If we fail to 
justify simplicity considerations on the basis of the context in which we make use of them, we may have no 
non-circular justification: "just as the question 'why be rational?' may have no non-circular answer, the same may be 
true of the question 'why should simplicity be considered in evaluating the plausibility of hypotheses?'" (Sober 2001) 

Richard Swinburne 

Richard Swinburne argues for simplicity on logical grounds: "...other things being equal... the simplest hypothesis 
proposed as an explanation of phenomena is more likely to be the true one than is any other available hypothesis, 
that its predictions are more likely to be true than those of any other available hypothesis, and that it is an ultimate a 
priori epistemic principle that simplicity is evidence for truth" (Swinburne 1997). 

He maintains that we have an innate bias towards simplicity and that simplicity considerations are part and parcel of 
common sense. Since our choice of theory cannot be determined by data (see Underdetermination and Quine-Duhem 
thesis), we must rely on some criterion to determine which theory to use. Since it is absurd to have no logical method 
by which to settle on one hypothesis amongst an infinite number of equally data-compliant hypotheses, we should 
choose the simplest theory: "...either science is irrational [in the way it judges theories and predictions probable] or 
the principle of simplicity is a fundamental synthetic a priori truth" (Swinburne 1997). 

Applications 

Science and the scientific method 

In science, Occam's razor is used as a heuristic (rule of thumb) to guide scientists in the development of theoretical 
models rather than as an arbiter between published models. In physics, parsimony was an important heuristic in 

the formulation of special relativity by Albert Einstein , the development and application of the principle of 

least action by Pierre Louis Maupertuis and Leonhard Euler, and the development of quantum mechanics by 
Louis de Broglie, Richard Feynman, and Julian Schwinger. In chemistry, Occam's razor is often an 

important heuristic when developing a model of a reaction mechanism. However, while it is useful as a 

heuristic in developing models of reaction mechanisms, it has been shown to fail as a criterion for selecting among 
published models. 

In the scientific method, parsimony is an epistemological, metaphysical or heuristic preference, not an irrefutable 
principle of logic, and certainly not a scientific result. As a logical principle, Occam's razor would demand 

that scientists accept the simplest possible theoretical explanation for existing data. However, science has shown 
repeatedly that future data often supports more complex theories than existing data. Science tends to prefer the 
simplest explanation that is consistent with the data available at a given time, but history shows that these simplest 



Occam's razor 12 

explanations often yield to complexities as new data become available. Science is open to the possibility that 

future experiments might support more complex theories than demanded by current data and is more interested in 
designing experiments to discriminate between competing theories than favoring one theory over another based 
merely on philosophical principles. 

When scientists use the idea of parsimony, it only has meaning in a very specific context of inquiry. A number of 
background assumptions are required for parsimony to connect with plausibility in a particular research problem. 
The reasonableness of parsimony in one research context may have nothing to do with its reasonableness in another. 
It is a mistake to think that there is a single global principle that spans diverse subject matter. 

As a methodological principle, the demand for simplicity suggested by Occam's razor cannot be generally sustained. 
Occam's razor cannot help toward a rational decision between competing explanations of the same empirical facts. 
One problem in formulating an explicit general principle is that complexity and simplicity are perspective notions 
whose meaning depends on the context of application and the user's prior understanding. In the absence of an 
objective criterion for simplicity and complexity, Occam's razor itself does not support an objective epistemology. 

The problem of deciding between competing explanations for empirical facts cannot be solved by formal tools. 
Simplicity principles can be useful heuristics in formulating hypotheses, but they do not make a contribution to the 
selection of theories. A theory that is compatible with one person's world view will be considered simple, clear, 
logical, and evident, whereas what is contrary to that world view will quickly be rejected as an overly complex 
explanation with senseless additional hypotheses. Occam's razor, in this way, becomes a "mirror of prejudice." 

It has been suggested that Occam's razor is a widely accepted example of extraevidential consideration, even though 
it is entirely a metaphysical assumption. There is little empirical evidence that the world is actually simple or that 
simple accounts are more likely than complex ones to be true. 

Most of the time, Occam's razor is a conservative tool, cutting out crazy, complicated constructions and assuring that 
hypotheses are grounded in the science of the day, thus yielding 'normal' science: models of explanation and 
prediction. There are, however, notable exceptions where Occam's razor turns a conservative scientist into a reluctant 
revolutionary. For example, Max Planck interpolated between the Wien and Jeans radiation laws used an Occam's 
razor logic to formulate the quantum hypothesis, and even resisting that hypothesis as it became more obvious that it 

<• [5] 

was correct. 

ro] 

However, on many occasions Occam's razor has stifled or delayed scientific progress. For example, appeals to 
simplicity were used to deny the phenomena of meteorites, ball lightning, continental drift, and reverse transcriptase. 
It originally rejected DNA as the carrier of genetic information in favor of proteins, since proteins provided the 
simpler explanation. Theories that reach far beyond the available data are rare, but general relativity provides one 
example. 

In hindsight, one can argue that it is simpler to consider DNA as the carrier of genetic information, because it uses a 
smaller number of building blocks (four nitrogenous bases). However, during the time that proteins were the favored 
genetic medium, it seemed like a more complex hypothesis to confer genetic information in DNA rather than 
proteins. 

One can also argue (also in hindsight) for atomic building blocks for matter, because it provides a simpler 
explanation for the observed reversibility of both mixing and chemical reactions as simple separation and 
re-arrangements of the atomic building blocks. However, at the time, the atomic theory was considered more 
complex because it inferred the existence of invisible particles which had not been directly detected. Ernst Mach and 

the logical positivists rejected the atomic theory of John Dalton, until the reality of atoms was more evident in 

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Brownian motion, as explained by Albert Einstein. 

In the same way, hindsight argues that postulating the aether is more complex than transmission of light through a 
vacuum. However, at the time, all known waves propagated through a physical medium, and it seemed simpler to 
postulate the existence of a medium rather than theorize about wave propagation without a medium. Likewise, 



Occam's razor 13 

Newton's idea of light particles seemed simpler than Young's idea of waves, so many favored it; however in this 
case, as it turned out, neither the wave- nor the particle-explanation alone suffices, since light behaves like waves as 
well as like particles (wave— particle duality). 

Three axioms presupposed by the scientific method are realism (the existence of objective reality), the existence of 
observable natural laws, and the constancy of observable natural law. Rather than depend on provability of these 
axioms, science depends on the fact that they have not been objectively falsified. Occam's razor and parsimony 
support, but do not prove these general axioms of science. The general principle of science is that theories (or 
models) of natural law must be consistent with repeatable experimental observations. This ultimate arbiter (selection 
criterion) rests upon the axioms mentioned above. 

There are many examples where Occam's razor would have picked the wrong theory given the available data. 
Simplicity principles are useful philosophical preferences for choosing a more likely theory from among several 
possibilities that are each consistent with available data. However, anyone invoking Occam's razor to support a 
model should be aware that additional data may well falsify the model currently favored by Occam's razor. One 
accurate observation of a white crow falsifies the theory that "all crows are black". Likewise, a single instance of 
Occam s razor picking a wrong theory falsifies the razor as a general principle . Note however that this only 
applies if the razor is meant to pick the correct theory for all time; if this is not the case, and it is only applied to pick 
the simplest theory which fits all the currently known data and it is understood that, should new data arise, the razor 
will have to be reapplied, then the principle keeps its validity. 

If multiple models of natural law make exactly the same testable predictions, they are equivalent and there is no need 
for parsimony to choose one that is preferred. For example, Newtonian, Hamiltonian, and Lagrangian classical 
mechanics are equivalent. Physicists have no interest in using Occam's razor to say the other two are wrong. 
Likewise, there is no demand for simplicity principles to arbitrate between wave and matrix formulations of quantum 
mechanics. Science often does not demand arbitration or selection criteria between models which make the same 
testable predictions. 

Biology 

Biologists or philosophers of biology use Occam's razor in either of two contexts both in evolutionary biology: the 
units of selection controversy and systematics. George C. Williams in his book Adaptation and Natural Selection 
(1966) argues that the best way to explain altruism among animals is based on low level (i.e. individual) selection as 
opposed to high level group selection. Altruism is defined as behavior that is beneficial to the group but not to the 
individual, and group selection is thought by some to be the evolutionary mechanism that selects for altruistic traits. 
Others posit individual selection as the mechanism which explains altruism solely in terms of the behaviors of 
individual organisms acting in their own self interest without regard to the group. The basis for Williams's contention 
is that of the two, individual selection is the more parsimonious theory. In doing so he is invoking a variant of 
Occam's razor known as Lloyd Morgan's Canon: "In no case is an animal activity to be interpreted in terms of higher 
psychological processes, if it can be fairly interpreted in terms of processes which stand lower in the scale of 
psychological evolution and development" (Morgan 1903). 

However, more recent biological analyses, such as Richard Dawkins's The Selfish Gene, have contended that 
Williams's view is not the simplest and most basic. Dawkins argues the way evolution works is that the genes that 
are propagated in most copies will end up determining the development of that particular species, i.e., natural 
selection turns out to select specific genes, and this is really the fundamental underlying principle, that automatically 
gives individual and group selection as emergent features of evolution. 

Zoology provides an example. Muskoxen, when threatened by wolves, will form a circle with the males on the 
outside and the females and young on the inside. This as an example of a behavior by the males that seems to be 
altruistic. The behavior is disadvantageous to them individually but beneficial to the group as a whole and was thus 
seen by some to support the group selection theory. 



Occam's razor 14 

However, a much better explanation immediately offers itself once one considers that natural selection works on 
genes. If the male musk ox runs off, leaving his offspring to the wolves, his genes will not be propagated. If however 
he takes up the fight his genes will live on in his offspring. And thus the "stay-and-fight" gene prevails. This is an 
example of kin selection. An underlying general principle thus offers a much simpler explanation, without retreating 
to special principles as group selection. 

Systematics is the branch of biology that attempts to establish genealogical relationships among organisms. It is also 
concerned with their classification. There are three primary camps in systematics; cladists, pheneticists, and 
evolutionary taxonomists. The cladists hold that genealogy alone should determine classification and pheneticists 
contend that similarity over propinquity of descent is the determining criterion while evolutionary taxonomists claim 
that both genealogy and similarity count in classification. 

It is among the cladists that Occam's razor is to be found, although their term for it is cladistic parsimony. Cladistic 
parsimony (or maximum parsimony) is a method of phylogenetic inference in the construction of cladograms. 
Cladograms are branching, tree-like structures used to represent lines of descent based on one or more evolutionary 
change(s). Cladistic parsimony is used to support the hypothesis(es) that require the fewest evolutionary changes. 
For some types of tree, it will consistently produce the wrong results regardless of how much data is collected (this is 
called long branch attraction). For a full treatment of cladistic parsimony, see Elliott Sober's Reconstructing the Past: 
Parsimony, Evolution, and Inference (1988). For a discussion of both uses of Occam's razor in Biology see Elliott 
Sober's article Let's Razor Ockham's Razor (1990). 

Other methods for inferring evolutionary relationships use parsimony in a more traditional way. Likelihood methods 
for phylogeny use parsimony as they do for all likelihood tests, with hypotheses requiring few differing parameters 
(i.e., numbers of different rates of character change or different frequencies of character state transitions) being 
treated as null hypotheses relative to hypotheses requiring many differing parameters. Thus, complex hypotheses 
must predict data much better than do simple hypotheses before researchers reject the simple hypotheses. Recent 
advances employ information theory, a close cousin of likelihood, which uses Occam's Razor in the same way. 

Francis Crick has commented on potential limitations of Occam's razor in biology. He advances the argument that 
because biological systems are the products of (an on-going) natural selection, the mechanisms are not necessarily 
optimal in an obvious sense. He cautions: "While Ockham's razor is a useful tool in the physical sciences, it can be a 
very dangerous implement in biology. It is thus very rash to use simplicity and elegance as a guide in biological 
research." 

Medicine 

When discussing Occam's razor in contemporary medicine, doctors and philosophers of medicine speak of diagnostic 
parsimony. Diagnostic parsimony advocates that when diagnosing a given injury, ailment, illness, or disease a doctor 
should strive to look for the fewest possible causes that will account for all the symptoms. This philosophy is one of 
several demonstrated in the popular medical adage "when you hear hoofbeats, think horses, not zebras". While 
diagnostic parsimony might often be beneficial, credence should also be given to the counter-argument modernly 
known as Hickam's dictum, which succinctly states that "patients can have as many diseases as they damn well 
please". It is often statistically more likely that a patient has several common diseases, rather than having a single 
rarer disease which explains their myriad symptoms. Also, independently of statistical likelihood, some patients do 
in fact turn out to have multiple diseases, which by common sense nullifies the approach of insisting to explain any 
given collection of symptoms with one disease. These misgivings emerge from simple probability theory — which is 
already taken into account in many modern variations of the razor — and from the fact that the loss function is much 
greater in medicine than in most of general science. Because misdiagnosis can result in the loss of a person's health 
and potentially life, it is considered better to test and pursue all reasonable theories even if there is some theory that 
appears the most likely. 



Occam's razor 15 

Diagnostic parsimony and the counter-balance it finds in Hickam's dictum have very important implications in 
medical practice. Any set of symptoms could be indicative of a range of possible diseases and disease combinations; 
though at no point is a diagnosis rejected or accepted just on the basis of one disease appearing more likely than 
another, the continuous flow of hypothesis formulation, testing and modification benefits greatly from estimates 
regarding which diseases (or sets of diseases) are relatively more likely to be responsible for a set of symptoms, 
given the patient's environment, habits, medical history and so on. For example, if a hypothetical patient's 
immediately apparent symptoms include fatigue and cirrhosis and they test negative for Hepatitis C, their doctor 
might formulate a working hypothesis that the cirrhosis was caused by their drinking problem, and then seek 
symptoms and perform tests to formulate and rule out hypotheses as to what has been causing the fatigue; but if the 
doctor were to further discover that the patient's breath inexplicably smells of garlic and they are suffering from 
pulmonary edema, they might decide to test for the relatively rare condition of Selenium poisoning. 

Prior to effective anti-retroviral therapy for HIV it was frequently stated that the most obvious implication of 
Occam's razor, that of cutting down the number of postulated diseases to a minimum, does not apply to patients with 
AIDS, as they frequently did have multiple infectious processes going on at the same time. While the probability of 
multiple diseases being higher certainly reduces the degree to which this kind of analysis is useful, it does not go all 
the way to invalidating it altogether; even in such a patient, it would make more sense to first test a theory 
postulating three diseases to be the cause of the symptoms than a theory postulating seven. 

Religion 

In the philosophy of religion, Occam's razor is sometimes applied to the existence of God; if the concept of God does 
not help to explain the universe, it is argued, God is irrelevant and should be cut away (Schmitt 2005). It is argued to 
imply that, in the absence of compelling reasons to believe in God, disbelief should be preferred. Such arguments are 
based on the assertion that belief in God requires more complex assumptions to explain the universe than non-belief. 

The history of theistic thought has produced many arguments attempting to show that this is not the case — that the 
difficulties encountered by a theory without God are equal to or greater than those encountered by a theory 
postulating one. The cosmological argument, for example, states that the universe must be the result of a "first cause" 
and that that first cause can be thought of as God. Similarly, the teleological argument credits the appearance of 
design and order in the universe to supernatural intelligence. Many people believe in miracles or have what they call 
religious experiences, and creationists consider divine design to be more believable than naturalistic explanations for 
the diversity and history of life on earth. 

Many scientists generally do not accept these arguments, and prefer to rely on explanations that deal with the same 
phenomena within the confines of existing scientific models. Among leading scientists defined as members of the 

National Academy of Sciences, in the United States, 72.2% expressed disbelief and 93% expressed disbelief or doubt 

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in the existence of a personal god in a survey conducted in 1998 (an ongoing survey being conducted by Elaine 

Ecklund of Rice University since 2004 indicates that this figure drops to as low as 38% when social scientists are 

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included and the definition of "God" is expanded to allow a non-personal god as per Pantheism or Deism). The 

typical scientific view challenges the validity of the teleological argument by the effects of emergence, leading to the 

creation-evolution controversy; likewise, religious experiences have naturalistic explanations in the psychology of 

religion. Other theistic arguments, such as the argument from miracles, are sometimes pejoratively said to be arguing 

for a mere God of the gaps; whether or not God actually works miracles, any explanation that "God did it" must fit 

the facts and make accurate predictions better than more parsimonious guesses like "something did it", or else 

Occam's razor still cuts God out. 

Rather than argue for the necessity of God, some theists consider their belief to be based on grounds independent of, 
or prior to, reason, making Occam's razor irrelevant. This was the stance of S0ren Kierkegaard, who viewed belief in 
God as a leap of faith which sometimes directly opposed reason (McDonald 2005); this is also the same basic view 
of Clarkian Presuppositional apologetics, with the exception that Clark never thought the leap of faith was contrary 



Occam's razor 16 

to reason. (See also: Fideism). In a different vein, Alvin Plantinga and others have argued for reformed 
epistemology, the view that God's existence can properly be assumed as part of a Christian's epistemological 
structure. (See also: Basic beliefs). Yet another school of thought, Van Tillian Presuppositional apologetics, claims 
that God's existence is the transcendentally necessary prior condition to the intelligibility of all human experience 
and thought. In other words, proponents of this view hold that there is no other viable option to ultimately explain 
any fact of human experience or knowledge, let alone a simpler one. It can be noted that these views tend to relate 
only to the Christian religion and non-Western understandings of God are not considered here. 

Considering that the razor is often wielded as an argument against theism, it is somewhat ironic that Ockham himself 
was a theist. He considered some Christian sources to be valid sources of factual data, equal to both logic and sense 
perception. He wrote, "No plurality should be assumed unless it can be proved (a) by reason, or (b) by experience, or 
(c) by some infallible authority"; referring in the last clause "to the Bible, the Saints and certain pronouncements of 
the Church" (Hoffmann 1997). In Ockham's view, an explanation which does not harmonize with reason, experience 
or the aforementioned sources cannot be considered valid. 

Philosophy of mind 

Probably the first person to make use of the principle was Ockham himself. He writes "The source of many errors in 
philosophy is the claim that a distinct signified thing always corresponds to a distinct word in such a way that there 
are as many distinct entities being signified as there are distinct names or words doing the signifying." (Summula 
Philosophiae Naturalis III, chap. 7, see also Summa Totus Logicae Bk I, C.51). We are apt to suppose that a word 
like "paternity" signifies some "distinct entity", because we suppose that each distinct word signifies a distinct entity. 
This leads to all sorts of absurdities, such as "a column is to the right by to-the-rightness", "God is creating by 
creation, is good by goodness, is just by justice, is powerful by power", "an accident inheres by inherence", "a 
subject is subjected by subjection", "a suitable thing is suitable by suitability", "a chimera is nothing by nothingness", 
"a blind thing is blind by blindness", " a body is mobile by mobility". We should say instead that a man is a father 
because he has a son (Summa C.51). 

Another application of the principle is to be found in the work of George Berkeley (1685—1753). Berkeley was an 
idealist who believed that all of reality could be explained in terms of the mind alone. He famously invoked Occam's 
razor against Idealism's metaphysical competitor, materialism, claiming that matter was not required by his 
metaphysic and was thus eliminable. 

In the 20th century Philosophy of Mind, Occam's razor found a champion in J. J. C. Smart, who in his article 
"Sensations and Brain Processes" (1959) claimed Occam's razor as the basis for his preference of the mind-brain 
identity theory over mind body dualism. Dualists claim that there are two kinds of substances in the universe: 
physical (including the body) and mental, which is nonphysical. In contrast identity theorists claim that everything is 
physical, including consciousness, and that there is nothing nonphysical. The basis for the materialist claim is that of 
the two competing theories, dualism and mind-brain identity, the identity theory is the simpler since it commits to 
fewer entities. Smart was criticized for his use of the razor and ultimately retracted his advocacy of it in this context. 

Paul Churchland (1984) cites Occam's razor as the first line of attack against dualism, but admits that by itself it is 
inconclusive. The deciding factor for Churchland is the greater explanatory prowess of a materialist position in the 
Philosophy of Mind as informed by findings in neurobiology. 

Dale Jacquette (1994) claims that Occam's razor is the rationale behind eliminativism and reductionism in the 
philosophy of mind. Eliminativism is the thesis that the ontology of folk psychology including such entities as 
"pain", "joy", "desire", "fear", etc., are eliminable in favor of an ontology of a completed neuroscience. 



Occam's razor 17 

Probability theory and statistics 

One intuitive justification of Occam's Razor's admonition against unnecessary hypotheses is a direct result of basic 
probability theory. By definition, all assumptions introduce possibilities for error; If an assumption does not improve 
the accuracy of a theory, its only effect is to increase the probability that the overall theory is wrong. 

There are various papers in scholarly journals deriving formal versions of Occam's razor from probability theory and 
applying it in statistical inference, and also of various criteria for penalizing complexity in statistical inference. 
Recent papers have suggested a connection between Occam's razor and Kolmogorov complexity. 

One of the problems with the original formulation of the principle is that it only applies to models with the same 
explanatory power (i.e. prefer the simplest of equally good models). A more general form of Occam's razor can be 
derived from Bayesian model comparison and Bayes factors, which can be used to compare models that don't fit the 
data equally well. These methods can sometimes optimally balance the complexity and power of a model. Generally 
the exact Ockham factor is intractable but approximations such as Akaike Information Criterion, Bayesian 
Information Criterion, Variational Bayes, False discovery rate and Laplace approximation are used. Many artificial 
intelligence researchers are now employing such techniques. 

William H. Jefferys and James O. Berger (1991) generalise and quantify the original formulation's "assumptions" 
concept as the degree to which a proposition is unnecessarily accommodating to possible observable data. The model 
they propose balances the precision of a theory's predictions against their sharpness; theories which sharply made 
their correct predictions are preferred over theories which would have accommodated a wide range of other possible 
results. This, again, reflects the mathematical relationship between key concepts in Bayesian inference (namely 
marginal probability, conditional probability and posterior probability). 

The statistical view leads to a more rigorous formulation of the razor than previous philosophical discussions. In 
particular, it shows that 'simplicity' must first be defined in some way before the razor may be used, and that this 
definition will always be subjective. For example, in the Kolmogorov-Chaitin Minimum description length 
approach, the subject must pick a Turing machine whose operations describe the basic operations believed to 
represent 'simplicity' by the subject. However one could always choose a Turing machine with a simple operation 
that happened to construct one's entire theory and would hence score highly under the razor. This has led to two 
opposing views of the objectivity of Occam's razor. 

Subjective razor 

The Turing machine can be thought of as embodying a Bayesian prior belief over the space of rival theories. Hence 
Occam's razor is not an objective comparison method, and merely reflects the subject's prior beliefs. One's choice of 
exactly which razor to use is culturally relative. 

Objective razor 

The minimum instruction set of a Universal Turing machine requires approximately the same length description 
across different formulations, and is small compared to the Kolmogorov complexity of most practical theories. 
Marcus Hutter has used this consistency to define a "natural" Turing machine of small size as the proper basis for 
excluding arbitrarily complex instruction sets in the formulation of razors. Describing the program for the universal 
program as the "hypothesis", and the representation of the evidence as program data, it has been formally proven 

under ZF that "the sum of the log universal probability of the model plus the log of the probability of the data given 

T271 
the model should be minimized." 

One possible conclusion from mixing the concepts of Kolmogorov complexity and Occam's Razor is that an ideal 
data compressor would also be a scientific explanation/formulation generator. Some attempts have been made to 
re-derive known laws from considerations of simplicity or compressibility. 

According to Jiirgen Schmidhuber, the appropriate mathematical theory of Occam's razor already exists, namely, 
Ray Solomonoffs theory of optimal inductive inference and its extensions 



Occam's razor 18 

Variations 

The principle is most often expressed as Entia non sunt multiplicanda praeter necessitatem, or "Entities should not 
be multiplied beyond necessity", but this sentence was written by later authors and is not found in Ockham's 
surviving writings. This also applies to non est ponenda pluritas sine necessitate, which translates literally into 
English as "pluralities ought not be posited without necessity". It has inspired numerous expressions including 
"parsimony of postulates", the "principle of simplicity", the "KISS principle" (Keep It Simple, Stupid). 

Other common restatements are: 

Entities are not to be multiplied without necessity, 
and 

The simplest answer is usually the correct answer. 

A restatement of Occam's razor, in more formal terms, is provided by information theory in the form of minimum 
message length (MML). Tests of Occam's razor on decision tree models which initially appeared critical have been 
shown to actually work fine when re- visited using MML. Other criticisms of Occam's razor and MML (e.g., a binary 
cut-point segmentation problem) have again been rectified when — crucially — an inefficient coding scheme is made 
more efficient. 

"When deciding between two models which make equivalent predictions, choose the simpler one," makes the point 
that a simpler model that doesn't make equivalent predictions is not among the models that this criterion applies to in 
the first place. 

Leonardo da Vinci (1452—1519) lived after Ockham's time and has a variant of Occam's razor. His variant 
short-circuits the need for sophistication by equating it to simplicity. 

Simplicity is the ultimate sophistication. 
Another related quote is attributed to Albert Einstein 

Make everything as simple as possible, but not simpler. 
Occam's razor is now usually stated as follows: 

Of two equivalent theories or explanations, all other things being equal, the simpler one is to be preferred. 

As this is ambiguous, Isaac Newton's version may be better: 

We are to admit no more causes of natural things than such as are both true and sufficient to explain their 
appearances. 

In the spirit of Occam's razor itself, the rule is sometimes stated as: 

The simplest explanation is usually the best. 
Another common statement of it is: 

The simplest explanation that covers all the facts is usually the best. 

Controversial aspects of the Razor 

Occam's razor is not an embargo against the positing of any kind of entity, or a recommendation of the simplest 

[331 
theory come what may (note that simplest theory is something like "only I exist" or "nothing exists"). 

The other things in question are the evidential support for the theory. Therefore, according to the principle, a 
simpler but less correct theory should not be preferred over a more complex but more correct one. It is this fact 
which gives the lie to the common misinterpretation of Occam's Razor that "the simplest" one is usually the correct 
one. 

For instance, classical physics is simpler than more recent theories; nonetheless it should not be preferred over them, 
because it is demonstrably wrong in certain respects. 



Occam's razor 19 

Occam's razor is used to adjudicate between theories that have already passed 'theoretical scrutiny' tests, and which 

[351 
are equally well-supported by the evidence. Furthermore, it may be used to prioritize empirical testing between 

two equally plausible but unequally testable hypotheses; thereby minimizing costs and wastes while increasing 

chances of falsification of the simpler-to-test hypothesis. 

Another contentious aspect of the Razor is that a theory can become more complex in terms of its structure (or 
syntax), while its ontology (or semantics) becomes simpler, or vice versa. The theory of relativity is often given 
as an example of the proliferation of complex words to describe a simple concept. 

Galileo Galilei lampooned the misuse of Occam's Razor in his Dialogue. The principle is represented in the dialogue 
by Simplicio. The telling point that Galileo presented ironically was that if you really wanted to start from a small 
number of entities, you could always consider the letters of the alphabet as the fundamental entities, since you could 
certainly construct the whole of human knowledge out of them. 

Anti-razors 

Occam's razor has met some opposition from people who have considered it too extreme or rash. Walter of Chatton 
was a contemporary of William of Ockham (1287—1347) who took exception to Occam's razor and Ockham's use of 
it. In response he devised his own anti-razor. "If three things are not enough to verify an affirmative proposition 
about things, a fourth must be added, and so on". Although there has been a number of philosophers who have 
formulated similar anti-razors since Chatton's time, no one anti-razor has perpetuated in as much notoriety as 
Occam's razor, although this could be the case of the Late Renaissance Italian motto of unknown attribution Se non e 
vero, e ben trovato ("Even if it is not true, it is well conceived") when referred to a particularly artful explanation. 

Anti-razors have also been created by Gottfried Wilhelm Leibniz (1646—1716), Immanuel Kant (1724—1804), and 
Karl Menger. Leibniz's version took the form of a principle of plenitude, as Arthur Lovejoy has called it, the idea 

being that God created the most varied and populous of possible worlds. Kant felt a need to moderate the effects of 

T371 
Occam's Razor and thus created his own counter-razor: "The variety of beings should not rashly be diminished." 

Einstein supposedly remarked, "Everything should be made as simple as possible, but not simpler." 

Karl Menger found mathematicians to be too parsimonious with regard to variables so he formulated his Law 
Against Miserliness which took one of two forms: "Entities must not be reduced to the point of inadequacy" and "It 
is vain to do with fewer what requires more". See "Ockham's Razor and Chatton's Anti-Razor" (1984) by Armand 
Maurer. A less serious, but (some might say) even more extremist anti-razor is 'Pataphysics, the "science of 
imaginary solutions" invented by Alfred Jarry (1873—1907). Perhaps the ultimate in anti-reductionism, 'Pataphysics 
seeks no less than to view each event in the universe as completely unique, subject to no laws but its own. Variations 
on this theme were subsequently explored by the Argentinean writer Jorge Luis Borges in his story/mock-essay Tlon, 
Uqbar, Orbis Tertius. There is also Crabtree's Bludgeon, which takes a cynical view that 'No set of mutually 
inconsistent observations can exist for which some human intellect cannot conceive a coherent explanation, however 
complicated.' 

While not technically contradicting the razor's notion that (other things being equal) "the simplest explanation is 
always the best", the reverse corollary — that the best explanation is not always the simplest — is well expressed by 
the Sir Arthur Conan Doyle character, Sherlock Holmes, in The Sign of the Four, especially in the following famous 
quote: "When you have eliminated the impossible, whatever remains, however improbable, must be the truth. " 



Occam's razor 20 

See also 

Algorithmic information theory 

Bayesian inference 

Buridan's ass 

Ceteris paribus 

Common sense 

Cladistics 

Crab tree's Bludgeon 

Curve fitting 

Data compression 

Eliminative materialism 

Egyptian fractions 

Falsifiability 

Greedy reductionism 

Hanlon's razor 

Isotelism 

KISS principle 

Kolmogorov complexity 

Metaphysical naturalism 

Minimum description length 

Minimum message length 

Model selection 

Morgan's canon 

Murphy's law 

Occam programming language 

Overfitting 

Parsimony 

Philosophy of science 

Poverty of the stimulus 

Principle of least astonishment 

Rationalism 

Reference class problem 

Scientific method 

Scientific reductionism 

Simplicity 

Turtles all the way down 



Occam's razor 21 

Further reading 

• Ariew, Roger (1976). Ockham's Razor: A Historical and Philosophical Analysis of Ockham's Principle of 
Parsimony. Champaign-Urbana, University of Illinois. 

Charlesworth, M. J. (1956). "Aristotle's Razor". Philosophical Studies (Ireland) 6: 105—112. 
Churchland, Paul M. (1984). Matter and Consciousness. Cambridge, Massachusetts: MIT Press. ISBN. 
Crick, Francis H. C. (1988). What Mad Pursuit: A Personal View of Scientific Discovery. New York, New York: 
Basic Books. ISBN. 

Dowe, David L.; Steve Gardner, Graham Oppy (December 2007). "Bayes not Bust! Why Simplicity is no 

T21 T391 

Problem for Bayesians" . British J. for the Philosophy of Science 58: 46pp. doi:10.1093/bjps/axm033. 

Retrieved 2007-09-24. 

Duda, Richard O.; Peter E. Hart, David G Stork (2000). Pattern Classification (2nd ed.). Wiley-Interscience. 

pp. 487-489. ISBN. 

Epstein, Robert (1984). "The Principle of Parsimony and Some Applications in Psychology". Journal of Mind 

Behavior 5: 119-130. 

Hoffmann, Roald; Vladimir I. Minkin, Barry K. Carpenter (1997). "Ockham's Razor and Chemistry" 

HYLE — International Journal for the Philosophy of Chemistry 3: 3—28. Retrieved 2006-04-14. 

Jacquette, Dale (1994). Philosophy of Mind. Engleswoods Cliffs, New Jersey: Prentice Hall. pp. 34-36. ISBN. 

Jaynes, Edwin Thompson (1994). "Model Comparison and Robustness" . Probability Theory: The Logic of 

Science 

Jefferys, William H.; Berger, James O. (1991). "Ockham's Razor and Bayesian Statistics (Preprint available as 

[431 
"Sharpening Occam's Razor on a Bayesian Strop)"," . American Scientist 80: 64—72. 

Katz, Jerrold (1998). Realistic Rationalism. MIT Press. 

Kneale, William; Martha Kneale (1962). The Development of Logic. London: Oxford University Press, pp. 243. 

ISBN. 

T441 
MacKay, David J. C. (2003). Information Theory, Inference and Learning Algorithms . Cambridge University 

Press. ISBN. 

Maurer, A. (1984). "Ockham's Razor and Chatton's Anti-Razor". Medieval Studies 46: 463-475. 

T451 
McDonald, William (2005). "S0ren Kierkegaard" . Stanford Encyclopedia of Philosophy. Retrieved 

2006-04-14. 

Menger, Karl (1960). "A Counterpart of Ockham's Razor in Pure and Applied Mathematics: Ontological Uses". 

Synthese 12: 415. doi:10.1007/BF00485426. 

Morgan, C. Lloyd (1903). "Other Minds than Ours" . An Introduction to Comparative Psychology (2nd 

ed.). London: W. Scott, pp. 59. Retrieved 2006-04-15. 

Nolan, D. (1997). "Quantitative Parsimony". British Journal for the Philosophy of Science 48 (3): 329—343. 

doi: 10. 1093/bjps/48. 3.329. 

Pegis, A. C, translator (1945). Basic Writings of St. Thomas Aquinas . New York: Random House, pp. 129. 

Popper, Karl (1992). "7. Simplicity". The Logic of Scientific Discovery (2nd ed.). London: Routledge. 

pp. 121-132. 

Rodriguez-Fernandez, J. L. (1999). "Ockham's Razor". Endeavour 23: 121—125. 

doi: 10. 1016/S0160-9327(99)01 199-0. 

T481 T491 

Schmitt, Gavin C. (2005). "Ockham's Razor Suggests Atheism" . Archived from the original on 

2007-02-11. Retrieved 2006-04-15. 

Smart, J. J. C. (1959). "Sensations and Brain Processes". Philosophical Review 68: 141—156. 

doi: 10.2307/2182164. 

Sober, Elliott (1975). Simplicity. Oxford: Oxford University Press. 

Sober, Elliott (1981). "The Principle of Parsimony". British Journal for the Philosophy of Science 32: 145—156. 

doi:10.1093/bjps/32.2.145. 



Occam's razor 22 

• Sober, Elliott (1990). "Let's Razor Ockham's Razor", in Dudley Knowles. Explanation and its Limits. Cambridge: 
Cambridge University Press, pp. 73—94. ISBN. 

• Sober, Elliott (2001). "What is the Problem of Simplicity?" [50] . in Zellner et al.. Retrieved 2006-04-15. 

• Swinburne, Richard (1997). Simplicity as Evidence for Truth. Milwaukee, Wisconsin: Marquette University 
Press. 

• Thorburn, W. M. (1918). "The Myth of Occam's Razor" [51] . Mind 27 (107): 345-353. 
doi: 10. 1093/mind/XXVII.3.345. 

• Williams, George C. (1966). Adaptation and natural selection: A Critique of some Current Evolutionary Thought. 
Princeton, New Jersey: Princeton University Press. ISBN. 

External links 

[52] 

• What is Occam's Razor? This essay distinguishes Occam's Razor (used for theories with identical predictions) 

from the Principle of Parsimony (which can be applied to theories with different predictions). 

[531 

• Skeptic's Dictionary: Occam 's Razor 

• Ockham's Razor , an essay at The Galilean Library on the historical and philosophical implications by Paul 
New all. 

• The Razor in the Toolbox: The history, use, and abuse of Occam's Razor , by Robert Novella 

• NIPS 2001 Workshop "Foundations of Occam's Razor and parsimony in learning" 



Simplicity at Stanford Encyclopedia of Philosophy 



[57] 

• Occam's Razor on PlanetMath 

[50] 

• Humorous corollary "Rev. Nocents' Toothbrush" (science vs. religion) 

References 

[I] "Occam's razor" (http://www.merriam-webster.com/dictionary/Occam's razor). Merriam-Webster's Collegiate Dictionary (1 1th ed.). New 
York: Merriam- Webster. 2003. ISBN 0-87779-809-5. . 

[2] http://www.britannica.com/EBchecked/topic/424706/Ockhams-razor { {Clarifyldate=August 2009lreason=This is not a proper reference 

citation. Use [[Template: Cite web theory 
[3] Hawking (2003). On the Shoulders of Giants (http://books.google.com/books?id=0eRZr_HK0LgC&pg=PA731). Running Press, p. 731. 

ISBN 076241698x. . 
[4] Hugh G. Gauch, Scientific Method in Practice, Cambridge University Press, 2003, ISBN 0521017084, 9780521017084 
[5] Roald Hoffmann, Vladimir I. Minkin, Barry K. Carpenter, Ockham's Razor and Chemistry, HYLE — International lournal for Philosophy of 

Chemistry, Vol. 3, pp. 3-28, (1997). 
[6] Alan Baker, Simplicity, Stanford Encyclopedia of Philosophy, (2004) http://plato.stanford.edu/entries/simplicity/ 
[7] Courtney A, Courtney M: Comments Regarding "On the Nature Of Science", Physics in Canada, Vol. 64, No. 3 (2008), p7-8. 
[8] Dieter Gernert, Ockham's Razor and Its Improper Use, Journal of Scientific Exploration, Vol. 21, No. 1, pp. 135-140, (2007). 
[9] Elliott Sober, Let's Razor Occam's Razor, p. 73-93, from Dudley Knowles (ed.) Explanation and Its Limits, Cambridge University Press 

(1994). 
[10] http://uk.geocities.eom/frege@btinternet.com/latin/mythofockham.htm 

[II] Inline Latin translations added 

[12] Alhazen; Smith, A. Mark (2001). Alhacen's Theory of Visual Perception: A Critical Edition, with English Translation and Commentary of 
the First Three Books of Alhacen's De Aspectibus, the Medieval Latin Version oflbn al-Haytham's Kitab al-Manazir. DIANE Publishing, 
pp. 372 & 408. ISBN 0871699141. 

[13] Pegis 1945 

[14] Albert Einstein, Does the Inertia of a Body Depend Upon Its Energy Content? Albert Einstein, Annalen der Physik 18: 639—641, (1905). 

[15] L. Nash, The Nature of the Natural Sciences, Boston: Little, Brown (1963). 

[16] P.L.M. de Maupertuis, Memoires de l'Academie Royale, 423 (1744). 

[17] L. de Broglie, Annates de Physique, 3/10, 22-128 (1925). 

[18] R.P. Feynman, R.B. Leighton, M. Sands, The Feynman Lectures on Physics, vol. II, Addison- Wesley, Reading, (1964). 

[19] R.A. Jackson, Mechanism: An Introduction to the Study of Organic Reactions, Clarendon, Oxford, 1972. 

[20] B.K. Carpenter, Determination of Organic Reaction Mechanism, Wiley-Interscience, New York, 1984. 

[21] Science, 263, 641-646 (1994) 

[22] Ernst Mach, The Stanford Encyclopedia of Philosophy, http://plato.stanford.edu/entries/ernst-mach/ 



Occam's razor 



23 



[23] Crick 1988, p. 146. 

[24] Larson and Witham, 1998 "Leading Scientists Still Reject God" (http://www.stephenjaygould.org/ctrl/news/file002.html) 

[25] Ref to survey at Livescience (http://www.livescience.com/strangenews/05081 l_scientists_god.html) article from Physorg.com (http:// 

www.physorg.com/pdf5785.pdf) 
[26] Algorithmic Information Theory (http://www.hutterl.net/ait.htm) 
[27] Paul M. B. Vitanyi and Ming Li; IEEE Transactions on Information Theory, Volume 46, Issue 2, Mar 2000 Page(s):446— 464, "Minimum 

Description Length Induction, Bayesianism and Kolmogorov Complexity". 
[28] 'Occam's Razor as a formal basis for a physical theory' by Andrei N. Soklakov (http://arxiv.org/pdf/math-ph/0009007) 
[29] 'Why Occam's Razor' by Russell Standish (http://arxiv.org/abs/physics/0001020) 

[30] Ray Solomonoff (1964): A formal theory of inductive inference. Part I. Information and Control, 7: 1-22, 1964 
[31] J. Schmidhuber (2006) The New AI: General & Sound & Relevant for Physics. In B. Goertzel and C. Pennachin, eds.: Artificial General 

Intelligence, p. 177-200 http://arxiv.org/abs/cs.AI/0302012 
[32] (http://users.openface.ca/~cobe/occams-razor/interpretations.html) 
[33] ["But Ockham's razor does not say that the more simple a hypothesis, the better." http://www.skepdic.com/occam.html Skeptic's 

Dictionary] 
[34] "when you have two competing theories which make exactly the same predictions, the one that is simpler is the better." Usenet Physics 

FAQs (http://math.ucr.edu/home/baez/physics/) 
[35] "Today, we think of the principle of parsimony as a heuristic device. We don't assume that the simpler theory is correct and the more 

complex one false. We know from experience that more often than not the theory that requires more complicated machinations is wrong. Until 

proved otherwise, the more complex theory competing with a simpler explanation should be put on the back burner, but not thrown onto the 

trash heap of history until proven false." ( The Skeptic's dictionary (http://www.skepdic.com/occam.html)) 
[36] "While these two facets of simplicity are frequently conflated, it is important to treat them as distinct. One reason for doing so is that 

considerations of parsimony and of elegance typically pull in different directions. Postulating extra entities may allow a theory to be 

formulated more simply, while reducing the ontology of a theory may only be possible at the price of making it syntactically more complex." 

Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/entries/simplicity/) 
[37] Original Latin: Entium varietates non temere esse minuendas. Kant, Immanuel (1950): The Critique of Pure Reason, transl. Kemp Smith, 

London. Available here: (http://www.hkbu.edu.hk/~ppp/cpr/toc.html) 
[38] Shapiro, Fred R., ed. (2006), The Yale Book of Quotations, Yale Press, ISBN 9780300107982 (http://books.google.com/books/ 

yup?hl=en&q=simple+as+possible&vid=ISBN9780300107982) 
[39] http://bjps.oxfordjournals.org/ 
[40] http://www.hyle.Org/journal/issues/3/hoffman.htm 
[41] http://omega.math.albany.edu:8008/ETJ-PS/cc24f.ps 
[42] http://omega.math. albany.edu:8008/JaynesBook. html 
[43] http://quasar.as.utexas.edu/papers/ockham.pdf 
[44] http://www.inference.phy.cam.ac.uk/mackay/itila/book.html 
[45] http://plato.stanford.edu/entries/kierkegaard/ 

[46] http://spartan.ac.brocku.ca/~lward7Morgan/Morgan_1903/Morgan_1903_03.html 
[47] http://spartan.ac.brocku.ca/~lward7Morgan/Morgan_1903/Morgan_1903_toc.html 
[48] http://web.archive.Org/web/20070211004045/http://framingbusiness.net/php/2005/ockhamatheism.php 
[49] http://framingbusiness.net/php/2005/ockhamatheism.php 
[50] http://philosophy.wisc.edu/sober/TILBURG.pdf 
[51] http://en.wikisource.org/wiki/The_Myth_of_Occam%27s_Razor 
[52] http://www.physics.adelaide.edu.au/~dkoks/Faq/General/occam.html 
[53] http://skepdic.com/occam.html 

[54] http://www.galilean-library.org/manuscript.php?postid=43832 
[55] http://www. theness.com/ articles. asp?id=71 
[56] http://rii.ricoh.com/~stork/OccamWorkshop.html 
[57] http://planetmath.org/?op=getobj&from=objects&id=6371 
[58] http://www.rainbowtel.net/~bryants/toothbrush.htm 



Complexity 



24 



Complexity 



In general usage, complexity tends to be used to characterize something with many parts in intricate arrangement. 
The study of these complex linkages is the main goal of network theory and network science. In science there are at 
this time a number of approaches to characterizing complexity, many of which are reflected in this article. In a 
business context, complexity management is the methodology to minimize value-destroying complexity and 
efficiently control value-adding complexity in a cross-functional approach. 

Definitions are often tied to the concept of a 'system' — a set of parts or elements which have relationships among 
them differentiated from relationships with other elements outside the relational regime. Many definitions tend to 
postulate or assume that complexity expresses a condition of numerous elements in a system and numerous forms of 
relationships among the elements. At the same time, what is complex and what is simple is relative and changes with 
time. 

Some definitions key on the question of the probability of encountering a given condition of a system once 
characteristics of the system are specified. Warren Weaver has posited that the complexity of a particular system is 
the degree of difficulty in predicting the properties of the system if the properties of the system's parts are given. In 
Weaver's view, complexity comes in two forms: disorganized complexity, and organized complexity. Weaver's 
paper has influenced contemporary thinking about complexity. 

The approaches which embody concepts of systems, multiple elements, multiple relational regimes, and state spaces 
might be summarized as implying that complexity arises from the number of distinguishable relational regimes (and 
their associated state spaces) in a defined system. 

Some definitions relate to the algorithmic basis for the expression of a complex phenomenon or model or 
mathematical expression, as is later set out herein. 



Disorganized complexity 
vs. organized complexity 

One of the problems in addressing 
complexity issues has been 
distinguishing conceptually between 
the large number of variances in 
relationships extant in random 
collections, and the sometimes large, 
but smaller, number of relationships 
between elements in systems where 
constraints (related to correlation of 
otherwise independent elements) 
simultaneously reduce the variations 
from element independence and create 
distinguishable regimes of 

more-uniform, or correlated, 

relationships, or interactions. 

Weaver perceived and addressed this problem, in at least a preliminary way, in drawing a distinction between 
'disorganized complexity' and 'organized complexity'. 




•Map of Complexity Science. *HERE FOR WEB VERSION OF MAP L J The web 

version of this map provides internet links to many of the leading scholars and areas 

of research in complexity science. 



Complexity 



25 



In Weaver's view, disorganized complexity results from the particular system 
having a very large number of parts, say millions of parts, or many more. 
Though the interactions of the parts in a 'disorganized complexity' situation can 
be seen as largely random, the properties of the system as a whole can be 
understood by using probability and statistical methods. 

A prime example of disorganized complexity is a gas in a container, with the gas 
molecules as the parts. Some would suggest that a system of disorganized 
complexity may be compared, for example, with the (relative) simplicity of the 
planetary orbits — the latter can be known by applying Newton's laws of motion, 
though this example involved highly correlated events. 

Organized complexity, in Weaver's view, resides in nothing else than the 
non-random, or correlated, interaction between the parts. These non-random, or 
correlated, relationships create a differentiated structure which can, as a system, 
interact with other systems. The coordinated system manifests properties not 
carried by, or dictated by, individual parts. The organized aspect of this form of 
complexity vis a vis other systems than the subject system can be said to 
"emerge," without any "guiding hand." 



HOW TO README 

TTie abowe map is a conceptual and historical overview or 
complexity science. 

"[tie Map is to be read as follows; 

First, the Map is roughly historiraLworkingasa limelineibai is 
divided into five major periods that ore unread from left to 
right; 1 J old-school, i] per caption, 31 the new science of 
complexity, 4) a work in progress, and 5] recent developments, 

Each fields of si udy is represented as d-o-u ble-lined -ellipse, with 
a double-lined arrow moving from left to the right, The 
relative size of these ellipses Is meaninolesSjand is strictly a 
function of trie space needed to write the name of each field 
Double-lined arrows represent the- trajectory of each field of 
study 5 pace con5lrainl5 requited that ihe length of these 
arrows be limited: reader sho^d trVrelore assume that all of 
them extend outward to 2M6. 

Tile decision Where to place the various fields Of research 
respective to one another is somewhat arbitrary. However, we 
did try to position relative to some degree of intellectual 
similarity. For example, those sciences oriented toward the 
study of systems are heated at the top of the map; the 
sciences that te-nd to extend outward from or around cyber- 
netics and artificial intelligence and are oriented toward the 
development of tomptHational method are located at the 
bottom. 

Areas of research identified for each field of Study are repre- 
sented as single-lined circles. As with I he fieldsof study, the 
size of these circles is strictly a function of the space needed to 
write *e different names. 

The intel lectual links. amongst the fields of study and amongst 
■he areas of research are represented with a bold single-lined 
snow. Trie head of the arrow indicates the direction of the 
relarlonsh Ip. In some cases, the relationship Is mutual. To keep 
the map simple, rather than draw this link to the trajectory for 
a field of study or area of research (as in the case of the recip- 
rocal relationsri Ip between co m plexrty science and agent- 
based modeling!, we draw it to the ellipse representing the 
field of study orarea of research. 

For each area of research we a Iso Include a short list of the 
leading scholars. This list is not exhaustive; but il is representa- 
tive, based on number of citation*, -general recognitkin,and 
importance In the historical development of the area of 
research. For each scholar we providethe following Informa- 
tiomname. most widely known contributlon.and links to key 
areas of research The links amongst ihe scholars and I heir 
respective areas of research are represented by a dashed Irn e. 
One will also note that the names of the scholars differ In font 
size. This was done to demonstrate their relative Importance 
within com ptasdty science and i 1 1 s soetolQgy of complexity. 

Because of the diversity of research in complexity science, we 
focused on the key topics In the field. 



MAP LEGEND. 



The number of parts does not have to be very large for a particular system to have emergent properties. A system of 
organized complexity may be understood in its properties (behavior among the properties) through modeling and 
simulation, particularly modeling and simulation with computers. An example of organized complexity is a city 



neighborhood as a living mechanism, with the neighborhood people among the system's parts 



[4] 



Sources and factors of complexity 

The source of disorganized complexity is the large number of parts in the system of interest, and the lack of 
correlation between elements in the system. 

There is no consensus at present on general rules regarding the sources of organized complexity, though the lack of 
randomness implies correlations between elements. See e.g. Robert Ulanowicz's treatment of ecosystems. 
Consistent with prior statements here, the number of parts (and types of parts) in the system and the number of 
relations between the parts would have to be non-trivial — however, there is no general rule to separate "trivial" from 
"non-trivial". 

Complexity of an object or system is a relative property. For instance, for many functions (problems), such a 
computational complexity as time of computation is smaller when multitape Turing machines are used than when 
Turing machines with one tape are used. Random Access Machines allow one to even more decrease time 
complexity (Greenlaw and Hoover 1998: 226), while inductive Turing machines can decrease even the complexity 
class of a function, language or set (Burgin 2005). This shows that tools of activity can be an important factor of 
complexity. 



Complexity 26 

Specific meanings of complexity 

In several scientific fields, "complexity" has a specific meaning : 

• In computational complexity theory, the amounts of resources required for the execution of algorithms is studied. 
The most popular types of computational complexity are the time complexity of a problem equal to the number of 
steps that it takes to solve an instance of the problem as a function of the size of the input (usually measured in 
bits), using the most efficient algorithm, and the space complexity of a problem equal to the volume of the 
memory used by the algorithm (e.g., cells of the tape) that it takes to solve an instance of the problem as a 
function of the size of the input (usually measured in bits), using the most efficient algorithm. This allows to 
classify computational problems by complexity class (such as P, NP ... ). An axiomatic approach to computational 
complexity was developed by Manuel Blum. It allows one to deduce many properties of concrete computational 
complexity measures, such as time complexity or space complexity, from properties of axiomatically defined 
measures. 

• In algorithmic information theory, the Kolmogorov complexity (also called descriptive complexity, algorithmic 
complexity or algorithmic entropy) of a string is the length of the shortest binary program which outputs that 
string. Different kinds of Kolmogorov complexity are studied: the uniform complexity, prefix complexity, 
monotone complexity, time-bounded Kolmogorov complexity, and space-bounded Kolmogorov complexity. An 
axiomatic approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark 
Burgin in the paper presented for publication by Andrey Kolmogorov (Burgin 1982). The axiomatic approach 
encompasses other approaches to Kolmogorov complexity. It is possible to treat different kinds of Kolmogorov 
complexity as particular cases of axiomatically defined generalized Kolmogorov complexity. Instead, of proving 
similar theorems, such as the basic invariance theorem, for each particular measure, it is possible to easily deduce 
all such results from one corresponding theorem proved in the axiomatic setting. This is a general advantage of 
the axiomatic approach in mathematics. The axiomatic approach to Kolmogorov complexity was further 
developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and 
Burgin, 2003). 

• In information processing, complexity is a measure of the total number of properties transmitted by an object and 
detected by an observer. Such a collection of properties is often referred to as a state. 

• In business, complexity describes the variances and their consequences in various fields such as product portfolio, 
technologies, markets and market segments, locations, manufacturing network, customer portfolio, IT systems, 
organization, processes etc. 

• In physical systems, complexity is a measure of the probability of the state vector of the system. This should not 
be confused with entropy; it is a distinct mathematical measure, one in which two distinct states are never 
conflated and considered equal, as is done for the notion of entropy statistical mechanics. 

• In mathematics, Krohn-Rhodes complexity is an important topic in the study of finite semigroups and automata. 

• In software engineering, programming complexity is a measure of the interactions of the various elements of the 
software. This differs from the computational complexity described above in that it is a measure of the design of 
the software. 

There are different specific forms of complexity: 

• In the sense of how complicated a problem is from the perspective of the person trying to solve it, limits of 
complexity are measured using a term from cognitive psychology, namely the hrair limit. 

• Complex adaptive system denotes systems which have some or all of the following attributes 

• The number of parts (and types of parts) in the system and the number of relations between the parts is 
non-trivial — however, there is no general rule to separate "trivial" from "non-trivial;" 

• The system has memory or includes feedback; 

• The system can adapt itself according to its history or feedback; 



Complexity 27 

• The relations between the system and its environment are non-trivial or non-linear; 

• The system can be influenced by, or can adapt itself to, its environment; and 

• The system is highly sensitive to initial conditions. 

Study of complexity 

Complexity has always been a part of our environment, and therefore many scientific fields have dealt with complex 
systems and phenomena. Indeed, some would say that only what is somehow complex — what displays variation 
without being random — is worthy of interest. 

The use of the term complex is often confused with the term complicated. In today's systems, this is the difference 
between myriad connecting "stovepipes" and effective "integrated" solutions. This means that complex is the 
opposite of independent, while complicated is the opposite of simple. 

While this has led some fields to come up with specific definitions of complexity, there is a more recent movement 
to regroup observations from different fields to study complexity in itself, whether it appears in anthills, human 
brains, or stock markets. One such interndisciplinary group of fields is relational order theories. 

Complexity topics 
Complex behaviour 

The behaviour of a complex system is often said to be due to emergence and self-organization. Chaos theory has 
investigated the sensitivity of systems to variations in initial conditions as one cause of complex behaviour. 

Complex mechanisms 

Recent developments around artificial life, evolutionary computation and genetic algorithms have led to an 
increasing emphasis on complexity and complex adaptive systems. 

Complex simulations 

In social science, the study on the emergence of macro-properties from the micro-properties, also known as 
macro-micro view in sociology. The topic is commonly recognized as social complexity that is often related to the 
use of computer simulation in social science, i.e.: computational sociology. 

Complex systems 

Systems theory has long been concerned with the study of complex systems (In recent times, complexity theory and 
complex systems have also been used as names of the field). These systems can be biological, economic, 
technological, etc. Recently, complexity is a natural domain of interest of the real world socio-cognitive systems and 
emerging systemics research. Complex systems tend to be high-dimensional, non-linear and hard to model. In 
specific circumstances they may exhibit low dimensional behaviour. 

Complexity in data 

In information theory, algorithmic information theory is concerned with the complexity of strings of data. 

Complex strings are harder to compress. While intuition tells us that this may depend on the codec used to compress 
a string (a codec could be theoretically created in any arbitrary language, including one in which the very small 
command "X" could cause the computer to output a very complicated string like '18995316'"), any two 
Turing-complete languages can be implemented in each other, meaning that the length of two encodings in different 
languages will vary by at most the length of the "translation" language - which will end up being negligible for 
sufficiently large data strings. 



Complexity 28 

These algorithmic measures of complexity tend to assign high values to random noise. However, those studying 
complex systems would not consider randomness as complexity. 

Information entropy is also sometimes used in information theory as indicative of complexity. 

Applications of complexity 

Computational complexity theory is the study of the complexity of problems - that is, the difficulty of solving them. 
Problems can be classified by complexity class according to the time it takes for an algorithm - usually a computer 
program - to solve them as a function of the problem size. Some problems are difficult to solve, while others are 
easy. For example, some difficult problems need algorithms that take an exponential amount of time in terms of the 
size of the problem to solve. Take the travelling salesman problem, for example. It can be solved in time 0(n^2 n ) 
(where n is the size of the network to visit - let's say the number of cities the travelling salesman must visit exactly 
once). As the size of the network of cities grows, the time needed to find the route grows (more than) exponentially. 
Even though a problem may be computationally solvable in principle, in actual practice it may not be that simple. 
These problems might require large amounts of time or an inordinate amount of space. Computational complexity 
may be approached from many different aspects. Computational complexity can be investigated on the basis of time, 
memory or other resources used to solve the problem. Time and space are two of the most important and popular 
considerations when problems of complexity are analyzed. 

There exist a certain class of problems that although they are solvable in principle they require so much time or 
space that it is not practical to attempt to solve them. These problems are called intractable. 

There is another form of complexity called hierarchical complexity. It is orthogonal to the forms of complexity 
discussed so far, which are called horizontal complexity 

See also 

Chaos theory 

Command and Control Research Program 

Complexity theory (disambiguation page) 

Cyclomatic complexity 

Evolution of complexity 

Game complexity 

Holism in science 

Interconnectedness 

Model of hierarchical complexity 

Names of large numbers 

Network science 

Network theory 

Occam's razor 

Process architecture 

Programming Complexity 

Sociology and complexity science 

Systems theory 

Variety (cybernetics) 

Volatility, uncertainty, complexity and ambiguity 



Complexity 29 

Further reading 

Lewin, Roger (1992). Complexity: Life at the Edge of Chaos. New York: Macmillan Publishing Co. 

ISBN 9780025704855. 

Waldrop, M. Mitchell (1992). Complexity: The Emerging Science at the Edge of Order and Chaos. New York: 

Simon & Schuster. ISBN 9780671767891. 

rsi 
Czerwinski, Tom; David Alberts (1997). Complexity, Global Politics, and National Security . National Defense 

University. ISBN 9781579060466. 

Czerwinski, Tom (1998). Coping with the Bounds: Speculations on Nonlinearity in Military Affairs . CCRP. 

ISBN 9781414503158 (from Pavilion Press, 2004). 

Lissack, Michael R.; Johan Roos (2000). The Next Common Sense, The e-Manager's Guide to Mastering 

Complexity. Intercultural Press. ISBN 9781857882353. 

Sole, R. V.; B. C. Goodwin (2002). Signs of Life: How Complexity Pervades Biology. Basic Books. 

ISBN 9780465019281. 

Moffat, James (2003). Complexity Theory and Network Centric Warfare [10] . CCRP. ISBN 9781893723115. 

Smith, Edward (2006). Complexity, Networking, and Effects Based Approaches to Operations . CCRP. 

ISBN 9781893723184. 

ri2i 
Heylighen, Francis (2008), "Complexity and Self-Organization , in Bates, Marcia J.; Maack, Mary Niles, 

Encyclopedia of Library and Information Sciences, CRC, ISBN 9780849397127 

Greenlaw, N. and Hoover, H.J. Fundamentals of the Theory of Computation, Morgan Kauffman Publishers, San 

Francisco, 1998 

Blum, M. (1967) On the Size of Machines, Information and Control, v. 11, pp. 257-265 

Burgin, M. (1982) Generalized Kolmogorov complexity and duality in theory of computations, Notices of the 

Russian Academy of Sciences, v. 25, No. 3, pp. 19-23 

Mark Burgin (2005), Super-recursive algorithms, Monographs in computer science, Springer. 

Burgin, M. and Debnath, N. Hardship of Program Utilization and User-Friendly Software, in Proceedings of the 

International Conference "Computer Applications in Industry and Engineering", Las Vegas, Nevada, 2003, 

pp. 314-317 

Debnath, N.C. and Burgin, M., (2003) Software Metrics from the Algorithmic Perspective, in Proceedings of the 

ISCA 18th International Conference "Computers and their Applications", Honolulu, Hawaii, pp. 279-282 

Meyers, R.A., (2009) "Encyclopedia of Complexity and Systems Science", ISBN 978-0-387-75888-6 

Caterina Liberati, J. Andrew Howe, Hamparsum Bozdogan, Data Adaptive Simultaneous Parameter and Kernel 

ri3i 

Selection in Kernel Discriminant Analysis Using Information Complexity , Journal of Pattern Recognition 

Research, JPRR [14] , Vol 4, No 1, 2009. 

Gershenson, C. and F. Heylighen (2005). How can we think the complex? In Richardson, Kurt (ed.) 

Managing Organizational Complexity: Philosophy, Theory and Application, Chapter 3. Information Age 

Publishing. 



Complexity 30 

External links 

• Quantifying Complexity Theory - classification of complex systems 

ri7i 

• Complexity Measures - an article about the abundance of not-that-useful complexity measures. 

n si 

• UC Four Campus Complexity Videoconferences - Human Sciences and Complexity 

ri9i 

• Complexity Digest - networking the complexity community 

• The Santa Fe Institute - engages in research in complexity related topics 

[211 

• Exploring Complexity in Science and Technology - A introductory course about complex system by Melanie 

Mitchell 

References 

[I] Weaver, Warren (1948), "Science and Complexity" (http://www.ceptualinstitute.com/genre/weaver/weaver-1947b.htm), American 
Scientist 36: 536 (Retrieved on 2007-1 1-21.), 

[2] Johnson, Steven (2001). Emergence: the connected lives of ants, brains, cities, and software. New York: Scribner. p. 46. 

ISBN 0-684-86875-X.. 
[3] http://www.art-sciencefactory.com/complexity-map_feb09.html'"CLICK 
[4] Jacobs, Jane (1961). The Death and Life of Great American Cities. New York: Random House. 
[5] Ulanowicz, Robert, "Ecology, the Ascendant Perspective", Columbia, 1997 
[6] Johnson, Neil F. (2007). Two's Company, Three is Complexity: A simple guide to the science of all sciences. Oxford: Oneworld. 

ISBN 978-1-85168-488-5. 
[7] Lissack, Michael R.; Johan Roos (2000). The Next Common Sense, The e- Manager's Guide to Mastering Complexity. Intercultural Press. 

ISBN 9781857882353. 
[8] http://www.dodccrp.org/files/Alberts_Complexity_Global.pdf 
[9] http://www.dodccrp.org/files/Czerwinski_Coping.pdf 
[10] http://www.dodccrp.org/files/Moffat_Complexity.pdf 

[II] http://www.dodccrp.org/files/Smith_Complexity.pdf 
[12] http://pespmcl.vub.ac.be/Papers/ELIS-Complexity.pdf 
[13] http://jprr.Org/index.php/jprr/article/view/l 17 

[14] http://www.jprr.org 

[15] http://uk.arxiv.org/abs/nlin.AO/0402023 

[16] http://www.calresco.org/lucas/quantify.htm 

[17] http://cscs.umich.edu/~crshalizi/notebooks/complexity-measures.html 

[18] http://eclectic.ss.uci.edu/~drwhite/center/cac.html 

[19] http://comdig.unam.mx/ 

[20] http://www.santafe.edu/ 

[21] http://web.cecs.pdx.edu/~mm/ExploringComplexityFall2009/index.html 



Nonlinear system 3 1 



Nonlinear system 



This article describes the use of the term nonlinearity in mathematics. For other meanings, see nonlinearity 
(disambiguation). 

In mathematics, a nonlinear system is a system which is not linear, that is, a system which does not satisfy the 
superposition principle, or whose output is not proportional to its input. Less technically, a nonlinear system is any 
problem where the variable(s) to be solved for cannot be written as a linear combination of independent components. 
A nonhomogeneous system, which is linear apart from the presence of a function of the independent variables, is 
nonlinear according to a strict definition, but such systems are usually studied alongside linear systems, because they 
can be transformed to a linear system of multiple variables. 

Nonlinear problems are of interest to engineers, physicists and mathematicians because most physical systems are 
inherently nonlinear in nature. Nonlinear equations are difficult to solve and give rise to interesting phenomena such 
as chaos. The weather is famously nonlinear, where simple changes in one part of the system produce complex 
effects throughout. 

Definition 

In mathematics, a linear function (or map) fix) is one which satisfies both of the following properties: 

* additivity, f( x + y) = f(x) + f(y); 

• homogeneity, f(ax) — af(x). 

(Additivity implies homogeneity for any rational a, and, for continuous functions, for any real a. For a complex a, 
homogeneity does not follow from additivity; for example, an antilinear map is additive but not homogeneous.) 

An equation written as 

m = c 

is called linear if fix) is a linear map (as defined above) and nonlinear otherwise. The equation is called 
homogeneous if Q = Q . 

The definition fix) = Cis very general in that a; can be any sensible mathematical object (number, vector, 
function, etc), and the function fix) can literally be any mapping, including integration or differentiation with 
associated constraints (such as boundary values). If fix) contains differentiation of x, the result will be a 
differential equation. 

Nonlinear algebraic equations 

Generally, nonlinear algebraic problems are often exactly solvable, and if not they usually can be thoroughly 
understood through qualitative and numeric analysis. As an example, the equation 

x 2 + x - 1 = 
may be written as 

fix) = C where f(x) = x + x and C = 1 
and is nonlinear because fix) satisfies neither additivity nor homogeneity (the nonlinearity is due to the x ^). 
Though nonlinear, this simple example may be solved exactly (via the quadratic formula) and is very well 
understood. On the other hand, the nonlinear equation 

x 5 - x - 1 = 
is not exactly solvable (see quintic equation), though it may be qualitatively analyzed and is well understood, for 
example through making a graph and examining the roots of fix) — C = 0- 



Nonlinear system 32 

Nonlinear recurrence relations 

A nonlinear recurrence relation defines successive terms of a sequence as a nonlinear function of preceding terms. 
Examples of nonlinear recurrence relations are the logistic map and the relations that define the various Hofstadter 
sequences. 

Nonlinear differential equations 

A system of differential equations is said to be nonlinear if it is not a linear system. Problems involving nonlinear 
differential equations are extremely diverse, and methods of solution or analysis are problem dependent. Examples 
of nonlinear differential equations are the Navier— Stokes equations in fluid dynamics, the Lotka— Volterra equations 
in biology, and the Black— Scholes PDE in finance. 

One of the greatest difficulties of nonlinear problems is that it is not generally possible to combine known solutions 
into new solutions. In linear problems, for example, a family of linearly independent solutions can be used to 
construct general solutions through the superposition principle. A good example of this is one-dimensional heat 
transport with Dirichlet boundary conditions, the solution of which can be written as a time-dependent linear 
combination of sinusoids of differing frequencies; this makes solutions very flexible. It is often possible to find 
several very specific solutions to nonlinear equations, however the lack of a superposition principle prevents the 
construction of new solutions. 

Ordinary differential equations 

First order ordinary differential equations are often exactly solvable by separation of variables, especially for 
autonomous equations. For example, the nonlinear equation 

du 2 

dx 

will easily yield u = (x + C) _1 as a general solution. The equation is nonlinear because it may be written as 

du o 

— +u 2 = 
dx 

and the left-hand side of the equation is not a linear function of u and its derivatives. Note that if the u 2 term were 
replaced with u, the problem would be linear (the exponential decay problem). 

Second and higher order ordinary differential equations (more generally, systems of nonlinear equations) rarely yield 
closed form solutions, though implicit solutions and solutions involving nonelementary integrals are encountered. 

Common methods for the qualitative analysis of nonlinear ordinary differential equations include: 

• Examination of any conserved quantities, especially in Hamiltonian systems. 

• Examination of dissipative quantities (see Lyapunov function) analogous to conserved quantities. 

• Linearization via Taylor expansion. 

• Change of variables into something easier to study. 

• Bifurcation theory. 

• Perturbation methods (can be applied to algebraic equations too). 



Nonlinear system 



33 



Partial differential equations 

The most common basic approach to studying nonlinear partial differential equations is to change the variables (or 
otherwise transform the problem) so that the resulting problem is simpler (possibly even linear). Sometimes, the 
equation may be transformed into one or more ordinary differential equations, as seen in the similarity transform or 
separation of variables, which is always useful whether or not the resulting ordinary differential equation(s) is 
solvable. 

Another common (though less mathematic) tactic, often seen in fluid and heat mechanics, is to use scale analysis to 
simplify a general, natural equation in a certain specific boundary value problem. For example, the (very) nonlinear 
Navier-Stokes equations can be simplified into one linear partial differential equation in the case of transient, 
laminar, one dimensional flow in a circular pipe; the scale analysis provides conditions under which the flow is 
laminar and one dimensional and also yields the simplified equation. 

Other methods include examining the characteristics and using the methods outlined above for ordinary differential 
equations. 



Example: pendulum 

A classic, extensively studied nonlinear problem is the dynamics 
of a pendulum under influence of gravity. Using Lagrangian 
mechanics, it may be shown that the motion of a pendulum can 
be described by the dimensionless nonlinear equation 



rigid massless rod 




gravity 



Illustration of a pendulum. 



Nonlinear system 



34 



8 ~ 71, unstable, exponential 
(hyperbolic sinusoidal) departure 
from the stationary point. 



9 ~ njl, free 
fall (parabolic). 




vv 



= 0, simple 
harmonic motion 
(sinusoidal). 



Linearizations of a pendulum. 



dt 2 



sin(0) = 



where gravity points "downwards" and is the angle the pendulum forms with its rest position, as shown in the 

figure at right. One approach to "solving" this equation is to use d6 / dt as an integrating factor, which would 

eventually yield 

d6 

t + d 



IC + 2 cos(0) 

which is an implicit solution involving an elliptic integral. This "solution" generally does not have many uses 
because most of the nature of the solution is hidden in the nonelementary integral (nonelementary even if C = 0). 

Another way to approach the problem is to linearize any nonlinearities (the sine function term in this case) at the 
various points of interest through Taylor expansions. For example, the linearization at Q = Q, called the small 
angle approximation, is 

d 2 e 



dt 2 



+ = 



since sin((?) ~ 9 for Q ~ 0- This is a simple harmonic oscillator corresponding to oscillations of the pendulum 
near the bottom of its path. Another linearization would be at Q = -^ , corresponding to the pendulum being straight 
up: 

d 2 9 



dt 2 



+ 7T-6 = 



since sin((9) Ri 7T — for Q ~ ^ . The solution to this problem involves hyperbolic sinusoids, and note that 
unlike the small angle approximation, this approximation is unstable, meaning that |0| will usually grow without 
limit, though bounded solutions are possible. This corresponds to the difficulty of balancing a pendulum upright, it is 
literally an unstable state. 



Nonlinear system 35 

One more interesting linearization is possible around Q = 7r/2. around which sin((9) ~ 1 : 

This corresponds to a free fall problem. A very useful qualitative picture of the pendulum's dynamics may be 
obtained by piecing together such linearizations, as seen in the figure at right. Other techniques may be used to find 
(exact) phase portraits and approximate periods. 

Metaphorical use 

Engineers often use the term nonlinear to refer to irrational or erratic behavior, with the implication that the person 
who "goes nonlinear" is on the edge of losing control or even having a nervous breakdown. 

Types of nonlinear behaviors 

• Indeterminism - the behavior of a system cannot be predicted. 

• Multistability - alternating between two or more exclusive states. 

• Aperiodic oscillations - functions that do not repeat values after some period (otherwise known as chaotic 
oscillations or chaos). 

Examples of nonlinear equations 

• AC power flow model 

• Ball and beam system 

• Bellman equation for optimal policy 

• Boltzmann transport equation 

• Colebrook equation 

• General relativity 

• Ginzburg-Landau equation 

• Navier-Stokes equations of fluid dynamics 

• Korteweg— de Vries equation 

• nonlinear optics 

• nonlinear Schrodinger equation 

• Richards equation for unsaturated water flow 

• Robot unicycle balancing 

• Sine-Gordon equation 

• Landau-Lifshitz equation 

• Ishimori equation 

• Van der Pol equation 

• Lienard equation 

• Vlasov equation 

See also the list of non-linear partial differential equations 



Nonlinear system 36 

See also 

• Aleksandr Mikhailovich Lyapunov 

• Dynamical system 

• Volterra series 

• Vector soliton 

Further reading 

• Diederich Hinrichsen and Anthony J. Pritchard (2005). Mathematical Systems Theory I - Modelling, State Space 
Analysis, Stability and Robustness. Springer Verlag. ISBN 0-978-3-540-441250. 

• Jordan, D. W.; Smith, P. (2007). Nonlinear Ordinary Differential Equations (fourth ed.). Oxford Univeresity 
Press. ISBN 978-0-19-9208241. 

• Khalil, Hassan K. (2001). Nonlinear Systems. Prentice Hall. ISBN 0-13-067389-7. 

• Kreyszig, Erwin (1998). Advanced Engineering Mathematics. Wiley. ISBN 0-471-15496-2. 

• Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second 
Edition. Springer. ISBN 0-387-984895. 

External links 

T31 
A collection of non-linear models and demo applets (in Monash University's Virtual Lab) 

mi 

Command and Control Research Program (CCRP) 

New England Complex Systems Institute: Concepts in Complex Systems 

Nonlinear Dynamics I: Chaos at MIT's OpenCourseWare 

Nonlinear Models [ ] Nonlinear Model Database of Physical Systems (MATLAB) 

The Center for Nonlinear Studies at Los Alamos National Laboratory 

FyDiK Software for simulations of nonlinear dynamical systems 

References 

[1] David Tong: Lectures on Classical Dynamics (http://www.damtp.cam.ac.uk/user/tong/dynamics.html) 

[2] Frank Rose (1990). West of Eden: The End of Innocence at Apple Computer (http://books. google. com/books?id=0DHnCxjX_t8C& 

pg=PA259&dq=go-nonlinear&lr=&as_brr=3&as_pt=ALLTYPES&ei=uJxJSfDvBouYMpOzwJAP). Frank Rose, publisher. 

ISBN 9780140093728. . 
[3] http://vlab.infotech.monash.edu.au/simulations/non-linear/ 
[4] http://www.dodccrp.org/ 

[5] http://necsi.org/guide/concepts/linearnonlinear.html 

[6] http://ocw.mit.edu/OcwWeb/Earth— Atmospheric— and-Planetary-Sciences/12-006JFall-2006/CourseHome/index.htm 
[7] http://ocw.mit.edu/OcwWeb/index.htm 
[8] http://www.hedengren.net/research/models.htm 
[9] http://cnls.lanl.gov/ 
[10] http://fydik.kitnarf.cz/ 



Kolmogorov complexity 



37 



Kolmogorov complexity 



In algorithmic information theory (a subfield of computer science), the Kolmogorov complexity of an object, such 
as a piece of text, is a measure of the computational resources needed to specify the object. 

Kolmogorov complexity is also known as descriptive complexity, Kolmogorov-Chaitin complexity, stochastic 
complexity, algorithmic entropy, or program-size complexity. 

For example, consider the following two strings of length 64, each containing only lowercase letters, numbers, and 
spaces: 

abababababababababababababababababababababababababababababababab 
4cl J5b2p0cv4wlx8rx2y39umgw5q8 5s7uraqb jf dppa0q7nieieqe9noc4cvaf zf 

The first string has a short English-language description, namely "ab 32 times", which consists of 11 characters. The 
second one has no obvious simple description (using the same character set) other than writing down the string itself, 
which has 64 characters. 

More formally, the complexity of a string is 
the length of the string's shortest description 
in some fixed universal description 
language. The sensitivity of complexity 
relative to the choice of description 
language is discussed below. It can be 
shown that the Kolmogorov complexity of 
any string cannot be too much larger than 
the length of the string itself. Strings whose 
Kolmogorov complexity is small relative to 
the string's size are not considered to be 
complex. The notion of Kolmogorov 
complexity is surprisingly deep and can be 
used to state and prove impossibility results 
akin to Godel's incompleteness theorem and 
Turing's halting problem. 




Definition 



This image illustrates part of the Mandelbrot set fractal. Simply storing the 24-bit 

color of each pixel in this image would require 1.62 million bits; but a small 

computer program can reproduce these 1.62 million bits using the definition of the 

Mandelbrot set. Thus, the Kolmogorov complexity of the raw file encoding this 

bitmap is much less than 1.62 million. 



To define Kolmogorov complexity, we must first specify a description language for strings. Such a description 
language can be based on any programming language, such as Lisp, Pascal, or Java Virtual Machine bytecode. If P is 
a program which outputs a string x, then P is a description of x. The length of the description is just the length of P as 
a character string. In determining the length of P, the lengths of any subroutines used in P must be accounted for. 
The length of any integer constant n which occurs in the program P is the number of bits required to represent n, that 
is (roughly) log n. 

We could alternatively choose an encoding for Turing machines, where an encoding is a function which associates to 
each Turing Machine M a bitstring <M>. If M is a Turing Machine which on input w outputs string x, then the 
concatenated string <M> w is a description of x. For theoretical analysis, this approach is more suited for 
constructing detailed formal proofs and is generally preferred in the research literature. The binary lambda calculus 
may provide the simplest definition of complexity yet. In this article we will use an informal approach. 

Any string s has at least one description, namely the program 



Kolmogorov complexity 38 

function GenerateFixedString ( ) 
return s 

If a description of s, d(s), is of minimal length — i.e. it uses the fewest number of characters — it is called a minimal 
description of s. Then the length of d(s) — i.e. the number of characters in the description — is the Kolmogorov 
complexity of s, written K(s). Symbolically, 

K(s) = \d(s)\. 
We now consider how the choice of description language affects the value of K and show that the effect of changing 
the description language is bounded. 

Theorem. If K and K are the complexity functions relative to description languages L and L , then there is a 
constant c (which depends only on the languages L and L ) such that 

Vs {K^s} - K 2 (s)\ <c. 
Proof. By symmetry, it suffices to prove that there is some constant c such that for all bitstrings s, 

Ki(s) < K 2 (s) + c. 
To see why this is so, suppose there is a program in the language L which acts as an interpreter for L : 

function InterpretLanguage (string p) 

where p is a program in L . The interpreter is characterized by the following property: 

Running InterpretLanguage on input/? returns the result of running/?. 

Thus if P is a program in L which is a minimal description of s, then InterpretLanguage(P) returns the string s. The 
length of this description of s is the sum of 

1 . The length of the program InterpretLanguage, which we can take to be the constant c. 

2. The length of P which by definition is K (s). 

This proves the desired upper bound. 
See also invariance theorem. 

History and context 

Algorithmic information theory is the area of computer science that studies Kolmogorov complexity and other 
complexity measures on strings (or other data structures). 

The concept and theory of Kolmogorov Complexity is based on a crucial theorem first discovered by Ray 
Solomonoff who published it in 1960, describing it in "A Preliminary Report on a General Theory of Inductive 
Inference" (see ref) as part of his invention of Algorithmic Probability. He gave a more complete description in his 
1964 publications, "A Formal Theory of Inductive Inference," Part 1 and Part 2 in Information and Control (see ref). 

Andrey Kolmogorov later independently published this theorem in Problems Inform. Transmission, 1, (1965), 1-7. 
Gregory Chaitin also presents this theorem in J. ACM, 16 (1969). Chaitin's paper was submitted October 1966, 
revised in December 1968 and cites both Solomonoff s and Kolmogorov's papers. 

The theorem says that among algorithms that decode strings from their descriptions (codes) there exists an optimal 
one. This algorithm, for all strings, allows codes as short as allowed by any other algorithm up to an additive 
constant that depends on the algorithms, but not on the strings themselves. Solomonoff used this algorithm, and the 
code lengths it allows, to define a string's "universal probability' on which inductive inference of a string's 
subsequent digits can be based. Kolmogorov used this theorem to define several functions of strings: complexity, 
randomness, and information. 

When Kolmogorov became aware of Solomonoffs work, he acknowledged Solomonoff s priority (IEEE Trans. 
Inform Theory, 14:5(1968), 662-664). For several years, Solomonoffs work was better known in the Soviet Union 



Kolmogorov complexity 39 

than in the Western World. The general consensus in the scientific community, however, was to associate this type 
of complexity with Kolmogorov, who was concerned with randomness of a sequence while Algorithmic Probability 
became associated with Solomonoff, who focused on prediction using his invention of the universal a priori 
probability distribution. 

There are several other variants of Kolmogorov complexity or algorithmic information. The most widely used one is 
based on self-delimiting programs and is mainly due to Leonid Levin (1974). 

An axiomatic approach to Kolmogorov complexity based on Blum axioms (Blum 1967) was introduced by Mark 
Burgin in the paper presented for publication by Andrey Kolmogorov (Burgin 1982). This approach was further 
developed in the book (Burgin 2005) and applied to software metrics (Burgin and Debnath, 2003; Debnath and 
Burgin, 2003). 

Naming this concept "Kolmogorov complexity" is an example of the Matthew effect. 

Basic results 

In the following, we will fix one definition and simply write K(s) for the complexity of the string s. 

It is not hard to see that the minimal description of a string cannot be too much larger than the string itself: the 
program GenerateFixedString above that outputs s is a fixed amount larger than s. 

Theorem. There is a constant c such that 

Vs K(s) < \s\ +c. 

Incomputability of Kolmogorov complexity 

The first result is that there is no way to effectively compute K. 

Theorem. ^Tis not a computable function. 

In other words, there is no program which takes a string s as input and produces the integer K(s) as output. We show 
this by contradiction by making a program that creates a string that should only be able to be created by a longer 
program. Suppose there is a program 

function KolmogorovComplexity (string s) 

that takes as input a string s and returns K(s). Now consider the program 

function GenerateComplexString (int n) 
for i = 1 to infinity: 

for each string s of length exactly i 
if KolmogorovComplexity ( s) >= n 
return s 
quit 

This program calls KolmogorovComplexity as a subroutine. This program tries every string, starting with the 
shortest, until it finds a string with complexity at least n, then returns that string. Therefore, given any positive 
integer n, it produces a string with Kolmogorov complexity at least as great as n. The program itself has a fixed 
length U. The input to the program GenerateComplexString is an integer «; here, the size of n is measured by the 
number of bits required to represent n which is log An). Now consider the following program: 

function GenerateParadoxicalString ( ) 
return GenerateComplexString (n) 

This program calls GenerateComplexString as a subroutine and also has a free parameter n . This program outputs a 
string s whose complexity is at least n . By an auspicious choice of the parameter n we will arrive at a contradiction. 



Kolmogorov complexity 40 

To choose this value, note s is described by the program GenerateParadoxicalString whose length is at most 

U + log 2 (n ) + C 
where C is the "overhead" added by the program GenerateParadoxicalString. Since n grows faster than log An), there 
exists a value n such that 

U + log 2 (n ) + C < n . 
But this contradicts the definition of having a complexity at least n . That is, by the definition of K(s), the string s 
returned by GenerateParadoxicalString is only supposed to be able to be generated by a program of length n or 
longer, but GenerateParadoxicalString is shorter than n . Thus the program named "KolmogorovComplexity" cannot 
actually computably find the complexity of arbitrary strings. 

This is proof by contradiction where the contradiction is similar to the Berry paradox: "Let n be the smallest positive 
integer that cannot be defined in fewer than twenty English words." It is also possible to show the uncomputability of 
K by reduction from the uncomputability of the halting problem H, since K and H are Turing-equivalent.[l] 

In the programming languages community there is a corollary known as the Full employment theorem, stating there 
is no perfect size-optimizing compiler. 

Chain rule for Kolmogorov complexity 

The chain rule for Kolmogorov complexity states that 

K(X,Y) = K(X) + K(Y\X) + 0(\og(K(XX)))- 

It states that the shortest program that reproduces X and Y is no more than a logarithmic term larger than a program 
to reproduce X and a program to reproduce Y given X. Using this statement one can define an analogue of mutual 
information for Kolmogorov complexity. 

Compression 

It is straightforward to compute upper bounds for K(s)'- simply compress the string swith some method, 

implement the corresponding decompressor in the chosen language, concatenate the decompressor to the compressed 

string, and measure the resulting string's length. 

A string s is compressible by a number c if it has a description whose length does not exceed \s\ — C- This is 

equivalent to saying K(s) < \s\ — C- Otherwise s is incompressible by c. A string incompressible by 1 is said to 

be simply incompressible; by the pigeonhole principle, incompressible strings must exist, since there are 2" bit 

strings of length n but only 2 n — 1 shorter strings, that is strings of length ji _ X or less. 

For the same reason, most strings are complex in the sense that they cannot be significantly compressed: K(s)i$ 

not much smaller than \s\ , the length of s in bits. To make this precise, fix a value of n. There are 2™ bitstrings of 

length n. The uniform probability distribution on the space of these bitstrings assigns exactly equal weight 2 _n to 

each string of length n. 

Theorem. With the uniform probability distribution on the space of bitstrings of length n, the probability that a 

string is incompressible by c is at least \ _ 2 _c+1 4- 2~ n ■ 

To prove the theorem, note that the number of descriptions of length not exceeding n — c is given by the geometric 

series: 

1 + 2 + 2 2 + ■ • ■ + T~ c = 2 n ~ c+1 - 1 

There remain at least 

T - 2 n ~ c+1 + 1 

many bitstrings of length n that are incompressible by c. To determine the probability divide by 2" • 



Kolmogorov complexity 4 1 

This theorem is the justification for various challenges in comp. compression FAQ . Despite this result, it is 
sometimes claimed by certain individuals (considered cranks) that they have produced algorithms which uniformly 
compress data without loss. See lossless data compression. 

Chaitin's incompleteness theorem 

We know that, in the set of all possible strings, most strings are complex in the sense that they cannot be described in 
any significantly "compressed" way. However, it turns out that the fact that a specific string is complex cannot be 
formally proved, if the string's length is above a certain threshold. The precise formalization is as follows. First fix a 
particular axiomatic system S for the natural numbers. The axiomatic system has to be powerful enough so that to 
certain assertions A about complexity of strings one can associate a formula F in S. This association must have the 
following property: if F is provable from the axioms of S, then the corresponding assertion A is true. This 
"formalization" can be achieved either by an artificial encoding such as a Godel numbering or by a formalization 
which more clearly respects the intended interpretation of S. 

Theorem. There exists a constant L (which only depends on the particular axiomatic system and the choice of 
description language) such that there does not exist a string s for which the statement 

K{s) > L 
(as formalized in S) can be proven within the axiomatic system S. 
Note that by the abundance of nearly incompressible strings, the vast majority of those statements must be true. 

The proof of this result is modeled on a self-referential construction used in Berry's paradox. The proof is by 
contradiction. If the theorem were false, then 

Assumption (X): For any integer n there exists a string s for which there is a proof in S of the formula 
"K{s) > n" (which we assume can be formalized in S). 

We can find an effective enumeration of all the formal proofs in S by some procedure 

function NthProof (int n) 

which takes as input n and outputs some proof. This function enumerates all proofs. Some of these are proofs for 
formulas we do not care about here (examples of proofs which will be listed by the procedure NthProof are the 
various known proofs of the law of quadratic reciprocity, those of Fermat's little theorem or the proof of Fermat's last 
theorem all translated into the formal language of S). Some of these are complexity formulas of the form K{s) > n 
where s and n are constants in the language of S. There is a program 

function NthProof ProvesComplexityFormula (int n) 

which determines whether the n proof actually proves a complexity formula K(s) > L. The strings s and the integer 
L in turn are computable by programs: 

function StringNthProof (int n) 

function ComplexityLowerBoundNthProof (int n) 

Consider the following program 

function GenerateProvablyComplexString (int n) 
for i = 1 to infinity: 

if NthProof ProvesComplexityFormula (i) and 
ComplexityLowerBoundNthProof (i) > n 
return StringNthProof (i) 
quit 



Kolmogorov complexity 42 

Given an n, this program tries every proof until it finds a string and a proof in the formal system S of the formula 
K(s) > L for some L>n. The program terminates by our Assumption (X). Now this program has a length U. There is 
an integer n such that U + log An ) + C < n , where C is the overhead cost of 

function GenerateProvablyParadoxicalString ( ) 
return GenerateProvablyComplexString ( n ) 
quit 

The program GenerateProvablyParadoxicalString outputs a string s for which there exists an L such that K(s) > L can 
be formally proved in S with L > n . In particular K{s) > n is true. However, s is also described by a program of 
length U + log An ) + C so its complexity is less than n . This contradiction proves Assumption (X) cannot hold. 

Similar ideas are used to prove the properties of Chaitin's constant. 

Minimum message length 

The minimum message length principle of statistical and inductive inference and machine learning was developed by 
C.S. Wallace and D.M. Boulton in 1968. MML is Bayesian (it incorporates prior beliefs) and information-theoretic. 
It has the desirable properties of statistical invariance (the inference transforms with a re-parametrisation, such as 
from polar coordinates to Cartesian coordinates), statistical consistency (even for very hard problems, MML will 
converge to any underlying model) and efficiency (the MML model will converge to any true underlying model 
about as quickly as is possible). C.S. Wallace and D.L. Dowe showed a formal connection between MML and 
algorithmic information theory (or Kolmogorov complexity) in 1999. 

Kolmogorov randomness 

Kolmogorov randomness (also called algorithmic randomness) defines a string (usually of bits) as being random if 
and only if it is shorter than any computer program that can produce that string. This definition of randomness is 
critically dependent on the definition of Kolmogorov complexity. To make this definition complete, a computer has 
to be specified, usually a Turing machine. According to the above definition of randomness, a random string is also 
an "incompressible" string, in the sense that it is impossible to give a representation of the string using a program 
whose length is shorter than the length of the string itself. However, according to this definition, most strings shorter 
than a certain length end up to be (Chaitin-Kolmogorovically) random because the best one can do with very small 
strings is to write a program that simply prints these strings. 

See also 

• Berlekamp— Massey algorithm 

• Data compression 

• Inductive inference 

• Important publications in algorithmic information theory 

• Levenshtein distance 

• Grammar induction 



Kolmogorov complexity 43 

References 

• Blum, M. (1967), "On the Size of Machines", Information and Control, v. 11, pp. 257—265. 

• Burgin, M. (1982), "Generalized Kolmogorov complexity and duality in theory of computations", Notices of the 
Russian Academy of Sciences, v. 25, No. 3, pp. 19—23. 

• Mark Burgin (2005), Super-recursive algorithms, Monographs in computer science, Springer. 

• Burgin, M. and Debnath, N. "Hardship of Program Utilization and User-Friendly Software", in Proceedings of the 
International Conference "Computer Applications in Industry and Engineering", Las Vegas, Nevada, 2003, pp. 
314-317. 

• Cover, Thomas M. and Thomas, Joy A., Elements of information theory, 1st Edition. New York: 
Wiley-Interscience, 1991. ISBN 0-471-06259-6. 2nd Edition. New York: Wiley-Interscience, 2006. ISBN 
0-471-24195-4. 

• Debnath, N.C. and Burgin, M. (2003), "Software Metrics from the Algorithmic Perspective", in Proceedings of 
the ISCA 18th International Conference "Computers and their Applications", Honolulu, Hawaii, pp. 279—282. 

• Kolmogorov, Andrei N. (1963). "On Tables of Random Numbers". Sankhya Ser. A. 25: pp. 369-375. MR178484 

• Kolmogorov, Andrei N. (1998). "On Tables of Random Numbers". Theoretical Computer Science 207 (2): 
pp. 387-395. doi:10.1016/S0304-3975(98)00075-9. MR1643414 

• Lajos, Ronyai and Gabor, Ivanyos and Reka, Szabo, Algoritmusok. TypoTeX, 1999. ISBN 963-2790-14-6 

• Solomonoff, Ray, "A Preliminary Report on a General Theory of Inductive Inference , Report V-131, Zator 
Co., Cambridge, Ma. Feb 4, 1960. 

• Solomonoff, Ray, "A Formal Theory of Inductive Inference", Information and Control, Part I , Vol 7, No. 1 pp 
1-22, March 1964 and Part II [6] , Vol 7, No. 2 pp 224-254, June 1964. 

• Li, Ming and Vitanyi, Paul, An Introduction to Kolmogorov Complexity and Its Applications, Springer, 1997. 
Introduction chapter full-text . 

• Yu Manin, A Course in Mathematical Logic, Springer- Verlag, 1977. 

• Sipser, Michael, Introduction to the Theory of Computation, PWS Publishing Company, 1997. ISBN 
0-534-95097-3. 

External links 

ro] 

The Legacy of Andrei Nikolaevich Kolmogorov 

Chaitin's online publications 

Solomonoff s IDSIA page [10] 

Generalizations of algorithmic information by J. Schmidhuber 

Ming Li and Paul Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications, 2nd Edition, 

Springer Verlag, 1997. [12] 

ri3i 

Tromp's lambda calculus computer model offers a concrete definition of K() 

Universal AI based on Kolmogorov Complexity ISBN 3-540-22139-5 by M. Hutter: ISBN 3-540-22139-5 

Minimum Message Length and Kolmogorov Complexity (by C.S. Wallace and D.L. Dowe , Computer 

Journal, Vol. 42, No. 4, 1999). 

David Dowe s Minimum Message Length (MML) and Occam's razor pages. 

P. Grunwald, M. A. Pitt and I. J. Myung (ed.), Advances in Minimum Description Length: Theory and 

Applications [19] , M.I.T. Press, April 2005, ISBN 0-262-07262-9. 



Kolmogorov complexity 44 

References 

[I] http://www.daimi.au.dk/~bromille/DC05/Kolmogorov.pdf 
[2] http://www.faqs.org/faqs/compression-faq/partl/ 

[3] http://www.csse.monash.edu.au/~dld/CSWallacePublications/ 

[4] http://world.std.com/~rjs/zl38.pdf 

[5] http://world. std. com/~rjs/ 1 964pt 1 . pdf 

[6] http://world. std. com/~rjs/ 1 964pt2. pdf 

[7] http://citeseer.ist.psu.edu/li97introduction.html 

[8] http://www.kolmogorov.com/ 

[9] http://www.cs.umaine.edu/~chaitin/ 

[10] http://www.idsia.ch/~juergen/ray.html 

[II] http://www.idsia.ch/~juergen/kolmogorov.html 
[12] http://homepages.cwi.nl/~paulv/kolmogorov.html 
[13] http://homepages.cwi.nl/~tromp/cl/cl.html 

[14] http://www3.oup.co.uk/computer_journal/hdb/Volume_42/Issue_04/pdf/420270.pdf 

[15] http://www.csse.monash.edu.au/~dld/CSWallacePublications 

[16] http://www.csse.monash.edu.au/~dld 

[17] http://www.csse.monash.edu.au/~dld/MML.html 

[18] http://www.csse.monash.edu.au/~dld/Occam.html 

[19] http://mitpress.mit.edu/catalog/item/default.asp?sid=4C100C6F-2255-40FF-A2ED-02FC49FEBE7C&ttype=2&tid=10478 



Godel's incompleteness theorems 



Godel's incompleteness theorems are two theorems of mathematical logic that establish inherent limitations of all 
but the most trivial axiomatic systems for mathematics. The theorems, proven by Kurt Godel in 1931, are important 
both in mathematical logic and in the philosophy of mathematics. The two results are widely interpreted as showing 
that Hilbert's program to find a complete and consistent set of axioms for all of mathematics is impossible, thus 
giving a negative answer to Hilbert's second problem. 

The first incompleteness theorem states that no consistent system of axioms whose theorems can be listed by an 
"effective procedure" (essentially, a computer program) is capable of proving all facts about the natural numbers. For 
any such system, there will always be statements about the natural numbers that are true, but that are unprovable 
within the system. The second incompleteness theorem shows that if such a system is also capable of proving certain 
basic facts about the natural numbers, then one particular arithmetic truth the system cannot prove is the consistency 
of the system itself. 

Background 

In mathematical logic, a theory is a set of sentences expressed in a formal language. Some statements in a theory are 
included without proof (these are the axioms of the theory), and others (the theorems) are included because they are 
implied by the axioms. 

Because statements of a formal theory are written in symbolic form, it is possible to mechanically verify that a 
formal proof from a finite set of axioms is valid. This task, known as automatic proof verification, is closely related 
to automated theorem proving; the difference is that instead of constructing a new proof, the proof verifier simply 
checks that a provided formal proof (or, in some cases, instructions that can be followed to create a formal proof) is 
correct. This is not merely hypothetical; systems such as Isabelle are used today to formalize proofs and then check 
their validity. 

Many theories of interest include an infinite set of axioms, however. To verify a formal proof when the set of axioms 
is infinite, it must be possible to determine whether a statement that is claimed to be an axiom is actually an axiom. 
This issue arises in first order theories of arithmetic, such as Peano arithmetic, because the principle of mathematical 
induction is expressed as an infinite set of axioms (an axiom schema). 



Godel's incompleteness theorems 45 

A formal theory is said to be effectively generated if its set of axioms is a recursively enumerable set. This means 
that there is a computer program that, in principle, could enumerate all the axioms of the theory without listing any 
statements that are not axioms. This is equivalent to the ability to enumerate all the theorems of the theory without 
enumerating any statements that are not theorems. For example, each of the theories of Peano arithmetic and 
Zermelo— Fraenkel set theory has an infinite number of axioms and each is effectively generated. 

In choosing a set of axioms, one goal is to be able to prove as many correct results as possible, without proving any 
incorrect results. A set of axioms is complete if, for any statement in the axioms' language, either that statement or its 
negation is provable from the axioms. A set of axioms is (simply) consistent if there is no statement such that both 
the statement and its negation are provable from the axioms. In the standard system of first-order logic, an 
inconsistent set of axioms will prove every statement in its language (this is sometimes called the principle of 
explosion), and is thus automatically complete. A set of axioms that is both complete and consistent, however, 
proves a maximal set of non-contradictory theorems. Godel's incompleteness theorems show that in certain cases it is 
not possible to obtain an effectively generated, complete, consistent theory. 

First incompleteness theorem 

Godel's first incompleteness theorem states that: 

Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and 
complete. In particular, for any consistent, effectively generated formal theory that proves certain basic 
arithmetic truths, there is an arithmetical statement that is true, but not provable in the theory (Kleene 1967, 
p. 250). 

The true but unprovable statement referred to by the theorem is often referred to as "the Godel sentence" for the 
theory. It is not unique; there are infinitely many statements in the language of the theory that share the property of 
being true but unprovable. 

For each consistent formal theory T having the required small amount of number theory, the corresponding Godel 
sentence G asserts: "G cannot be proved to be true within the theory 7™. If G were provable under the axioms and 
rules of inference of T, then T would have a theorem, G, which effectively contradicts itself, and thus the theory T 
would be inconsistent. This means that if the theory T is consistent then G cannot be proved within it. This means 
that G's claim about its own unprovability is correct; in this sense G is not only unprovable but true. Thus 
provability -within-the-theory-T is not the same as truth; the theory 7 is incomplete. 

If G is true: G cannot be proved within the theory, and the theory is incomplete. If G is false: then G can be proved 
within the theory and then the theory is inconsistent, since G is both provable and refutable from T. 

Each theory has its own Godel statement. It is possible to define a larger theory T that contains the whole of T, plus 
G as an additional axiom. This will not result in a complete theory, because Godel's theorem will also apply to T\ 
and thus 7" cannot be complete. In this case, G is indeed a theorem in 7", because it is an axiom. Since G states only 
that it is not provable in T, no contradiction is presented by its provability in 7". However, because the 
incompleteness theorem applies to 7": there will be a new Godel statement G' for 7", showing that 7" is also 
incomplete. G' will differ from G in that G' will refer to 7", rather than T. 

To prove the first incompleteness theorem, Godel represented statements by numbers. Then the theory at hand, 
which is assumed to prove certain facts about numbers, also proves facts about its own statements. Questions about 
the provability of statements are represented as questions about the properties of numbers, which would be decidable 
by the theory if it were complete. In these terms, the Godel sentence states that no natural number exists with a 
certain, strange property. A number with this property would encode a proof of the inconsistency of the theory. If 
there were such a number then the theory would be inconsistent, contrary to the consistency hypothesis. So, under 
the assumption that the theory is consistent, there is no such number. 



Godel's incompleteness theorems 46 

Meaning of the first incompleteness theorem 

Godel's first incompleteness theorem shows that any consistent formal system that includes enough of the theory of 
the natural numbers is incomplete; there are true statements expressible in its language that are unprovable. Thus no 
formal system (satisfying the hypotheses of the theorem) that aims to characterize the natural numbers can actually 
do so, as there will be true number-theoretical statements which that system cannot prove. This fact is sometimes 
thought to have severe consequences for the program of logicism proposed by Gottlob Frege and Bertrand Russell, 
which aimed to define the natural numbers in terms of logic (Hellman 1981, p. 451—468). Some (like Bob Hale and 
Crispin Wright) believe that it is not a problem for logicism because the incompleteness theorems apply equally to 
second order logic as they do to arithmetic. It is only those who believe that the natural numbers are to be defined in 
terms of first order logic — which is consistent and complete — who have this problem. 

The existence of an incomplete formal system is in itself not particularly surprising. A system may be incomplete 
simply because not all the necessary axioms have been discovered. For example, Euclidean geometry without the 
parallel postulate is incomplete; it is not possible to prove or disprove the parallel postulate from the remaining 
axioms. 

Godel's theorem shows that, in theories that include a small portion of number theory, a complete and consistent 
finite list of axioms can never be created, nor even an infinite list that can be enumerated by a computer program. 
Each time a new statement is added as an axiom, there are other true statements that still cannot be proved, even with 
the new axiom. If an axiom is ever added that makes the system complete, it does so at the cost of making the system 
inconsistent. 

It is possible to have a complete and consistent list of axioms that cannot be enumerated by a computer program. For 
example, one might take all true statements about the natural numbers to be axioms (and no false statements). But 
then there is no mechanical way to decide, given a statement about the natural numbers, whether it is an axiom or 
not, and thus no effective way to verify a formal proof in this theory. 

Many logicians believe that Godel's incompleteness theorems struck a fatal blow to David Hilbert's second problem, 
which asked for a finitary consistency proof for mathematics. The second incompleteness theorem, in particular, is 
often viewed as making the problem impossible. Not all mathematicians agree with this analysis, however, and the 
status of Hilbert's second problem is not yet decided (see "Modern viewpoints on the status of the problem"). 

Relation to the liar paradox 

The liar paradox is the sentence "This sentence is false." An analysis of the liar sentence shows that it cannot be true 
(for then, as it asserts, it is false), nor can it be false (for then, it is true). A Godel sentence G for a theory T makes a 
similar assertion to the liar sentence, but with truth replaced by provability: G says "G is not provable in the theory 
77' The analysis of the truth and provability of G is a formalized version of the analysis of the truth of the liar 
sentence. 

It is not possible to replace "not provable" with "false" in a Godel sentence because the predicate "Q is the Godel 
number of a false formula" cannot be represented as a formula of arithmetic. This result, known as Tarski's 
undefinability theorem, was discovered independently by Godel (when he was working on the proof of the 
incompleteness theorem) and by Alfred Tarski. 



Godel's incompleteness theorems 47 

Original statements 

The first incompleteness theorem first appeared as "Theorem VI" in Godel's 1931 paper On Formally Undecidable 
Propositions in Principia Mathematica and Related Systems I. The second incompletness theorem appeared as 
"Theorem XI" in the same paper. 

Extensions of Godel's original result 

Godel demonstrated the incompleteness of the theory of Principia Mathematica, a particular theory of arithmetic, but 
a parallel demonstration could be given for any effective theory of a certain expressiveness. Godel commented on 
this fact in the introduction to his paper, but restricted the proof to one system for concreteness. In modern 
statements of the theorem, it is common to state the effectiveness and expressiveness conditions as hypotheses for 
the incompleteness theorem, so that it is not limited to any particular formal theory. The terminology used to state 
these conditions was not yet developed in 1931 when Godel published his results. 

Godel's original statement and proof of the incompleteness theorem requires the assumption that the theory is not just 
consistent but co-consistent. A theory is to-consistent if it is not to-inconsistent, and is co-inconsistent if there is a 
predicate P such that for every specific natural number n the theory proves ~P{n), and yet the theory also proves that 
there exists a natural number n such that P(n). That is, the theory says that a number with property P exists while 
denying that it has any specific value. The co-consistency of a theory implies its consistency, but consistency does 
not imply co-consistency. J. Barkley Rosser (1936) strengthened the incompleteness theorem by finding a variation 
of the proof (Rosser's trick) that only requires the theory to be consistent, rather than co-consistent. This is mostly of 
technical interest, since all true formal theories of arithmetic (theories whose axioms are all true statements about 
natural numbers) are co-consistent, and thus Godel's theorem as originally stated applies to them. The stronger 
version of the incompleteness theorem that only assumes consistency, rather than co-consistency, is now commonly 
known as Godel's incompleteness theorem and as the Godel— Rosser theorem. 

Second incompleteness theorem 

Godel's second incompleteness theorem can be stated as follows: 

For any formal effectively generated theory T including basic arithmetical truths and also certain truths about 
formal provability, T includes a statement of its own consistency if and only if T is inconsistent. 

This strengthens the first incompleteness theorem, because the statement constructed in the first incompleteness 
theorem does not directly express the consistency of the theory. The proof of the second incompleteness theorem is 
obtained, essentially, by formalizing the proof of the first incompleteness theorem within the theory itself. 

A technical subtlety in the second incompleteness theorem is how to express the consistency of T as a formula in the 
language of T. There are many ways to do this, and not all of them lead to the same result. In particular, different 
formalizations of the claim that T is consistent may be inequivalent in T, and some may even be provable. For 
example, first-order Peano arithmetic (PA) can prove that the largest consistent subset of PA is consistent. But since 
PA is consistent, the largest consistent subset of PA is just PA, so in this sense PA "proves that it is consistent". 
What PA does not prove is that the largest consistent subset of PA is, in fact, the whole of PA. (The term "largest 
consistent subset of PA" is rather vague, but what is meant here is the largest consistent initial segment of the axioms 
of PA ordered according to some criteria; for example, by "Godel numbers", the numbers encoding the axioms as per 
the scheme used by Godel mentioned above). 

In the case of Peano arithmetic, or any familiar explicitly axiomatized theory T, it is possible to canonically define a 
formula Con(7) expressing the consistency of T; this formula expresses the property that "there does not exist a 
natural number coding a sequence of formulas, such that each formula is either one of the axioms of T, a logical 
axiom, or an immediate consequence of preceding formulas according to the rules of inference of first-order logic, 
and such that the last formula is a contradiction". 



Godel's incompleteness theorems 48 

The formalization of Con(7) depends on two factors: formalizing the notion of a sentence being derivable from a set 
of sentences and formalizing the notion of being an axiom of T. Formalizing derivability can be done in canonical 
fashion: given an arithmetical formula A(x) defining a set of axioms, one can canonically form a predicate Prov (P) 
which expresses that P is provable from the set of axioms defined by A(x). In addition, Prov (P) must satisfy the 
so-called Hilbert— Bernays provability conditions: 

1 . If T proves P, then T proves Prov (P). 

2. Tproves 1.; that is, P proves that if P proves P, then Tproves Prov (P). 

3. T proves that if T proves that (P — > Q) then T proves that provability of P implies provability of Q. 

Implications for consistency proofs 

Godel's second incompleteness theorem also implies that a theory T satisfying the technical conditions outlined 
above cannot prove the consistency of any theory T which proves the consistency of T . This is because such a 
theory T can prove that if T proves the consistency of T , then T is in fact consistent. For the claim that T is 
consistent has form "for all numbers n, n has the decidable property of not being a code for a proof of contradiction 
in T ". If T were in fact inconsistent, then T would prove for some n that n is the code of a contradiction in T . But 
if T also proved that T is consistent (that is, that there is no such n), then it would itself be inconsistent. This 
reasoning can be formalized in T to show that if T is consistent, then T is consistent. Since, by second 
incompleteness theorem, T does not prove its consistency, it cannot prove the consistency of T either. 

This corollary of the second incompleteness theorem shows that there is no hope of proving, for example, the 
consistency of Peano arithmetic using any finitistic means that can be formalized in a theory the consistency of 
which is provable in Peano arithmetic. For example, the theory of primitive recursive arithmetic (PRA), which is 
widely accepted as an accurate formalization of finitistic mathematics, is provably consistent in PA. Thus PRA 
cannot prove the consistency of PA. This fact is generally seen to imply that Hilbert's program, which aimed to 
justify the use of "ideal" (infinitistic) mathematical principles in the proofs of "real" (finitistic) mathematical 
statements by giving a finitistic proof that the ideal principles are consistent, cannot be carried out. 

The corollary also indicates the epistemological relevance of the second incompleteness theorem. As Georg Kreisel 
remarked, it would actually provide no interesting information if a theory T proved its consistency. This is because 
inconsistent theories prove everything, including their consistency. Thus a consistency proof of Tin T would give us 
no clue as to whether T really is consistent; no doubts about the consistency of T would be resolved by such a 
consistency proof. The interest in consistency proofs lies in the possibility of proving the consistency of a theory T in 
some theory T' which is in some sense less doubtful than T itself, for example weaker than T. For many naturally 
occurring theories T and 7", such as 7= Zermelo— Fraenkel set theory and T = primitive recursive arithmetic, the 
consistency of 7" is provable in T, and thus 7" can't prove the consistency of T by the above corollary of the second 
incompleteness theorem. 

The second incompleteness theorem does not rule out consistency proofs altogether, only consistency proofs that 
could be formalized in the theory that is proved consistent. For example, Gerhard Gentzen proved the consistency of 
Peano arithmetic (PA) using the assumption that a certain ordinal called e is actually wellfounded; see Gentzen's 
consistency proof. Gentzen's theorem spurred the development of ordinal analysis in proof theory. 

Examples of undecidable statements 

There are two distinct senses of the word "undecidable" in mathematics and computer science. The first of these is 
the proof-theoretic sense used in relation to Godel's theorems, that of a statement being neither provable nor 
refutable in a specified deductive system. The second sense, which will not be discussed here, is used in relation to 
computability theory and applies not to statements but to decision problems, which are countably infinite sets of 
questions each requiring a yes or no answer. Such a problem is said to be undecidable if there is no computable 
function that correctly answers every question in the problem set (see undecidable problem). 



Godel's incompleteness theorems 49 

Because of the two meanings of the word undecidable, the term independent is sometimes used instead of 
undecidable for the "neither provable nor refutable" sense. The usage of "independent" is also ambiguous, however. 
Some use it to mean just "not provable", leaving open whether an independent statement might be refuted. 

Undecidability of a statement in a particular deductive system does not, in and of itself, address the question of 
whether the truth value of the statement is well-defined, or whether it can be determined by other means. 
Undecidability only implies that the particular deductive system being considered does not prove the truth or falsity 
of the statement. Whether there exist so-called "absolutely undecidable" statements, whose truth value can never be 
known or is ill-specified, is a controversial point in the philosophy of mathematics. 

The combined work of Godel and Paul Cohen has given two concrete examples of undecidable statements (in the 
first sense of the term): The continuum hypothesis can neither be proved nor refuted in ZFC (the standard 
axiomatization of set theory), and the axiom of choice can neither be proved nor refuted in ZF (which is all the ZFC 
axioms except the axiom of choice). These results do not require the incompleteness theorem. Godel proved in 1940 
that neither of these statements could be disproved in ZF or ZFC set theory. In the 1960s, Cohen proved that neither 
is provable from ZF, and the continuum hypothesis cannot be proven from ZFC. 

In 1973, the Whitehead problem in group theory was shown to be undecidable, in the first sense of the term, in 
standard set theory. 

In 1977, Paris and Harrington proved that the Paris-Harrington principle, a version of the Ramsey theorem, is 
undecidable in the first-order axiomatization of arithmetic called Peano arithmetic, but can be proven to be true in 
the larger system of second-order arithmetic. Kirby and Paris later showed Goodstein's theorem, a statement about 
sequences of natural numbers somewhat simpler than the Paris-Harrington principle, to be undecidable in Peano 
arithmetic. 

Kruskal's tree theorem, which has applications in computer science, is also undecidable from Peano arithmetic but 
provable in set theory. In fact Kruskal's tree theorem (or its finite form) is undecidable in a much stronger system 
codifying the principles acceptable on the basis of a philosophy of mathematics called predicativism. The related but 
more general graph minor theorem (2003) has consequences for computational complexity theory. 

Gregory Chaitin produced undecidable statements in algorithmic information theory and proved another 
incompleteness theorem in that setting. Chaitin's theorem states that for any theory that can represent enough 
arithmetic, there is an upper bound c such that no specific number can be proven in that theory to have Kolmogorov 
complexity greater than c. While Godel's theorem is related to the liar paradox, Chaitin's result is related to Berry's 
paradox. 

Limitations of Godel's theorems 

The conclusions of Godel's theorems only hold for the formal theories that satisfy the necessary hypotheses. Not all 
axiom systems satisfy these hypotheses, even when these systems have models that include the natural numbers as a 
subset. For example, there are first-order axiomatizations of Euclidean geometry, of real closed fields, and of 
arithmetic in which multiplication is not provably total; none of these meet the hypotheses of Godel's theorems. The 
key fact is that these axiomatizations are not expressive enough to define the set of natural numbers or develop basic 
properties of the natural numbers. Regarding the third example, Dan E. Willard (Willard 2001) has studied many 
weak systems of arithmetic which do not satisfy the hypotheses of the second incompleteness theorem, and which 
are consistent and capable of proving their own consistency (see self-verifying theories). 

Godel's theorems only apply to effectively generated (that is, recursively enumerable) theories. If all true statements 
about natural numbers are taken as axioms for a theory, then this theory is a consistent, complete extension of Peano 
arithmetic (called true arithmetic) for which none of Godel's theorems hold, because this theory is not recursively 
enumerable. 



Godel's incompleteness theorems 50 

The second incompleteness theorem only shows that the consistency of certain theories cannot be proved from the 
axioms of those theories themselves. It does not show that the consistency cannot be proved from other (consistent) 
axioms. For example, the consistency of the Peano arithmetic can be proved in Zermelo— Fraenkel set theory (ZFC), 
or in theories of arithmetic augmented with transfinite induction, as in Gentzen's consistency proof. 

Relationship with computability 

The incompleteness theorem is closely related to several results about undecidable sets in recursion theory. 

Stephen Cole Kleene (1943) presented a proof of Godel's incompleteness theorem using basic results of 
computability theory. One such result shows that the halting problem is unsolvable: there is no computer program 
that can correctly determine, given a program P as input, whether P eventually halts when run with no input. Kleene 
showed that the existence of a complete effective theory of arithmetic with certain consistency properties would 
force the halting problem to be decidable, a contradiction. This method of proof has also been presented by 
Shoenfield (1967, p. 132); Charlesworth (1980); and Hopcroft and Ullman (1979). 

Franzen (2005, p. 73) explains how Matiyasevich's solution to Hilbert's 10th problem can be used to obtain a proof 
to Godel's first incompleteness theorem. Matiyasevich proved that there is no algorithm that, given a multivariate 
polynomial p(x , x ,...,x ) with integer coefficients, determines whether there is an integer solution to the equation p 
= 0. Because polynomials with integer coefficients, and integers themselves, are directly expressible in the language 
of arithmetic, if a multivariate integer polynomial equation p = does have a solution in the integers then any 
sufficiently strong theory of arithmetic Twill prove this. Moreover, if the theory 7 is co-consistent, then it will never 
prove that some polynomial equation has a solution when in fact there is no solution in the integers. Thus, if T were 
complete and co-consistent, it would be possible to algorithmically determine whether a polynomial equation has a 
solution by merely enumerating proofs of T until either "p has a solution" or "p has no solution" is found, in 
contradiction to Matiyasevich's theorem. 

Smorynski (1971, p. 842) shows how the existence of recursively inseparable sets can be used to prove the first 
incompleteness theorem. This proof is often extended to show that systems such as Peano arithmetic are essentially 
undecidable (see Kleene 1967, p. 274). 

Proof sketch for the first theorem 

Throughout the proof we assume a formal system is fixed and satisfies the necessary hypotheses. The proof has three 
essential parts. The first part is to show that statements can be represented by natural numbers, known as Godel 
numbers, and that properties of the statements can be detected by examining their Godel numbers. This part 
culminates in the construction of a formula expressing the idea that a statement is provable in the system. The second 
part of the proof is to construct a particular statement that, essentially, says that it is unprovable. The third part of the 
proof is to analyze this statement to show that it is neither provable nor disprovable in the system. 

Arithmetization of syntax 

The main problem in fleshing out the proof described above is that it seems at first that to construct a statement p 
that is equivalent to "p cannot be proved", p would have to somehow contain a reference to p, which could easily 
give rise to an infinite regress. Godel's ingenious trick, which was later used by Alan Turing in his work on the 
Entscheidungsproblem, is to represent statements as numbers, which is often called the arithmetization of syntax. 
This allows a self-referential formula to be constructed in a way that avoids any infinite regress of definitions. 

To begin with, every formula or statement that can be formulated in our system gets a unique number, called its 
Godel number. This is done in such a way that it is easy to mechanically convert back and forth between formulas 
and Godel numbers. It is similar, for example, to the way English sentences are encoded as sequences (or "strings") 
of numbers using ASCII: such a sequence is considered as a single (if potentially very large) number. Because our 



Godel's incompleteness theorems 51 

system is strong enough to reason about numbers, it is now also possible to reason about formulas within the system. 

A formula F(x) that contains exactly one free variable x is called a statement form or class-sign. As soon as x is 
replaced by a specific number, the statement form turns into a bona fide statement, and it is then either provable in 
the system, or not. For certain formulas one can show that for every natural number n, F(n) is true if and only if it 
can be proven (the precise requirement in the original proof is weaker, but for the proof sketch this will suffice). In 
particular, this is true for every specific arithmetic operation between a finite number of natural numbers, such as 
"2x3=6". 

Statement forms themselves are not statements and therefore cannot be proved or disproved. But every statement 
form F(x) can be assigned with a Godel number which we will denote by G(F). The choice of the free variable used 
in the form F{x) is not relevant to the assignment of the Godel number G(F). 

Now comes the trick: The notion of provability itself can also be encoded by Godel numbers, in the following way. 
Since a proof is a list of statements which obey certain rules, we can define the Godel number of a proof. Now, for 
every statement p, we may ask whether a number x is the Godel number of its proof. The relation between the Godel 
number of p and x, the Godel number of its proof, is an arithmetical relation between two numbers. Therefore there 
is a statement form Bew(x) that uses this arithmetical relation to state that a Godel number of a proof of x exists: 

Bew(;y) = 3 x ( y is the Godel number of a formula and x is the Godel number of a proof of the formula 
encoded by y). 

The name Bew is short for beweisbar, the German word for "provable"; this name was originally used by Godel to 
denote the provability formula just described. Note that "Bew(j)" is merely an abbreviation that represents a 
particular, very long, formula in the original language of T; the string "Bew" itself is not claimed to be part of this 
language. 

An important feature of the formula Bew(y) is that if a statement p is provable in the system then Bew(G(/?)) is also 
provable. This is because any proof of p would have a corresponding Godel number, the existence of which causes 
Bew(G(/?)) to be satisfied. 

Diagonalization 

The next step in the proof is to obtain a statement that says it is unprovable. Although Godel constructed this 
statement directly, the existence of at least one such statement follows from the diagonal lemma, which says that for 
any sufficiently strong formal system and any statement form F there is a statement p such that the system proves 

p <-> F(G(p)). 

We obtain p by letting F be the negation of Bew(x); thus p roughly states that its own Godel number is the Godel 
number of an unprovable formula. 

The statement p is not literally equal to ~Bew(G(p)); rather, p states that if a certain calculation is performed, the 
resulting Godel number will be that of an unprovable statement. But when this calculation is performed, the resulting 
Godel number turns out to be the Godel number of p itself. This is similar to the following sentence in English: 

", when preceded by itself in quotes, is unprovable.", when preceded by itself in quotes, is unprovable. 

This sentence does not directly refer to itself, but when the stated transformation is made the original sentence is 
obtained as a result, and thus this sentence asserts its own unprovability. The proof of the diagonal lemma employs a 
similar method. 



Godel's incompleteness theorems 52 

Proof of independence 

We will now assume that our axiomatic system is co-consistent. We let p be the statement obtained in the previous 
section. 

If p were provable, then Bew(G(/?)) would be provable, as argued above. But p asserts the negation of Bew(G(/?)). 
Thus our system would be inconsistent, proving both a statement and its negation. This contradiction shows that p 
cannot be provable. 

If the negation of p were provable, then Bew(G(/?)) would be provable (because p was constructed to be equivalent 
to the negation of Bew(G(/?))). However, for each specific number x, x cannot be the Godel number of the proof of p, 
because p is not provable (from the previous paragraph). Thus on one hand the system proves there is a number with 
a certain property (that it is the Godel number of the proof of p), but on the other hand, for every specific number x, 
we can prove that it does not have this property. This is impossible in an co-consistent system. Thus the negation of p 
is not provable. 

Thus the statement/? is undecidable: it can neither be proved nor disproved within the system. 

It should be noted that p is not provable (and thus true) in every consistent system. The assumption of co-consistency 
is only required for the negation of p to be not provable. Thus: 

• In an co-consistent formal system, we may prove neither p nor its negation, and so p is undecidable. 

• In a consistent formal system we may either have the same situation, or we may prove the negation of p; In the 
later case, we have a statement ("not/?") which is false but provable. 

Note that if one tries to "add the missing axioms" to avoid the undecidability of the system, then one has to add 
either p or "not p" as axioms. But then the definition of "being a Godel number of a proof" of a statement changes, 
which means that the statement form Bew(x) is now different. Thus when we apply the diagonal lemma to this new 
form Bew, we obtain a new statement p, different from the previous one, which will be undecidable in the new 
system if it is co-consistent. 

Proof via Berry's paradox 

George Boolos (1989) sketches an alternative proof of the first incompleteness theorem that uses Berry's paradox 
rather than the liar paradox to construct a true but unprovable formula. A similar proof method was independently 
discovered by Saul Kripke (Boolos 1998, p. 383). Boolos's proof proceeds by constructing, for any computably 
enumerable set S of true sentences of arithmetic, another sentence which is true but not contained in S. This gives the 
first incompleteness theorem as a corollary. According to Boolos, this proof is interesting because it provides a 
"different sort of reason" for the incompleteness of effective, consistent theories of arithmetic (Boolos 1998, p. 388). 

Formalized proofs 

Formalized proofs of versions of the incompleteness theorem have been developed by N. Shankar in 1986 using 
Nqthm (Shankar 1994) and by R. O'Connor in 2003 using Coq (O'Connor 2005). 

Proof sketch for the second theorem 

The main difficulty in proving the second incompleteness theorem is to show that various facts about provability 
used in the proof of the first incompleteness theorem can be formalized within the system using a formal predicate 
for provability. Once this is done, the second incompleteness theorem essentially follows by formalizing the entire 
proof of the first incompleteness theorem within the system itself. 

Let p stand for the undecidable sentence constructed above, and assume that the consistency of the system can be 
proven from within the system itself. We have seen above that if the system is consistent, then p is not provable. The 
proof of this implication can be formalized within the system, and therefore the statement "p is not provable", or "not 



Godel's incompleteness theorems 53 

P(p)" can be proven in the system. 

But this last statement is equivalent to p itself (and this equivalence can be proven in the system), so p can be proven 
in the system. This contradiction shows that the system must be inconsistent. 

Discussion and implications 

The incompleteness results affect the philosophy of mathematics, particularly versions of formalism, which use a 
single system formal logic to define their principles. One can paraphrase the first theorem as saying the following: 

We can never find an all-encompassing axiomatic system which is able to prove all mathematical truths, but 
no falsehoods. 

On the other hand, from a strict formalist perspective this paraphrase would be considered meaningless because it 
presupposes that mathematical "truth" and "falsehood" are well-defined in an absolute sense, rather than relative to 
each formal system. 

The following rephrasing of the second theorem is even more unsettling to the foundations of mathematics: 

If an axiomatic system can be proven to be consistent and complete from within itself, then it is inconsistent. 

Therefore, to establish the consistency of a system S, one needs to use some other more powerful system T, but a 
proof in T is not completely convincing unless T's consistency has already been established without using S. 

At first, Godel's theorems seemed to leave some hope — it was thought that it might be possible to produce a general 
algorithm that indicates whether a given statement is undecidable or not, thus allowing mathematicians to bypass the 
undecidable statements altogether. However, the negative answer to the Entscheidungsproblem, obtained in 1936, 
showed that no such algorithm exists. 

There are some who hold that a statement that is unprovable within a deductive system may be quite provable in a 
metalanguage. And what cannot be proven in that metalanguage can likely be proven in a meta-metalanguage, 
recursively, ad infinitum, in principle. By invoking such a system of typed metalanguages, along with an axiom of 
Reducibility — which by an inductive assumption applies to the entire stack of languages — one may, for all 
practical purposes, overcome the obstacle of incompleteness. 

Note that Godel's theorems only apply to sufficiently strong axiomatic systems. "Sufficiently strong" means that the 
theory contains enough arithmetic to carry out the coding constructions needed for the proof of the first 
incompleteness theorem. Essentially, all that is required are some basic facts about addition and multiplication as 
formalized, for example, in Robinson arithmetic Q. There are even weaker axiomatic systems that are consistent and 
complete, for instance Presburger arithmetic which proves every true first-order statement involving only addition. 

The axiomatic system may consist of infinitely many axioms (as first-order Peano arithmetic does), but for Godel's 
theorem to apply, there has to be an effective algorithm which is able to check proofs for correctness. For instance, 
one might take the set of all first-order sentences which are true in the standard model of the natural numbers. This 
system is complete; Godel's theorem does not apply because there is no effective procedure that decides if a given 
sentence is an axiom. In fact, that this is so is a consequence of Godel's first incompleteness theorem. 

Another example of a specification of a theory to which Godel's first theorem does not apply can be constructed as 
follows: order all possible statements about natural numbers first by length and then lexicographically, start with an 
axiomatic system initially equal to the Peano axioms, go through your list of statements one by one, and, if the 
current statement cannot be proven nor disproven from the current axiom system, add it to that system. This creates a 
system which is complete, consistent, and sufficiently powerful, but not computably enumerable. 

Godel himself only proved a technically slightly weaker version of the above theorems; the first proof for the 
versions stated above was given by J. Barkley Rosser in 1936. 

In essence, the proof of the first theorem consists of constructing a statement p within a formal axiomatic system that 
can be given a meta-mathematical interpretation of: 



Godel's incompleteness theorems 54 

p = "This statement cannot be proven in the given formal theory" 

As such, it can be seen as a modern variant of the Liar paradox, although unlike the classical paradoxes it is not 
really paradoxical. 

If the axiomatic system is consistent, Godel's proof shows that p (and its negation) cannot be proven in the system. 
Therefore p is true (p claims to be not provable, and it is not provable) yet it cannot be formally proved in the 
system. If the axiomatic system is to-consistent, then the negation of p cannot be proven either, and so p is 
undecidable. In a system which is not to-consistent (but consistent), either we have the same situation, or we have a 
false statement which can be proven (namely, the negation of p). 

Adding p to the axioms of the system would not solve the problem: there would be another Godel sentence for the 
enlarged theory. Theories such as Peano arithmetic, for which any computably enumerable consistent extension is 
incomplete, are called essentially incomplete. 

Minds and machines 

Authors including J. R. Lucas have debated what, if anything, Godel's incompleteness theorems imply about human 
intelligence. Much of the debate centers on whether the human mind is equivalent to a Turing machine, or by the 
Church— Turing thesis, any finite machine at all. If it is, and if the machine is consistent, then Godel's incompleteness 
theorems would apply to it. 

Hilary Putnam (1960) suggested that while Godel's theorems cannot be applied to humans, since they make mistakes 
and are therefore inconsistent, it may be applied to the human faculty of science or mathematics in general. If we are 
to assume that it is consistent, then either we cannot prove its consistency, or it cannot be represented by a Turing 
machine. 

Appeals to the incompleteness theorems in other fields 

Appeals and analogies are sometimes made to the incompleteness theorems in support of arguments that go beyond 
mathematics and logic. A number of authors have commented negatively on such extensions and interpretations, 
including Torkel Franzen (2005); Alan Sokal and Jean Bricmont (1999); and Ophelia Benson and Jeremy Stangroom 
(2006). Bricmont and Stangroom (2006, p. 10), for example, quote from Rebecca Goldstein's comments on the 
disparity between Godel's avowed Platonism and the anti-realist uses to which his ideas are sometimes put. Sokal 
and Bricmont (1999, p. 187) criticize Regis Debray's invocation of the theorem in the context of sociology; Debray 
has defended this use as metaphorical (ibid.). Carl Hewitt (2008, 2009) has shown that Godel's result can be 
extended to prove incompleteness theorems in certain non-classical, paraconsistent logics with proposed applications 
in software engineering. 

See also 

Godel's completeness theorem 
Lob's Theorem 
Provability logic 
Mlinchhausen Trilemma 
Non-standard model of arithmetic 
Tarski's indefinability theorem 
Minds, Machines and Godel 
Third Man Argument 



Godel's incompleteness theorems 55 

References 
Articles by Godel 

• 1931, tj ber formal unentscheidbare Sdtze der Principia Mathematica und verwandter Systeme, I. Monatshefte filr 
Mathematik und Physik 38: 173-98. 

• Hirzel, Martin, 2000, On formally undecidable propositions of Principia Mathematica and related systems I. 

.A modern translation by the author. 

• 1951, Some basic theorems on the foundations of mathematics and their implications in Solomon Feferman, ed., 
1995. Collected works / Kurt Godel, Vol. III. Oxford University Press: 304-23. 

Translations, during his lifetime, of Godel's paper into English 

None of the following agree in all translated words and in typography. The typography is a serious matter, because 
Godel expressly wished to emphasize "those metamathematical notions that had been defined in their usual sense 
before . . ."(van Heijenoort 1967:595). Three translations exist. Of the first John Dawson states that: "The Meltzer 
translation was seriously deficient and received a devastating review in the Journal of Symbolic Logic; "Godel also 
complained about Braithwaite's commentary (Dawson 1997:216). "Fortunately, the Meltzer translation was soon 
supplanted by a better one prepared by Elliott Mendelson for Martin Davis's anthology The Undecidable ... he 
found the translation "not quite so good" as he had expected . . . [but because of time constraints he] agreed to its 
publication" (ibid). (In a footnote Dawson states that "he would regret his compliance, for the published volume was 
marred throughout by sloppy typography and numerous misprints" (ibid)). Dawson states that "The translation that 
Godel favored was that by Jean van Heijenoort"(ibid). For the serious student another version exists as a set of 
lecture notes recorded by Stephen Kleene and J. B. Rosser "during lectures given by Godel at to the Institute for 
Advanced Study during the spring of 1934" (cf commentary by Davis 1965:39 and beginning on p. 41); this version 
is titled "On Undecidable Propositions of Formal Mathematical Systems". In their order of publication: 

• B. Meltzer (translation) and R. B. Braithwaite (Introduction), 1962. On Formally Undecidable Propositions of 
Principia Mathematica and Related Systems, Dover Publications, New York (Dover edition 1992), ISBN 
0-486-66980-7 (pbk.) This contains a useful translation of Godel's German abbreviations on pp. 33—34. As noted 
above, typography, translation and commentary is suspect. Unfortunately, this translation was reprinted with all 
its suspect content by 

• Stephen Hawking editor, 2005. God Created the Integers: The Mathematical Breakthroughs That Changed 
History, Running Press, Philadelphia, ISBN 0-7624-1922-9. Godel's paper appears starting on p. 1097, with 
Hawking's commentary starting on p. 1089. 

• Martin Davis editor, 1965. The Undecidable: Basic Papers on Undecidable Propositions, Unsolvable problems 
and Computable Functions, Raven Press, New York, no ISBN. Godel's paper begins on page 5, preceded by one 
page of commentary. 

• Jean van Heijenoort editor, 1967, 3rd edition 1967. From Frege to Godel: A Source Book in Mathematical Logic, 
1979-1931, Harvard University Press, Cambridge Mass., ISBN 0-674-32449-8 (pbk). van Heijenoort did the 
translation. He states that "Professor Godel approved the translation, which in many places was accommodated to 
his wishes. "(p. 595). Godel's paper begins on p. 595; van Heijenoort's commentary begins on p. 592. 

• Martin Davis editor, 1965, ibid. "On Undecidable Propositions of Formal Mathematical Systems." A copy with 
Godel's corrections of errata and Godel's added notes begins on page 41, preceded by two pages of Davis's 
commentary. Until Davis included this in his volume this lecture existed only as mimeographed notes. 



Godel's incompleteness theorems 56 

Articles by others 

George Boolos, 1989, "A New Proof of the Godel Incompleteness Theorem", Notices of the American 

Mathematical Society v. 36, pp. 388—390 and p. 676, reprinted in in Boolos, 1998, Logic, Logic, and Logic, 

Harvard Univ. Press. ISBN 674 53766 1 

Arthur Charlesworth, 1980, "A Proof of Godel's Theorem in Terms of Computer Programs," Mathematics 

Magazine, v. 54 n. 3, pp. 109-121. JStor [4] 

Solomon Feferman, 1984, Toward Useful Type-Free Theories, I , Journal of Symbolic Logic, v. 49 n. 1, 

pp. 75-111. 

Jean van Heijenoort, 1963. "Godel's Theorem" in Edwards, Paul, ed., Encyclopedia of Philosophy, Vol. 3. 

Macmillan: 348-57. 

Geoffrey Hellman, How to Godel a Frege-Russell: Godel's Incompleteness Theorems and Logicism. Nous, Vol. 

15, No. 4, Special Issue on Philosophy of Mathematics. (Nov., 1981), pp. 451-468. 

David Hilbert, 1900, "Mathematical Problems. English translation of a lecture delivered before the 

International Congress of Mathematicians at Paris, containing Hilbert's statement of his Second Problem. 

Kikuchi, Makoto; Tanaka, Kazuyuki (1994), "On formalization of model-theoretic proofs of Godel's theorems", 

Notre Dame Journal of Formal Logic 35 (3): 403-412, doi:10.1305/ndjfl/1040511346, MR1326122, 

ISSN 0029-4527 

Stephen Cole Kleene, 1943, "Recursive predicates and quantifiers," reprinted from Transactions of the American 

Mathematical Society, v. 53 n. 1, pp. 41—73 in Martin Davis 1965, The Undecidable (loc. cit.) pp. 255—287. 

Hilary Putnam, 1960, Minds and Machines in Sidney Hook, ed., Dimensions of Mind: A Symposium. New York 

University Press. Reprinted in Anderson, A. R., ed., 1964. Minds and Machines. Prentice-Hall: 77. 

Russell O'Connor (2005), "Essential Incompleteness of Arithmetic Verified by Coq" , Lecture Notes in 

Computer Science, 3603, pp. 245-260 

John Barkley Rosser, 1936, "Extensions of some theorems of Godel and Church," reprinted from the Journal of 

Symbolic Logic vol. 1 (1936) pp. 87—91, in Martin Davis 1965, The Undecidable (loc. cit.) pp. 230—235. 

John Barkley Rosser, 1939, "An Informal Exposition of proofs of Godel's Theorem and Church's Theorem", 

Reprinted from the Journal of Symbolic Logic, vol. 4 (1939) pp. 53—60, in Martin Davis 1965, The Undecidable 

(loc. cit.) pp. 223-230 

C. Smoryhski, "The incompleteness theorems", in J. Barwise, ed., Handbook of Mathematical Logic, 

North-Holland 1982 ISBN 978-0444863881, pp. 821-866. 

Dan E. Willard (2001), "Self-Verifying Axiom Systems, the Incompleteness Theorem and Related Reflection 

ro] 

Principles ", Journal of Symbolic Logic, v. 66 n. 2, pp. 536—596. doi: 10.2307/2695030 

Richard Zach, 2005, "Paper on the incompleteness theorems" in Grattan-Guinness, I., ed., Landmark Writings in 

Western Mathematics. Elsevier: 917-25. 

Books about the theorems 

• Domeisen, Norbert, 1990. Logik der Antinomien. Bern: Peter Lang. 142 S. 1990. ISBN 3-261-04214-1. 
Zentralblatt MATH [9] 

• Torkel Franzen, 2005. Godel's Theorem: An Incomplete Guide to its Use and Abuse. A.K. Peters. ISBN 
1568812388 MR2007d:03001 

• Douglas Hofstadter, 1979. Godel, Escher, Bach: An Eternal Golden Braid. Vintage Books. ISBN 0465026850. 
1999 reprint: ISBN 0465026567. MR80j:03009 

• Douglas Hofstadter, 2007. 1 Am a Strange Loop. Basic Books. ISBN 9780465030781. ISBN 0465030785. 
MR2008g:00004 

• Stanley Jaki, OSB, 2005. The drama of the quantities. Real View Books. 

• J.R. Lucas, FBA, 1970. The Freedom of the Will. Clarendon Press, Oxford, 1970. 



Godel's incompleteness theorems 57 

Ernest Nagel, James Roy Newman, Douglas Hofstadter, 2002 (1958). Godel's Proof, revised ed. ISBN 

0814758169. MR2002i:03001 

Rudy Rucker, 1995 (1982). Infinity and the Mind: The Science and Philosophy of the Infinite. Princeton Univ. 

Press. MR84d:03012 

Smith, Peter, 2007. An Introduction to Godel's Theorems. Cambridge University Press. MathSciNet 

N. Shankar, 1994. Metamathematics, Machines and Godel's Proof Volume 38 of Cambridge tracts in theoretical 

computer science. ISBN 0521585333 

Raymond Smullyan, 1991. Godel's Incompleteness Theorems. Oxford Univ. Press. 

— , 1994. Diagonalization and Self-Reference. Oxford Univ. Press. MR96c:03001 

Hao Wang, 1997. A Logical Journey: From Godel to Philosophy. MIT Press. ISBN 0262231891 MR97m:01090 

Miscellaneous references 

John W. Dawson, Jr., 1997. Logical Dilemmas: The Life and Work of Kurt Godel, A.K. Peters, Wellesley Mass, 

ISBN 1-56881-256-6. 

Goldstein, Rebecca, 2005, Incompleteness: the Proof and Paradox of Kurt Godel, W. W. Norton & Company. 

ISBN 0-393-05169-2 

Carl Hewitt, 2008, Large-scale Organizational Computing requires Unstratified Reflection and Strong 

ri3i 

Paraconsistency , Coordination, Organizations, Institutions, and Norms in Agent Systems III, Springer- Verlag. 

Carl Hewitt, 2009, Common sense for concurrency and inconsistency tolerance using Direct Logic ArXiv 

0812.4852. 

John Hopcroft and Jeffrey Ullman 1979, Introduction to Automata theory, Addison-Wesley, ISBN 

0-201-02988-X 

Stephen Cole Kleene, 1967, Mathematical Logic. Reprinted by Dover, 2002. ISBN 0-486-42533-9 

Alan Sokal and Jean Bricmont, 1999, Fashionable Nonsense: Postmodern Intellectuals' Abuse of Science, 

Picador. ISBN 0-31-220407-8 

Joseph R. Shoenfield (1967), Mathematical Logic. Reprinted by A.K. Peters for the Association of Symbolic 

Logic, 2001. ISBN 978-156881 135-2 

Jeremy Stangroom and Ophelia Benson, Why Truth Matters, Continuum. ISBN 0-82-649528-1 

External links 

Stanford Encyclopedia of Philosophy: "Kurt Godel — by Juliette Kennedy. 



MacTutor biographies: 

[17] 



Kurt Godel. [16] 



• Gerhard Gentzen. 

noi 

• What is Mathematics:G6del's Theorem and Around by Karlis Podnieks. An online free book. 

ri9i 

World's shortest explanation of Godel's theorem using a printing machine as an example. 



Godel's incompleteness theorems 58 

References 

[I] The word "true" is used disquotationally here: the Godel sentence is true in this sense because it "asserts its own unprovability and it is indeed 
unprovable" (Smoryhski 1977 p. 825; also see Franzen 2005 pp. 28—33). It is also possible to read "G is true" in the formal sense that 
primitive recursive arithmetic proves the implication Con(7*)— >G , where Con(T) is a canonical sentence asserting the consistency of T 
(Smorynski 1977 p. 840, Kikuchi and Tanaka 1994 p. 403) 

[2] For example, the conjunction of the Godel sentence and any logically valid sentence will have this property. 

[3] http://www.research.ibm.eom/people/h/hirzel/papers/canonOO-goedel.pdf 

[4] http://links.jstor.org/sici?sici=0025-570X%28198105%2954%3A3%3C109%3AAPOGTI%3E2.0.CO%3B2-l&size=LARGE& 

origin=JSTOR-enlargePage 
[5] http://links.jstor.org/sici?sici=0022-4812(198403)49%3Al%3C75%3ATUTTI%3E2.0.CO;2-D 
[6] http://alephO.clarku.edU/~djoyce/hilbert/problems.html#prob2 
[7] http://arxiv.org/abs/cs/0505034 

[8] http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.jsl/ 11 83746459 

[9] http://www.emis. de/cgi-bin/zmen/ZMATH/en/quick.html?first=l&maxdocs=3&type=html&an=0724.03003&format=complete 
[10] http://www.realviewbooks.com/ 

[II] http://www.godelbook.net/ 

[12] http://www.ams.org/mathscinet/search/publdoc. html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all& 

pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&s4=Smith%2C%20Peter&s5=&s6=&s7=&s8=All&yearRangeFirst=& 
yearRangeSecond=&yrop=eq&r=2&mx-pid=2384958 

[13] http://organizational.carlhewitt.info/ 

[14] http://arxiv.org/abs/0812.4852 

[15] http://plato.stanford.edu/entries/goedel/ 

[16] http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Godel.html 

[17] http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Gentzen.html 

[18] http://www. ltn.lv/~podnieks/index. html 

[19] http://blog.plover.com/math/Gdl-Smullyan.html 



Tarski's undefinability theorem 



Tarski's undefinability theorem, stated and proved by Alfred Tarski in 1936, is an important limitative result in 
mathematical logic, the foundations of mathematics, and in formal semantics. Informally, the theorem states that 
arithmetical truth cannot be defined in arithmetic. 

The theorem applies more generally to any sufficiently strong formal system, showing that truth in the standard 
model of the system cannot be defined within the system. 

History 

In 1931, Kurt Godel published his famous incompleteness theorems, which he proved in part by showing how to 
represent syntax within (first-order) arithmetic. Each expression of the language of arithmetic is assigned a distinct 
number. This procedure is known variously as Godel-numbering, coding, and more generally, as arithmetization. 

In particular, various sets of expressions are coded as sets of numbers. It turns out that for various syntactic 
properties (such as being a formula, being a sentence, etc.), these sets are computable. Moreover, any computable set 
of numbers can be defined by some arithmetical formula. For example, there are formulas in the language of 
arithmetic defining the set of codes for arithmetic sentences, and for provable arithmetic sentences. 

The indefinability theorem shows that this encoding cannot be done for semantical concepts such as truth. It shows 
that no sufficiently rich interpreted language can represent its own semantics. A corollary is that any metalanguage 
capable of expressing the semantics of some object language must have expressive power exceeding that of the 
object language. The metalanguage includes primitive notions, axioms, and rules absent from the object language, so 
that there are theorems provable in the metalanguage not provable in the object language. 

The indefinability theorem is conventionally attributed to Alfred Tarski. Godel also discovered the indefinability 
theorem in 1930, while proving his incompleteness theorems published in 1931, and well before the 1936 



Tarski's undefinability theorem 59 

publication of Tarski's work (Murawski 1998). While Godel never published anything bearing on his independent 
discovery of indefinability, he did describe it in a 1931 letter to John von Neumann. Tarski had obtained almost all 
results of his 1936 paper Der Wahrheitsbe griff in den formalisierten Sprachen between 1929 and 1931, and spoke 
about them to Polish audiences. However, as he emphasized in the paper, the indefinability theorem was the only one 
exception. According to the footnote of the indefinability theorem (Satz I) of the 1936 paper, the theorem and the 
sketch of the proof were added to the paper only after the paper was sent to print. When he presented the paper to the 
Warsaw Academy of Science on March 21 1931, he wrote only some conjectures instead of the results after his own 
investigations and partly after Godel's short report on the incompleteness theorems "Einige metamathematische 
Resultate iiber Entscheidungsdefinitheit und Widerspruchsfreiheit", Akd. der Wiss. in Wien, 1930. 

Statement of the theorem 

We will first state a simplified version of Tarski's theorem, then state and prove in the next section the theorem 
Tarski actually proved in 1936. Let L be the language of first-order arithmetic, and let N be the standard structure for 
L. Thus (L, AO is the "interpreted first-order language of arithmetic." Let T denote the set of L-sentences true in N, 
and T* the set of code numbers of the sentences in T. The following theorem answers the question: Can T* be 
defined by a formula of first-order arithmetic? 

Tarski's undefinability theorem: There is no L-formula True{x) which defines T*. That is, there is no L-formula 
True{x) such that for every L-formula x, True(x) <-> x is true. 

Informally, the theorem says that given some formal arithmetic, the concept of truth in that arithmetic is not 
definable using the expressive means that arithmetic affords. This implies a major limitation on the scope of 
"self-representation." It is possible to define a formula True{x) whose extension is T*, but only by drawing on a 
metalanguage whose expressive power goes beyond that of L, second-order arithmetic for example. 

The theorem just stated is a corollary of Post's theorem about the arithmetical hierarchy, proved some years after 
Tarski (1936). A semantic proof of Tarski's theorem from Post's theorem is obtained by reductio ad absurdum as 
follows. Assuming T* is arithmetically definable, there is a natural number n such that T* is definable by a formula 
at level $]° of the arithmetical hierarchy. However, T* is Yft complete for all k. Thus the arithmetical hierarchy 

collapses at level n, contradicting Post's theorem. 

General form of the theorem 

Tarski proved a stronger theorem than the one stated above, using an entirely syntactical method. The resulting 
theorem applies to any formal language with negation, and with sufficient capability for self-reference that Godel's 
Diagonal Lemma holds. First-order arithmetic satisfies these preconditions, but the theorem applies to much more 
general formal systems. 

Proof of Tarski's undefinability theorem in its most general form, by reductio ad absurdum. Suppose that an L- 
formula True{x) defines T*. In particular, if A is a sentence of arithmetic then True{"A") is true in N iff A is true in N. 
Hence for all A, the Tarski ^-sentence True{"A") <-> A is true in N, But Godel's diagonal lemma yields a 
counterexample to this equivalence: the "Liar" sentence S such that S <-» -iTrue("S") holds. Thus no L-formula 
True{x) can define T*. QED. 

The formal machinery of this proof is wholly elementary except for the diagonalization the diagonal lemma requires. 
The proof of that Lemma is likewise surprisingly simple; for example, it does not invoke recursive functions in any 
way. The proof does assume that every L-formula has a Godel number, but the specifics of a coding method are not 
required. Hence Tarski's theorem is much easier to motivate and prove than the more celebrated theorems of Godel 
about the metamathematical properties of first-order arithmetic. 



Tarski's undefinability theorem 60 

Discussion 

Smullyan (1991, 2001) has argued forcefully that Tarski's undefinability theorem deserves much of the attention 
garnered by Godel's incompleteness theorems. That the latter theorems have much to say about all of mathematics 
and more controversially, about a range of philosophical issues (e.g., Lucas 1961) is less than evident. Tarski's 
theorem, on the other hand, is not directly about mathematics but about the inherent limitations of any formal 
language sufficiently expressive to be of real interest. Such languages are necessarily capable of enough 
self-reference for the diagonal lemma to apply to them. The broader philosophical import of Tarski's theorem is more 
strikingly evident. 

An interpreted language is strongly-semantically-self-representational exactly when the language contains predicates 
and function symbols defining all the semantic concepts specific to the language. Hence the required functions 
include the "semantic valuation function" mapping a formula A to its truth value I All, and the "semantic denotation 
function" mapping a term t to the object it denotes. Tarski's theorem then generalizes as follows: No sufficiently 
powerful language is strongly-semantically-self-representational. 

The undefinability theorem does not prevent truth in one theory from being defined in a stronger theory. For 
example, the set of (codes for) formulas of first-order Peano arithmetic that are true in N is definable by a formula in 
second order arithmetic. Similarly, the set of true formulas of the standard model of second order arithmetic (or n-th 
order arithmetic for any n) can be defined by a formula in first-order ZFC. 

References 

• Bell, J.L., and Machover, M., 1977. A Course in Mathematical Logic. North-Holland. 

• George Boolos, John Burgess, and Richard Jeffrey, 2002. Computability and Logic, 4th ed. Cambridge University 
Press. 

• Lucas, J. R., 1961, "Mind, Machines, and Godel, [1] " Philosophy 36: 1 12-27. 

• Murawski, Roman, Undefinability of truth. The problem of the priority: Tarski vs. Godel , History and 
Philosophy of Logic 19 (1998), 153-160 

• Raymond Smullyan, 1991. Godel's Incompleteness Theorems. Oxford Univ. Press. 

• Raymond Smullyan, 2001, "Godel's Incompleteness Theorems" in Goble, Lou, ed., The Blackwell Guide to 
Philosophical Logic. Blackwell: 72-89. 

• Alfred Tarski, 1983, "The Concept of Truth in Formalized Languages" in Corcoran, J., ed., Logic, Semantics and 
Metamathematics. Indianapolis: Hackett. The English translation of Tarski's 1936 Der Wahrheitsbegriff in den 
formalisierten Sprachen. 

References 

[1] http://users.ox.ac.uk/~jrlucas/Godel/mmg.html 
[2] http://www.staff.amu.edu.pl/~rmur/hpll.ps 



Model of hierarchical complexity 61 



Model of hierarchical complexity 



The model of hierarchical complexity, is a framework for scoring how complex a behavior is. It quantifies the 
order of hierarchical complexity of a task based on mathematical principles of how the information is organized and 
of information science. This model has been developed by Michael Commons and others since the 1980s. 

Overview 

The Model of Hierarchical Complexity, which has been presented as a formal theory , is a framework for scoring 
how complex a behavior is. Developed by Michael Commons , it quantifies the order of hierarchical complexity of 
a task based on mathematical principles of how the information is organized , and of information science .Its 
forerunner was the General Stage Model . It is a model in mathematical psychology. 

Behaviors that may be scored include those of individual humans or their social groupings (e.g., organizations, 
governments, societies), animals, or machines. It enables scoring the complexity of human reasoning in any domain. 
It is cross-culturally valid. The reason it applies cross-culturally is that the scoring is based on the mathematical 
complexity of the hierarchical organization of information. Scoring does not depend upon the content of the 
information (e.g., what is done, said, written, or analyzed) but upon how the information is organized. 

The MHC is a non-mentalistic model of developmental stages. It specifies 14 orders of hierarchical complexity and 
their corresponding stages. It is different from previous proposals about developmental stage applied to humans 
Instead of attributing behavioral changes across a person's age to the development of mental structures or schema, 
this model posits that task sequences form hierarchies that become increasingly complex. Because less complex 
tasks must be completed and practiced before more complex tasks can be acquired, this accounts for the 
developmental changes seen, for example, in individual persons' performance of tasks. (For example, a person 
cannot perform arithmetic until the numeral representations of numbers are learned. A person cannot multiply 
numbers until addition is learned). Furthermore, previous theories of stage have confounded the stimulus and 
response in assessing stage by simply scoring responses and ignoring the task or stimulus. The Model of Hierarchical 
Complexity separates the task or stimulus from the performance. The participant's performance on a task of a given 
complexity represents the stage of developmental complexity. 

Vertical complexity of tasks performed 

One major basis for this developmental theory is task analysis. The study of ideal tasks, including their instantiation 
in the real world, has been the basis of the branch of stimulus control called psychophysics. Tasks are defined as 
sequences of contingencies, each presenting stimuli and each requiring a behavior or a sequence of behaviors that 
must occur in some non-arbitrary fashion. The complexity of behaviors necessary to complete a task can be specified 
using the horizontal complexity and vertical complexity definitions described below. Behavior is examined with 
respect to the analytically-known complexity of the task. 

Tasks are quantal in nature. They are either completed correctly or not completed at all. There is no intermediate 
state. For this reason, the Model characterizes all stages as hard and distinct. The orders of hierarchical complexity 
are quantized like the electron atomic orbitals around the nucleus. Each task difficulty has an order of hierarchical 
complexity required to complete it correctly. Since tasks of a given order of hierarchical complexity require actions 
of a given order of hierarchical complexity to perform them, the stage of the participant's performance is equivalent 
to the order of complexity of the successfully completed task. The quantal feature of tasks is thus particularly 
instrumental in stage assessment because the scores obtained for stages are likewise discrete. 

Every task contains a multitude of subtasks (Overton, 1990). When the subtasks are carried out by the participant in 
a required order, the task in question is successfully completed. Therefore, the model asserts that all tasks fit in some 
sequence of tasks, making it possible to precisely determine the hierarchical order of task complexity. Tasks vary in 



Model of hierarchical complexity 62 

complexity in two ways: either as horizontal (involving classical information); or as vertical (involving hierarchical 
information). 

Horizontal complexity 

Classical information describes the number of "yes-no" questions it takes to do a task. For example, if one asked a 
person across the room whether a penny came up heads when they flipped it, their saying "heads" would transmit 1 
bit of "horizontal" information. If there were 2 pennies, one would have to ask at least two questions, one about each 
penny. Hence, each additional 1-bit question would add another bit. Let us say they had a four-faced top with the 
faces numbered 1, 2, 3, and 4. Instead of spinning it, they tossed it against a backboard as one does with dice in a 
game of craps. Again, there would be 2 bits. One could ask them whether the face had an even number. If it did, one 
would then ask if it were a 2. Horizontal complexity, then, is the sum of bits required by just such tasks as these. 

Vertical complexity 

Hierarchical complexity refers to the number of recursions that the coordinating actions must perform on a set of 
primary elements. Actions at a higher order of hierarchical complexity: (a) are defined in terms of actions at the next 
lower order of hierarchical complexity; (b) organize and transform the lower-order actions (see Figure 2); (c) 
produce organizations of lower-order actions that are new and not arbitrary, and cannot be accomplished by those 
lower-order actions alone. Once these conditions have been met, we say the higher-order action coordinates the 
actions of the next lower order. 

To illustrate how lower actions get organized into more hierarchically complex actions, let us turn to a simple 
example. Completing the entire operation 3 x (4 + 1) constitutes a task requiring the distributive act. That act 
non-arbitrarily orders adding and multiplying to coordinate them. The distributive act is therefore one order more 
hierarchically complex than the acts of adding and multiplying alone; it indicates the singular proper sequence of the 
simpler actions. Although simply adding results in the same answer, people who can do both display a greater 
freedom of mental functioning. Thus, the order of complexity of the task is determined through analyzing the 
demands of each task by breaking it down into its constituent parts. 

The hierarchical complexity of a task refers to the number of concatenation operations it contains, that is, the number 
of recursions that the coordinating actions must perform. An order-three task has three concatenation operations. A 
task of order three operates on a task of order two and a task of order two operates on a task of order one (a simple 
task). 

Stages of development 

The notion of stages is fundamental in the description of human, organismic, and [7] machine evolution. Previously 
it has been defined in some ad hoc ways. Here, it is described formally in terms of the Model of Hierarchical 
Complexity (MHC). 

Formal definition of stage 

Since actions are defined inductively, so is the function h, known as the order of the hierarchical complexity. To 
each action A, we wish to associate a notion of that action's hierarchical complexity, h(A). Given a collection of 
actions A and a participant S performing A, the stage of performance of S on A is the highest order of the actions in 
A completed successfully at least once, i.e., it is stage (S, A) = max{/z(A) I A (epsilon) A and A completed 
successfully by S}. Thus, the notion of stage is discontinuous, having the same gaps as the orders of hierarchical 
complexity. This is in agreement with previous definitions (Commons et al., 1998; Commons & Miller, 2001; 
Commons & Pekker, 2007). 



Model of hierarchical complexity 



63 



Because MHC stages are conceptualized in terms of the hierarchical complexity of tasks rather than in terms of 
mental representations (as in Piaget's stages), the highest stage represents successful performances on the most 
hierarchically complex tasks rather than intellectual maturity. Table 1 gives descriptions of each stage. 



Stages of hierarchical complexity 

Table 1 . Stages described in the Model of Hierarchical Complexity 



Order or Stage 


What they do 


How they do it 


End result 


- calculatory 


Exact computation only, no 
generalization 


Human-made programs manipulate 0, 1, not 2 or 

3. 


Minimal human result. Literal, unreasoning 
computer programs act in a way analogous 
to this stage. 


1 - sensory or 
motor 


Discriminate in a rote 
fashion, stimuli 
generalization, move 


Move limbs, lips, toes, eyes, elbows, head View 
objects or move 


Discriminative establishing and conditioned 
reinforcing stimuli 


2 - circular 
sensory-motor 


Form open-ended proper 
classes 


Reach, touch, grab, shake objects, circular babble 


Open ended proper classes, phonemes, 
archiphonemes 


3 - sensory-motor 


Form concepts 


Respond to stimuli in a class successfully 


Morphemes, concepts 


4 - nominal 


Find relations among 
concepts. Use names 


Find relations among concepts Use names 


Single words: ejaculatives & exclamations, 
verbs, nouns, number names, letter names 


5 - sentential 


Imitate and acquire 
sequences Follows short 
sequential acts 


Generalize match-dependent task actions. Chain 
words 


Various forms of pronouns: subject (I), 
object (me), possessive adjective (my), 
possessive pronoun (mine), and reflexive 
(myself) for various persons (I, you, he, she, 
it, we, y'all, they) 


6 - preoperational 


Make simple deductions. 
Follow lists of sequential 
acts. Tell stories. 


Count event events and objects Connect the dots 
Combine numbers and simple propositions 


Connectives: as, when, then, why, before; 
products of simple operations 


7 - primary 


Simple logical deduction 
and empirical rules 
involving time sequence 
Simple arithmetic 


Adds, subtracts, multiplies, divides, counts, 
proves, does series of tasks on own 


Times, places, counts acts, actors, 
arithmetic outcome, sequence from 
calculation 


8 - concrete 


Carry out full arithmetic, 
form cliques, plan deals 


Does long division, short division, follows 
complex social rules, ignores simple social rules, 
takes and coordinates perspective of other and self 


Interrelations, social events, what happened 
among others, reasonable deals, history, 
geography 


9 - abstract 


Discriminate variables 
such as stereotypes; logical 
quantification; (none, 
some, all) 


Form variables out of finite classes Make and 
quantify propositions 


Variable time, place, act, actor, state, type; 
quantifiers (all, none, some) ; categorical 
assertions (e.g., "We all die") 


10 - formal 


Argue using empirical or 
logical evidence Logic is 
linear, 1 dimensional 


Solve problems with one unknown using algebra, 
logic and empiricism 


Relationships (for example: causality) are 
formed out of variables; words: linear, 
logical, one dimensional, if then, thus, 
therefore, because; correct scientific 
solutions 


11- systematic 


Construct multivariate 
systems and matrices 


Coordinates more than one variable as input. 
Consider relationships in contexts. 


Events and concepts situated in a 
multivariate context; systems are formed 
out of relations; systems: legal, societal, 
corporate, economic, national 



Model of hierarchical complexity 



64 



12 - metasystematic 


Construct multi-systems 


Create metasystems out of systems Compare 


Metasystems and supersystems are formed 




and metasystems out of 


systems and perspectives Name properties of 


out of systems of relationships 




disparate systems 


systems: e.g. homomorphic, isomorphic, 
complete, consistent (such as tested by 
consistency proofs), commensurable 




1 3 - paradigmatic 


Fit metasystems together to 


Synthesize metasystems 


Paradigms are formed out of multiple 




form new paradigms 




metasystems 


14- 


Fit paradigms together to 


Form new fields by crossing paradigms 


New fields are formed out of multiple 


cross-paradigmatic 


form new fields 




paradigms 



Relationship with Piaget's theory 



There are some commonalities between the Piagetian and Commons' notions of stage and many more things that are 
different. In both, one finds: 

1. Higher order actions defined in terms of lower order actions. This forces the hierarchical nature of the relations 
and makes the higher order tasks include the lower ones and requires that lower order actions are hierarchically 
contained within the relative definitions of the higher order tasks. 

2. Higher order of complexity actions organize those lower order actions. This makes them more powerful. Lower 
order actions are organized by the actions with a higher order of complexity, i.e., the more complex tasks. 

What Commons et al. (1998) have added includes: 

3. Higher order of complexity actions organize those lower order actions in a non-arbitrary way. 

This makes it possible for the Model's application to meet real world requirements, including the empirical and 
analytic. Arbitrary organization of lower order of complexity actions, possible in the Piagetian theory, despite the 
hierarchical definition structure, leaves the functional correlates of the interrelationships of tasks of differential 
complexity formulations ill-defined. 

Moreover, the model is consistent with the neo-Piagetian theories of cognitive development. According to these 
theories, progression to higher stages or levels of cognitive development is caused by increases in processing 
efficiency and working memory capacity. That is, higher order stages place increasingly higher demands on these 
functions of information processing, so that their order of appearance reflects the information processing possibilities 
at successive ages (Demetriou, 1998). 

The following dimensions are inherent in the application: 

1 . Task and performance are separated 

2. All tasks have an order of hierarchical complexity 

3. There is only one sequence of orders of hierarchical complexity. 

4. Hence, there is structure of the whole for ideal tasks and actions 

5. There are gaps between the orders of hierarchical complexity 

6. Stage is defined as the most hierarchically complex task solved. 

7. There are gaps in Rasch Scaled Stage of Performance. 

8. Performance stage is different task area to task area. 

9. There is no structure of the whole — horizontal decalage — for performance. It is not inconsistency in thinking 
within a developmental stage. Decalage is the normal modal state of affairs. 



Model of hierarchical complexity 65 

Orders and corresponding stages 

The MHC specifies 15 orders of hierarchical complexity and their corresponding stages, showing that each of 
Piaget's substages, in fact, are hard stages. Commons also adds four postformal stages: Systematic stage 11, 
Metasystematic stage 12, Paradigmatic stage 13, and Crossparadigmatic stage 14. It may be the Piaget's consolidate 
formal stage is the same as the systematic stage. There is one other difference in the orders and stages. At the 
suggestion of Biggs and Biggs, the sentential stage 5 was added. The sequence is as follows: (0) computory, (1) 
sensory & motor, (2) circular sensory-motor, (3) sensory-motor, (4) nominal, the new (5) sentential, (6) 
preoperational, (7) primary, (8) concrete, (9) abstract, (10) formal, and the four postformal: (11) systematic, (12) 
metasystematic, (13) paradigmatic, and (14) cross-paradigmatic. The first four stages (0-3) correspond to Piaget's 
sensorimotor stage at which infants and very young children perform. The sentential stage was added at Fischer's 
suggestion (1981, personal communication) citing Biggs & Collis (1982). Adolescents and adults can perform at any 
of the subsequent stages. MHC stages 4 through 5 correspond to Piaget's pre-operational stage; 6 through 8 
correspond to his concrete operational stage; and 9 through 1 1 correspond to his formal operational stage. 

The three highest stages in the MHC are not represented in Piaget's model. These stages from the Model of 
Hierarchical Complexity have extensively influenced the field of Positive Adult Development. Few individuals 
perform at stages above formal operations. More complex behaviors characterize multiple system models (Kallio, 
1995; Kallio & Helkama, 1991). Some adults are said to develop alternatives to, and perspectives on, formal 
operations. They use formal operations within a "higher" system of operations and transcend the limitations of formal 
operations. In any case, these are all ways in which these theories argue for and present converging evidence that 
adults are using forms of reasoning that are more complex than formal operations with which Piaget's model ended. 

Empirical research using the model 

The MHC has a broad range of applicability. The mathematical foundation of the model makes it an excellent 
research tool to be used by anyone examining performance that is organized into stages. It is designed to assess 
development based on the order of complexity which the individual utilizes to organize information. The MHC 
offers a singular mathematical method of measuring stages in any domain because the tasks presented can contain 
any kind of information. The model thus allows for a standard quantitative analysis of developmental complexity in 
any cultural setting. Other advantages of this model include its avoidance of mentalistic or contextual explanations, 
as well as its use of purely quantitative principles which are universally applicable in any context. 

The following can use the Model of Hierarchical Complexity to quantitatively assess developmental stages: 

Cross-cultural developmentalists; 

Animal developmentalists; 

Evolutionary psychologists; 

Organizational psychologists; 

Developmental political psychologists; 

Learning theorists; 

Perception researchers; 

History of science historians; 

Educators; 

Therapists; 

Anthropologists. 

The following list shows the large range of domains to which the Model has been applied. In one representative 
study, Commons, Goodheart, and Dawson (1997) found, using Rasch (1980) analysis, that hierarchical complexity 
of a given task predicts stage of a performance, the correlation being r = .92. Correlations of similar magnitude have 
been found in a number of the studies. 



Model of hierarchical complexity 66 

List of examples 

List of examples of tasks studied using the Model of Hierarchical Complexity or Fischer's Skill Theory (1980): 

Algebra (Commons, in preparation) 

Animal stages (Commons & Miller, 2004) 

Atheism (Commons -Miller, 2005) 

Attachment and Loss (Commons, 1991; Miller & Lee, 2000) 

Balance beam and pendulum (Commons, Goodheart, & Bresette, 1995; Commons, Pekker, et al., 2007) 

Contingencies of reinforcement (Commons, in preparation) 

Counselor stages (Lovell, 2004) 

Empathy of Hominids (Commons & Wolfsont, 2002) 

Epistemology (Kitchener & King, 1990; Kitchener & Fischer, 1990) 

Evaluative reasoning (Dawson, 2000) 

Four Story problem (Commons, Richards & Kuhn, 1982; Kallio & Helkama, 1991) 

Good Education (Dawson-Tunik, 2004) 

Good Interpersonal (Armon, 1989) 

Good Work (Armon, 1993) 

Honesty and Kindness (Lamborn, Fischer & Pipp, 1994) 

Informed consent (Commons & Rodriguez, 1990, 1993; Commons, Goodheart, Rodriguez, & Gutheil, 2006; 

Commons, Rodriguez, Adams, Goodheart, Gutheil, & Cyr, 2007). 

Language stages (Commons, et al., 2007) 

Leadership before and after crises (Oliver, 2004) 

Loevinger's Sentence Completion task (Cook-Greuter, 1990) 

Moral Judgment, (Armon & Dawson, 1997; Dawson, 2000) 

Music (Beethoven) (Funk, 1989) 

Orienteering (Commons, in preparation) 

Physics tasks (Inhelder & Piaget, 1958) 

Political development (Sonnert & Commons, 1994) 

Relationships (Armon, 1984a, 1984b) 

Report patient's prior crimes (Commons, Lee, Gutheil, et al., 1995) 

Social perspective-taking (Commons & Rodriguez, 1990; 1993) 

Spirituality (Miller & Cook-Greuter, 2000) 

Tool Making of Hominids (Commons & Miller 2002) 

Views of the Agood life® (Armon, 1984c; Danaher, 1993; Dawson, 2000; Lam, 1995) 

Workplace culture (Commons, Krause, Fayer, & Meaney, 1993) 

Workplace organization (Bowman, 1996a, 1996b) 

Writing (Commons & DeVos, 1985) 



Model of hierarchical complexity 67 

References 

[1] Commons & Pekker, 2007 

[2] (Commons, Trudeau, Stein, Richards, & Krause, 1998) 

[3] (Coombs, Dawes, & Tversky, 1970) 

[4] (Commons & Richards, 1984a, 1984b; Lindsay & Norman, 1977; Commons & Rodriguez, 1990, 1993) 

[5] (Commons & Richards, 1984a, 1984b) 

[6] (e.g., Inhelder & Piaget, 1958) 

[7] http://www.computing.dcu.ie/~humphrys/Notes/GA/advanced.topics.html 

Copyright permissions 

Portions of this article are from "Applying the Model of Hierarchical Complexity" by Commons, M. L., Miller, P. 
M., Goodheart, E. A., Danaher-Gilpin, D., Locicero, A., Ross, S. N. Unpublished manuscript. Copyright 2007 by 
Dare Association, Inc. Available from Dare Institute, commons@tiac.net. Reproduced and adapted with permission 
of the publisher. Portions of this article are also from "Introduction to the Model of Hierarchical Complexity" by M. 
L. Commons, in the Behavioral Development Bulletin, 13, 1-6 (http://www.behavioral-development-bulletin.com/ 
). Copyright 2007 Martha Pelaez. Reproduced with permission of the publisher. 

Literature 

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cognitive development (pp. 357—380). New York: Praeger. 

• Armon, C. (1984c). Ideals of the good life and moral judgment: Evaluative reasoning in children and adults. 
Moral Education Forum, 9(2). 

• Armon, C. (1989). Individuality and autonomy in adult ethical reasoning. In M. L. Commons, J. D. Sinnott, F. A. 
Richards, & C. Armon (Eds.), Adult development, Vol. 1. Comparisons and applications of adolescent and adult 
developmental models, (pp. 179—196). New York: Praeger. 

• Armon, C. (1993). The nature of good work: A longitudinal study. In J. Demick & P. M. Miller (Eds.), 
Development in the workplace (pp. 21—38). Hillsdale, NJ: Erlbaum. 

• Armon, C. & Dawson, T. L. (1997). Developmental trajectories in moral reasoning across the life-span. Journal 
of Moral Education, 26, 433-453. 

• Biggs, J. & Collis, K. (1982). A system for evaluating learning outomes: The SOLO Taxonomy. New York: 
Academic Press. 

• Bowman, A. K. (1996b). Examples of task and relationship 4b, 5a, 5b statements for task performance, 
atmosphere, and preferred atmosphere. In M. L. Commons, E. A. Goodheart, T. L. Dawson, P. M. Miller, & D. L. 
Danaher, (Eds.) The general stage scoring system (GSSS). Presented at the Society for Research in Adult 
Development, Amherst, MA. 

• Commons, M. L. (1991). A comparison and synthesis of Kohlberg's cognitive-developmental and Gewirtz's 
learning-developmental attachment theories. In J. L. Gewirtz & W. M. Kurtines (Eds.), Intersections with 
attachment (pp. 257-291). Hillsdale, NJ: Erlbaum. 

• Commons, M. L., Goodheart, E. A., & Bresette, L. M. with Bauer, N. F., Farrell, E. W., McCarthy, K. G, 
Danaher, D. L., Richards, F. A., Ellis, J. B., O'Brien, A. M., Rodriguez, J. A., and Schraeder, D. (1995). Formal, 
systematic, and metasystematic operations with a balance-beam task series: A reply to Kallio's claim of no 
distinct systematic stage. Adult Development, 2 (3), 193-199. 

• Commons, M. L., Goodheart, E. A., & Dawson T. L. (1997). Psychophysics of Stage: Task Complexity and 
Statistical Models. Paper presented at the International Objective Measurement Workshop at the Annual 
Conference of the American Educational Research Association, Chicago, IL. 

• Commons, M. L., Goodheart, E. A., Pekker, A., Dawson, T. L., Draney, K., & Adams, K. M. (2007). Using 
Rasch scaled stage scores to validate orders of hierarchical complexity of balance beam task sequences. In E. V. 



Model of hierarchical complexity 68 

Smith, Jr. & R. M. Smith (Eds.). Rasch measurement: Advanced and specialized applications (pp. 121—147). 

Maple Grove, MN: JAM Press. 

Commons, M. L., Goodheart, E. A., Rodriguez, J. A., Gutheil, T. G. (2006). Informed Consent: Do you know it 

when you see it? Psychiatric Annals, June, 430-435. 

Commons, M. L., Krause, S. R., Fayer, G. A., & Meaney, M. (1993). Atmosphere and stage development in the 

workplace. In J. Demick & P. M. Miller (Eds.). Development in the workplace (pp. 199—220). Hillsdale, NJ: 

Lawrence Erlbaum Associates, Inc. 

Commons, M. L., Lee, P., Gutheil, T. G, Goldman, M., Rubin, E. & Appelbaum, P. S. (1995). Moral stage of 

reasoning and the misperceived "duty" to report past crimes (misprision). International Journal of Law and 

Psychiatry, 18(4), 415-424. 

Commons, M. L., & Miller, P. A. (2001). A quantitative behavioral model of developmental stage based upon 

hierarchical complexity theory. Behavior Analyst Today, 2(3), 222-240. 

Commons, M. L., Miller, P. M. (2002). A complete theory of human evolution of intelligence must consider stage 

changes: A commentary on Thomas Wynn's Archeology and Cognitive Evolution. Behavioral and Brain 

Sciences. 25(3), 404-405. 

Commons, M. L., & Miller, P. M. (2004). Development of behavioral stages in animals. In Marc Bekoff (Ed.). 

Encyclopedia of animal behavior, (pp. 484—487). Westport, CT: Greenwood Publishing Group. 

Commons, M. L., & Pekker, A. (2007). Hierarchical Complexity: A Formal Theory. Manuscript submitted for 

publication. 

Commons, M. L., & Richards, F. A. (1984a). A general model of stage theory. In M. L. Commons, F. A. 

Richards, & C. Armon (Eds.), Beyond formal operations: Vol. 1. Late adolescent and adult cognitive development 

(pp. 120-140). New York: Praeger. 

Commons, M. L., & Richards, F. A. (1984b). Applying the general stage model. In M. L. Commons, F. A. 

Richards, & C. Armon (Eds.), Beyond formal operations: Vol. 1. Late adolescent and adult cognitive development 

(pp. 141-157). New York: Praeger. 

Commons, M. L., Richards, F. A., & Kuhn, D. (1982). Systematic and metasystematic reasoning: A case for a 

level of reasoning beyond Piaget's formal operations. Child Development, 53, 1058-1069. 

Commons, M. L., Rodriguez, J. A. (1990). AEqual access" without "establishing" religion: The necessity for 

assessing social perspective-taking skills and institutional atmosphere. Developmental Review, 10, 323-340. 

Commons, M. L., Rodriguez, J. A. (1993). The development of hierarchically complex equivalence classes. 

Psychological Record, 43, 667-697. 

Commons, M. L., Rodriguez, J. A. (1990). "Equal access" without "establishing" religion: The necessity for 

assessing social perspective-taking skills and institutional atmosphere. Developmental Review, 10, 323-340. 

Commons, M. L., Trudeau, E. J., Stein, S. A., Richards, F. A., & Krause, S. R. (1998). The existence of 

developmental stages as shown by the hierarchical complexity of tasks. Developmental Review, 8(3), 237-278. 

Commons, M. L., & De Vos, I. B. (1985). How researchers help writers. Unpublished manuscript available from 

Commons@tiac.net. 

Commons-Miller, N. H. K. (2005). The stages of atheism. Paper presented at the Society for Research in Adult 

Development, Atlanta, GA. 

Cook-Greuter, S. R. (1990). Maps for living: Ego-development theory from symbiosis to conscious universal 

embeddedness. In M. L. Commons, J. D. Sinnott, F. A. Richards, & C. Armon (Eds.). Adult Development: Vol. 2, 

Comparisons and applications of adolescent and adult developmental models (pp. 79—104). New York: Praeger. 

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Englewood Cliffs, New Jersey: Prentice-Hall. 

Danaher, D. (1993). Sex role differences in ego and moral development: Mitigation with maturity. Unpublished 

dissertation, Harvard Graduate School of Education. 



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Dawson, T. L. (2000). Moral reasoning and evaluative reasoning about the good life. Journal of Applied 

Measurement, 1 (372-397). 

Dawson Tunik, T. L. (2004). "A good education is" The development of evaluative thought across the life span. 

Genetic, Social, and General Psychology Monographs, 130,4 112. 

Demetriou, A. (1998). Cognitive development. In A. Demetriou, W. Doise, K. F. M. van Lieshout (Eds.), 

Life-span developmental psychology (pp. 179-269). London: Wiley. 

Fischer, K. W. (1980). A theory of cognitive development: The control and construction of hierarchies of skills. 

Psychological Review, 87(6), 477-531. 

Funk, J. D. (1989). Postformal cognitive theory and developmental stages of musical composition. In M. L. 

Commons, J. D. Sinnott, F. A. Richards & C. Armon (Eds.), Adult Development: (Vol. I) Comparisons and 

applications of developmental models (pp. 3—30). Westport, CT: Praeger. 

Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence: An essay on the 

development of formal operational structures. (A. Parsons, & S. Seagrim, Trans.). New York: Basic Books 

(originally published 1955). 

Kallio, E. (1995). Systematic Reasoning: Formal or postformal cognition? Journal of Adult Development, 2, 187 

192. 

Kallio, E., & Helkama, K. (1991). Formal operations and postformal reasoning: A replication. Scandinavian 

Journal of Psychology. 32(1), 18 21. 

Kitchener, K. S., & King, P. M. (1990). Reflective judgement: Ten years of research. In M. L. Commons, C. 

Armon, L. Kohlberg, F. A. Richards, T. A. Grotzer, & J. D. Sinnott (Eds.), Beyond formal operations: Vol. 2. 

Models and methods in the study of adolescent and adult thought (pp. 63—78). New York: Praeger. 

Kitchener, K. S. & Fischer, K. W. (1990). A skill approach to the development of reflective thinking. In D. Kuhn 

(Ed.), Developmental perspectives on teaching and learning thinking skills . Contributions to Human 

Development: Vol. 21 (pp. 48—62). 

Lam, M. S. (1995). Women and men scientists' notions of the good life: A developmental approach. Unpublished 

doctoral dissertation, University of Massachusetts, Amherst, MA. 

Lamborn, S., Fischer, K.W., & Pipp, SL. (1994). Constructive criticism and social lies: A developmental 

sequence for understanding honesty and kindness in social relationships. Developmental Psychology, 30, 495 508. 

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Edition), New York: Academic Press. 

Lovell, C. W. (1999). Development and disequilibration: Predicting counselor trainee gain and loss scores on the 

Supervisee Levels Questionnaire. Journal of Adult Development. 

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Rowman & Littlefield. 

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attachment objects. Paper presented at the Jean Piaget Society. Montreal, Quebec, Canada. 

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discrepancies in moral reasoning, explanatory style, and rumination. Unpublished doctoral dissertation, Fielding 

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Press. 

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Individual, 4(1), 31-55. 



Model of hierarchical complexity 70 

External links 

• Dare Association, Inc. (http://dareassociation.org) display text. 

• Behavioral Development Bulletin (http://www.behavioraldevelopmentbulletin.com) display text. 

• Society for Research in Adult Development (http://adultdevelopment.org) display text 



Complexity theory 



Complexity theory may refer to: 

• The study of complex systems. 

• Computational complexity theory, a field in theoretical computer science and mathematics dealing with the 
resources required during computation to solve a given problem. 

• The theoretical treatment of Kolmogorov complexity of a string studied in algorithmic information theory by 
identifying the length of the shortest binary program which can output that string. 

• Complexity theory and organizations or complexity theory and strategy, which have been influential in strategic 
management and organizational studies and incorporate the study of complex adaptive systems. 

• Complexity economics, the application of complexity theory to economics. 

See also 

• Systems theory (or systemics or general systems theory), an interdisciplinary field including engineering, biology, 
and philosophy that incorporates science to study large systems 

• Complexity 



Complex adaptive system 



71 



Complex adaptive system 



Complex adaptive systems are special cases of complex systems. They are complex in that they are diverse and 
made up of multiple interconnected elements (and so a part of network science) and adaptive in that they have the 
capacity to change and learn from experience. The term complex adaptive systems (CAS) was coined at the 
interdisciplinary Santa Fe Institute (SFI), by John H. Holland, Murray Gell-Mann and others. 



Changing 

External 

Environment 




Complex Adaptive Behavior 



Changing 

External 

Environment 




Overview 

The term complex adaptive systems, or 
complexity science, is often used to 
describe the loosely organized 
academic field that has grown up 
around the study of such systems. 
Complexity science is not a single 
theory — it encompasses more than one 
theoretical framework and is highly 
interdisciplinary, seeking the answers 
to some fundamental questions about 
living, adaptable, changeable systems. 

Examples of complex adaptive systems 

include the stock market, social insect 

and ant colonies, the biosphere and the 

ecosystem, the brain and the immune 

system, the cell and the developing 

embryo, manufacturing businesses and any human social group-based endeavour in a cultural and social system such 

as political parties or communities. There are close relationships between the field of CAS and artificial life. In both 

areas the principles of emergence and self-organization are very important. 

The ideas and models of CAS are essentially evolutionary, grounded in modern biological views on adaptation and 
evolution. The theory of complex adaptive systems bridges developments of systems theory with the ideas of 
generalized Darwinism, which suggests that Darwinian principles of evolution can explain a range of complex 
material phenomena, from cosmic to social objects. 



Changing 

External 

Environment 



Simple Self -Organized 
Local Relationships 




Changing 

External 

Environment 



Complex Adaptive System 



Definitions 

A CAS is a complex, self-similar collection of interacting adaptive agents. The study of CAS focuses on complex, 
emergent and macroscopic properties of the system. Various definitions have been offered by different researchers: 

• John H. Holland 

A Complex Adaptive System (CAS) is a dynamic network of many agents (which may represent cells, species, 
individuals, firms, nations) acting in parallel, constantly acting and reacting to what the other agents are doing. 
The control of a CAS tends to be highly dispersed and decentralized. If there is to be any coherent behavior in 
the system, it has to arise from competition and cooperation among the agents themselves. The overall 

behavior of the system is the result of a huge number of decisions made every moment by many individual 

. [l] 
agents. 

A CAS behaves/evolves according to three key principles: order is emergent as opposed to predetermined (c.f. 
Neural Networks), the system's history is irreversible, and the system's future is often unpredictable. The basic 



Complex adaptive system 72 

building blocks of the CAS are agents. Agents scan their environment and develop schema representing 
interpretive and action rules. These schema are subject to change and evolution. 

• Other definitions 

Macroscopic collections of simple (and typically nonlinear) interacting units that are endowed with the ability 
to evolve and adapt to a changing environment. 

General properties 

What distinguishes a CAS from a pure multi-agent system (MAS) is the focus on top-level properties and features 
like self-similarity, complexity, emergence and self-organization. A MAS is simply defined as a system composed of 
multiple, interacting agents. In CASs, the agents as well as the system are adaptive: the system is self-similar. A 
CAS is a complex, self-similar collectivity of interacting adaptive agents. Complex Adaptive Systems are 
characterised by a high degree of adaptive capacity, giving them resilience in the face of perturbation. 

Other important properties are adaptation (or homeostasis), communication, cooperation, specialization, spatial and 
temporal organization, and of course reproduction. They can be found on all levels: cells specialize, adapt and 
reproduce themselves just like larger organisms do. Communication and cooperation take place on all levels, from 
the agent to the system level. The forces driving co-operation between agents in such a system can be analysed with 
game theory. Many of the issues of complexity science and new tools for the analysis of complexity are being 
developed within network science. 

Properties 

mi 
Complex adaptive systems have many properties and the most important are: 

• Emergence: Rather than being planned or controlled the agents in the system interact in apparently random ways. 
From all these interactions patterns emerge which informs the behaviour of the agents within the system and the 
behaviour of the system itself. For example a termite hill is a wondrous piece of architecture with a maze of 
interconnecting passages, large caverns, ventilation tunnels and much more. Yet there is no grand plan, the hill 
just emerges as a result of the termites following a few simple local rules. 

• Co-evolution: All systems exist within their own environment and they are also part of that environment. 
Therefore, as their environment changes they need to change to ensure best fit. But because they are part of their 
environment, when they change, they change their environment, and as it has changed they need to change again, 
and so it goes on as a constant process. Some people draw a distinction between complex adaptive systems and 
complex evolving systems. Where the former continuously adapt to the changes around them but do not learn 
from the process. And where the latter learn and evolve from each change enabling them to influence their 
environment, better predict likely changes in the future, and prepare for them accordingly. 

• Sub optimal: A complex adaptive systems does not have to be perfect in order for it to thrive within its 
environment. It only has to be slightly better than its competitors and any energy used on being better than that is 
wasted energy. A complex adaptive systems once it has reached the state of being good enough will trade off 
increased efficiency every time in favour of greater effectiveness. 

• Requisite Variety: The greater the variety within the system the stronger it is. In fact ambiguity and paradox 
abound in complex adaptive systems which use contradictions to create new possibilities to co-evolve with their 
environment. Democracy is a good example in that its strength is derived from its tolerance and even insistence in 
a variety of political perspectives. 

• Connectivity: The ways in which the agents in a system connect and relate to one another is critical to the 
survival of the system, because it is from these connections that the patterns are formed and the feedback 
disseminated. The relationships between the agents are generally more important than the agents themselves. 



Complex adaptive system 73 

• Simple Rules: Complex adaptive systems are not complicated. The emerging patterns may have a rich variety, 
but like a kaleidoscope the rules governing the function of the system are quite simple. A classic example is that 
all the water systems in the world, all the streams, rivers, lakes, oceans, waterfalls etc with their infinite beauty, 
power and variety are governed by the simple principle that water finds its own level. 

• Iteration: Small changes in the initial conditions of the system can have significant effects after they have passed 
through the emergence - feedback loop a few times (often referred to as the butterfly effect). A rolling snowball 
for example gains on each roll much more snow than it did on the previous roll and very soon a fist sized 
snowball becomes a giant one. 

• Self Organising: There is no hierarchy of command and control in a complex adaptive system. There is no 
planning or managing, but there is a constant re-organising to find the best fit with the environment. A classic 
example is that if one were to take any western town and add up all the food in the shops and divide by the 
number of people in the town there will be near enough two weeks supply of food, but there is no food plan, food 
manager or any other formal controlling process. The system is continually self organising through the process of 
emergence and feedback. 

• Edge of Chaos: Complexity theory is not the same as chaos theory, which is derived from mathematics. But 
chaos does have a place in complexity theory in that systems exist on a spectrum ranging from equilibrium to 
chaos. A system in equilibrium does not have the internal dynamics to enable it to respond to its environment and 
will slowly (or quickly) die. A system in chaos ceases to function as a system. The most productive state to be in 
is at the edge of chaos where there is maximum variety and creativity, leading to new possibilities. 

• Nested Systems: Most systems are nested within other systems and many systems are systems of smaller 
systems. If we take the example in self organising above and consider a food shop. The shop is itself a system 
with its staff, customers, suppliers, and neighbours. It also belongs within the food system of that town and the 
larger food system of that country. It belongs to the retail system locally and nationally and the economy system 
locally and nationally, and probably many more. Therefore it is part of many different systems most of which are 
themselves part of other systems. 



Complex adaptive system 



74 



Management 

When used in the management of people, CAS includes [1] setting appropriate containers, [2] understanding 
significant differences, and [3] facilitating transformation exchanges. In a CAS, managers set guidelines for workers 
to interpret, and use to self-organize. 



Evolution of complexity 

Living organisms are complex adaptive 
systems. Although complexity is hard to 
quantify in biology, evolution has produced 
some remarkably complex organisms. 
This observation has led to the common 
misconception of evolution being 
progressive and leading towards what are 
viewed as "higher organisms 



« [6] 



If this were generally true, evolution would 
possess an active trend towards complexity. 
As shown below, in this type of process the 

value of the most common amount of 

171 
complexity would increase over time. 

Indeed, some artificial life simulations have 

suggested that the generation of CAS is an 



E 
'I- 



1 



Passive trend 



m 



On 



inescapable feature of evolution 



[8] [9] 



Minimal 
complexity 



Complexity 




Active trend 



dllh 



Complexity 



Passive versus active trends in the evolution of complexity. CAS at the beginning 

of the processes are colored red. Changes in the number of systems are shown by 

the height of the bars, with each set of graphs moving up in a time series. 



However, the idea of a general trend 

towards complexity in evolution can also be 

171 
explained through a passive process. This 

involves an increase in variance but the most common value, the mode, does not change. Thus, the maximum level 

of complexity increases over time, but only as an indirect product of there being more organisms in total. This type 

of random process is also called a bounded random walk. 

In this hypothesis, the apparent trend towards more complex organisms is an illusion resulting from concentrating on 
the small number of large, very complex organisms that inhabit the right-hand tail of the complexity distribution and 
ignoring simpler and much more common organisms. This passive model emphasizes that the overwhelming 
majority of species are microscopic prokaryotes, which comprise about half the world's biomass and constitute 
the vast majority of Earth's biodiversity. Therefore, simple life remains dominant on Earth, and complex life 
appears more diverse only because of sampling bias. 

This lack of an overall trend towards complexity in biology does not preclude the existence of forces driving systems 
towards complexity in a subset of cases. These minor trends are balanced by other evolutionary pressures that drive 
systems towards less complex states. 



Complex adaptive system 75 

See also 



Agent-based model 

Artificial life 

Center for Complex Systems and Brain Sciences 

[13] 
Center for Social Dynamics & Complexity (CSDC) at Arizona State University 

Cognitive Science 

Command and Control Research Program 

Complex system 



Computational Sociology 
Enterprise systems engineering 
Generative sciences 
Santa Fe Institute 
Simulated reality 
Sociology and complexity 
science 
Swarm Development Group 



Literature 

114] 

Ahmed E, Elgazzar AS, Hegazi AS (28 June 2005). "An overview of complex adaptive systems" . Mansoura 
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Dooley, K., Complexity in Social Science glossary a research training project of the European Commission. 
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Gell-Mann, Murray (1994). The quark and the jaguar: adventures in the simple and the complex. San Francisco: 
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Holland, John H. (1992). Adaptation in natural and artificial systems: an introductory analysis with applications 
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• Pharaoh, M.C. (online). Looking to systems theory for a reductive explanation of phenomenal experience and 

ri9] 

evolutionary foundations for higher order thought Retrieved 15 January 2008. 

External links 

Complexity Digest comprehensive digest of latest CAS related news and research. 

T21] 
DNA Wales Research Group Current Research in Organisational change CAS/CES related news and free 

research data. Also linked to the Business Doctor & BBC documentary series 

[221 

A description of complex adaptive systems on the Principia Cybernetica Web. 

1231 
Quick reference single-page description of the 'world' of complexity and related ideas hosted by the Center for 

the Study of Complex Systems at the University of Michigan. 

Complex systems research network 

1251 
The Open Agent-Based Modeling Consortium 



Complex adaptive system 76 

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doi: 10.1 103/PhysRevLett.84.6130. PMID 10991141. 
[9] Adami C, Ofria C, Collier TC (2000). "Evolution of biological complexity" (http://www.pnas.Org/cgi/content/full/97/9/4463). Proc. 

Natl. Acad. Sci. U.S.A. 97 (9): 4463-8. doi:10.1073/pnas.97.9.4463. PMID 10781045. PMC 18257. . 
[10] Oren A (2004). "Prokaryote diversity and taxonomy: current status and future challenges" (http://www.pubmedcentral.nih.gov/ 

articlerender.fcgi?tool=pmcentrez&artid=1693353). Philos. Trans. R. Soc. Lond., B, Biol. Sci. 359 (1444): 623-38. 

doi:10.1098/rstb.2003.1458. PMID 15253349. PMC 1693353. 

[II] Whitman W, Coleman D, Wiebe W (1998). "Prokaryotes: the unseen majority" (http://www.pnas.org/cgi/content/full/95/12/6578). 
Proc Natl Acad Sci USA 95 (12): 6578-83. doi:10.1073/pnas.95. 12.6578. PMID 9618454. PMC 33863. . 

[12] Schloss P, Handelsman J (2004). "Status of the microbial census" (http://mmbr.asm. org/cgi/pmidlookup?view=long&pmid=15590780). 

Microbiol Mol Biol Rev 68 (4): 686-91. doi:10.1128/MMBR.68.4.686-691.2004. PMID 15590780. PMC 539005. . 

[13] http://csdc.asu.edu/ 

[14] http://arxiv.org/abs/nlin/0506059 

[15] http://www.hpl.hp.com/techreports/2004/HPL-2004-187.html 

[16] http://www.foresight.gov.uk/OurWork/CompletedProjects/IIS/Docs/ComplexityandEmergentBehaviour.asp 

[17] http://www.foresight.gov.uk/ 

[18] http ://www. kk.org/outofcontrol/contents. php 

[19] http://homepage.ntlworld.com/rn.pharoah/ 

[20] http://www.comdig.org/ 

[21] http://www.dnawales.co.uk/ 

[22] http://pespmcl.vub.ac.be/CAS.html 

[23] http://bactra.org/notebooks/complexity.html 

[24] http://www.complexsystems.net.au/ 

[25] http://www.openabm.org/site/ 



77 



System Theories and Dynamics 



System 



SURROUNDINGS 



SYSTEM 



System (from Latin systema, in turn from Greek 

avarrijia systema) is a set of interacting or 
interdependent entities forming an integrated whole. 

The concept of an 'integrated whole' can also be stated 
in terms of a system embodying a set of relationships 
which are differentiated from relationships of the set to 
other elements, and from relationships between an 
element of the set and elements not a part of the 
relational regime. 

The scientific research field which is engaged in the 
study of the general properties of systems include 
systems theory, cybernetics, dynamical systems and 
complex systems. They investigate the abstract 
properties of the matter and organization, searching 
concepts and principles which are independent of the 
specific domain, substance, type, or temporal scales of existence. 

Most systems share common characteristics, including: 

• Systems have structure, defined by parts and their composition; 

• Systems have behavior, which involves inputs, processing and outputs of material, energy or information; 

• Systems have interconnect! vity: the various parts of a system have functional as well as structural relationships 
between each other. 

• Systems have by themselves functions or groups of functions 

The term system may also refer to a set of rules that governs behavior or structure. 




BOUNDARY 

A schematic representation of a closed system and its boundary 



History 

The word system in its meaning here, has a long history which can be traced back to Plato (Philebus), Aristotle 
(Politics) and Euclid (Elements). It had meant "total", "crowd" or "union" in even more ancient times, as it derives 
from the verb sunistemi, uniting, putting together. 

In the 19th century the first to develop the concept of a "system" in the natural sciences was the French physicist 
Nicolas Leonard Sadi Carnot who studied thermodynamics. In 1824 he studied what he called the working substance 
(system), i.e. typically a body of water vapor, in steam engines, in regards to the system's ability to do work when 
heat is applied to it. The working substance could be put in contact with either a boiler, a cold reservoir (a stream of 
cold water), or a piston (to which the working body could do work by pushing on it). In 1850, the German physicist 
Rudolf Clausius generalized this picture to include the concept of the surroundings and began to use the term 
"working body" when referring to the system. 

One of the pioneers of the general systems theory was the biologist Ludwig von Bertalanffy. In 1945 he introduced 
models, principles, and laws that apply to generalized systems or their subclasses, irrespective of their particular 



kind, the nature of their component elements, and the relation or forces' between them. 



[1] 



System 78 

Significant development to the concept of a system was done by Norbert Wiener and Ross Ashby who pioneered the 
use of mathematics to study systems . 

In the 1980s the term complex adaptive system was coined at the interdisciplinary Santa Fe Institute by John H. 
Holland, Murray Gell-Mann and others. 

System concepts 

Environment and boundaries 

Systems theory views the world as a complex system of interconnected parts. We scope a system by defining 
its boundary; this means choosing which entities are inside the system and which are outside - part of the 
environment. We then make simplified representations (models) of the system in order to understand it and to 
predict or impact its future behavior. These models may define the structure and/or the behavior of the system. 

Natural and man-made systems 

There are natural and man-made (designed) systems. Natural systems may not have an apparent objective but 
their outputs can be interpreted as purposes. Man-made systems are made with purposes that are achieved by 
the delivery of outputs. Their parts must be related; they must be "designed to work as a coherent entity" - else 
they would be two or more distinct systems 

Theoretical Framework 

An open system exchanges matter and energy with its surroundings. Most systems are open systems; like a 
car, coffeemaker, or computer. A closed system exchanges energy, but not matter, with its environment; like 
Earth or the project Biosphere2 or 3. An isolated system exchanges neither matter nor energy with its 
environment; a theoretical example of which would be the universe. 

Process and transformation process 

A system can also be viewed as a bounded transformation process, that is, a process or collection of processes 
that transforms inputs into outputs. Inputs are consumed; outputs are produced. The concept of input and 
output here is very broad. E.g., an output of a passenger ship is the movement of people from departure to 
destination. 

Subsystem 

A subsystem is a set of elements, which is a system itself, and a part of a larger system. 

Types of systems 

Evidently, there are many types of systems that can be analyzed both quantitatively and qualitatively. For example, 

mi 
with an analysis of urban systems dynamics, [A.W. Steiss] defines five intersecting systems, including the 

physical subsystem and behavioral system. For sociological models influenced by systems theory, where Kenneth D. 

Bailey defines systems in terms of conceptual, concrete and abstract systems; either isolated, closed, or open, 

Walter F. Buckley defines social systems in sociology in terms of mechanical, organic, and process models. Bela 

H. Banathy cautions that with any inquiry into a system that understanding the type of system is crucial and 

defines Natural and Designed systems. 

In offering these more global definitions, the author maintains that it is important not to confuse one for the other. 
The theorist explains that natural systems include sub-atomic systems, living systems, the solar system, the galactic 
system and the Universe. Designed systems are our creations, our physical structures, hybrid systems which include 
natural and designed systems, and our conceptual knowledge. The human element of organization and activities are 
emphasized with their relevant abstract systems and representations. A key consideration in making distinctions 
among various types of systems is to determine how much freedom the system has to select purpose, goals, methods, 
tools, etc. and how widely is the freedom to select itself distributed (or concentrated) in the system. 



System 79 

ro] 

George J. Klir maintains that no "classification is complete and perfect for all purposes," and defines systems in 
terms of abstract, real, and conceptual physical systems, bounded and unbounded systems, discrete to continuous, 
pulse to hybrid systems, et cetera. The interaction between systems and their environments are categorized in terms 
of relatively closed, and open systems. It seems most unlikely that an absolutely closed system can exist or, if it did, 
that it could be known by us. Important distinctions have also been made between hard and soft systems. Hard 
systems are associated with areas such as systems engineering, operations research and quantitative systems analysis. 
Soft systems are commonly associated with concepts developed by Peter Checkland and Brian Wilson through Soft 
Systems Methodology (SSM) involving methods such as action research and emphasizing participatory designs. 
Where hard systems might be identified as more "scientific," the distinction between them is actually often hard to 
define. 

Cultural system 

A cultural system may be defined as the interaction of different elements of culture. While a cultural system is quite 
different from a social system, sometimes both systems together are referred to as the sociocultural system. A major 
concern in the social sciences is the problem of order. One way that social order has been theorized is according to 
the degree of integration of cultural and social factors. 

Economic system 

An economic system is a mechanism (social institution) which deals with the production, distribution and 
consumption of goods and services in a particular society. The economic system is composed of people, institutions 
and their relationships to resources, such as the convention of property. It addresses the problems of economics, like 
the allocation and scarcity of resources. 

Application of the system concept 

Systems modeling is generally a basic principle in engineering and in social sciences. The system is the 
representation of the entities under concern. Hence inclusion to or exclusion from system context is dependent of the 
intention of the modeler. 

No model of a system will include all features of the real system of concern, and no model of a system must include 
all entities belonging to a real system of concern. 

Systems in information and computer science 

In computer science and information science, system could also be a method or an algorithm. Again, an example 
will illustrate: There are systems of counting, as with Roman numerals, and various systems for filing papers, or 
catalogues, and various library systems, of which the Dewey Decimal System is an example. This still fits with the 
definition of components which are connected together (in this case in order to facilitate the flow of information). 

System can also be used referring to a framework, be it software or hardware, designed to allow software programs 
to run, see platform. 



System 80 

Systems in engineering and physics 

In engineering and physics, a physical system is the portion of the universe that is being studied (of which a 
thermodynamic system is one major example). Engineering also has the concept of a system that refers to all of the 
parts and interactions between parts of a complex project. Systems engineering refers to the branch of engineering 
that studies how this type of system should be planned, designed, implemented, built, and maintained. 

Systems in social and cognitive sciences and management research 

Social and cognitive sciences recognize systems in human person models and in human societies. They include 
human brain functions and human mental processes as well as normative ethics systems and social/cultural 
behavioral patterns. 

In management science, operations research and organizational development (OD), human organizations are viewed 
as systems (conceptual systems) of interacting components such as subsystems or system aggregates, which are 
carriers of numerous complex processes and organizational structures. Organizational development theorist Peter 
Senge developed the notion of organizations as systems in his book The Fifth Discipline. 

Systems thinking is a style of thinking/reasoning and problem solving. It starts from the recognition of system 
properties in a given problem. It can be a leadership competency. Some people can think globally while acting 
locally. Such people consider the potential consequences of their decisions on other parts of larger systems. This is 
also a basis of systemic coaching in psychology. 

Organizational theorists such as Margaret Wheatley have also described the workings of organizational systems in 
new metaphoric contexts, such as quantum physics, chaos theory, and the self-organization of systems. 

Systems applied to strategic thinking 

In 1988, military strategist, John A. Warden III introduced his Five Ring System model in his book, The Air 
Campaign contending that any complex system could be broken down into five concentric rings. Each 
ring — Leadership, Processes, Infrastructure, Population and Action Units — could be used to isolate key elements of 
any system that needed change. The model was used effectively by Air Force planners in the First Gulf War. 

ri2i ri3i 

, .In the late 1990s, Warden applied this five ring model to business strategy 

See also 

Examples of systems Theories about systems Related topics 



List of systems 

(Wiki Project) 

Complex system 

Formal system 

Information system 

Meta-system 

Solar System 

Systems in human anatomy 



Chaos theory • Glossary of systems theory 

Cybernetics • Complexity theory and 

Systems ecology organizations 

Systems engineering • Network 

Systems psychology • System of systems (engineering) 

Systems theory • Systems art 



System 81 

Further reading 

• Alexander Backlund (2000). "The definition of system". In: Kybernetes Vol. 29 nr. 4, pp. 444—451. 

• Kenneth D. Bailey (1994). Sociology and the New Systems Theory: Toward a Theoretical Synthesis. New York: 
State of New York Press. 

• Bela H. Banathy (1997). "A Taste of Systemics" [14] , ISSS The Primer Project. 

• Walter F. Buckley (1967). Sociology and Modern Systems Theory, New Jersey: Englewood Cliffs. 

• Peter Checkland (1997). Systems Thinking, Systems Practice. Chichester: John Wiley & Sons, Ltd. 

• Robert L. Flood (1999). Rethinking the Fifth Discipline: Learning within the unknowable. London: Routledge. 

• George J. Klir (1969). Approach to General Systems Theory, 1969. 

• Brian Wilson (1980). Systems: Concepts, methodologies and Applications , John Wiley 

• Brian Wilson (2001). Soft Systems Methodology — Conceptual model building and its contribution, J.H.Wiley. 

• Beynon-Davies P. (2009). Business Information + Systems. Palgrave, Basingstoke. ISBN 978-0-230-20368-6 

External links 

• Definitions of Systems and Models by Michael Pidwirny, 1999-2007. 

• Definitionen von "System" (1572-2002) [16] by Roland Muller, 2001-2007 (most in German). 

References 

[I] 1945, Zu einer allgemeinen Systemlehre, Blatter filr deutsche Philosophie, 3/4. (Extract in: Biologia Generalis, 19 (1949), 139-164. 
[2] 1948, Cybernetics: Or the Control and Communication in the Animal and the Machine. Paris, France: Librairie Hermann & Cie, and 

Cambridge, MA: MIT Press.Cambridge, MA: MIT Press. 
[3] 1956. An Introduction to Cybernetics (http://pespmcl.vub.ac.be/ASHBBOOK.html), Chapman & Hall. 
[4] Steiss 1967, p.8-18. 
[5] Bailey, 1994. 
[6] Buckley, 1967. 
[7] Banathy, 1997. 
[8] Klir 1969, pp. 69-72 
[9] Checkland 1997; Flood 1999. 
[10] Warden, John A. Ill (1988). The Air Campaign: Planning for Combat. Washington, D.C.: National Defense University Press. 

ISBN 9781583481004. 

[II] Warden, John A. Ill (September 1995). "Chapter 4: Air theory for the 21st century" (http://www.airpower.maxwell.af.mil/airchronicles/ 
battle/chp4.html) (in Air and Space Power Journal). Battlefield of the Future: 21st Century Warfare Issues. United States Air Force. . 
Retrieved December 26, 2008. 

[12] Warden, John A. Ill (1995). "Enemy as a System" (http://www.airpower.maxwell.af.mil/airchronicles/apj/apj95/spr95_files/warden. 

htm). Airpower Journal Spring (9): 40-55. . Retrieved 2009-03-25. 
[13] Russell, Leland A.; Warden, John A. (2001). Winning in FastTime: Harness the Competitive Advantage of Prometheus in Business and in 

Life. Newport Beach, CA: GEO Group Press. ISBN 0971269718. 
[14] http://www.newciv.org/ISSS_Primer/asem04bb.html 
[15] http://www.physicalgeography.net/fundamentals/4b.html 
[16] http://www.muellerscience.com/SPEZIALITAETEN/System/System_Definitionen.htm 



Causal loop diagram 



82 



Causal loop diagram 



A causal loop diagram (CLD) is a diagram that 
aids in visualizing how interrelated variables affect 
one another. The diagram consists of a set of nodes 
representing the variables connected together. The 
relationships between these variables, represented 
by arrows, can be labelled as positive or negative. 

Example of positive reinforcing loop: 

The amount of the Bank Balance will affect the 

amount of the Earned Interest, as represented by the 

top blue arrow, pointing from Bank Balance to 

Earned Interest. 

Since an increase in Bank balance results in an 

increase in Earned Interest, this link is positive, 

which is denoted with a ""+"". 

The Earned interest gets added to the Bank balance, 

also a positive link, represented by the bottom blue 

arrow. 

The causal effect between these nodes forms a 

positive reinforcing loop, represented by the green 

arrow, which is denoted with an "R". 



Dynamic causal loop diagram 




ctH.u 
10 
Bank balance 



Example of positive reinforcing loop: Bank balance and Earned interest 



Positive and negative causal links 



• Positive causal link means that the two nodes change in the same direction, i.e. if the node in which the link 
starts decreases, the other node also decreases. Similarly, if the node in which the link starts increases, the other 
node increases. 

• Negative causal link means that the two nodes change in opposite directions, i.e. if the node in which the link 
starts increases, then the other node decreases, and vice versa. 

Example 




Dynamic causal loop diagram: positive and 
negative links 



Causal loop diagram 



83 



Reinforcing and balancing loops 

To determine if a causal loop is reinforcing or balancing, one can start with an assumption, e.g. "Node 1 increases" 
and follow the loop around. The loop is: 

• reinforcing if, after going around the loop, one ends up with the same result as the initial assumption. 

• balancing if the result contradicts the initial assumption. 

Or to put it in other words: 

• reinforcing loops have an even number of negative links (zero also is even, see example above) 

• balancing loops have an uneven number of negative links. 

Identifying reinforcing and balancing loops is an important step for identifying Reference Behaviour Patterns, i.e. 
possible dynamic behaviours of the system. 

• Reinforcing loops are associated with exponential increases/decreases. 

• Balancing loops are associated with reaching a plateau. 

If the system has delays (often denoted by drawing a short line across the causal link), the system might fluctuate. 

Example 




Causal loop diagram of Adoption model, used to 
demonstrate systems dynamics 




Causal loop diagram of a model examining the 
growth or decline of a life insurance company 



See also 

• System dynamics 

• Positive feedback 

• Negative feedback 



External links 



[i] 



System Models & Simulation - Shows a causal-loop diagram of a dynamic system that is parameterized with 
data and equations, then simulated and graphed. 
WikiSD [2] the System Dynamics Society [3] Wiki 



Causal loop diagram 



84 



References 

[1] http://www.excelsoftware.com/systemmodels.html 
[2] http://www.systemdynamics.org/wiki/index.php/Main_ 
[3] http://systemdynamics.org/ 



Phase space 



In mathematics and physics, a phase space, 

introduced by Willard Gibbs in 1901 , is a 
space in which all possible states of a 
system are represented, with each possible 
state of the system corresponding to one 
unique point in the phase space. For 
mechanical systems, the phase space usually 
consists of all possible values of position 
and momentum variables. A plot of position 
and momentum variables as a function of 
time is sometimes called a phase plot or a 
phase diagram. Phase diagram, however, is 
more usually reserved in the physical 
sciences for a diagram showing the various 
regions of stability of the thermodynamic 
phases of a chemical system, which consists 
of pressure, temperature, and composition. 



0.10 



0,05 



0,00 



-0,05 



-0,10 




0,5 0,6 0,7 0,8 0,9 

Phase space of a dynamical system with focal stability. 



In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional 
space. For every possible state of the system, or allowed combination of values of the system's parameters, a point is 
plotted in the multidimensional space. Often this succession of plotted points is analogous to the system's state 
evolving over time. In the end, the phase diagram represents all that the system can be, and its shape can easily 
elucidate qualities of the system that might not be obvious otherwise. A phase space may contain very many 
dimensions. For instance, a gas containing many molecules may require a separate dimension for each particle's x, y 
and z positions and momenta as well as any number of other properties. 

In classical mechanics the phase space co-ordinates are the generalized coordinates q. and their conjugate generalized 
momenta p.. The motion of an ensemble of systems in this space is studied by classical statistical mechanics. The 
local density of points in such systems obeys Liouville's Theorem, and so can be taken as constant. Within the 
context of a model system in classical mechanics, the phase space coordinates of the system at any given time are 
composed of all of the system's dynamical variables. Because of this, it is possible to calculate the state of the system 
at any given time in the future or the past, through integration of Hamilton's or Lagrange's equations of motion. 
Furthermore, because each point in phase space lies on exactly one phase trajectory, no two phase trajectories can 
intersect. 

For simple systems, such as a single particle moving in one dimension for example, there may be as few as two 
degrees of freedom, (typically, position and velocity), and a sketch of the phase portrait may give qualitative 
information about the dynamics of the system, such as the limit-cycle of the Van der Pol oscillator shown in the 
diagram. 



Phase space 



85 



f**rt pal.il JhVKndn F:. ■,:,l-l: l 



t « 



Here, the horizontal axis gives the position 
and vertical axis the velocity. As the system 
evolves, its state follows one of the lines 
(trajectories) on the phase diagram. 

Classic examples of phase diagrams from 
chaos theory are : 

• the Lorenz attractor 

• parameter plane of complex quadratic 
polynomials with Mandelbrot set. 

Quantum mechanics 

In quantum mechanics, the coordinates p 

and q of phase space become hermitian 

operators in a Hilbert space, but may 

alternatively retain their classical 

interpretation, provided functions of them compose in novel algebraic ways (through Groenewold's 1946 star 

product). Every quantum mechanical observable corresponds to a unique function or distribution on phase space, and 

vice versa, as specified by Hermann Weyl (1927) and supplemented by John von Neumann (1931); Eugene Wigner 

(1932); and, in a grand synthesis, by H J Groenewold (1946). With J E Moyal (1949), these completed the 

foundations of phase-space quantization, a logically autonomous reformulation of quantum mechanics. Its modern 

abstractions include deformation quantization and geometric quantization. 




Phase portrait of the Van der Pol oscillator 



Thermodynamics and statistical mechanics 

In thermodynamics and statistical mechanics contexts, the term phase space has two meanings: It is used in the same 
sense as in classical mechanics. If a thermodynamical system consists of N particles, then a point in the 
d/V-dimensional phase space describes the dynamical state of every particle in that system, as each particle is 
associated with three position variables and three momentum variables. In this sense, a point in phase space is said to 
be a microstate of the system. N is typically on the order of Avogadro's number, thus describing the system at a 
microscopic level is often impractical. This leads us to the use of phase space in a different sense. 

The phase space can refer to the space that is parametrized by the macroscopic states of the system, such as pressure, 
temperature, etc. For instance, one can view the pressure-volume diagram or entropy-temperature diagrams as 
describing part of this phase space. A point in this phase space is correspondingly called a macrostate. There may 
easily be more than one microstate with the same macrostate. For example, for a fixed temperature, the system could 
have many dynamic configurations at the microscopic level. When used in this sense, a phase is a region of phase 
space where the system in question is in, for example, the liquid phase, or solid phase, etc. 

Since there are many more microstates than macrostates, the phase space in the first sense is usually a manifold of 
much larger dimensions than the second sense. Clearly, many more parameters are required to register every detail of 
the system up to the molecular or atomic scale than to simply specify, say, the temperature or the pressure of the 
system. 



Phase space 86 

See also 

Classical mechanics 

Dynamical system 

Molecular dynamics 

Hamiltonian mechanics 

Lagrangian mechanics 

Cotangent bundle 

Symplectic manifold 

Phase plane 

Phase space method 

Parameter space 

Optical Phase Space 

State space (controls) for information about state space (similar to phase state) in control engineering. 

State space (physics) for information about state space in physics 

State space for information about state space with discrete states in computer science. 

References 

[1] Findlay, Alex. The Phase Rule and its Applications. 3rd edition, pg 8. Longmans, Green and Co. 191 1. 



Negative feedback 



Negative feedback 



87 



Negative feedback occurs when the output of a system acts to oppose 
changes to the input of the system; with the result that the changes are 
attenuated. If the overall feedback of the system is negative, then the 
system will tend to be stable. 

Overview 

In many physical and biological systems, qualitatively different 
influences can oppose each other. For example, in biochemistry, one 
set of chemicals drives the system in a given direction, whereas 
another set of chemicals drives it in an opposing direction. If one, or 
both of these opposing influences are non-linear, equilibrium point(s) 
result. 

In biology, this process (generally biochemical) is often referred to as 
homeostasis; whereas in mechanics, the more common term is 
equilibrium. 

In engineering, mathematics and the physical and biological sciences, 
common terms for the points around which the system gravitates 
include: attractors, stable states, eigenstates/eigenfunctions, 
equilibrium points, and setpoints. 

Negative refers to the sign of the multiplier in mathematical models for 
feedback. In delta notation, - ^ output is added to or mixed into the 
input. In multivariate systems, vectors help to illustrate how several 
influences can both partially complement and partially oppose each 
other. 







e 



Hypothalamus 
I I CRH 



t 



Pituitary Gland 



Iacth 



Adrenal Cortex 

Glucocorticoids 



t-J 



Most endocrine hormones are controlled by a 

physiologic negative feedback inhibition loop, 

such as the glucocorticoids secreted by the 

adrenal cortex. The hypothalamus secretes 

corticotropin-releasing hormone (CRH), which 

directs the anterior pituitary gland to secrete 
adrenocortocotropic hormone (ACTH). In turn, 

ACTH directs the adrenal cortex to secrete 
glucocorticoids, such as Cortisol. Glucocorticoids 

not only perform their respective functions 

throughout the body but also negatively affect the 

release of further stimulating secretions of both 

the hypothalamus and the pituitary gland, 

effectively reducing the output of glucocorticoids 

once a sufficient amount has been released. 



In contrast, positive feedback is feedback in which the system responds so as to act to increase the magnitude of any 
particular perturbation, resulting in amplification of the original signal instead of stabilization. Any system where 
there is a net positive feedback will result in a runaway situation. Both positive and negative feedback require a 
feedback loop to operate. 

Negative feedback is used to describe the act of reversing any discrepancy between desired and actual output. 

Examples 



Mechanical Engineering 

Negative feedback was first implemented in the 16th Century with the invention of the centrifugal governor. Its 
operation is most easily seen in its use by James Watt to control the speed of his steam engine. Two heavy balls on 
an upright frame rotate at the same speed as the engine. As their speed increases they move outwards due to the 
centrifugal force. This causes them to lift a mechanism which closes the steam inlet valve and the engine slows. 
When the speed of the engine falls too far, the balls will move in the opposite direction and open the steam valve. 



Negative feedback 

Control 

Examples of the use of negative feedback to control its system are: thermostat control, phase-locked loop, hormonal 
regulation (see diagram above), and temperature regulation in animals. 

A simple and practical example is a thermostat. When the temperature in a heated room reaches a certain upper limit 
the room heating is switched off so that the temperature begins to fall. When the temperature drops to a lower limit, 
the heating is switched on again. Provided the limits are close to each other, a steady room temperature is 
maintained. The same applies to a cooling system, such as an air conditioner, a refrigerator, or a freezer. 

Biology & Chemistry 

Some biological systems exhibit negative feedback such as the baroreflex in blood pressure regulation and 
erythropoiesis. Many biological process (e.g., in the human anatomy) use negative feedback. Examples of this are 
numerous, from the regulating of body temperature, to the regulating of blood glucose levels. The disruption of 
feedback loops can lead to undesirable results: in the case of blood glucose levels, if negative feedback fails, the 
glucose levels in the blood may begin to rise dramatically, thus resulting in diabetes. 

For hormone secretion regulated by the negative feedback loop: when gland X releases hormone X, this stimulates 
target cells to release hormone Y. When there is an excess of hormone Y, gland X "senses" this and inhibits its 
release of hormone X. 

Economics 

In economics, automatic stabilisers are government programs which work as negative feedback to dampen 
fluctuations in real GDP. 

Electronic amplifiers 

The negative feedback amplifier was invented by Harold Stephen Black at Bell Laboratories in 1927, and patented 
by him in 1934. Fundamentally, all electronic devices (e.g. vacuum tubes, bipolar transistors, MOS transistors) 
exhibit some nonlinear behavior. Negative feedback corrects this by trading unused gain for higher linearity (lower 
distortion). An amplifier with too large an open-loop gain, possibly in a specific frequency range, will additionally 
produce too large a feedback signal in that same range. This feedback signal, when subtracted from the original 
input, will act to reduce the original input, also by "too large" an amount. This "too small" input will be amplified 
again by the "too large" open-loop gain, creating a signal that is "just right". The net result is a flattening of the 
amplifier's gain over all frequencies (desensitising). Though much more accurate, amplifiers with negative feedback 
can become unstable if not designed correctly, causing them to oscillate. Harry Nyquist of Bell Laboratories 
managed to work out a theory about how to make this behaviour stable. 

Negative feedback is used in this way in many types of amplification systems to stabilize and improve their 
operating characteristics (see e.g., operational amplifiers). 



Negative feedback 

See also 

• Asymptotic gain model 

• Biofeedback 

• Control theory 

• Cybernetics 

• Nyquist stability criterion 

• Stability criterion 

• Step response 

• Global Warming 

External links 

• http://www.bioiogy-oniine.0rg/4/1_physioiogicai_homeostasis.htm 

References 

[1] Raven, PH; Johnson, GB. Biology, Fifth Edition, Boston: Hill Companies, Inc. 1999. page 1058. 

Information flow diagram 

An information flow diagram (IFD) is an illustration of information flow throughout an organisation. An IFD 
shows the relationship between external and internal information flows between an organisation. It also shows the 
relationship between the internal departments and sub-systems. 

An Information Flow Diagram is information about a system laid out in diagramatic form. Usually using "blobs" to 
explain in more details the system and sub-systems to elemental parts. Following on from this you can add in lines of 
how the information travels from one system to another. This is used in businesses, Government agencies, television 
and cinematic processes. 

Overview 

Peter Checkland, a British management scientist, identifies that information flows between the different elements 
that compose the system. He also defines a system as a 'community situated within an environment'. 

IFD is useful for specifying the boundaries and scope of the system. The IFD shows the boundaries and scope of the 
system, it's interactions with its external entities, and the main flows of information within the system and within any 
complex subsystems. 

The Information Flow Diagram (IFD) is one of the simplest tools used to document findings from the requirements 
determination process. They are used for a number of purposes: 1. to document the main flows of information 
around the organisation; 2. for the analyst to check that he/she has understood those flows and that none has been 
omitted; 3. the analyst may use them during the fact-finding process itself as an accurate and efficient way to 
document findings as they are identified; 4. as a high-level (not detailed) tool to document information flows within 
the organisation as a whole or a lower-level tool to document an individual functional area of the business. 



Information flow diagram 



90 



See also 

• Information systems 

• Systems thinking 



System theory 



Systems theory is an interdisciplinary theory about the nature of complex systems in nature, society, and science, 
and is a framework by which one can investigate and/or describe any group of objects that work together to produce 
some result. This could be a single organism, any organization or society, or any electro-mechanical or informational 
artifact. As a technical and general academic area of study it predominantly refers to the science of systems that 
resulted from Bertalanffy's General System Theory (GST), among others, in initiating what became a project of 
systems research and practice. Systems theoretical approaches were later appropriated in other fields, such as in the 
structural functionalist sociology of Talcott Parsons and Niklas Luhmann. 



Overview 




Margaret Mead was an influential figure in systems 
theory. 



Contemporary ideas from systems theory have grown with 

diversified areas, exemplified by the work of Bela H. Banathy, 

ecological systems with Howard T. Odum, Eugene Odum and Fritjof 

Capra, organizational theory and management with individuals such 

as Peter Senge, interdisciplinary study with areas like Human 

Resource Development from the work of Richard A. Swanson, and 

insights from educators such as Debora Hammond and Alfonso 

Montuori. As a transdisciplinary, interdisciplinary and 

multiperspectival domain, the area brings together principles and 

concepts from ontology, philosophy of science, physics, computer 

science, biology, and engineering as well as geography, sociology, 

political science, psychotherapy (within family systems therapy) and 

economics among others. Systems theory thus serves as a bridge for interdisciplinary dialogue between autonomous 

areas of study as well as within the area of systems science itself. 

In this respect, with the possibility of misinterpretations, von Bertalanffy believed a general theory of systems 
"should be an important regulative device in science," to guard against superficial analogies that "are useless in 
science and harmful in their practical consequences." Others remain closer to the direct systems concepts developed 
by the original theorists. For example, Ilya Prigogine, of the Center for Complex Quantum Systems at the University 
of Texas, Austin, has studied emergent properties, suggesting that they offer analogues for living systems. The 
theories of autopoiesis of Francisco Varela and Humberto Maturana are a further development in this field. 
Important names in contemporary systems science include Russell Ackoff, Bela H. Banathy, Anthony Stafford Beer, 
Peter Checkland, Robert L. Flood, Fritjof Capra, Michael C. Jackson, Edgar Morin and Werner Ulrich, among 
others. 

With the modern foundations for a general theory of systems following the World Wars, Ervin Laszlo, in the preface 
for Bertalanffy's book Perspectives on General System Theory, maintains that the translation of "general system 

121 

theory" from German into English has "wrought a certain amount of havoc" . The preface explains that the 
original concept of a general system theory was "Allgemeine Systemtheorie (or Lehre)", pointing out the fact that 

"Theorie" (or "Lehre") just as "Wissenschaft" (translated Scholarship), "has a much broader meaning in German than 

i 121 

the closest English words theory and science'" . With these ideas referring to an organized body of knowledge 



System theory 91 

and "any systematically presented set of concepts, whether they are empirical, axiomatic, or philosophical", "Lehre" 
is associated with theory and science in the etymology of general systems, but also does not translate from the 
German very well; "teaching" is the "closest equivalent", but "sounds dogmatic and off the mark" . While many of 
the root meanings for the idea of a "general systems theory" might have been lost in the translation and many were 
led to believe that the systems theorists had articulated nothing but a pseudoscience, systems theory became a 
nomenclature that early investigators used to describe the interdependence of relationships in organization by 
defining a new way of thinking about science and scientific paradigms. 

A system from this frame of reference is composed of regularly interacting or interrelating groups of activities. For 

example, in noting the influence in organizational psychology as the field evolved from "an individually oriented 

industrial psychology to a systems and developmentally oriented organizational psychology," it was recognized that 

organizations are complex social systems; reducing the parts from the whole reduces the overall effectiveness of 

organizations . This is at difference to conventional models that center on individuals, structures, departments and 

units separate in part from the whole instead of recognizing the interdependence between groups of individuals, 

mi 
structures and processes that enable an organization to function. Laszlo explains that the new systems view of 

organized complexity went "one step beyond the Newtonian view of organized simplicity" in reducing the parts from 

the whole, or in understanding the whole without relation to the parts. The relationship between organizations and 

their environments became recognized as the foremost source of complexity and interdependence. In most cases the 

whole has properties that cannot be known from analysis of the constituent elements in isolation. Bela H. Banathy, 

who argued - along with the founders of the systems society - that "the benefit of humankind" is the purpose of 

science, has made significant and far-reaching contributions to the area of systems theory. For the Primer Group at 

ISSS, Banathy defines a perspective that iterates this view: 

The systems view is a world-view that is based on the discipline of SYSTEM INQUIRY. Central to systems 
inquiry is the concept of SYSTEM. In the most general sense, system means a configuration of parts 
connected and joined together by a web of relationships. The Primer group defines system as a family of 
relationships among the members acting as a whole. Von Bertalanffy defined system as "elements in standing 
relationship. 
_[5] 

Similar ideas are found in learning theories that developed from the same fundamental concepts, emphasizing that 
understanding results from knowing concepts both in part and as a whole. In fact, Bertalanffy's organismic 
psychology paralleled the learning theory of Jean Piaget. Interdisciplinary perspectives are critical in breaking 
away from industrial age models and thinking where history is history and math is math segregated from the arts and 
music separate from the sciences and never the twain shall meet . The influential contemporary work of Peter 

ro] 

Senge provides detailed discussion of the commonplace critique of educational systems grounded in conventional 
assumptions about learning, including the problems with fragmented knowledge and lack of holistic learning from 
the "machine-age thinking" that became a "model of school separated from daily life." It is in this way that systems 
theorists attempted to provide alternatives and an evolved ideation from orthodox theories with individuals such as 
Max Weber, Emile Durkheim in sociology and Frederick Winslow Taylor in scientific management, which were 
grounded in classical assumptions . The theorists sought holistic methods by developing systems concepts that 
could be integrated with different areas. 

The contradiction of reductionism in conventional theory (which has as its subject a single part) is simply an 
example of changing assumptions. The emphasis with systems theory shifts from parts to the organization of parts, 
recognizing interactions of the parts are not "static" and constant but "dynamic" processes. Conventional closed 
systems were questioned with the development of open systems perspectives. The shift was from absolute and 
universal authoritative principles and knowledge to relative and general conceptual and perceptual knowledge 
still in the tradition of theorists that sought to provide means in organizing human life. Meaning, the history of ideas 
that preceded were rethought not lost. Mechanistic thinking was particularly critiqued, especially the industrial-age 



System theory 92 

mechanistic metaphor of the mind from interpretations of Newtonian mechanics by Enlightenment philosophers and 
later psychologists that laid the foundations of modern organizational theory and management by the late 19th 
century . Classical science had not been overthrown, but questions arose over core assumptions that historically 
influenced organized systems, within both social and technical sciences. 

History 

TIMELINE 

Precursors 

• Herbert Spencer (1820-1903), Vilfredo Pareto (1848-1923), Emile Durkheim (1858-1917), Alexander 
Bogdanov (1873-1928), Nicolai Hartmann (1882-1950), Robert Maynard Hutchins (1929-1951), among others 

Pioneers 

• 1946-1953 Macy conferences 

• 1948 Norbert Wiener publishes Cybernetics or Control and Communication in the Animal and the Machine 

• 1954 Ludwig von Bertalanffy, Anatol Rapoport, Ralph W. Gerard, Kenneth Boulding establish Society for the 
Advancement of General Systems Theory, in 1956 renamed to Society for General Systems Research. 

• 1955 W. Ross Ashby publishes Introduction to Cybernetics 

• 1968 Ludwig von Bertalanffy publishes General System theory: Foundations, Development, Applications 

Developments 

• 1970- 1980s Second-order cybernetics developed by Heinz von Foerster, Gregory Bateson, Humberto Maturana 
and others 

• 1970s Catastrophe theory (Rene Thom, E.C. Zeeman) Dynamical systems in mathematics. 

• 1980s Chaos theory David Ruelle, Edward Lorenz, Mitchell Feigenbaum, Steve Smale, James A. Yorke 

• 1986 Context theory Anthony Wilden 

• 1988 International Society for Systems Science 

• 1990 Complex adaptive systems (CAS) John H. Holland, Murray Gell-Mann, W. Brian Arthur 

Whether considering the first systems of written communication with Sumerian cuneiform to Mayan numerals, or 
the feats of engineering with the Egyptian pyramids, systems thinking in essence dates back to antiquity. 

Differentiated from Western rationalist traditions of philosophy, C. West Churchman often identified with the I 

ri2i 
Ching as a systems approach sharing a frame of reference similar to pre-Socratic philosophy and Heraclitus 

Von Bertalanffy traced systems concepts to the philosophy of GW. von Leibniz and Nicholas of Cusa's coincidentia 

oppositorum. While modern systems are considerably more complicated, today's systems are embedded in history. 

Systems theory as an area of study specifically developed following the World Wars from the work of Ludwig von 
Bertalanffy, Anatol Rapoport, Kenneth E. Boulding, William Ross Ashby, Margaret Mead, Gregory Bateson, C. 
West Churchman and others in the 1950s, specifically catalyzed by the cooperation in the Society for General 
Systems Research. Cognizant of advances in science that questioned classical assumptions in the organizational 
sciences, Bertalanffy's idea to develop a theory of systems began as early as the interwar period, publishing "An 
Outline for General Systems Theory" in the British Journal for the Philosophy of Science, Vol 1, No. 2, by 1950. 
Where assumptions in Western science from Greek thought with Plato and Aristotle to Newton's Principia have 
historically influenced all areas from the hard to social sciences (see David Easton's seminal development of the 
"political system" as an analytical construct), the original theorists explored the implications of twentieth century 
advances in terms of systems. 

Subjects like complexity, self-organization, connectionism and adaptive systems had already been studied in the 
1940s and 1950s. In fields like cybernetics, researchers like Norbert Wiener, William Ross Ashby, John von 
Neumann and Heinz von Foerster examined complex systems using mathematics. John von Neumann discovered 
cellular automata and self-reproducing systems, again with only pencil and paper. Aleksandr Lyapunov and Jules 



System theory 93 

Henri Poincare worked on the foundations of chaos theory without any computer at all. At the same time Howard T. 
Odum, the radiation ecologist, recognised that the study of general systems required a language that could depict 
energetics and kinetics at any system scale. Odum developed a general systems, or Universal language, based on the 
circuit language of electronics to fulfill this role, known as the Energy Systems Language. Between 1929-1951, 
Robert Maynard Hutchins at the University of Chicago had undertaken efforts to encourage innovation and 
interdisciplinary research in the social sciences, aided by the Ford Foundation with the interdisciplinary Division of 

ri3i 

the Social Sciences established in 1931 . Numerous scholars had been actively engaged in ideas before 
(Tectology of Alexander Bogdanov published in 1912-1917 is a remarkable example), but in 1937 von Bertalanffy 
presented the general theory of systems for a conference at the University of Chicago. 

The systems view was based on several fundamental ideas. First, all phenomena can be viewed as a web of 
relationships among elements, or a system. Second, all systems, whether electrical, biological, or social, have 
common patterns, behaviors, and properties that can be understood and used to develop greater insight into the 
behavior of complex phenomena and to move closer toward a unity of science. System philosophy, methodology and 
application are complementary to this science .By 1956, the Society for General Systems Research was 
established, renamed the International Society for Systems Science in 1988. The Cold War affected the research 
project for systems theory in ways that sorely disappointed many of the seminal theorists. Some began to recognize 
theories defined in association with systems theory had deviated from the initial General Systems Theory (GST) 

ri4i 

view . The economist Kenneth Boulding, an early researcher in systems theory, had concerns over the 



manipulation of systems concepts. Boulding concluded from the effects of the Cold War that abuses of power always 
prove consequential and that systems theory might address such issues . Since the end of the Cold War, there has 
been a renewed interest in systems theory with efforts to strengthen an ethical view. 

Developments in system theories 
General systems research and systems inquiry 

Many early systems theorists aimed at finding a general systems theory that could explain all systems in all fields of 
science. The term goes back to Bertalanffy's book titled "General System theory: Foundations, Development, 
Applications" from 1968 . Von Bertalanffy tells that he developed the "allgemeine Systemtheorie" since 1937 in 
talks and since 1946 with publications. 

Von Bertalanffy's objective was to bring together under one heading the organismic science that he had observed in 
his work as a biologist. His desire was to use the word "system" to describe those principles which are common to 
systems in general. In GST, he writes: 

...there exist models, principles, and laws that apply to generalized systems or their subclasses, irrespective of 
their particular kind, the nature of their component elements, and the relationships or "forces" between them. It 
seems legitimate to ask for a theory, not of systems of a more or less special kind, but of universal principles 
applying to systems in general. 
_[17] 

Ervin Laszlo in the preface of von Bertalanffy's book Perspectives on General System Theory.. 

Thus when von Bertalanffy spoke of Allgemeine Systemtheorie it was consistent with his view that he was 
proposing a new perspective, a new way of doing science. It was not directly consistent with an interpretation 
often put on "general system theory", to wit, that it is a (scientific) "theory of general systems." To criticize it 
as such is to shoot at straw men. Von Bertalanffy opened up something much broader and of much greater 
significance than a single theory (which, as we now know, can always be falsified and has usually an 
ephemeral existence): he created a new paradigm for the development of theories. 



System theory 94 

Ludwig von Bertalanffy outlines systems inquiry into three major domains: Philosophy, Science, and Technology. In 
his work with the Primer Group, Bela H. Banathy generalized the domains into four integratable domains of 
systemic inquiry: 



Domain Description 

Philosophy the ontology, epistemology, and axiology of systems; 



Theory a set of interrelated concepts and principles applying to all systems 

Methodology the set of models, strategies, methods, and tools that instrumentalize systems theory and philosophy 



Application the application and interaction of the domains 

These operate in a recursive relationship, he explained. Integrating Philosophy and Theory as Knowledge, and 
Method and Application as action, Systems Inquiry then is knowledgeable action. 

Cybernetics 

The term cybernetics derives from a Greek word which meant steersman, and which is the origin of English words 
such as "govern". Cybernetics is the study of feedback and derived concepts such as communication and control in 
living organisms, machines and organisations. Its focus is how anything (digital, mechanical or biological) processes 
information, reacts to information, and changes or can be changed to better accomplish the first two tasks. 

The terms "systems theory" and "cybernetics" have been widely used as synonyms. Some authors use the term 
cybernetic systems to denote a proper subset of the class of general systems, namely those systems that include 
feedback loops. However Gordon Pask's differences of eternal interacting actor loops (that produce finite products) 
makes general systems a proper subset of cybernetics. According to Jackson (2000), von Bertalanffy promoted an 
embryonic form of general system theory (GST) as early as the 1920s and 1930s but it was not until the early 1950s 
it became more widely known in scientific circles. 

Threads of cybernetics began in the late 1800s that led toward the publishing of seminal works (e.g., Wiener's 
Cybernetics in 1948 and von Bertalanffy's General Systems Theory in 1968). Cybernetics arose more from 
engineering fields and GST from biology. If anything it appears that although the two probably mutually influenced 
each other, cybernetics had the greater influence. Von Bertalanffy (1969) specifically makes the point of 
distinguishing between the areas in noting the influence of cybernetics: "Systems theory is frequently identified with 
cybernetics and control theory. This again is incorrect. Cybernetics as the theory of control mechanisms in 
technology and nature is founded on the concepts of information and feedback, but as part of a general theory of 
systems;" then reiterates: "the model is of wide application but should not be identified with 'systems theory' in 
general", and that "warning is necessary against its incautious expansion to fields for which its concepts are not 
made." (17-23). Jackson (2000) also claims von Bertalanffy was informed by Alexander Bogdanov's three volume 
Tectology that was published in Russia between 1912 and 1917, and was translated into German in 1928. He also 
states it is clear to Gorelik (1975) that the "conceptual part" of general system theory (GST) had first been put in 
place by Bogdanov. The similar position is held by Mattessich (1978) and Capra (1996). Ludwig von Bertalanffy 
never even mentioned Bogdanov in his works, which Capra (1996) finds "surprising". 

Cybernetics, catastrophe theory, chaos theory and complexity theory have the common goal to explain complex 
systems that consist of a large number of mutually interacting and interrelated parts in terms of those interactions. 
Cellular automata (CA), neural networks (NN), artificial intelligence (AI), and artificial life (ALife) are related 
fields, but they do not try to describe general (universal) complex (singular) systems. The best context to compare 
the different "C"-Theories about complex systems is historical, which emphasizes different tools and methodologies, 
from pure mathematics in the beginning to pure computer science now. Since the beginning of chaos theory when 
Edward Lorenz accidentally discovered a strange attractor with his computer, computers have become an 
indispensable source of information. One could not imagine the study of complex systems without the use of 



System theory 95 

computers today. 

Complex adaptive systems 

Complex adaptive systems are special cases of complex systems. They are complex in that they are diverse and made 
up of multiple interconnected elements and adaptive in that they have the capacity to change and learn from 
experience. The term complex adaptive systems was coined at the interdisciplinary Santa Fe Institute (SFI), by John 
H. Holland, Murray Gell-Mann and others. However, the approach of the complex adaptive systems does not take 
into account the adoption of information which enables people to use it. 

CAS ideas and models are essentially evolutionary. Accordingly, the theory of complex adaptive systems bridges 
developments of the system theory with the ideas of 'generalized Darwinism', which suggests that Darwinian 
principles of evolution help explain a wide range of phenomena. 

Applications of system theories 
Living systems theory 

Living systems theory is an offshoot of von Bertalanffy's general systems theory, created by James Grier Miller, 
which was intended to formalize the concept of "life". According to Miller's original conception as spelled out in his 
magnum opus Living Systems, a "living system" must contain each of 20 "critical subsystems", which are defined by 
their functions and visible in numerous systems, from simple cells to organisms, countries, and societies. In Living 
Systems Miller provides a detailed look at a number of systems in order of increasing size, and identifies his 
subsystems in each. 

James Grier Miller (1978) wrote a 1,102 -page volume to present his living systems theory. He constructed a general 
theory of living systems by focusing on concrete systems — nonrandom accumulations of matter-energy in physical 
space-time organized into interacting, interrelated subsystems or components. Slightly revising the original model a 
dozen years later, he distinguished eight "nested" hierarchical levels in such complex structures. Each level is 
"nested" in the sense that each higher level contains the next lower level in a nested fashion. 

Organizational theory 



System theory 



The systems framework is also fundamental to organizational theory as 
organizations are complex dynamic goal-oriented processes. One of the early 
thinkers in the field was Alexander Bogdanov, who developed his Tectology, a 
theory widely considered a precursor of von Bertalanffy's GST, aiming to model 
and design human organizations (see Mattessich 1978, Capra 1996). Kurt Lewin 
was particularly influential in developing the systems perspective within 
organizational theory and coined the term "systems of ideology", from his 

frustration with behavioral psychologies that became an obstacle to sustainable 

r2ii 
work in psychology . Jay Forrester with his work in dynamics and management 

alongside numerous theorists including Edgar Schein that followed in their 

tradition since the Civil Rights Era have also been influential. 

The systems to organizations relies heavily upon achieving negative entropy 

through openness and feedback. A systemic view on organizations is 

transdisciplinary and integrative. In other words, it transcends the perspectives of 

individual disciplines, integrating them on the basis of a common "code", or more 

exactly, on the basis of the formal apparatus provided by systems theory. The 

systems approach gives primacy to the interrelationships, not to the elements of the 

system. It is from these dynamic interrelationships that new properties of the 

system emerge. In recent years, systems thinking has been developed to provide techniques for studying systems in 

holistic ways to supplement traditional reductionistic methods. In this more recent tradition, systems theory in 

organizational studies is considered by some as a humanistic extension of the natural sciences. 




Kurt Lewin attended the Macy 

conferences and is commonly 

identified as the founder of the 

movement to study groups 

scientifically. 



Software and computing 

In the 1960s, systems theory was adopted by the post John Von Neumann computing and information technology 
field and, in fact, formed the basis of structured analysis and structured design (see also Larry Constantine, Tom 
DeMarco and Ed Yourdon). It was also the basis for early software engineering and computer-aided software 
engineering principles. 

By the 1970s, General Systems Theory (GST) was the fundamental underpinning of most commercial software 
design techniques, and by the 1980, W. Vaughn Frick and Albert F. Case, Jr. had used GST to design the "missing 
link" transformation from system analysis (defining what's needed in a system) to system design (what's actually 
implemented) using the Yourdon/DeMarco notation. These principles were incorporated into computer-aided 
software engineering tools delivered by Nastec Corporation, Transform Logic, Inc., KnowledgeWare (see Fran 
Tarkenton and James Martin), Texas Instruments, Arthur Andersen and ultimately IBM Corporation. 



Sociology and Sociocybernetics 

Systems theory has also been developed within sociology. An important figure in the sociological systems 
perspective as developed from GST is Walter Buckley (who from Bertalanffy's theory). Niklas Luhmann (see 
Luhmann 1994) is also predominant in the literatures for sociology and systems theory. Miller's living systems 
theory was particularly influential in sociology from the time of the early systems movement. Models for dynamic 
equilibrium in systems analysis that contrasted classical views from Talcott Parsons and George Homans were 
influential in integrating concepts with the general movement. With the renewed interest in systems theory on the 
rise since the 1990s, Bailey (1994) notes the concept of systems in sociology dates back to Auguste Comte in the 
19th century, Herbert Spencer and Vilfredo Pareto, and that sociology was readying into its centennial as the new 
systems theory was emerging following the World Wars. To explore the current inroads of systems theory into 
sociology (primarily in the form of complexity science) see sociology and complexity science. 



System theory 97 

In sociology, members of Research Committee 5 1 of the International Sociological Association (which focuses on 
sociocybernetics), have sought to identify the sociocybernetic feedback loops which, it is argued, primarily control 
the operation of society. On the basis of research largely conducted in the area of education, Raven (1995) has, for 
example, argued that it is these sociocybernetic processes which consistently undermine well intentioned public 
action and are currently heading our species, at an exponentially increasing rate, toward extinction. See 
sustainability. He suggests that an understanding of these systems processes will allow us to generate the kind of 
(non "common-sense") targeted interventions that are required for things to be otherwise - i.e. to halt the destruction 
of the planet. 

System dynamics 

System Dynamics was founded in the late 1950s by Jay W. Forrester of the MIT Sloan School of Management with 
the establishment of the MIT System Dynamics Group. At that time, he began applying what he had learned about 
systems during his work in electrical engineering to everyday kinds of systems. Determining the exact date of the 
founding of the field of system dynamics is difficult and involves a certain degree of arbitrariness. Jay W. Forrester 
joined the faculty of the Sloan School at MIT in 1956, where he then developed what is now System Dynamics. The 
first published article by Jay W. Forrester in the Harvard Business Review on "Industrial Dynamics", was published 
in 1958. The members of the System Dynamics Society have chosen 1957 to mark the occasion as it is the year in 
which the work leading to that article, which described the dynamics of a manufacturing supply chain, was done. 

As an aspect of systems theory, system dynamics is a method for understanding the dynamic behavior of complex 
systems. The basis of the method is the recognition that the structure of any system — the many circular, 
interlocking, sometimes time-delayed relationships among its components — is often just as important in 
determining its behavior as the individual components themselves. Examples are chaos theory and social dynamics. 
It is also claimed that, because there are often properties-of-the-whole which cannot be found among the 
properties-of-the-elements, in some cases the behavior of the whole cannot be explained in terms of the behavior of 
the parts. An example is the properties of these letters which when considered together can give rise to meaning 
which does not exist in the letters by themselves. This further explains the integration of tools, like language, as a 
more parsimonious process in the human application of easiest path adaptability through interconnected systems. 

Systems engineering 

Systems Engineering is an interdisciplinary approach and means for enabling the realization and deployment of 
successful systems. It can be viewed as the application of engineering techniques to the engineering of systems, as 

["221 

well as the application of a systems approach to engineering efforts. Systems Engineering integrates other 
disciplines and specialty groups into a team effort, forming a structured development process that proceeds from 

concept to production to operation and disposal. Systems Engineering considers both the business and the technical 

T231 
needs of all customers, with the goal of providing a quality product that meets the user needs. 

Systems psychology 

Systems psychology is a branch of psychology that studies human behaviour and experience in complex systems. It 
is inspired by systems theory and systems thinking, and based on the theoretical work of Roger Barker, Gregory 
Bateson, Humberto Maturana and others. It is an approach in psychology, in which groups and individuals, are 
considered as systems in homeostasis. Systems psychology "includes the domain of engineering psychology, but in 

addition is more concerned with societal systems and with the study of motivational, affective, cognitive and group 

T241 
behavior than is engineering psychology." In systems psychology "characteristics of organizational behaviour for 



example individual needs, rewards, expectations, and attributes of the people interacting with the systems are 

T251 
considered in the process in order to create an effective system". . The Systems psychology includes an illusion of 

homeostatic systems, although most of the living systems are in a continuous disequilibrium of various degrees. 



System theory 

See also 



List of types of systems theory 

Cybernetics 

Emergence 

Glossary of systems theory 

Holism 

Meta-systems 

Open and Closed Systems in Social Science 

Social rule system theory 

Sociology and complexity science 

Systemantics 



System engineering 

Systems psychology 

Systemics 

Systems theory in archaeology 

Systems theory in anthropology 

Systems theory in political science 

Systems thinking 

World-systems theory 

Systematics - study of multi-term systems 



Further reading 

Ackoff, R. (1978). The art of problem solving. New York: Wiley. 

Ash, M.G. (1992). "Cultural Contexts and Scientific Change in Psychology: Kurt Lewin in Iowa." American 

Psychologist, Vol. 47, No. 2, pp. 198-207. 

Bailey, K.D. (1994). Sociology and the New Systems Theory: Toward a Theoretical Synthesis. New York: State 

of New York Press. 

Banathy, B (1996) Designing Social Systems in a Changing World New York Plenum 

Banathy, B. (1991) Systems Design of Education. Englewood Cliffs: Educational Technology Publications 

Banathy, B. (1992) A Systems View of Education. Englewood Cliffs: Educational Technology Publications. 

ISBN 0-87778-245-8 

Banathy, B.H. (1997). "A Taste of Systemics" [14] , The Primer Project, Retrieved May 14, (2007) 

Bateson, G. (1979). Mind and nature: A necessary unity. New York: Ballantine 

Bausch, Kenneth C. (2001) The Emerging Consensus in Social Systems Theory, Kluwer Academic New York 

ISBN 0-306-46539-6 

Ludwig von Bertalanffy (1968). General System Theory: Foundations, Development, Applications New York: 

George Braziller 

Bertalanffy, L. von. (1950). "An Outline of General System Theory." British Journal for the Philosophy of 

Science, Vol. 1, No. 2. 

Bertalanffy, L. von. (1955). "An Essay on the Relativity of Categories." Philosophy of Science, Vol. 22, No. 4, 

pp. 243-263. 

Bertalanffy, Ludwig von. (1968). Organismic Psychology and Systems Theory. Worchester: Clark University 

Press. 

Bertalanffy, Ludwig Von. (1974). Perspectives on General System Theory Edited by Edgar Taschdjian. George 

Braziller, New York. 

Buckley, W. (1967). Sociology and Modern Systems Theory. New Jersey: Englewood Cliffs. 

Mario Bunge (1979) Treatise on Basic Philosophy, Volume 4. Ontology II A World of Systems. Dordrecht, 

Netherlands: D. Reidel. 

Capra, F. (1997). The Web of Life-A New Scientific Understanding of Living Systems, Anchor ISBN 

978-0385476768 

Checkland, P. (1981). Systems thinking, Systems practice. New York: Wiley. 

Checkland, P. 1997. Systems Thinking, Systems Practice. Chichester: John Wiley & Sons, Ltd. 

Churchman, C.W. (1968). The systems approach. New York: Laurel. 

Churchman, C.W. (1971). The design of inquiring systems. New York: Basic Books. 

Corning, P. (1983) The Synergism Hupothesis: A Theory of Progressive Evolution. New York: McGraw Hill 



System theory 99 

• Davidson, Mark. (1983). Uncommon Sense: The Life and Thought ofLudwig von Bertalanffy, Father of General 
Systems Theory. Los Angeles: J. P. Tarcher, Inc. 
Durand, D. La systemique, Presses Universitaires de France 

Flood, R.L. 1999. Rethinking the Fifth Discipline: Learning within the unknowable." London: Routledge. 
Charles Francois. (2004). Encyclopedia of Systems and Cybernetics, Introducing the 2nd Volume [26] and further 
links to the ENCYCLOPEDIA, K G Saur, Munich [27] see also [28] 
Kahn, Herman. (1956). Techniques of System Analysis, Rand Corporation 

Laszlo, E. (1995). The Interconnected Universe. New Jersey, World Scientific. ISBN 981-02-2202-5 

T291 
Francois, C. (1999). Systemics and Cybernetics in a Historical Perspective 

Jantsch, E. (1980). The Self Organizing Universe. New York: Pergamon. 

Gorelik, G. (1975) Reemergence of Bogdanov's Tektology in. Soviet Studies of Organization, Academy of 

Management Journal. 18/2, pp. 345—357 

Hammond, D. 2003. The Science of Synthesis. Colorado: University of Colorado Press. 

Hinrichsen, D. and Pritchard, A.J. (2005) Mathematical Systems Theory. New York: Springer. ISBN 

978-3-540-44125-0 

Hull, D.L. 1970. "Systemic Dynamic Social Theory." Sociological Quarterly, Vol. 11, Issue 3, pp. 351—363. 

Hyotyniemi, H. (2006). Neocybernetics in Biological Systems . Espoo: Helsinki University of Technology, 

Control Engineering Laboratory. 

Jackson, M.C. 2000. Systems Approaches to Management. London: Springer. 

Klir, G.J. 1969. An Approach to General Systems Theory. New York: Van Nostrand Reinhold Company. 

Ervin Laszlo 1972. The Systems View of the World. New York: George Brazilier. 

Laszlo, E. (1972a). The systems view of the world. The natural philosophy of the new developments in the 

sciences. New York: George Brazillier. ISBN 0-8076-0636-7 

Laszlo, E. (1972b). Introduction to systems philosophy. Toward a new paradigm of contemporary thought. San 

Francisco: Harper. 

Laszlo, Ervin. 1996. The Systems View of the World. Hampton Press, NJ. (ISBN 1-57273-053-6). 

Lemkow, A. (1995) The Wholeness Principle: Dynamics of Unity Within Science, Religion & Society. Quest 

Books, Wheaton. 

Niklas Luhmann (1996), "Social Systems", Stanford University Press, Palo Alto, CA 

Mattessich, R. (1978) Instrumental Reasoning and Systems Methodology: An Epistemology of the Applied and 

Social Sciences. Reidel, Boston 

Minati, Gianfranco. Collen, Arne. (1997) Introduction to Systemics Eagleye books. ISBN 0-924025-06-9 

Montuori, A. (1989). Evolutionary Competence. Creating the Future. Amsterdam: Gieben. 

Morin, E. (2008). On Complexity. Cresskill, NJ: Hampton Press. 

Odum, H. (1994) Ecological and General Systems: An introduction to systems ecology, Colorado University 

Press, Colorado. 

Olmeda, Christopher J. (1998). Health Informatics: Concepts of Information Technology in Health and Human 

Services. Delfin Press. ISBN 0982144210 

Owens, R.G. (2004). Organizational Behavior in Education: Adaptive Leadership and School Reform, Eighth 

Edition. Boston: Pearson Education, Inc. 

Pharaoh, M.C. (online). Looking to systems theory for a reductive explanation of phenomenal experience and 

evolutionary foundations for higher order thought Retrieved Dec. 14 2007. 

Schein, E.H. (1980). Organizational Psychology, Third Edition. New Jersey: Prentice-Hall. 

Peter Senge (1990). The Fifth Discipline. The art and practice of the learning organization. New York: 

Doubleday. 

Senge, P., Ed. (2000). Schools That Learn: A Fifth Discipline Fieldbook for Educators, Parents, and Everyone 

Who Cares About Education. New York: Doubleday Dell Publishing Group. 



System theory 



100 



Snooks, G.D. (2008). "A general theory of complex living systems: Exploring the demand side of dynamics", 

Complexity, 13: 12-20. 

Steiss, A.W. (1967). Urban Systems Dynamics. Toronto: Lexington Books. 

Gerald Weinberg. (1975). An Introduction to General Systems Thinking (1975 ed., Wiley-Interscience) (2001 ed. 

Dorset House). 

Wiener, N. (1967). The human use of human beings. Cybernetics and Society. New York: Avon. 

Young, O. R., "A Survey of General Systems Theory", General Systems, vol. 9 (1964), pages 61—80. (overview 

about different trends and tendencies, with bibliography) 



External links 

T311 

• Systems theory at Principia Cybernetica Web 

Organizations 

T321 

• International Society for the System Sciences 



New England Complex Systems Institute 
System Dynamics Society 



[33] 



References 

[1] Bertalanffy (1950: 142) 

[2] (Laszlo 1974) 

[3] (Schein 1980: 4-11) 

[4] Laslo(1972: 14-15) 

[5] (Banathy 1997: f 22) 

[6] 1968, General System theory: Foundations, Development, Applications, New York: George Braziller, revised edition 1976: ISBN 

0-8076-0453-4 

[7] (see Steiss 1967; Buckley, 1967) 

[8] Peter Senge (2000: 27-49) 

[9] (Bailey 1994: 3-8; see also Owens 2004) 



[10 

[11 
[12 
[13 
[14 
[15 
[16 

[17 
[18 
[19 

[20 
[21 

[22 

[23 
[24 
[25 
[26 
[27 
[28 
[29 
[30 
[31 
[32 
[33 



(Bailey 1994: 3-8) 

(Bailey 1994; Flood 1997; Checkland 1999; Laszlo 1972) 

(Hammond 2003: 12-13) 

Hammond 2003: 5-9 

Hull 1970 

(Hammond 2003: 229-233) 

Karl Ludwig von Bertalanffy: ... aber vom Menschen wissen wir nichts, (English title: Robots, Men and Minds), translated by Dr. 
Hans-Joachim Flechtner. page 115. Econ Verlag GmbH (1970), Duesseldorf, Wien. 1st edition. 

(GST p.32) 

perspectives_on_general_system_theory [ProjectsISSS] (http://projects.isss.org/perspectives_on_general_system_theory) 

von Bertalanffy, Ludwig, (1974) Perspectives on General System Theory Edited by Edgar Taschdjian. George Braziller, New York 

main_systemsinquiry [ProjectsISSS] (http://projects.isss.org/Main/SystemsInquiry) 

(see Ash 1992: 198-207) 

Thome, Bernhard (1993). Systems Engineering: Principles and Practice of Computer-based Systems Engineering. Chichester: John Wiley ■ 
Sons. ISBN 0-471-93552-2. 

INCOSE. "What is Systems Engineering" (http://www.incose.org/practice/whatissystemseng.aspx). . Retrieved 2006-1 1-26. 

Lester R. Bittel and Muriel Albers Bittel (1978), Encyclopedia of Professional Management, McGraw-Hill, ISBN 0070054789, p.498. 

Michael M. Behrmann (1984), Handbook of Microcomputers in Special Education. College Hill Press. ISBN 093301435X. Page 212. 

http://benking.de/systems/encyclopedia/concepts-and-models.htm 

http://benking.de/encyclopedia/ 

http://wwwu.uni-klu.ac.at/gossimit/ifsr/francois/encyclopedia.htm 

http://www.uni-klu.ac.at/~gossimit/ifsr/francois/papers/systemics_and_cybernetics_in_a_historical_perspective.pdf 

http://neocybernetics.com/reportl51/ 

http://pespmcl.vub.ac.be/SYSTHEOR.html 

http://projects.isss.org/Main/Primer 

http://www.necsi.org/ 



System theory 101 

[34] http://www.systemdynamics.org/ 



Systems thinking 



Systems thinking is the process of understanding how things influence one another within a whole. In nature 
systems thinking examples include ecosystems in which various elements such as air, water, movement, plant and 
animals work together to survive or perish. In organizations, systems consist of people, structures, and processes that 
work together to make an organization healthy or unhealthy. 

Systems thinking has been defined as an approach to problem solving, by viewing "problems" as parts of an overall 
system, rather than reacting to specific part, outcomes or events and potentially contributing to further development 
of unintended consequences. Systems thinking is not one thing but a set of habits or practices within a framework 
that is based on the belief that the component parts of a system can best be understood in the context of relationships 
with each other and with other systems, rather than in isolation. Systems thinking focuses on cyclical rather than 
linear cause and effect. 

In science systems, it is argued that the only way to fully understand why a problem or element occurs and persists is 
to understand the parts in relation to the whole. Standing in contrast to Descartes's scientific reductionism and 
philosophical analysis, it proposes to view systems in a holistic manner. Consistent with systems philosophy, 
systems thinking concerns an understanding of a system by examining the linkages and interactions between the 
elements that compose the entirety of the system. 

Science systems thinking attempts to illustrate that events are separated by distance and time and that small catalytic 
events can cause large changes in complex systems. Acknowledging that an improvement in one area of a system 
can adversely affect another area of the system, it promotes organizational communication at all levels in order to 
avoid the silo effect. Systems thinking techniques may be used to study any kind of system — natural, scientific, 
engineered, human, or conceptual. 

The concept of a system 

Science systems thinkers consider that: 

• a system is a dynamic and complex whole, interacting as a structured functional unit; 

• energy, material and information flow among the different elements that compose the system; 

• a system is a community situated within an environment; 

• energy, material and information flow from and to the surrounding environment via semi-permeable membranes 
or boundaries; 

• systems are often composed of entities seeking equilibrium but can exhibit oscillating, chaotic, or exponential 
behavior. 

A holistic system is any set (group) of interdependent or temporally interacting parts. Parts are generally systems 
themselves and are composed of other parts, just as systems are generally parts or holons of other systems. 

Science systems and the application of science systems thinking has been grouped into three categories based on the 
techniques used to tackle a system: 

• Hard systems — involving simulations, often using computers and the techniques of operations research. Useful 
for problems that can justifiably be quantified. However it cannot easily take into account unquantifiable variables 
(opinions, culture, politics, etc), and may treat people as being passive, rather than having complex motivations. 

• Soft systems — For systems that cannot easily be quantified, especially those involving people holding multiple 
and conflicting frames of reference. Useful for understanding motivations, viewpoints, and interactions and 
addressing qualitative as well as quantitative dimensions of problem situations. Soft systems are a field that 
utilizes foundation methodological work developed by Peter Checkland, Brian Wilson and their colleagues at 



Systems thinking 102 

Lancaster University. Morphological analysis is a complementary method for structuring and analysing 
non-quantifiable problem complexes. 

• Evolutionary systems — Bela H. Banathy developed a methodology that is applicable to the design of complex 
social systems. This technique integrates critical systems inquiry with soft systems methodologies. Evolutionary 
systems, similar to dynamic systems are understood as open, complex systems, but with the capacity to evolve 
over time. Banathy uniquely integrated the interdisciplinary perspectives of systems research (including chaos, 
complexity, cybernetics), cultural anthropology, evolutionary theory, and others. 

The systems approach 

The systems thinking approach incorporates several tenets: 

Interdependence of objects and their attributes - independent elements can never constitute a system 

Holism - emergent properties not possible to detect by analysis should be possible to define by a holistic approach 

Goal seeking - systemic interaction must result in some goal or final state 

Inputs and Outputs - in a closed system inputs are determined once and constant; in an open system additional 

inputs are admitted from the environment 

Transformation of inputs into outputs - this is the process by which the goals are obtained 

Entropy - the amount of disorder or randomness present in any system 

Regulation - a method of feedback is necessary for the system to operate predictably 

Hierarchy - complex wholes are made up of smaller subsystems 

Differentiation - specialized units perform specialized functions 

Equifinality - alternative ways of attaining the same objectives (convergence) 

Multifinality - attaining alternative objectives from the same inputs (divergence) 

Some examples: 

• Rather than trying to improve the braking system on a car by looking in great detail at the material composition of 
the brake pads (reductionist), the boundary of the braking system may be extended to include the interactions 
between the: 

brake disks or drums 

brake pedal sensors 

hydraulics 

driver reaction time 

tires 

road conditions 

weather conditions 

time of day 

• Using the tenet of "Multifinality", a supermarket could be considered to be: 

• a "profit making system" from the perspective of management and owners 

• a "distribution system" from the perspective of the suppliers 

• an "employment system" from the perspective of employees 

• a "materials supply system" from the perspective of customers 

• an "entertainment system" from the perspective of loiterers 

• a "social system" from the perspective of local residents 

• a "dating system" from the perspective of single customers 

As a result of such thinking, new insights may be gained into how the supermarket works, why it has problems, how 
it can be improved or how changes made to one component of the system may impact the other components. 



Systems thinking 



103 



Applications 

Science systems thinking is increasingly being used to tackle a wide variety of subjects in fields such as computing 
engineering, epidemiology, information science, health, manufacture, management, and the environment. 

Some examples: 

Organizational architecture 

Job design 

Team Population and Work Unit Design 

Linear and Complex Process Design 

Supply Chain Design 

Business continuity planning with FMEA protocol 

Critical Infrastructure Protection via FBI Infragard 

Delphi method — developed by RAND for USAF 

Futures studies — Thought leadership mentoring 

The public sector including examples at The Systems Thinking Review [4] 

Leadership development 

Oceanography — forecasting complex systems behavior 

Permaculture 

Quality function deployment (QFD) 

Quality management — Hoshin planning methods 

Quality storyboard — StoryTech framework (LeapfrogU-EE) 

Software quality 

Program management 

Project management 

MECE - McKinsey Way 



See also 



Boundary critique 
Crossdisciplinarity 
Holistic management 
Information Flow Diagram 
Interdisciplinary 
Multi disciplinary 
Negative feedback 
Soft systems methodology 
Synergetics (Fuller) 
System dynamics 



Systematics - study of multi-term systems 

Systemics 

Systems engineering 

Systems intelligence 

Systems philosophy 

Systems theory 

Systems science 

Transdisciplinary 

Terms used in systems theory 



Systems thinking 104 

Bibliography 

Russell L. Ackoff (1999) Ackoff's Best: His Classic Writings on Management. (Wiley) ISBN 0-471-31634-2 

Russell L. Ackoff (2010) Systems Thinking for Curious Managers [6] . (Triarchy Press). ISBN 978-0-9562631-5-5 

Bela H. Banathy (1996) Designing Social Systems in a Changing World (Contemporary Systems Thinking). 

(Springer) ISBN 0-306-45251-0 

Bela H. Banathy (2000) Guided Evolution of Society: A Systems View (Contemporary Systems Thinking). 

(Springer) ISBN 0-306-46382-2 

Ludwig von Bertalanffy (1976 - revised) General System theory: Foundations, Development, Applications. 

(George Braziller) ISBN 0-807-60453-4 

Fritjof Capra (1997) The Web of Life (HarperCollins) ISBN 0-00-654751-6 

Peter Checkland (1981) Systems Thinking, Systems Practice. (Wiley) ISBN 0-471-27911-0 

Peter Checkland, Jim Scholes (1990) Soft Systems Methodology in Action. (Wiley) ISBN 0-471-92768-6 

Peter Checkland, Jim Sue Holwell (1998) Information, Systems and Information Systems. (Wiley) ISBN 

0-471-95820-4 

Peter Checkland, John Poulter (2006) Learning for Action. (Wiley) ISBN 0-470-02554-9 

C. West Churchman (1984 - revised) The Systems Approach. (Delacorte Press) ISBN 0-440-38407-9. 

John Gall (2003) The Systems Bible: The Beginner's Guide to Systems Large and Small. (General Systemantics 

Pr/Liberty) ISBN 0-961-82517-0 

Jamshid Gharajedaghi (2005) Systems Thinking: Managing Chaos and Complexity - A Platform for Designing 

Business Architecture. (Butterworth-Heinemann) ISBN 0-750-67973-5 

Charles Francois (ed) (1997), International Encyclopedia of Systems and Cybernetics, Munchen: K. G Saur. 

Charles L. Hutchins (1996) Systemic Thinking: Solving Complex Problems CO:PDS ISBN 1-888017-51-1 

Bradford Keeney (2002 - revised) Aesthetics of Change. (Guilford Press) ISBN 1-572-30830-3 

Donella Meadows (2008) Thinking in Systems - A primer (Earthscan) ISBN 978-1-84407-726-7 

John Seddon (2008) Systems Thinking in the Public Sector [7] . (Triarchy Press). ISBN 978-0-9550081-8-4 

Peter M. Senge (1990) The Fifth Discipline - The Art & Practice of The Learning Organization. (Currency 

Doubleday) ISBN 0-385-26095-4 

Lars Skyttner (2006) General Systems Theory: Problems, Perspective, Practice (World Scientific Publishing 

Company) ISBN 9-812-56467-5 

Frederic Vester (2007) The Art of interconnected Thinking. Ideas and Tools for tackling with Complexity (MCB) 

ISBN 3-939-31405-6 

Gerald M. Weinberg (2001 - revised) An Introduction to General Systems Thinking. (Dorset House) ISBN 

0-932-63349-8 

Brian Wilson (1990) Systems: Concepts, Methodologies and Applications, 2nd ed. (Wiley) ISBN 0-471-92716-3 

Brian Wilson (2001) Soft Systems Methodology: Conceptual Model Building and its Contribution. (Wiley) ISBN 

0-471-89489-3 



Systems thinking 105 

External links 

• International Society for the Systems Sciences (ISSS) on Wikipedia, 

ro] 

• International Society for the System Sciences home page 

• UK Systems Society [9] 

• The Systems Thinker newsletter glossary 

• Dancing With Systems from Project Worldview 

ri2i 

• Systems-thinking.de : systems thinking links displayed as a network 

ri3i 

• Systems Thinking 

References 

[I] http://www.watersfoundation.org/index.cfm?fuseaction=materials.main 

[2] Capra, F. (1996) The web of life: a new scientific understanding of living systems (1st Anchor Books ed). New York: Anchor Books, p. 30 
[3] Skyttner, Lars (2006). General Systems Theory: Problems, Perspective, Practice. World Scientific Publishing Company. 

ISBN 9-812-56467-5. 
[4] http://www.thesystemsthinkingreview.co.uk/ 
[5] http://www.qualitydigest.com/may97/html/hoshin.html 

[6] http://triarchypress.com/pages/Systems_Thinking_for_Curious_Managers.htm 
[7] http://www.triarchypress.co.uk/pages/book5.htm 
[8] http://isss.org/world/ 
[9] http://www.ukss.org.uk 
[10] http://www.thesystemsthinker.com/systemsthinkinglearn.html 

[II] http://www.projectworldview.org/wvthemel3.htm 
[12] http ://www. systems-thinking, de/ 

[13] http://www.thinking.net/Systems_Thinking/systems_thinking.html 



System dynamics 



106 



System dynamics 



System dynamics is an approach to 
understanding the behaviour of 
complex systems over time. It deals 
with internal feedback loops and time 
delays that affect the behaviour of the 
entire system. What makes using 
system dynamics different from other 
approaches to studying complex 
systems is the use of feedback loops 
and stocks and flows. These elements 
help describe how even seemingly 
simple systems display baffling 
nonlinearity. 

Overview 

System dynamics is a methodology 
and computer simulation modeling 
technique for framing, understanding, 
and discussing complex issues and 
problems. Originally developed in the 
1950s to help corporate managers 
improve their understanding of 

industrial processes, system dynamics 

is currently being used throughout the 

public and private sector for policy analysis and design 




Dynamic stock and flow diagram of model New product adoption (model from article by 

John Sterman 2001) 



[2] 



System dynamics is an aspect of systems theory as a method for understanding the dynamic behavior of complex 
systems. The basis of the method is the recognition that the structure of any system — the many circular, 
interlocking, sometimes time-delayed relationships among its components — is often just as important in 
determining its behavior as the individual components themselves. Examples are chaos theory and social dynamics. 
It is also claimed that because there are often properties-of-the-whole which cannot be found among the 
properties-of-the-elements, in some cases the behavior of the whole cannot be explained in terms of the behavior of 
the parts. 



History 

System dynamics was created during the mid-1950s by Professor Jay Forrester of the Massachusetts Institute of 
Technology. In 1956, Forrester accepted a professorship in the newly-formed MIT School of Management. His 
initial goal was to determine how his background in science and engineering could be brought to bear, in some 
useful way, on the core issues that determine the success or failure of corporations. Forrester's insights into the 
common foundations that underlie engineering and management, which led to the creation of system dynamics, were 
triggered, to a large degree, by his involvement with managers at General Electric (GE) during the mid-1950s. At 
that time, the managers at GE were perplexed because employment at their appliance plants in Kentucky exhibited a 
significant three-year cycle. The business cycle was judged to be an insufficient explanation for the employment 



System dynamics 107 

instability. From hand simulations (or calculations) of the stock-flow-feedback structure of the GE plants, which 
included the existing corporate decision-making structure for hiring and layoffs, Forrester was able to show how the 

instability in GE employment was due to the internal structure of the firm and not to an external force such as the 

T21 
business cycle. These hand simulations were the beginning of the field of system dynamics. 

During the late 1950s and early 1960s, Forrester and a team of graduate students moved the emerging field of system 
dynamics from the hand-simulation stage to the formal computer modeling stage. Richard Bennett created the first 
system dynamics computer modeling language called SIMPLE (Simulation of Industrial Management Problems with 
Lots of Equations) in the spring of 1958. In 1959, Phyllis Fox and Alexander Pugh wrote the first version of 
DYNAMO (DYNAmic MOdels), an improved version of SIMPLE, and the system dynamics language became the 

industry standard for over thirty years. Forrester published the first, and still classic, book in the field titled Industrial 

T21 
Dynamics in 1961. 

From the late 1950s to the late 1960s, system dynamics was applied almost exclusively to corporate/managerial 
problems. In 1968, however, an unexpected occurrence caused the field to broaden beyond corporate modeling. John 
Collins, the former mayor of Boston, was appointed a visiting professor of Urban Affairs at MIT. The result of the 

Collins-Forrester collaboration was a book titled Urban Dynamics. The Urban Dynamics model presented in the 

T21 
book was the first major non-corporate application of system dynamics. 

The second major noncorporate application of system dynamics came shortly after the first. In 1970, Jay Forrester 
was invited by the Club of Rome to a meeting in Bern, Switzerland. The Club of Rome is an organization devoted to 
solving what its members describe as the "predicament of mankind" — that is, the global crisis that may appear 
sometime in the future, due to the demands being placed on the earth's carrying capacity (its sources of renewable 
and nonrenewable resources and its sinks for the disposal of pollutants) by the world's exponentially growing 
population. At the Bern meeting, Forrester was asked if system dynamics could be used to address the predicament 
of mankind. His answer, of course, was that it could. On the plane back from the Bern meeting, Forrester created the 
first draft of a system dynamics model of the world's socioeconomic system. He called this model WORLD 1. Upon 
his return to the United States, Forrester refined WORLD 1 in preparation for a visit to MIT by members of the Club 

of Rome. Forrester called the refined version of the model WORLD2. Forrester published WORLD2 in a book titled 

T21 
World Dynamics. 

Topics in systems dynamics 

The elements of system dynamics diagrams are feedback, accumulation of flows into stocks and time delays. 

As an illustration of the use of system dynamics, imagine an organisation that plans to introduce an innovative new 
durable consumer product. The organisation needs to understand the possible market dynamics in order to design 
marketing and production plans. 

Causal loop diagrams 

A causal loop diagram is a visual representation of the feedback loops in a system. The causal loop diagram of the 
new product introduction may look as follows: 




Adoption rate i R | Adopters 

Word of mouth 



Causal loop diagram of New product adoption 
model 



System dynamics 



108 



There are two feedback loops in this diagram. The positive reinforcement (labeled R) loop on the right indicates that 
the more people have already adopted the new product, the stronger the word-of-mouth impact. There will be more 
references to the product, more demonstrations, and more reviews. This positive feedback should generate sales that 
continue to grow. 

The second feedback loop on the left is negative reinforcement (or "balancing" and hence labeled B). Clearly growth 
can not continue forever, because as more and more people adopt, there remain fewer and fewer potential adopters. 

Both feedback loops act simultaneously, but at different times they may have different strengths. Thus one would 
expect growing sales in the initial years, and then declining sales in the later years. 




Causal loop diagram of New product adoption 
model with nodes values after calculus 



In this dynamic causal loop diagram : 

• stepl : (+) green arrows show that Adoption rate is function of Potential Adopters and Adopters 

• step2 : (-) red arrow shows that Potential adopters decreases by Adoption rate 

• step3 : (+) blue arrow shows that Adopters increases by Adoption rate 



Stock and flow diagrams 

The next step is to create what is termed a stock and flow diagram. A stock is the term for any entity that 
accumulates or depletes over time. A flow is the rate of change in a stock. 









{--.. TT fc. 


Stock 




<vJ A ^" 

Flow 


A flow changes the rate of ace 
stock 


umulation of th 


e 



In our example, there are two stocks: Potential adopters and Adopters. There is one flow: New adopters. For every 
new adopter, the stock of potential adopters declines by one, and the stock of adopters increases by one. 



System dynamics 



109 




Probability that 
"■contact has not yet- 
adopted 



Stock and flow diagram of New product adoption 
model 



Equations 

The real power of system dynamics is utilised through simulation. Although it is possible to perform the modeling in 
a spreadsheet, there are a variety of software packages that have been optimised for this. 

The steps involved in a simulation are: 

Define the problem boundary 

Identify the most important stocks and flows that change these stock levels 

Identify sources of information that impact the flows 

Identify the main feedback loops 

Draw a causal loop diagram that links the stocks, flows and sources of information 

Write the equations that determine the flows 

Estimate the parameters and initial conditions. These can be estimated using statistical methods, expert opinion, 

market research data or other relevant sources of information. 

Simulate the model and analyse results 



[3] 



In this example, the equations that change the two stocks via the flow are: 

Potential adopters = / -New adopters dt 

Jo 

Adopters = / New adopters dt 
Jo 

List of all the equations, in their order of e 

1) Probability that contact has not yet adopted = 



List of all the equations, in their order of execution in each year, from year 1 to 15: 

Potential adopters 



Potential adopters + Adopters 

2) Imitators = q ■ Adopters ■ Probability that contact has not yet adopted 

3) Innovators = p • Potential adopters 

4) New adopters = Innovators + Imitators 
4.1) Potential adopters — = New adopters 



4.2) Adopters + = New adopters 



System dynamics 



110 




Dynamic simulation results 

The dynamic simulation results show that the behaviour of the system would be to have growth in Adopters that 

follows a classical s-curve shape. 

The increase in Adopters is very slow initially, then exponential growth for a period, followed ultimately by 

saturation. 




Dynamic stock and flow diagram of New product 
adoption model 





















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3 

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7 

10 

12 
13 

15 




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397.05 


Stocks and flows values for years = to 15 





Application 

System dynamics has found application in a wide range of areas, for example population, ecological and economic 
systems, which usually interact strongly with each other. 

System dynamics have various "back of the envelope" management applications. They are a potent tool to: 

• Teach system thinking reflexes to persons being coached 

• Analyze and compare assumptions and mental models about the way things work 

• Gain qualitative insight into the workings of a system or the consequences of a decision 

• Recognize archetypes of dysfunctional systems in everyday practice 

Computer software is used to simulate a system dynamics model of the situation being studied. Running "what if" 
simulations to test certain policies on such a model can greatly aid in understanding how the system changes over 
time. System dynamics is very similar to systems thinking and constructs the same causal loop diagrams of systems 
with feedback. However, system dynamics typically goes further and utilises simulation to study the behaviour of 



systems and the impact of alternative policies 



[4] 



System dynamics has been used to investigate resource dependencies, and resulting problems, in product 
development. 



System dynamics 

Example 



ill 



..■>*■*» fcj&mmisskiru 
as camp«ti4^l<ifi 

liiy*H-i--i-: 
peitorrnancfl 

r 

Pwttofio 

nmrnalcfi 

SoJvtncy J 

mtroin 





Sick ante \ j 

^r Eipania 

mfe 



Afeiityfcxsfl* 
hig* 1 *« rwdrr 
vakifl- Bart^ oft 



Causal loop diagram of a model examining the growth or decline of a life insurance company. 



The figure above is a causal loop diagram of a system dynamics model created to examine forces that may be 
responsible for the growth or decline of life insurance companies in the United Kingdom. A number of this figure's 
features are worth mentioning. The first is that the model's negative feedback loops are identified by "C's," which 
stand for "Counteracting" loops. The second is that double slashes are used to indicate places where there is a 
significant delay between causes (i.e., variables at the tails of arrows) and effects (i.e., variables at the heads of 
arrows). This is a common causal loop diagramming convention in system dynamics. Third, is that thicker lines are 
used to identify the feedback loops and links that author wishes the audience to focus on. This is also a common 
system dynamics diagramming convention. Last, it is clear that a decision maker would find it impossible to think 



through the dynamic behavior inherent in the model, from inspection of the figure alone 



[7] 



Example of 4D piston motion 




Piston_motion_equations 



This animation was made with the 3D modeler of a system dynamics software. 

The calculated values are associated with parameters of the rod and crank. 

In this example the crank is driving, we vary both the speed of rotation, its radius and the length of the rod, the piston 

follows. 



System dynamics 

See also 



112 



Related subjects 

Causal loop diagram 
Ecosystem model 
Plateau Principle 
System Archetypes 
System Dynamics Society 
Twelve leverage points 
Wicked problems 
World3 

Population dynamics 
Predator-prey interaction 



Related fields 

• Dynamical systems theory 

• Operations research 

• Social dynamics 

• Systems theory 

• Systems thinking 

• TRIZ 



Related scientists 

Jay Forrester 
Dennis Meadows 
Donella Meadows 
Peter Senge 
John Sterman 



Further reading 



• Forrester, Jay W. (1961). Industrial Dynamics. Pegasus Communications. ISBN 1883823366. 

• Forrester, Jay W. (1969). Urban Dynamics. Pegasus Communications. ISBN 1883823390. 

• Meadows, Donella H. (1972). Limits to Growth. New York: University books. ISBN 0-87663-165-0. 

• Morecroft, John (2007). Strategic Modelling and Business Dynamics: A Feedback Systems Approach. John Wiley 
& Sons. ISBN 0470012862. 

• Roberts, Edward B. (1978). Managerial Applications of System Dynamics. Cambridge: MIT Press. ISBN 
026218088X. 

• Randers, Jorgen (1980). Elements of the System Dynamics Method. Cambridge: MIT Press. ISBN 0915299399. 

• Senge, Peter (1990). The Fifth Discipline. Currency. ISBN 0-385-26095-4. 

• Sterman, John D. (2000). Business Dynamics: Systems thinking and modeling for a complex world. McGraw Hill. 
ISBN 0-07-231135-5. 



External links 

• U.S. Department of Energy's Introduction to System Dynamics 

[9] 



[8] 



Desert Island Dynamics "An Annotated Survey of the Essential System Dynamics Literature" 



References 



[1] MIT System Dynamics in Education Project (SDEP) (http://sysdyn.clexchange.org) 

[2] Robert A. Taylor (2008). "Origin of System Dynamics: Jay W. Forrester and the History of System Dynamics" (http://www. 

systemdynamics.org/DL-IntroSysDyn/start.htm). In: U.S. Department of Energy's Introduction to System Dynamics. Retrieved 23 Oktober 

2008. 
[3] Sterman, John D. (2001). "System dynamics modeling: Tools for learning in a complex world". California management review 43 (4): 8—25. 
[4] System Dynamics Society (http://www.systemdynamics.org/) 
[5] Repenning, Nelson P. (2001). "Understanding fire fighting in new product development". The Journal of Product Innovation Management 

18: 285 - 300. doi:10.1016/S0737-6782(01)00099-6. 
[6] Neldon P. Repenning (1999). Resource dependence in product development improvement efforts, Massachusetts Institute of Technology 

Sloan School of Management Department of Operations Management/System Dynamics Group, dec 1999. 
[7] Robert A. Taylor (2008). "Feedback" (http://www.systemdynamics.org/DL-IntroSysDyn/start.htm). In: U.S. Department of Energy's 

Introduction to System Dynamics. Retrieved 23 October 2008. 
[8] http://www.systemdynamics.org/DL-IntroSysDyn/ 
[9] http://web.mit.edu/jsterman/www/DID.html 



Dynamics 113 



Dynamics 



Dynamics (from Greek Svva^uKog - dynamikos "powerful", from Svvafug - dynamis "power") may refer to: 

• Dynamics (music), In music, dynamics refers to the softness or loudness of a sound or note. The term is also 
applied to the written or printed musical notation used to indicate dynamics (also known as volume in a song) 

GBF Underground Mining 

Physics 

• Dynamics (physics), in physics, dynamics refers to time evolution of physical processes 

Field 

• Analytical dynamics refers to the motion of bodies as induced by external forces 

• Flight dynamics, the science of aircraft and spacecraft design 

• Force Dynamics 

• Fluid dynamics, the study of fluid flow 

• Computational fluid dynamics 

• Molecular dynamics, the study of motion on the molecular level 

• Langevin dynamics 

• Brownian dynamics 

• In quantum physics, dynamics may refer to how forces are quantized, as in quantum electrodynamics or quantum 
chromodynamics 

• Relativistic dynamics may refer to a combination of relativistic and quantum concepts 

• Stellar dynamics 

• System dynamics, the study of the behaviour of complex systems 

• Thermodynamics, a branch of physics that studies the relationships between heat and mechanical energy 

Branch 

• Aerodynamics, the study of gases in motion 

• Hydrodynamics, the study of liquids or water in motion 

• Neurodynamics, an area of research in the brain sciences which places a strong focus upon the spatio-temporal 
(dynamic) character of neural activity in describing brain function 

• Thermodynamics 

Sociology 

• Group dynamics, the study of social group processes 

• Population dynamics 

• Power dynamics, the dynamics of power, used in sociology 

• Psychodynamics, the study of the interrelationship of various parts of the mind, personality, or psyche as they 
relate to mental, emotional, or motivational forces especially at the subconscious level 

• Spiral Dynamics, a social development theory 

• Social dynamics (interdisciplinary) 



Dynamics 114 

Computer science and math 

• Dynamical system, in mathematics or complexity 

• Dynamic programming in computer science and control theory 

• Dynamic program analysis, in computer science, a set of methods for analyzing code that is performed with 
executing programs built from that software on a real or virtual processor 

• Symbolic dynamics 

Companies 

• Arrow Dynamics, rollercoast company 

• Boston Dynamics, robot design company 

• Crystal Dynamics, video game developer 

• General Dynamics 

Other 

• Microsoft Dynamics, a line of business software owned and developed by Microsoft 

See also 

• All pages beginning with "dynamic" 

• All pages with titles containing "dynamic" 

• Power (disambiguation) 

• Kinetics (disambiguation) 

• Dynamic Host Configuration Protocol 



115 



Mathematical Biology, Complex Systems 

Biology 

Mathematical biology 

Mathematical and theoretical biology is an interdisciplinary academic research field with a range of applications in 
biology, medicine and biotechnology. The field may be referred to as mathematical biology or biomathematics 

to stress the mathematical side, or as theoretical biology to stress the biological side. It includes at least four 
major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), 
bioinformatics and computational biomodelinglbiocomputing. 

Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, 
using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in 
biological, biomedical and biotechnology research. For example, in cell biology, protein interactions are often 
represented as "cartoon" models, which, although easy to visualize, do not accurately describe the systems studied. 
In order to do this, precise mathematical models are required. By describing the systems in a quantitative manner, 
their behavior can be better simulated, and hence properties can be predicted that might not be evident to the 
experimenter. 

Importance 

Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the 
field. Some reasons for this include: 

• the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand 
without the use of analytical tools, 

• recent development of mathematical tools such as chaos theory to help understand complex, nonlinear 
mechanisms in biology, 

• an increase in computing power which enables calculations and simulations to be performed that were not 
previously possible, and 

• an increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other 
complications involved in human and animal research. 

Areas of research 

Several areas of specialized research in mathematical and theoretical biology as well as external links 

to related projects in various universities are concisely presented in the following subsections, including also a large 
number of appropriate validating references from a list of several thousands of published authors contributing to this 
field. Many of the included examples are characterised by highly complex, nonlinear, and supercomplex 
mechanisms, as it is being increasingly recognised that the result of such interactions may only be understood 
through a combination of mathematical, logical, physical/chemical, molecular and computational models. Due to the 
wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between 
mathematicians, biomathematicians, theoretical biologists, physicists, biophysicists, biochemists, bioengineers, 
engineers, biologists, physiologists, research physicians, biomedical researchers, oncologists, molecular biologists, 
geneticists, embryologists, zoologists, chemists, etc. 



Mathematical biology 116 

Computer models and automata theory 

ro] 

A monograph on this topic summarizes an extensive amount of published research in this area up to 1987, 
including subsections in the following areas: computer modeling in biology and medicine, arterial system models, 
neuron models, biochemical and oscillation networks, quantum automata , quantum computers in molecular 
biology and genetics, cancer modelling, neural nets, genetic networks, abstract relational biology, 
metabolic-replication systems, category theory applications in biology and medicine, automata theory, cellular 
automata, tessallation models and complete self-reproduction , chaotic systems in organisms, relational 

biology and organismic theories. This published report also includes 390 references to peer-reviewed articles 

by a large number of authors. 

Modeling cell and molecular biology 

This area has received a boost due to the growing importance of molecular biology. 

ri9i 

Mechanics of biological tissues 

Theoretical enzymology and enzyme kinetics 

Cancer modelling and simulation 

["221 

Modelling the movement of interacting cell populations 

T231 
Mathematical modelling of scar tissue formation 

Mathematical modelling of intracellular dynamics 

T251 
Mathematical modelling of the cell cycle 

Modelling physiological systems 

• Modelling of arterial disease 

T271 

• Multi-scale modelling of the heart 

Molecular set theory 

Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical 

no] 

biology and especially in Mathematical Medicine. Molecular set theory (MST) is a mathematical formulation of 
the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical 
transformations represented by set-theoretical mappings between molecular sets. In a more general sense, MST is the 
theory of molecular categories defined as categories of molecular sets and their chemical transformations represented 
as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of 
clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to 

no] r9Ql 

Physiology, Clinical Biochemistry and Medicine. 

Population dynamics 

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back 
to the 19th century. The Lotka— Volterra predator-prey equations are a famous example. In the past 30 years, 
population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. 
Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population 
dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the 
study of infectious disease affecting populations. Various models of the spread of infections have been proposed and 
analyzed, and provide important results that may be applied to health policy decisions. 



Mathematical biology 117 

Mathematical methods 

A model of a biological system is converted into a system of equations, although the word 'model' is often used 
synonymously with the system of corresponding equations. The solution of the equations, by either analytical or 
numerical means, describes how the biological system behaves either over time or at equilibrium. There are many 
different types of equations and the type of behavior that can occur is dependent on both the model and the equations 
used. The model often makes assumptions about the system. The equations may also make assumptions about the 
nature of what may occur. 

Mathematical biophysics 

The earlier stages of mathematical biology were dominated by mathematical biophysics, described as the application 
of mathematics in biophysics, often involving specific physical/mathematical models of biosystems and their 
components or compartments. 

The following is a list of mathematical descriptions and their assumptions. 

Deterministic processes (dynamical systems) 

A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in 
time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space. 

• Difference equations/Maps — discrete time, continuous state space. 

• Ordinary differential equations — continuous time, continuous state space, no spatial derivatives. See also: 
Numerical ordinary differential equations. 

• Partial differential equations — continuous time, continuous state space, spatial derivatives. See also: Numerical 
partial differential equations. 

Stochastic processes (random dynamical systems) 

A random mapping between an initial state and a final state, making the state of the system a random variable with a 
corresponding probability distribution. 

• Non-Markovian processes — generalized master equation — continuous time with memory of past events, discrete 
state space, waiting times of events (or transitions between states) discretely occur and have a generalized 
probability distribution. 

• Jump Markov process — master equation — continuous time with no memory of past events, discrete state space, 
waiting times between events discretely occur and are exponentially distributed. See also: Monte Carlo method 
for numerical simulation methods, specifically continuous -time Monte Carlo which is also called kinetic Monte 
Carlo or the stochastic simulation algorithm. 

• Continuous Markov process — stochastic differential equations or a Fokker-Planck equation — continuous time, 
continuous state space, events occur continuously according to a random Wiener process. 

Spatial modelling 

One classic work in this area is Alan Turing's paper on morphogenesis entitled The Chemical Basis of 
Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society. 

• Travelling waves in a wound-healing assay 

• Swarming behaviour 

T321 

• A mechanochemical theory of morphogenesis 

[331 

• Biological pattern formation 

• Spatial distribution modeling using plot samples 



Mathematical biology 



118 



Phylogenetics 

Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and 

[351 
networks based on inherited characteristics 



Model example: the cell cycle 

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to 
cancers. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid 
results. Two research groups have produced several models of the cell cycle simulating several organisms. 

They have recently produced a generic eukaryotic cell cycle model which can represent a particular eukaryote 
depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due 
to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et 
al., 2006). 

By means of a system of ordinary differential equations these models show the change in time (dynamical system) of 
the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model 
describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process). 
To obtain these equations an iterative series of steps must be done: first the several models and observations are 
combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential 
equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme substrate 
reactions and Goldbeter— Koshland kinetics for ultrasensitive transcription factors, afterwards the parameters of the 
equations (rate constants, enzyme efficiency coefficients and Michealis constants) must be fitted to match 
observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring 
diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such 
as protein half-life and cell size. 

In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or 
by analysis. 

In a simulation, given a starting vector (list of the values of the variables), the progression of the system is calculated 
by solving the equations at each time-frame in small increments. 

In analysis, the proprieties of the 

equations are used to investigate the 

behavior of the system depending of 

the values of the parameters and 

variables. A system of differential 

equations can be represented as a 

vector field, where each vector 

described the change (in concentration 

of two or more protein) determining 

where and how fast the trajectory 

L 
(simulation) is heading. Vector fields 

can have several special points: a 

stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an 

unstable point, either a source or a saddle point which repels (forcing the concentrations to change away from a 

certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the 

concentrations oscillate). 

A better representation which can handle the large number of variables and parameters is called a bifurcation 



BIFURCATION DIAGRAM 

Fixed Points 




stable steady -state- 
Mass dictates tiic ;ici ve cydiiii levels because stable steady- 
states attract {rc:;s\ ■jo cisc-rvaUes) keeping [MPF] consiant 



• o Stable/Unstable limit cycle max/min: 

Tlhe system is in a loop so at that mass tho [f.lPFf will oscill: 

Singularities 

Saddle Node; 

5W' A ;,:,,(.'!■:> 2-r.u an :i;isuiblo stead y-slatos annihilate-, boyosid 
SN2 ivtiich Ihcis are no Btpafcrium points: those bifurcation 
events v.ilt trigger the exit framGf and G2 i e spec t!v sly 



we 



cell mass- (au.) 

dMass*di = S^o^-Mass [exponential growth) 
d[C3n2JM = (k s1 + k^ |SBF])mass - ka- [Gin 2] 

The parameter mass directly conlrolE cyelin levels, osjjfc-sslr j 

implicitly i(£ yet UnKhOwn mass dependant, pjntM mechanism 



Hopf Bifurcation 

A siable and an unslable steady-states annihilate .-esuhmg ^n 
an uiiii;ia:t- ihiit cycle [eigisiwalijiss have no Real part) 



SNIPER SNIPER Bifurcation 



diagram(Bifurcation theory): the presence of these special steady-state points at certain values of a parameter (e.g. 
mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a 



Mathematical biology 119 

bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the 
cell cycle has phases (partially corresponding to Gl and G2) in which mass, via a stable point, controls cyclin levels, 
and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a 
bifurcation event (Cell cycle checkpoint), the system cannot go back to the previous levels since at the current mass 
the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a 
checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a 
Hopf bifurcation and an infinite period bifurcation. 

See also 

Abstract relational biology 

Artificial life 

Biocybernetics 

Bioinformatics 

Biologically inspired computing 

Biosemiotics 

Biostatistics 

Cellular automata 

Coalescent theory 

Complex systems biology 

Computational biology 

Digital morphogenesis 

t~> • i • u- i 13] [42] [43] [44] [45] [46] 

Dynamical systems in biology 

Epidemiology 

Evolution theories and Population Genetics 

• Population genetics models 

• Molecular evolution theories 
Ewens's sampling formula 
Excitable medium 
Journal of Theoretical Biology 
Mathematical models 

Molecular modelling 
Molecular modelling on GPU 
Software for molecular modeling 
Metabolic -replication systems 
Models of Growth and Form 
Neighbour-sensing model 
Morphometries 

/-. • • ♦ /A c\ [48] [49] 

Orgamsmic systems (OS) 

r> t ■ [48] [42] [50] 

Orgamsmic supercategones 

Population dynamics of fisheries 

Protein folding, also blue Gene and folding© home 

Quantum computers 

Quantum genetics 

Relational biology 

T521 

Self-reproduction (also called self-replication in a more general context). 
Computational gene models 



Mathematical biology 120 

• Systems biology 

• Theoretical biology 

• Theoretical ecology 

• Topological models of morphogenesis 

• DNA topology 

• DNA sequencing theory 

For use of basic arithmetics in biology, see relevant topic, such as Serial dilution. 

• Biographies 

Charles Darwin 
D'Arcy Thompson 
Joseph Fourier 
Charles S. Peskin 
Nicolas Rashevsky 
Robert Rosen 
Rosalind Franklin 
Francis Crick 
Rene Thom 
Vito Volterra 

Societies and Institutes 

• Division of Mathematical Biology at NIMR 

• Society for Mathematical Biology 

• European Society for Mathematical and Theoretical Biology 

References 

• Nicolas Rashevsky. (1938)., Mathematical Biophysics. Chicago: University of Chicago Press. 

• Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0-471-73550-7 

• Israel, G., 2005, "Book on mathematical biology" in Grattan-Guinness, I., ed., Landmark Writings in Western 
Mathematics. Elsevier: 936-44. 

• Israel G (1988). "On the contribution of Volterra and Lotka to the development of modern biomathematics". 
History and Philosophy of the Life Sciences 10 (1): 37-49. PMID 3045853. 

• Scudo FM (March 1971). "Vito Volterra and theoretical ecology". Theoretical Population Biology 2 (1): 1—23. 
doi:10.1016/0040-5809(71)90002-5. PMID 4950157. 

• S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. 
Perseus, 2001, ISBN 0-7382-0453-6 

• N.G van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 
0-444-89349-0 

• I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine., Monograph, Ch.l 1 in M. 
Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol. 7: 1513-1577 (1987),Pergamon Press:New York, 
(updated by Hsiao Chen Lin in 2004 ISBN 0-08-036377-6 
P.G Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4 
L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6 

G Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2 
A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0-521-59946-6 
L.G Harrison, Kinetic theory of living pattern. C.U.P., 1993. ISBN 0-521-30691-4 



Mathematical biology 121 

• F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics. SIAM, 
Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0 

• D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3 

• J.D. Murray, Mathematical Biology. Springer- Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An 
Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications, 
2003 ISBN 0-387-95228-4. 

• E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7 

• S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8 

• L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 0-521-27477-X 

• L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8. 

Theoretical biology 

• Bonner, J. T. 1988. The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University 
Press. 

• Hertel, H. 1963. Structure, Form, Movement. New York: Reinhold Publishing Corp. 

• Mangel, M. 1990. Special Issue, Classics of Theoretical Biology (part 1). Bull. Math. Biol. 52(1/2): 1-318. 

• Mangel, M. 2006. The Theoretical Biologist's Toolbox. Quantitative Methods for Ecology and Evolutionary 
Biology. Cambridge University Press. 

• Prusinkiewicz, P. & Lindenmeyer, A. 1990. The Algorithmic Beauty of Plants. Berlin: Springer-Verlag. 

• Reinke, J. 1901. Einleitung in die theoretische Biologic Berlin: Verlag von Gebriider Paetel. 

• Thompson, D.W. 1942. On Growth and Form. 2nd ed. Cambridge: Cambridge University Press: 2. vols. 

• Uexkiill, J. v. 1920. Theoretische Biologic Berlin: Gebr. Paetel. 

• Vogel, S. 1988. Life's Devices: The Physical World of Animals and Plants. Princeton: Princeton University Press. 

• Waddington, C.H. 1968-1972. Towards a Theoretical Biology. 4 vols. Edinburg: Edinburg University Press. 



Hoppensteadt, F. (September 1995), "Getting Started in Mathematical Biology" , Notices of American 



Further reading 

• Hoppensteadt, F. (Sej 

Mathematical Society. 

[57] 

• Reed, M. C. (March 2004), "Why Is Mathematical Biology So Hard?" , Notices of American Mathematical 
Society. 

• May, R. M. (2004), "Uses and Abuses of Mathematics in Biology", Science 303 (5659): 790-793, 
doi: 10. 1 126/science. 1094442. 

• Murray, J. D. (1988), "How the leopard gets its spots?" [58] , Scientific American 258 (3): 80-87. 

• Schnell, S.; Grima, R.; Maini, P. K. (2007), "Multiscale Modeling in Biology" [59] , American Scientist 95: 
134-142. 

• Chen, Katherine C; Calzone, Laurence; Csikasz-Nagy, Attila (2004), "Integrative analysis of cell cycle control in 
budding yeast", Mol Biol Cell 15 (8): 3841-3862, doi:10.1091/mbc.E03-ll-0794. 

• Csikasz-Nagy, Attila; Battogtokh, Dorjsuren; Chen, Katherine C; Novak, Bela; Tyson, John J. (2006), "Analysis 
of a generic model of eukaryotic cell-cycle regulation", Biophys J. 90 (12): 4361—4379, 

doi: 10. 1529/biophysj. 106.08 1240. 

• Fuss, H; Dubitzky, Werner; Downes, C. Stephen; Kurth, Mary Jo (2005), "Mathematical models of cell cycle 
regulation", Brief Bioinform. 6 (2): 163-177, doi:10.1093/bib/6.2.163. 

• Lovrics, Anna; Csikasz-Nagy, Attila; Zselyl, Istvan Gy; Zador, Judit; Turanyi, Tamas; Novak, Bela (2006), 
"Time scale and dimension analysis of a budding yeast cell cycle model", BMC Bioinform. 9 (7): 494, 
doi:10.1186/1471-2105-7-494. 



Mathematical biology 122 

External links 

The Society for Mathematical Biology 

Theoretical and mathematical biology website 

Complexity Discussion Group 

UCLA Biocybernetics Laboratory 

TUCS Computational Biomodelling Laboratory 

Nagoya University Division of Biomodeling 

Technische Universiteit Biomodeling and Informatics 

BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology 

Bulletin of Mathematical Biology 

European Society for Mathematical and Theoretical Biology 

Journal of Mathematical Biology 

Biomathematics Research Centre at University of Canterbury 

T721 
Centre for Mathematical Biology at Oxford University 

T731 
Mathematical Biology at the National Institute for Medical Research 

T741 
Institute for Medical BioMathematics 

[751 
Mathematical Biology Systems of Differential Equations from EqWorld: The World of Mathematical 

Equations 

Systems Biology Workbench - a set of tools for modelling biochemical networks 

T771 
The Collection of Biostatistics Research Archive 

T781 

Statistical Applications in Genetics and Molecular Biology 

[791 

The International Journal of Biostatistics 

Theoretical Modeling of Cellular Physiology at Ecole Normale Superieure, Paris 

Lists of references 

A general list of Theoretical biology/Mathematical biology references, including an updated list of actively 
contributing authors 

roil 

A list of references for applications of category theory in relational biology 

[89] 

An updated list of publications of theoretical biologist Robert Rosen 

~ [OQ] 

Theory of Biological Anthropology (Documents No. 9 and 10 in English) 

Drawing the Line Between Theoretical and Basic Biology (a forum article by Isidro T. Savillo) 

Related journals 

roc] 

Acta Biotheoretica 

Bioinformatics 

Biological Theory 

BioSystems 

Bulletin of Mathematical Biology [68] 

Ecological Modelling 

Journal of Mathematical Biology 

Journal of Theoretical Biology 

Journal of the Royal Society Interface 

[92] 
Mathematical Biosciences 

Medical Hypotheses 

[94] 
Rivista di Biologia-Biology Forum 

Theoretical and Applied Genetics 

Theoretical Biology and Medical Modelling 



Mathematical biology 123 

[97] 

• Theoretical Population Biology 

• Theory in Biosciences (formerly: Biologisches Zentralblatt) 

Related societies 

• ESMTB: European Society for Mathematical and Theoretical Biology 

[99] 

• The Israeli Society for Theoretical and Mathematical Biology 

• Societe Francophone de Biologie Theorique 

• International Society for Biosemiotic Studies 

References 

[I] Mathematical and Theoretical Biology: A European Perspective (http://sciencecareers.sciencemag.org/career_development/ 
previous_issues/articles/2870/mathematical_and_theoretical_biology_a_european_perspective) 

[2] "There is a subtle difference between mathematical biologists and theoretical biologists. Mathematical biologists tend to be employed in 

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Dynamical systems theory 



126 



Dynamical systems theory 



Dynamical systems theory is an area of applied mathematics used to describe the behavior of complex dynamical 
systems, usually by employing differential equations or difference equations. When differential equations are 
employed, the theory is called continuous dynamical systems. When difference equations are employed, the theory is 
called discrete dynamical systems. When the time variable runs over a set which is discrete over some intervals and 
continuous over other intervals or is any arbitrary time-set such as a cantor set then one gets dynamic equations on 
time scales. Some situations may also be modelled by mixed operators such as differential-difference equations. 

This theory deals with the long-term qualitative behavior of dynamical systems, and the studies of the solutions to 
the equations of motion of systems that are primarily mechanical in nature; although this includes both planetary 
orbits as well as the behaviour of electronic circuits and the solutions to partial differential equations that arise in 
biology. Much of modern research is focused on the study of chaotic systems. 

This field of study is also called just Dynamical systems, Systems theory or longer as Mathematical Dynamical 
Systems Theory and the Mathematical theory of dynamical systems. 



Overview 

Dynamical systems theory and chaos theory deal with the 
long-term qualitative behavior of dynamical systems. 
Here, the focus is not on finding precise solutions to the 
equations defining the dynamical system (which is often 
hopeless), but rather to answer questions like "Will the 
system settle down to a steady state in the long term, and 
if so, what are the possible steady states?", or "Does the 
long-term behavior of the system depend on its initial 
condition?" 

An important goal is to describe the fixed points, or 
steady states of a given dynamical system; these are 
values of the variable which won't change over time. 
Some of these fixed points are attractive, meaning that if 
the system starts out in a nearby state, it will converge 
towards the fixed point. 




The Lorenz attractor is an example of a non-linear dynamical 
system. Studying this system helped give rise to Chaos theory. 



Similarly, one is interested in periodic points, states of the system which repeat themselves after several timesteps. 
Periodic points can also be attractive. Sarkovskii's theorem is an interesting statement about the number of periodic 
points of a one-dimensional discrete dynamical system. 

Even simple nonlinear dynamical systems often exhibit almost random, completely unpredictable behavior that has 
been called chaos. The branch of dynamical systems which deals with the clean definition and investigation of chaos 
is called chaos theory. 



Dynamical systems theory 127 

History 

The concept of dynamical systems theory has its origins in Newtonian mechanics. There, as in other natural sciences 
and engineering disciplines, the evolution rule of dynamical systems is given implicitly by a relation that gives the 
state of the system only a short time into the future. 

Before the advent of fast computing machines, solving a dynamical system required sophisticated mathematical 
techniques and could only be accomplished for a small class of dynamical systems. 

Some excellent presentations of mathematical dynamic system theory include Beltrami (1987), Luenberger (1979), 
Padula and Arbib (1974), and Strogatz (1994). [1] 

Concepts 
Dynamical systems 

The dynamical system concept is a mathematical formalization for any fixed "rule" which describes the time 
dependence of a point's position in its ambient space. Examples include the mathematical models that describe the 
swinging of a clock pendulum, the flow of water in a pipe, and the number of fish each spring in a lake. 

A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an 
appropriate state space. Small changes in the state of the system correspond to small changes in the numbers. The 
numbers are also the coordinates of a geometrical space — a manifold. The evolution rule of the dynamical system is 
a fixed rule that describes what future states follow from the current state. The rule is deterministic: for a given time 
interval only one future state follows from the current state. 

Dynamicism 

Dynamicism, also termed the dynamic hypothesis or the dynamic hypothesis in cognitive science or dynamic 
cognition, is a new approach in cognitive science exemplified by the work of philosopher Tim van Gelder. It argues 
that differential equations are more suited to modelling cognition than more traditional computer models. 

Nonlinear system 

In mathematics, a nonlinear system is a system which is not linear, i.e. a system which does not satisfy the 
superposition principle. Less technically, a nonlinear system is any problem where the variable(s) to be solved for 
cannot be written as a linear sum of independent components. A nonhomogenous system, which is linear apart from 
the presence of a function of the independent variables, is nonlinear according to a strict definition, but such systems 
are usually studied alongside linear systems, because they can be transformed to a linear system as long as a 
particular solution is known. 



Dynamical systems theory 128 

Related fields 
Arithmetic dynamics 

Arithmetic dynamics is a field that emerged in the 1990s that amalgamates two areas of mathematics, 
dynamical systems and number theory. Classically, discrete dynamics refers to the study of the iteration of 
self-maps of the complex plane or real line. Arithmetic dynamics is the study of the number-theoretic 
properties of integer, rational, p-adic, and/or algebraic points under repeated application of a polynomial or 
rational function. 

Chaos theory 

Chaos theory describes the behavior of certain dynamical systems — that is, systems whose state evolves with 
time — that may exhibit dynamics that are highly sensitive to initial conditions (popularly referred to as the 
butterfly effect). As a result of this sensitivity, which manifests itself as an exponential growth of perturbations 
in the initial conditions, the behavior of chaotic systems appears to be random. This happens even though these 
systems are deterministic, meaning that their future dynamics are fully defined by their initial conditions, with 
no random elements involved. This behavior is known as deterministic chaos, or simply chaos. 

Complex systems 

Complex systems is a scientific field, which studies the common properties of systems considered complex in 
nature, society and science. It is also called complex systems theory, complexity science, study of complex 
systems and/or sciences of complexity. The key problems of such systems are difficulties with their formal 
modeling and simulation. From such perspective, in different research contexts complex systems are defined 
on the base of their different attributes. 

The study of complex systems is bringing new vitality to many areas of science where a more typical 
reductionist strategy has fallen short. Complex systems is therefore often used as a broad term encompassing a 
research approach to problems in many diverse disciplines including neurosciences, social sciences, 
meteorology, chemistry, physics, computer science, psychology, artificial life, evolutionary computation, 
economics, earthquake prediction, molecular biology and inquiries into the nature of living cells themselves. 

Control theory 

Control theory is an interdisciplinary branch of engineering and mathematics, that deals with influencing the 
behavior of dynamical systems. 

Ergodic theory 

Ergodic theory is a branch of mathematics that studies dynamical systems with an invariant measure and 
related problems. Its initial development was motivated by problems of statistical physics. 

Functional analysis 

Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of 
vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in 
particular transformations of functions, such as the Fourier transform, as well as in the study of differential and 
integral equations. This usage of the word functional goes back to the calculus of variations, implying a 
function whose argument is a function. Its use in general has been attributed to mathematician and physicist 
Vito Volterra and its founding is largely attributed to mathematician Stefan Banach. 



Dynamical systems theory 129 

Graph dynamical systems 

The concept of graph dynamical systems (GDS) can be used to capture a wide range of processes taking place 
on graphs or networks. A major theme in the mathematical and computational analysis of GDS is to relate 
their structural properties (e.g. the network connectivity) and the global dynamics that result. 

Projected dynamical systems 

Projected dynamical systems is a mathematical theory investigating the behaviour of dynamical systems where 
solutions are restricted to a constraint set. The discipline shares connections to and applications with both the 
static world of optimization and equilibrium problems and the dynamical world of ordinary differential 
equations. A projected dynamical system is given by the flow to the projected differential equation. 

Symbolic dynamics 

Symbolic dynamics is the practice of modelling a topological or smooth dynamical system by a discrete space 
consisting of infinite sequences of abstract symbols, each of which corresponds to a state of the system, with 
the dynamics (evolution) given by the shift operator. 

System dynamics 

System dynamics is an approach to understanding the behaviour of complex systems over time. It deals with 
internal feedback loops and time delays that affect the behaviour of the entire system. What makes using 
system dynamics different from other approaches to studying complex systems is the use of feedback loops 
and stocks and flows. These elements help describe how even seemingly simple systems display baffling 
nonlinearity. 

Topological dynamics 

Topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic 
properties of dynamical systems are studied from the viewpoint of general topology. 

Applications 
In biomechanics 

In sports biomechanics, dynamical systems theory has emerged in the movement sciences as a viable framework for 
modeling athletic performance. From a dynamical systems perspective, the human movement system is a highly 
intricate network of co-dependent sub-systems (e.g. respiratory, circulatory, nervous, skeletomuscular, perceptual) 
that are composed of a large number of interacting components (e.g. blood cells, oxygen molecules, muscle tissue, 
metabolic enzymes, connective tissue and bone). In dynamical systems theory, movement patterns emerge through 
generic processes of self-organization found in physical and biological systems. 

In cognitive science 

Dynamical system theory has been applied in the field of neuroscience and cognitive development, especially in the 
neo-Piagetian theories of cognitive development. It is the belief that cognitive development is best represented by 
physical theories rather than theories based on syntax and AI. It also believes that differential equations are the most 
appropriate tool for modeling human behavior. These equations are interpreted to represent an agent's cognitive 
trajectory through state space. In other words, dynamicists argue that psychology should be (or is) the description 
(via differential equations) of the cognitions and behaviors of an agent under certain environmental and internal 
pressures. The language of chaos theory is also frequently adopted. 



Dynamical systems theory 130 

In it, the learner's mind reaches a state of disequilibrium where old patterns have broken down. This is the phase 

transition of cognitive development. Self organization (the spontaneous creation of coherent forms) sets in as activity 

levels link to each other. Newly formed macroscopic and microscopic structures support each other, speeding up the 

process. These links form the structure of a new state of order in the mind through a process called scalloping (the 

repeated building up and collapsing of complex performance.) This new, novel state is progressive, discrete, 

Mi 
idiosyncratic and unpredictable. 

Dynamic systems theory has recently been used to explain a long-unanswered problem in child development referred 
to as the A-not-B error. 

See also 

Related subjects 

List of dynamical system topics 

Baker's map 

Dynamical system (definition) 

Embodied Embedded Cognition 

Gingerbreadman map 

Halo orbit 

List of types of systems theory 

Oscillation 

Postcognitivism 

Recurrent neural network 

Combinatorics and dynamical systems 

Synergetics 

Related scientists 

People in systems and control 

Dmitri Anosov 

Vladimir Arnold 

Nikolay Bogolyubov 

Andrey Kolmogorov 

Nikolay Krylov 

Jilrgen Moser 

Yakov G. Sinai 

Stephen Smale 

Hillel Furstenberg 

Further reading 

• Frederick David Abraham (1990), A Visual Introduction to Dynamical Systems Theory for Psychology, 1990. 

• Beltrami, E. J. (1987). Mathematics for dynamic modeling. NY: Academic Press 

• Otomar Hajek (1968 }, Dynamical Systems in the Plane. 

• Luenberger, D. G. (1979). Introduction to dynamic systems. NY: Wiley. 

• Anthony N. Michel, Kaining Wang & Bo Hu (2001), Qualitative Theory of Dynamical Systems: The Role of 
Stability Preserving Mappings. 

• Padulo, L. & Arbib, M A. (1974). System Theory. Philadelphia: Saunders 

• Strogatz, S. H. (1994), Nonlinear dynamics and chaos. Reading, MA: Addison Wesley 



Dynamical systems theory 131 

External links 

• Dynamic Systems Encyclopedia of Cognitive Science entry. 

T71 

• Definition of dynamical system in Math World. 

ro] 

• DSWeb Dynamical Systems Magazine 

References 

[1] Jerome R. Busemeyer (2008), "Dynamic Systems" (http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html). To Appear 

in: Encyclopedia of cognitive science, Macmillan. Retrieved 8 May 2008. 
[2] MIT System Dynamics in Education Project (SDEP) (http://sysdyn.clexchange.org) 
[3] Paul S Glaziera, Keith Davidsb, Roger M Bartlettc (2003). "DYNAMICAL SYSTEMS THEORY: a Relevant Framework for 

Performance-Oriented Sports Biomechanics Research" (http://www.sportsci.org/jour/03/psg.htm). in: Sportscience 7. 

Accessdate=2008-05-08. 
[4] Lewis, Mark D. (2000-02-25). "The Promise of Dynamic Systems Approaches for an Integrated Account of Human Development" (http:// 

home.oise.utoronto.ca/~mlewis/Manuscripts/Promise.pdf) (PDF). Child Development 71 (1): 36^-3. doi:10.1111/1467-8624.00116. . 

Retrieved 2008-04-04. 
[5] Smith, Linda B.; Esther Thelen (2003-07-30). "Development as a dynamic system" (http://www.indiana.edu/~cogdev/labwork/ 

dynamicsystem.pdf) (PDF). TRENDS in Cognitive Sciences 7 (8): 343-8. doi:10.1016/S1364-6613(03)00156-6. . Retrieved 2008-04-04. 
[6] http://www.cogs.indiana.edu/Publications/techreps2000/241/241.html 
[7] http://mathworld.wolfram.com/DynamicalSystem.html 
[8] http://www.dynamicalsystems.org/ 



Living systems 



Living systems are open self-organizing systems that have the special characteristics of life and interact with their 
environment. This takes place by means of information and material-energy exchanges. 

Publications 

• Kenneth D. Bailey (2006). Living systems theory and social entropy theory. Systems Research and Behavioral 
Science, 22, 291-300. 

• James Grier Miller, (1978). Living systems. New York: McGraw-Hill. ISBN 0-87081-363-3 

• Humberto Maturana (1978), "Biology of language: The epistemology of reality, in Miller, George A., and 
Elizabeth Lenneberg (eds.), Psychology and Biology of Language and Thought: Essays in Honor of Eric 
Lenneberg. Academic Press: 27-63. 

External links 

T21 

• Joanna Macy pH.D. on Living systems 

References 

[1] http://www.enolagaia.com/M78BoL.html 
[2] http : //www .j oannamacy .net/html/living. html 



Complex Systems Biology (CSB) 



132 



Complex Systems Biology (CSB) 



Systems biology is a term used to 

describe a number of trends in 

bioscience research, and a movement 

which draws on those trends. 

Proponents describe systems biology as 

a biology-based inter-disciplinary study 

field that focuses on complex 

interactions in biological systems, 

claiming that it uses a new perspective 

(holism instead of reduction). 

Particularly from year 2000 onwards, 

the term is used widely in the 

biosciences, and in a variety of 

contexts. An often stated ambition of 

systems biology is the modeling and 

discovery of emergent properties, properties of a system whose theoretical description is only possible using 

techniques which fall under the remit of systems biology. 




Overview 

Systems biology can be considered from a number of different aspects: 

• As a field of study, particularly, the study of the interactions between the components of biological systems, and 
how these interactions give rise to the function and behavior of that system (for example, the enzymes and 
metabolites in a metabolic pathway). 

• As a paradigm, usually defined in antithesis to the so-called reductionist paradigm (biological organisation), 
although fully consistent with the scientific method. The distinction between the two paradigms is referred to in 
these quotations: 

"The reductionist approach has successfully identified most of the components and many of the interactions 
but, unfortunately, offers no convincing concepts or methods to understand how system properties emerge. ..the 
pluralism of causes and effects in biological networks is better addressed by observing, through quantitative 
measures, multiple components simultaneously and by rigorous data integration with mathematical models" 
Science 

"Systems biology. ..is about putting together rather than taking apart, integration rather than reduction. It 
requires that we develop ways of thinking about integration that are as rigorous as our reductionist 
programmes, but different. ..It means changing our philosophy, in the full sense of the term" Denis Noble 

• As a series operational protocols used for performing research, namely a cycle composed of theory, analytic 
or computational modelling to propose specific testable hypotheses about a biological system, experimental 
validation, and then using the newly acquired quantitative description of cells or cell processes to refine the 
computational model or theory. Since the objective is a model of the interactions in a system, the 
experimental techniques that most suit systems biology are those that are system-wide and attempt to be as 
complete as possible. Therefore, transcriptomics, metabolomics, proteomics and high-throughput techniques are 
used to collect quantitative data for the construction and validation of models. 

• As the application of dynamical systems theory to molecular biology. 



Complex Systems Biology (CSB) 133 

• As a socioscientific phenomenon defined by the strategy of pursuing integration of complex data about the 
interactions in biological systems from diverse experimental sources using interdisciplinary tools and personnel. 

This variety of viewpoints is illustrative of the fact that systems biology refers to a cluster of peripherally 
overlapping concepts rather than a single well-delineated field. However the term has widespread currency and 
popularity as of 2007, with chairs and institutes of systems biology proliferating worldwide. 

History 

Systems biology finds its roots in: 

• the quantitative modeling of enzyme kinetics, a discipline that flourished between 1900 and 1970, 

• the mathematical modeling of population growth, 

• the simulations developed to study neurophysiology, and 

• control theory and cybernetics. 

One of the theorists who can be seen as one of the precursors of systems biology - and who allegedly coined the term 
in 1928 - is Ludwig von Bertalanffy with his general systems theory . One of the first numerical simulations in 
biology was published in 1952 by the British neurophysiologists and Nobel prize winners Alan Lloyd Hodgkin and 
Andrew Fielding Huxley, who constructed a mathematical model that explained the action potential propagating 
along the axon of a neuronal cell. Their model described a cellular function emerging from the interaction between 
two different molecular components, a potassium and a sodium channels, and can therefore be seen as the beginning 
of computational systems biology. In 1960, Denis Noble developed the first computer model of the heart 
pacemaker. 

The formal study of systems biology, as a distinct discipline, was launched by systems theorist Mihajlo Mesarovic in 
1966 with an international symposium at the Case Institute of Technology in Cleveland, Ohio entitled "Systems 
Theory and Biology." 

The 1960s and 1970s saw the development of several approaches to study complex molecular systems, such as the 
Metabolic Control Analysis and the biochemical systems theory. The successes of molecular biology throughout the 
1980s, coupled with a skepticism toward theoretical biology, that then promised more than it achieved, caused the 
quantitative modelling of biological processes to become a somewhat minor field. 

However the birth of functional genomics in the 1990s meant that large quantities of high quality data became 
available, while the computing power exploded, making more realistic models possible. In 1997, the group of 
Masaru Tomita published the first quantitative model of the metabolism of a whole (hypothetical) cell. 

Around the year 2000, after Institutes of Systems Biology were established in Seattle and Tokyo, systems biology 
emerged as a movement in its own right, spurred on by the completion of various genome projects, the large increase 
in data from the omics (e.g. genomics and proteomics) and the accompanying advances in high-throughput 
experiments and bioinformatics. Since then, various research institutes dedicated to systems biology have been 

ri3i 

developed. As of summer 2006, due to a shortage of people in systems biology several doctoral training centres in 
systems biology have been established in many parts of the world. 



Complex Systems Biology (CSB) 



134 



Disciplines associated with systems biology 

According to the interpretation of System 
Biology as the ability to obtain, integrate and 
analyze complex data from multiple 
experimental sources using interdisciplinary 
tools, some typical technology platforms are: 

• Genomics: Organismal deoxyribonucleic 
acid(DNA) sequence, including 
intra-organisamal cell specific variation, (i.e. 
Telomere length variation etc). 

• Epigenomics / Epigenetics: Organismal and 
corresponding cell specific transcriptomic 
regulating factors not empirically coded in the 
genomic sequence, (i.e. DNA methylation, 
Histone Acetelation etc). 




Overview of signal transduction pathways 



• Transcriptomics: Organismal, tissue or whole cell gene expression measurements by DNA microarrays or serial 
analysis of gene expression 

• Interferomics: Organismal, tissue, or cell level transcript correcting factors (i.e. RNA interference) 

• Translatomics / Proteomics: Organismal, tissue, or cell level measurements of proteins and peptides via 
two-dimensional gel electrophoresis, mass spectrometry or multi-dimensional protein identification techniques 
(advanced HPLC systems coupled with mass spectrometry). Sub disciplines include phosphoproteomics, 
glycoproteomics and other methods to detect chemically modified proteins. 

• Metabolomics: Organismal, tissue, or cell level measurements of all small-molecules known as metabolites. 

• Glycomics: Organismal, tissue, or cell level measurements of carbohydrates. 

• Lipidomics: Organismal, tissue, or cell level measurements of lipids. 

In addition to the identification and quantification of the above given molecules further techniques analyze the 
dynamics and interactions within a cell. This includes: 

• Interactomics: Organismal, tissue, or cell level study of interactions between molecules. Currently the 
authoratative molecular discipline in this field of study is protein-protein interactions (PPI), although the working 
definition does not pre-clude inclusion of other molecular disciplines such as those defined here. 

• Fluxomics: Organismal, tissue, or cell level measurements of molecular dynamic changes over time. 

• Biomics: systems analysis of the biome. 

The investigations are frequently combined with large scale perturbation methods, including gene-based (RNAi, 
mis-expression of wild type and mutant genes) and chemical approaches using small molecule libraries. Robots and 
automated sensors enable such large-scale experimentation and data acquisition. These technologies are still 
emerging and many face problems that the larger the quantity of data produced, the lower the quality. A wide variety 
of quantitative scientists (computational biologists, statisticians, mathematicians, computer scientists, engineers, and 
physicists) are working to improve the quality of these approaches and to create, refine, and retest the models to 
accurately reflect observations. 

The systems biology approach often involves the development of mechanistic models, such as the reconstruction of 
dynamic systems from the quantitative properties of their elementary building blocks. For instance, a cellular 

network can be modelled mathematically using methods coming from chemical kinetics and control theory. Due to 
the large number of parameters, variables and constraints in cellular networks, numerical and computational 
techniques are often used. Other aspects of computer science and informatics are also used in systems biology. These 
include new forms of computational model, such as the use of process calculi to model biological processes, the 



Complex Systems Biology (CSB) 



135 



integration of information from the literature, using techniques of information extraction and text mining, the 
development of online databases and repositories for sharing data and models, approaches to database integration 
and software interoperability via loose coupling of software, websites and databases, or commercial suits, and the 
development of syntactically and semantically sound ways of representing biological models. 

See also 



Related fields 

Complex systems 

Complex systems biology 

Bioinformatics 

Biological network inference 

Biological systems engineering 

Biomedical cybernetics 

Biostatistics 

Extrapolation based molecular systems 

biology 

Theoretical Biophysics 

Relational Biology 

Translational Research 

Computational biology 

Computational systems biology 

Scotobiology 

Synthetic biology 

Systems biology modeling 

Systems ecology 

Systems immunology 



Related terms 

Life 

Biological organisation 

Artificial life 

Gene regulatory network 

Metabolic network modelling 

Living systems theory 

Network Theory of Aging 

Regulome 

Systems Biology Markup Language 

(SBML) 

Systems Biology Graphical Notation 

(SBGN) 

SBO 

Viable System Model 

Antireductionism 



Systems biologists 

• Category:Systems biologists 

Lists 

Category:Systems biologists 

List of systems biology conferences 

List of omics topics in biology 

List of publications in systems biology 

List of systems biology research groups 

List of systems biology visualization 

software 



Further reading 
Books 

Zeng BJ. Structurity - Pan-evolution theory of biosystems, Hunan Changsha Xinghai, May, 1994. 

Hiroaki Kitano (editor). Foundations of Systems Biology. MIT Press: 2001. ISBN 0-262-1 1266-3 

CP Fall, E Marland, J Wagner and JJ Tyson (Editors). "Computational Cell Biology." Springer Verlag: 2002 

ISBN 0-387-95369-8 

G Bock and JA Goode (eds)./« Silico" Simulation of Biological Processes, Novartis Foundation Symposium 247. 

John Wiley & Sons: 2002. ISBN 0-470-84480-9 

E Klipp, R Herwig, A Kowald, C Wierling, and H Lehrach. Systems Biology in Practice. Wiley- VCH: 2005. 

ISBN 3-527-31078-9 

L. Alberghina and H. Westerhoff (Editors) — Systems Biology: Definitions and Perspectives, Topics in Current 

Genetics 13, Springer Verlag (2005), ISBN 978-3540229681 

A Kriete, R Eils. Computational Systems Biology., Elsevier - Academic Press: 2005. ISBN 0-12-088786-X 

K. Sneppen and G Zocchi, (2005) Physics in Molecular Biology, Cambridge University Press, ISBN 

0-521-84419-3 

D. Noble, The Music of life. Biology beyond the genome Oxford University Press [16] 2006. ISBN 0199295735, 

ISBN 978-0199295739 

Z. Szallasi, J. Sterling, and V.Periwal (eds.) System Modeling in Cellular Biology: From Concepts to Nuts and 

Bolts (Hardcover), MIT Press: 2006, ISBN 0-262-19548-8 

1171 
B Palsson, Systems Biology - Properties of Reconstructed Networks. Cambridge University Press: 2006. ISBN 

978-0-521-85903-5 



Complex Systems Biology (CSB) 136 

• K Kaneko. Life: An Introduction to Complex Systems Biology. Springer: 2006. ISBN 3540326669 

• U Alon. An Introduction to Systems Biology: Design Principles of Biological Circuits. CRC Press: 2006. ISBN 
1-58488-642-0 - emphasis on Network Biology (For a comparative review of Alon, Kaneko and Palsson see 
Werner, E. (March 29, 2007). "All systems go" [18] (PDF). Nature 446: 493-494. doi:10.1038/446493a.) 

• Andriani Daskalaki (editor) "Handbook of Research on Systems Biology Applications in Medicine" Medical 
Information Science Reference, October 2008 ISBN 978-1-60566-076-9 

Journals 

ri9i 

• BMC Systems Biology - open access journal on systems biology 

• Molecular Systems Biology - open access journal on systems biology 

T211 

• IET Systems Biology - not open access journal on systems biology 

T221 

• WIRES Systems Biology and Medicine - open access review journal on systems biology and medicine 

[231 

• EURASIP Journal on Bioinformatics and Systems Biology 

T241 

• Systems and Synthetic Biology 

T251 

• International Journal of Computational Intelligence in Bioinformatics and Systems Biology 

Articles 

• Zeng BJ., On the concept of system biological engineering, Communication on Transgenic Animals, CAS, June, 
1994. 

• Zeng BJ., Transgenic expression system - goldegg plan (termed system genetics as the third wave of genetics), 
Communication on Transgenic Animals, CAS, Nov. 1994. 

• Zeng BJ., From positive to synthetic medical science, Communication on Transgenic Animals, CAS, Nov. 1995. 

• Binnewies, Tim Terence, Miller, WG, Wang, G. The complete genome sequence and analysis of the human 
pathogen Campylobacter lari . Published in journal: Foodborne Pathog Disease (ISSN 1535-3141) , vol: 5, 
issue: 4, pages: 371-386, 2008, Mary Ann Liebert, Inc. Publishers. 

• M. Tomita, Hashimoto K, Takahashi K, Shimizu T, Matsuzaki Y, Miyoshi F, Saito K, Tanida S, Yugi K, Venter 
JC, Hutchison CA. E-CELL: Software Environment for Whole Cell Simulation. Genome Inform Ser Workshop 
Genome Inform. 1997;8:147-155. [27] 

no] 

• ScienceMag.org - Special Issue: Systems Biology, Science, Vol 295, No 5560, March 1, 2002 

[291 

• Marc Vidal and Eileen E. M. Furlong. Nature Reviews Genetics 2004 From OMICS to systems biology 

• Marc Facciotti, Richard Bonneau, Leroy Hood and Nitin Baliga. Current Genomics 2004 Systems Biology 
Experimental Design - Considerations for Building Predictive Gene Regulatory Network Models for Prokaryotic 

c > [30] 

Systems 

• Katia Basso, Adam A Margolin, Gustavo Stolovitzky, Ulf Klein, Riccardo Dalla-Favera, Andrea Califano, (2005) 

"Reverse engineering of regulatory networks in human B cells" . Nat Genet;37(4):382-90 

[32] 

• Mario Jardon Systems Biology: An Overview - a review from the Science Creative Quarterly, 2005 

[33] 

• Johnjoe McFadden, Guardian.co.uk - 'The unselfish gene: The new biology is reasserting the primacy of the 

whole organism - the individual - over the behaviour of isolated genes', The Guardian (May 6, 2005) 

• Pharoah, M.C. (online). Looking to systems theory for a reductive explanation of phenomenal experience and 
evolutionary foundations for higher order thought Retrieved Jan, 15 2008. 

• WTEC Panel Report on International Research and Development in Systems Biology (2005) 

• E. Werner, "The Future and Limits of Systems Biology", Science STKE [35] 2005, pel6 (2005). 

• Francis J. Doyle and Jorg Stelling, "Systems interface biology" J. R. Soc. Interface Vol 3, No 10 2006 

• Kahlem, P. and Birney E. (2006). "Dry work in a wet world: computation in systems biology." Mol Syst Biol 2: 
40. [37] 

• E. Werner, "All systems go" [18] , "Nature" [38] vol 446, pp 493-494, March 29, 2007. (Review of three books 
(Alon, Kaneko, and Palsson) on systems biology.) 



Complex Systems Biology (CSB) 137 

[39] 

• Santiago Schnell, Ramon Grima, Philip K. Maini, "Multiscale Modeling in Biology" , American Scientist, Vol 

95, pages 134-142, March-April 2007. 

• TS Gardner, D di Bernardo, D Lorenz and JJ Collins. "Inferring genetic networks and identifying compound of 

action via expression profiling." Science 301: 102-105 (2003). 

[411 

• Jeffery C. Way and Pamela A. Silver, Why We Need Systems Biology 

T421 

• H.S. Wiley, "Systems Biology - Beyond the Buzz." The Scientist . June 2006. 

T431 

• Nina Flanagan, "Systems Biology Alters Drug Development." Genetic Engineering & Biotechnology News, 

January 2008 

External links 

• Applied BioDynamics Laboratory: Boston University 

[45] 

• Institute for Research in Immunology and Cancer (IRIC): Universite de Montreal 

• Systems Biology - BioChemWeb.org 

T471 

• Systems Biology Portal - administered by the Systems Biology Institute 

• Semantic Systems Biology 

[49] 

• SystemsX.ch - The Swiss Initiative in Systems Biology 

• Systems Biology at the Pacific Northwest National Laboratory 

References 

[I] Snoep J.L. and Westerhoff H.V.; Alberghina L. and Westerhoff H.V. (Eds.) (2005.). "From isolation to integration, a systems biology 
approach for building the Silicon Cell". Systems Biology: Definitions and Perspectives. Springer- Verlag. p. 7. 

[2] "Systems Biology - the 21st Century Science" (http://www.systemsbiology.org/Intro_to_ISB_and_Systems_Biology/ 

Systems_Biology_— _the_21st_Century_Science). . 
[3] Sauer, U. et al. (27 April 2007). "Getting Closer to the Whole Picture". Science 316: 550. doi: 10.1 126/science. 1 142502. PMTD 17463274. 
[4] Denis Noble (2006). The Music of Life: Biology beyond the genome. Oxford University Press. ISBN 978-0199295739. p21 
[5] "Systems Biology: Modelling, Simulation and Experimental Validation" (http://www.bbsrc.ac.uk/science/areas/ebs/themes/ 

main_sysbio.html). . 
[6] Kholodenko B.N., Bruggeman F.J., Sauro H.M.; Alberghina L. and Westerhoff H.V. (Eds.) (2005.). "Mechanistic and modular approaches to 

modeling and inference of cellular regulatory networks". Systems Biology: Definitions and Perspectives. Springer- Verlag. p. 143. 
[7] von Bertalanffy, Ludwig (1968). General System theory: Foundations, Development, Applications. George Braziller. 
[8] Hodgkin AL, Huxley AF (1952). "A quantitative description of membrane current and its application to conduction and excitation in nerve". J 

Physiol 117 (4): 500-544. PMID 12991237. 
[9] Le Novere, N (2007). "The long journey to a Systems Biology of neuronal function". BMC Systems Biology 1: 28. 

doi: 10.1 186/1752-0509-1-28. PMID 17567903. 
[10] Noble D (1960). "Cardiac action and pacemaker potentials based on the Hodgkin-Huxley equations". Nature 188: 495^497. 

doi:10.1038/188495b0. PMID 13729365. 

[II] Mesarovic, M. D. (1968). Systems Theory and Biology. Springer- Verlag. 

[12] "A Means Toward a New Holism" (http://www.jstor.Org/view/00368075/ap004022/00a00220/0). Science 161 (3836): 34-35. 

doi: 10.1 126/science.l61.3836.34. . 
[13] "Working the Systems" (http://sciencecareers.sciencemag.org/career_development/previous_issues/articles/2006_03_03/ 

working_the_sy stems/ (parent)/ 158).. 
[14] Gardner, TS; di Bernardo D, Lorenz D and Collins JI (4 luly 2003). "Inferring genetic networks and identifying compound of action via 

expression profiling". Science 301: 102-1005. doi:10.1126/science.l081900. PMID 12843395. 
[15] di Bernardo, D; Thompson MI, Gardner TS, Chobot SE, Eastwood EL, Wojtovich AP, Elliot SI, Schaus SE and Collins H (March 2005). 

"Chemogenomic profiling on a genome-wide scale using reverse-engineered gene networks". Nature Biotechnology 23: 377—383. 

doi:10.1038/nbtl075. PMID 15765094. 
[16] http://www. musicoflife. co. uk/ 
[17] http://gcrg.ucsd.edu/book/index.html 

[18] http://www.nature.com/nature/journal/v446/n7135/pdf/446493a.pdf 
[19] http://www.biomedcentral.com/bmcsystbiol 
[20] http://www.nature.com/msb 
[2 1 ] http ://www. ietdl.org/IET-SYB 
[22] http://wires.wiley.com/WileyCDA/WiresIournal/wisId-WSBM.html 



Complex Systems Biology (CSB) 



138 



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//www. hindawi.com/journals/bsb/ 

//www. springer.com/biomed/journal/11693 

//www. inderscience. com/browse/index. php?journalCODE=ijcibsb 

//www. bio. dtu.dk/English/Publications/l/all.aspx?lg=showcommon&id=23 1324 

//web. sfc.keio.ac.jp/~mt/mt-lab/publications/Paper/ecell/bioinfo99/btc007_gml. html 

//www. sciencemag.org/content/vol295/issue5560/ 

//www. nature.com/nrg/journal/v5/nlO/poster/omics/index.html 

//www.ingentaconnect.com/content/ben/cg/2004/00000005/00000007/art00002 

//www. ncbi.nlm.nih.gov/entrez/ query. fcgi?cmd=Retrieve&db=pubmed&dopt=Abstract&list_uids=15778709&query_hl=7 

//www. scq.ubc.ca/?p=253 

//www. guardian, co. uk/life/science/story/0, 12996, 1477776, OO.html 

//www. wtec.org/sysbio/welcome.htm 

//stke. sciencemag.org/content/vol2005/issue278/ 

//www.journals.royalsoc.ac.uk/openurl.asp?genre=article&doi=10. 1098/rsif.2006.0143 

//www. nature.com/doifinder/ 1 0. 1 038/msb41 00080 

//www. nature.com/nature/journal/v446/n7135/index. html 

//www. americanscientist.org/template/AssetDetail/assetid/54784 

//www. bu.edu/ abl/publications. html 

//cs.calstatela.edu/wiki/images/9/9b/Silver.pdf 

//www.the-scientist.com/ 2006/ 6/ 1/52/1/ 

//www. genengnews.com/articles/chi tern. aspx?aid=2337 

//www. bu.edu/abl/ 

//www.iric.ca 

//www. biochemweb.org/systems.shtml 

//www. systems-biology.org/ 

//www. semantic-systems-biology.org 

//www. systemsx.ch/ 

//www. sysbio.org/ 



Network theory 



For the anthropological theory, see Social network 

Network theory is an area of computer science and network science and part of graph theory. It has application in 
many disciplines including particle physics, computer science, biology, economics, operations research, and 
sociology. Network theory concerns itself with the study of graphs as a representation of either symmetric relations 
or, more generally, of asymmetric relations between discrete objects. Applications of network theory include 
logistical networks, the World Wide Web, gene regulatory networks, metabolic networks, social networks, 
epistemological networks, etc. See list of network theory topics for more examples. 

Network optimization 

Network problems that involve finding an optimal way of doing something are studied under the name of 
combinatorial optimization. Examples include network flow, shortest path problem, transport problem, 
transshipment problem, location problem, matching problem, assignment problem, packing problem, routing 
problem, Critical Path Analysis and PERT (Program Evaluation & Review Technique). 



Network analysis 



in 



Social network analysis 

Social network analysis maps relationships between individuals in social networks. L1J Such individuals are often 

[21 

persons, but may be groups (including cliques and cohesive blocks ), organizations, nation states, web sites, or 
citations between scholarly publications (scientometrics). 



Network theory 139 

Network analysis, and its close cousin traffic analysis, has significant use in intelligence. By monitoring the 
communication patterns between the network nodes, its structure can be established. This can be used for uncovering 
insurgent networks of both hierarchical and leaderless nature. 

Biological network analysis 

With the recent explosion of publicly available high throughput biological data, the analysis of molecular networks 
has gained significant interest. The type of analysis in this content are closely related to social network analysis, but 
often focusing on local patterns in the network. For example network motifs are small subgraphs that are 
over-represented in the network. Activity motifs are similar over-represented patterns in the attributes of nodes and 
edges in the network that are over represented given the network structure. 

Link analysis 

Link analysis is a subset of network analysis, exploring associations between objects. An example may be examining 
the addresses of suspects and victims, the telephone numbers they have dialed and financial transactions that they 
have partaken in during a given timeframe, and the familial relationships between these subjects as a part of police 
investigation. Link analysis here provides the crucial relationships and associations between very many objects of 
different types that are not apparent from isolated pieces of information. Computer-assisted or fully automatic 
computer-based link analysis is increasingly employed by banks and insurance agencies in fraud detection, by 
telecommunication operators in telecommunication network analysis, by medical sector in epidemiology and 
pharmacology, in law enforcement investigations, by search engines for relevance rating (and conversely by the 
spammers for spamdexing and by business owners for search engine optimization), and everywhere else where 
relationships between many objects have to be analyzed. 

Web link analysis 

Several Web search ranking algorithms use link-based centrality metrics, including (in order of appearance) 
Marchiori's Hyper Search, Google's PageRank, Kleinberg's HITS algorithm, and the TrustRank algorithm. Link 
analysis is also conducted in information science and communication science in order to understand and extract 
information from the structure of collections of web pages. For example the analysis might be of the interlinking 
between politicians' web sites or blogs. 

Centrality measures 

Information about the relative importance of nodes and edges in a graph can be obtained through centrality 
measures, widely used in disciplines like sociology. For example, eigenvector centrality uses the eigenvectors of the 
adjacency matrix to determine nodes that tend to be frequently visited. 

Spread of content in networks 

Content in a complex network can spread via two major methods: conserved spread and non-conserved spread. In 
conserved spread, the total amount of content that enters a complex network remains constant as it passes through. 
The model of conserved spread can best be represented by a pitcher containing a fixed amount of water being poured 
into a series of funnels connected by tubes . Here, the pitcher represents the original source and the water is the 
content being spread. The funnels and connecting tubing represent the nodes and the connections between nodes, 
respectively. As the water passes from one funnel into another, the water disappears instantly from the funnel that 
was previously exposed to the water. In non-conserved spread, the amount of content changes as it enters and passes 
through a complex network. The model of non-conserved spread can best be represented by a continuously running 
faucet running through a series of funnels connected by tubes . Here, the amount of water from the original source is 
infinite. Also, any funnels that have been exposed to the water continue to experience the water even as it passes into 



Network theory 140 

successive funnels. The non-conserved model is the most suitable for explaining the transmission of most infectious 
diseases. 

Related Applications 

Clay Shirky on institutions vs. collaboration Dan Pink on the surprising science of motivation 

See also 

• Complex network 

• Network science 

• Network topology 

• Small-world networks 

• Social circles 

• Scale-free networks 

• Sequential dynamical systems 

Implementations 

• Orange, a free data mining software suite, module orngNetwork 

• Pajek , program for (large) network analysis and visualization 

External links 

• netwiki Scientific wiki dedicated to network theory 

[71 

• New Network Theory International Conference on 'New Network Theory' 

ro] 

• Network Workbench : A Large-Scale Network Analysis, Modeling and Visualization Toolkit 

[91 

• Network analysis of computer networks 

• Network analysis of organizational networks 

• Network analysis of terrorist networks 

ri2i 

• Network analysis of a disease outbreak 

ri3i 

• Link Analysis: An Information Science Approach (book) 

• Connected: The Power of Six Degrees (documentary) 

References 

[I] Wasserman, Stanley and Katherine Faust. 1994. Social Network Analysis: Methods and Applications. Cambridge: Cambridge University 
Press. 

[2] http://intersci. ss.uci.edu/wiki/index.php/Cohesive_blocking 

[3] Newman, M., Barabasi, A.-L., Watts, D.J. [eds.] (2006) The Structure and Dynamics of Networks. Princeton, N.J.: Princeton University 

Press. 
[4] http://www.ailab.si/orange/doc/modules/orngNetwork.htm 
[5] http://pajek.imfm.si/doku.php 
[6] http://netwiki.amath.unc.edu/ 
[7] http://www.networkcultures.org/networktheory/ 
[8] http://nwb.slis.indiana.edu/ 
[9] http : // w w w . orgnet. com/S ocialLifeOfRouters . pdf 
[10] http ://www. orgnet.com/orgnetmap. pdf 

[II] http://firstmonday.org/htbin/cgiwrap/bin/ojs/index.php/fm/article/view/941/863 
[12] http://www.orgnet.com/AJPH2007.pdf 

[13] http://linkanalysis.wlv.ac.uk/ 

[14] http://gephi.org/2008/how-kevin-bacon-cured-cancer/ 



Cybernetics 



141 



Cybernetics 



Cybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related to 
control theory and systems theory. Both in its origins and in its evolution in the second-half of the 20th century, 
cybernetics is equally applicable to physical and social (that is, language-based) systems. 

Cybernetics is preeminent when the system under scrutiny is involved in a closed signal loop, where action by the 
system in an environment causes some change in the environment and that change is manifest to the system via 
information, or feedback, that causes the system to adapt to new conditions: the system changes its behavior. This 
"circular causal" relationship is necessary and sufficient for a cybernetic perspective. System Dynamics, a related 
field, originated with applications of electrical engineering control theory to other kinds of simulation models 
(especially business systems) by Jay Forrester at MIT in the 1950s. Convenient GUI system dynamics software 
developed into user friendly versions by the 1990s and have been applied to diverse systems. SD models solve the 
problem of simultaneity (mutual causation) by updating all variables in small time increments with positive and 
negative feedbacks and time delays structuring the interactions and control. The best known SD model is probably 
the 1972 The Limits to Growth. This model forecast that exponential growth would lead to economic collapse during 
the 21st century under a wide variety of growth scenarios. 

Contemporary cybernetics began as an 
interdisciplinary study connecting the 
fields of control systems, electrical 
network theory, mechanical 

engineering, logic modeling, 

evolutionary biology, neuroscience, 
anthropology, and psychology in the 
1940s, often attributed to the Macy 
Conferences. 



Other fields of study which have 
influenced or been influenced by 
cybernetics include game theory, 
system theory (a mathematical 
counterpart to cybernetics), sociology, 
psychology (especially 

neuropsychology, behavioral 

psychology, cognitive psychology), 




Example of cybernetic thinking. On the one hand a company is approached as a system in 
an environment. On the other hand cybernetic factory can be modeled as a control system. 



philosophy, and architecture 



[l] 



Overview 



,^i*toju 



The term cybernetics stems from the Greek icuPepvr|Tr|<; (kybernetes, steersman, governor, 
pilot, or rudder — the same root as government). Cybernetics is a broad field of study, but 
the essential goal of cybernetics is to understand and define the functions and processes of 
systems that have goals and that participate in circular, causal chains that move from action 
to sensing to comparison with desired goal, and again to action. Studies in cybernetics 
provide a means for examining the design and function of any system, including social 
systems such as business management and organizational learning, including for the 
purpose of making them more efficient and effective. 



■ST * 



\ 



''flOUt^ 



# 



Cybernetics 



142 



Cybernetics was defined by Norbert Wiener, in his book of that title, as the study of control and communication in 
the animal and the machine. Stafford Beer called it the science of effective organization and Gordon Pask extended it 
to include information flows "in all media" from stars to brains. It includes the study of feedback, black boxes and 
derived concepts such as communication and control in living organisms, machines and organizations including 
self-organization. Its focus is how anything (digital, mechanical or biological) processes information, reacts to 
information, and changes or can be changed to better accomplish the first two tasks .A more philosophical 
definition, suggested in 1956 by Louis Couffignal, one of the pioneers of cybernetics, characterizes cybernetics as 
"the art of ensuring the efficacy of action" . The most recent definition has been proposed by Louis Kauffman, 
President of the American Society for Cybernetics, "Cybernetics is the study of systems and processes that interact 
with themselves and produce themselves from themselves" 

Concepts studied by cyberneticists (or, as some prefer, cyberneticians) include, but are not limited to: learning, 
cognition, adaption, social control, emergence, communication, efficiency, efficacy and interconnect! vity. These 
concepts are studied by other subjects such as engineering and biology, but in cybernetics these are removed from 
the context of the individual organism or device. 

Other fields of study which have influenced or been influenced by cybernetics include game theory; system theory (a 
mathematical counterpart to cybernetics); psychology, especially neuropsychology, behavioral psychology and 
cognitive psychology; philosophy; anthropology; and even architecture. 



History 



The roots of cybernetic theory 

The word cybernetics was first used in the context of "the study of self-governance" by Plato in The Laws to signify 
the governance of people. The words govern and governor are related to the same Greek root through the Latin 
cognates gubernare and gubernator. The word "cybernetique" was also used in 1834 by the physicist Andre-Marie 
Ampere (1775—1836) to denote the sciences of government in his classification system of human knowledge. 

The first artificial automatic regulatory system, a water clock, was invented 
by the mechanician Ktesibios. In his water clocks, water flowed from a source 
such as a holding tank into a reservoir, then from the reservoir to the 
mechanisms of the clock. Ktesibios's device used a cone-shaped float to 
monitor the level of the water in its reservoir and adjust the rate of flow of the 
water accordingly to maintain a constant level of water in the reservoir, so 
that it neither overflowed nor was allowed to run dry. This was the first 
artificial truly automatic self-regulatory device that required no outside 
intervention between the feedback and the controls of the mechanism. 
Although they did not refer to this concept by the name of Cybernetics (they 
considered it a field of engineering), Ktesibios and others such as Heron and 
Su Song are considered to be some of the first to study cybernetic principles. 




James Watt 



The study of teleological mechanisms (from the Greek xekoq or telos for end, 
goal, or purpose) in machines with corrective feedback dates from as far back 
as the late 1700s when James Watt's steam engine was equipped with a 

governor, a centrifugal feedback valve for controlling the speed of the engine. Alfred Russel Wallace identified this 
as the principle of evolution in his famous 1858 paper. In 1868 James Clerk Maxwell published a theoretical article 
on governors, one of the first to discuss and refine the principles of self-regulating devices. Jakob von Uexkull 
applied the feedback mechanism via his model of functional cycle (Funktionskreis) in order to explain animal 
behaviour and the origins of meaning in general. 



Cybernetics 



143 



The early 20th century 

Contemporary cybernetics began as an interdisciplinary study connecting the fields of control systems, electrical 
network theory, mechanical engineering, logic modeling, evolutionary biology and neuroscience in the 1940s. 
Electronic control systems originated with the 1927 work of Bell Telephone Laboratories engineer Harold S. Black 
on using negative feedback to control amplifiers. The ideas are also related to the biological work of Ludwig von 
Bertalanffy in General Systems Theory. 

Early applications of negative feedback in electronic circuits included the control of gun mounts and radar antenna 
during WWII. Jay Forrester, a graduate student at the Servomechanisms Laboratory at MIT during WWII working 
with Gordon S. Brown to develop electronic control systems for the U.S. Navy, later applied these ideas to social 
organizations such as corporations and cities as an original organizer of the MIT School of Industrial Management at 
the MIT Sloan School of Management. Forrester is known as the founder of System Dynamics. 

W. Edwards Deming, the Total Quality Management guru for whom Japan named its top post- WWII industrial prize, 
was an intern at Bell Telephone Labs in 1927 and may have been influenced by network theory. Deming made 
"Understanding Systems" one of the four pillars of what he described as "Profound Knowledge" in his book "The 
New Economics." 

Numerous papers spearheaded the coalescing of the field. In 1935 Russian physiologist P.K. Anokhin published a 
book in which the concept of feedback ("back afferentation") was studied. The study and mathematical modelling of 
regulatory processes became a continuing research effort and two key articles were published in 1943. These papers 
were "Behavior, Purpose and Teleology" by Arturo Rosenblueth, Norbert Wiener, and Julian Bigelow; and the paper 
"A Logical Calculus of the Ideas Immanent in Nervous Activity" by Warren McCulloch and Walter Pitts. 

Cybernetics as a discipline was firmly established by Wiener, McCulloch and others, such as W. Ross Ashby and W. 
Grey Walter. 

Walter was one of the first to build autonomous robots as an aid to the study of animal behaviour. Together with the 
US and UK, an important geographical locus of early cybernetics was France. 

In the spring of 1947, Wiener was invited to a congress on harmonic analysis, held in Nancy, France. The event was 
organized by the Bourbaki, a French scientific society, and mathematician Szolem Mandelbrojt (1899-1983), uncle 
of the world-famous mathematician Benoit Mandelbrot. 

During this stay in France, Wiener received the offer to write a 
manuscript on the unifying character of this part of applied 
mathematics, which is found in the study of Brownian motion and in 
telecommunication engineering. The following summer, back in the 
United States, Wiener decided to introduce the neologism cybernetics 
into his scientific theory. The name cybernetics was coined to denote 
the study of "teleological mechanisms" and was popularized through 
his book Cybernetics, or Control and Communication in the Animal 
and Machine (Hermann & Cie, Paris, 1948). In the UK this became the 
focus for the Ratio Club. 

In the early 1940s John von Neumann, although better known for his 
work in mathematics and computer science, did contribute a unique 
and unusual addition to the world of cybernetics: Von Neumann 
cellular automata, and their logical follow up the Von Neumann 
Universal Constructor. The result of these deceptively simple 
thought-experiments was the concept of self replication which 




John von Neumann 



Cybernetics 



144 



cybernetics adopted as a core concept. The concept that the same properties of genetic reproduction applied to social 
memes, living cells, and even computer viruses is further proof of the somewhat surprising universality of cybernetic 
study. 

Wiener popularized the social implications of cybernetics, drawing analogies between automatic systems (such as a 
regulated steam engine) and human institutions in his best-selling The Human Use of Human Beings : Cybernetics 
and Society (Houghton-Mifflin, 1950). 

While not the only instance of a research organization focused on cybernetics, the Biological Computer Lab at the 
University of Illinois, Urbana/Champaign, under the direction of Heinz von Foerster, was a major center of 
cybernetic research for almost 20 years, beginning in 1958. 



The fall and rebirth of cybernetics 

For a time during the past 30 years, the field of cybernetics followed a boom-bust cycle of becoming more and more 
dominated by the subfields of artificial intelligence and machine-biological interfaces (ie. cyborgs) and when this 
research fell out of favor, the field as a whole fell from grace. 

In the 1970s new cyberneticians emerged in multiple fields, but 
especially in biology. The ideas of Maturana, Varela and Atlan, 
according to Dupuy (1986) "realized that the cybernetic metaphors of 
the program upon which molecular biology had been based rendered a 
conception of the autonomy of the living being impossible. 
Consequently, these thinkers were led to invent a new cybernetics, one 
more suited to the organizations which mankind discovers in nature - 
organizations he has not himself invented" . However, during the 
1980s the question of whether the features of this new cybernetics 
could be applied to social forms of organization remained open to 




debate 



[7] 



Francisco Varela 



In political science, Project Cybersyn attempted to introduce a 
cybernetically controlled economy during the early 1970s. In the 
1980s, according to Harries-Jones (1988) "unlike its predecessor, the 
new cybernetics concerns itself with the interaction of autonomous 
political actors and subgroups, and the practical and reflexive 
consciousness of the subjects who produce and reproduce the structure 
of a political community. A dominant consideration is that of 
recursiveness, or self-reference of political action both with regards to 

the expression of political consciousness and with the ways in which 

rxi 
systems build upon themselves". 

One characteristic of the emerging new cybernetics considered in that 

time by Geyer and van der Zouwen, according to Bailey (1994), was 

"that it views information as constructed and reconstructed by an 

individual interacting with the environment. This provides an 

epistemological foundation of science, by viewing it as observer-dependent. Another characteristic of the new 

cybernetics is its contribution towards bridging the "micro-macro gap". That is, it links the individual with the 

society" Another charateristic noted was the "transition from classical cybernetics to the new cybernetics [that] 

involves a transition from classical problems to new problems. These shifts in thinking involve, among others, (a) a 




Stuart A. Umpleby 



change from emphasis on the system being steered to the system doing the steering, and the factor which guides the 
steering decisions.; and (b) new emphasis on communication between several systems which are trying to steer each 



Cybernetics 



145 



other 



,[9] 



, The work of Gregory Bateson was also strongly influenced by cybernetics. 

Recent endeavors into the true focus of cybernetics, systems of control and emergent behavior, by such related fields 
as game theory (the analysis of group interaction), systems of feedback in evolution, and metamaterials (the study of 
materials with properties beyond the Newtonian properties of their constituent atoms), have led to a revived interest 



in this increasingly relevant field 



[2] 



Subdivisions of the field 

Cybernetics is an earlier but still-used generic term for many types of subject matter. These subjects also extend into 
many others areas of science, but are united in their study of control of systems. 



Pure cybernetics 

Pure cybernetics studies systems of control as a concept, attempting to discover the basic principles underlying such 
things as 

Artificial intelligence 
Robotics 
Computer Vision 
Control systems 
Emergence 
Learning organization 
New Cybernetics 
Second-order cybernetics 
Interactions of Actors Theory 

_ . _, ASIMO uses sensors and intelligent algorithms to 

Conversation Theory 

avoid obstacles and navigate stairs. 



1 1 m K I 


ft h "''-" 


tlBif 




{ 

9 if 


! ! f 


fX 7 
tl 7 


1 ' i 




mmmBk*^~ 



In biology 

Cybernetics in biology is the study of cybernetic systems present in biological organisms, primarily focusing on how 
animals adapt to their environment, and how information in the form of genes is passed from generation to 
generation . There is also a secondary focus on combining artificial systems with biological systems. 

Bioengineering 
Biocybernetics 
Bionics 
Homeostasis 
Medical cybernetics 
Synthetic Biology 
Systems Biology 



In computer science 




Thermal image of a cold-blooded tarantula on a 
warm-blooded human hand 



Computer science directly applies the concepts of cybernetics to the control of devices and the analysis of 
information. 

• Robotics 

• Decision support system 

• Cellular automaton 

• Simulation 



Cybernetics 



146 



In engineering 

Cybernetics in engineering is used to analyze cascading failures and System Accidents, in which the small errors and 
imperfections in a system can generate disasters. Other topics studied include: 

• Adaptive systems 

• Engineering cybernetics 

• Ergonomics 

• Biomedical engineering 

• Systems engineering 

In management 



Entrepreneurial cybernetics 
Management cybernetics 
Organizational cybernetics 

Operations research 
Systems engineering 




An artificial heart, a product of biomedical 
engineering. 



In mathematics 

Mathematical Cybernetics focuses on the factors of information, interaction of parts in systems, and the structure of 
systems. 

• Dynamical system 

• Information theory 

• Systems theory 

In psychology 

• Homunculus 

• Psycho-Cybernetics 

• Systems psychology 



In sociology 

By examining group behavior through the lens of cybernetics, sociology seeks the reasons for such spontaneous 
events as smart mobs and riots, as well as how communities develop rules, such as etiquette, by consensus without 
formal discussion.Affect Control Theory explains role behavior, emotions, and labeling theory in terms of 
homeostatic maintenance of sentiments associated with cultural categories. The most comprehensive attempt ever 
made in the social sciences to increase cybernetics in a generalized theory of society was made by Talcott Parsons. 
These and other cybernetic models in sociology are reviewed in a book edited by McClelland and Fararo 

• Affect Control Theory 

• Memetics 

• Sociocybernetics 



ill] 



Cybernetics 



147 



Related fields 
Complexity science 

Complexity science attempts to understand the nature of complex systems. 

• Complex Adaptive System 

• Complex systems 

• Complexity theory 

See also 



Artificial life 

Automation 

Brain-computer 

interface 

Chaos theory 

Connectionism 

Decision theory 



Family therapy 
Gaia hypothesis 
Industrial Ecology 
Intelligence amplification 
Management science 
Perceptual control theory 



Principia Cybernetica 
Project Cybersyn 
Viable System Model 
Semiotics 
Superorganisms 
Synergetics 



Further reading 



[12] 



• W. Ross Ashby (1956), Introduction to Cybernetics. Methuen, London, UK. PDF text 

• Stafford Beer (1974), Designing Freedom, John Wiley, London and New York, 1975. 

• Lars Bluma, (2005), Norbert Wiener und die Entstehung der Kybernetik im Zweiten Weltkrieg, Miinster. 

[291 

• Charles Francois (1999). "Systemics and cybernetics in a historical perspective . In: Systems Research and 

Behavioral Science. Vol 16, pp. 203-219 (1999) 

• Steve J. Heims (1980), John von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life 
and Death, 3. Aufl., Cambridge. 

• Steve J. Heims (1993), Constructing a Social Science for Postwar America. The Cybernetics Group, 1946-1953, 
Cambridge University Press, London, UK. 

• Helvey, T.C. The Age of Information: An Interdisciplinary Survey of Cybernetics. Englewood Cliffs, N.J.: 
Educational Technology Publications, 1971. 

ri3i 

• Francis Heylighen, and Cliff Joslyn (2001). "Cybernetics and Second Order Cybernetics , in: R.A. Meyers 
(ed.), Encyclopedia of Physical Science & Technology (3rd ed.), Vol. 4, (Academic Press, New York), p. 
155-170. 

• Heikki Hyotyniemi (2006). Neocybernetics in Biological Systems . Espoo: Helsinki University of Technology, 
Control Engineering Laboratory. 

• Hans Joachim Ilgauds (1980), Norbert Wiener, Leipzig. 

• John Johnston, (2008) "The Allure of Machinic Life: Cybernetics, Artificial Life, and the New AI", MIT Press 

• Eden Medina, "Designing Freedom, Regulating a Nation: Socialist Cybernetics in Allende's Chile." Journal of 

Latin American Studies 38 (2006):57 1-606. 

[141 

• Paul Pangaro (1990), "Cybernetics — A Definition", Eprint 

• Gordon Pask (1972), "Cybernetics , entry in Encyclopaedia Britannica 1972. 

• B.C. Patten, and E.P. Odum(1981), "The Cybernetic Nature of Ecosystems", The American Naturalist 118, 
886-895. 

• Heinz von Foerster, (1995), Ethics and Second-Order Cybernetics 

[171 

• Stuart Umpleby (1989), "The science of cybernetics and the cybernetics of science" , in: Cybernetics and 
Systems", Vol. 21, No. 1, (1990), pp. 109-121. 



Cybernetics 148 

• Norbert Wiener (1948), Cybernetics or Control and Communication in the Animal and the Machine, (Hermann & 
Cie Editeurs, Paris, The Technology Press, Cambridge, Mass., John Wiley & Sons Inc., New York, 1948). 

External links 

General 

no] 

• Principia Cybernetica Web 

ri9i 

• Web Dictionary of Cybernetics and Systems 

• Glossary Slideshow (136 slides) 

[211 

• Basics of Cybernetics 

["221 

• What is Cybernetics? Livas short introductory videos on YouTube 
Societies 

T231 

• American Society for Cybernetics 

T241 

• IEEE Systems, Man, & Cybernetics Society 

[251 

• The Cybernetics Society 

References 

[I] Tange, Kenzo (1966) "Function, Structure and Symbol". 

[2] Kelly, Kevin (1994). Out of control: The new biology of machines, social systems and the economic world. Boston: Addison- Wesley. 

ISBN 0-201-48340-8. OCLC 221860672 32208523 40868076 56082721 57396750. 
[3] Couffignal, Louis, "Essai d'une definition generale de la cybernetique", The First International Congress on Cybernetics, Namur, Belgium, 

June 26-29, 1956, Gauthier-Villars, Paris, 1958, pp. 46-54 
[4] CYBCON discusstion group 20 September 2007 18: 15 
[5] http://www.ece.uiuc.edu/pubs/bcl/mueller/index.htm 
[6] http://www.ece.uiuc.edu/pubs/bcl/hutchinson/index.htm 
[7] Jean-Pierre Dupuy, "The autonomy of social reality: on the contribution of systems theory to the theory of society" in: Elias L. Khalil & 

Kenneth E. Boulding eds., Evolution, Order and Complexity, 1986. 
[8] Peter Harries-Jones (1988), "The Self-Organizing Polity: An Epistemological Analysis of Political Life by Laurent Dobuzinskis" in: 

Canadian Journal of Political Science (Revue canadienne de science politique), Vol. 21, No. 2 (Jun., 1988), pp. 431-433. 
[9] Kenneth D. Bailey (1994), Sociology and the New Systems Theory: Toward a Theoretical Synthesis, p. 163. 
[10] Note: this does not refer to the concept of Racial Memory but to the concept of cumulative adaptation to a particular niche, such as the case 

of the pepper moth having genes for both light and dark environments. 

[II] McClelland, Kent A., and Thomas J. Fararo (Eds.). 2006. Purpose, Meaning, and Action: Control Systems Theories in Sociology. New 
York: Palgrave Macmillan. 

[12] http://pespmcl.vub.ac.be/books/IntroCyb.pdf 

[13] http://pespmcl.vub.ac.be/Papers/Cybernetics-EPST.pdf 

[ 14] http ://pangaro. com/published/cyber- macmillan. html 

[15] http://www.cybsoc.org/gcyb.htm 

[16] http://www.stanford.edu/group/SHR/4-2/text/foerster.html 

[17] ftp://ftp.vub.ac.be/pub/projects/Principia_Cybernetica/Papers_Umpleby/Science-Cybernetics.txt 

[18] http://pespmcl.vub.ac.be/DEFAULT.html 

[19] http://pespmcl.vub.ac.be/ASC/indexASC.html 

[20] http://www.gwu.edu/~asc/slide/sl.html 

[21] http://www.smithsrisca.demon.co.uk/cybernetics.html 

[22] http ://www. youtube.com/watch?v=_hj AXkNbPfk 

[23] http://www.asc-cybernetics.org/ 

[24] http://www.ieeesmc.org/ 

[25] http://www.cybsoc.org 



Control theory 



149 



Control theory 



Control theory is an interdisciplinary 
branch of engineering and 
mathematics, that deals with the 
behavior of dynamical systems. The 
desired output of a system is called the 
reference. When one or more output 
variables of a system need to follow a 
certain reference over time, a 
controller manipulates the inputs to a 
system to obtain the desired effect on the output of the system. 



M 


easured 




System 








Reference j- — 


error 


Controller 


input 


System 


System output 


-> 


\ 


























Measured output 



























The concept of the feedback loop to control the dynamic behavior of the system: this is 

negative feedback, because the sensed value is subtracted from the desired value to create 

the error signal which is amplified by the controller. 



Overview 

Control theory is 

• a theory that deals with influencing the behavior of dynamical systems 

• an interdisciplinary subfield of science, which originated in engineering and mathematics, and evolved into use by 
the social sciences, like psychology, sociology and criminology. 



An example 

Consider an automobile's cruise control, which is a device designed to maintain a constant vehicle speed; the desired 
or reference speed, provided by the driver. The system in this case is the vehicle. The system output is the vehicle 
speed, and the control variable is the engine's throttle position which influences engine torque output. 

A primitive way to implement cruise control is simply to lock the throttle position when the driver engages cruise 
control. However, on mountain terrain, the vehicle will slow down going uphill and accelerate going downhill. In 
fact, any parameter different than what was assumed at design time will translate into a proportional error in the 
output velocity, including exact mass of the vehicle, wind resistance, and tire pressure. This type of controller is 
called an open-loop controller because there is no direct connection between the output of the system (the vehicle's 
speed) and the actual conditions encountered; that is to say, the system does not and can not compensate for 
unexpected forces. 

In a closed-loop control system, a sensor monitors the output (the vehicle's speed) and feeds the data to a computer 
which continuously adjusts the control input (the throttle) as necessary to keep the control error to a minimum (that 
is, to maintain the desired speed). Feedback on how the system is actually performing allows the controller (vehicle's 
on board computer) to dynamically compensate for disturbances to the system, such as changes in slope of the 
ground or wind speed. An ideal feedback control system cancels out all errors, effectively mitigating the effects of 
any forces that might or might not arise during operation and producing a response in the system that perfectly 
matches the user's wishes. In reality, this cannot be achieved due to measurement errors in the sensors, delays in the 
controller, and imperfections in the control input. 



Control theory 



150 



History 

Although control systems of various types date back to antiquity, a 
more formal analysis of the field began with a dynamics analysis of the 
centrifugal governor, conducted by the physicist James Clerk Maxwell 
in 1868 entitled On Governors. This described and analyzed the 
phenomenon of "hunting", in which lags in the system can lead to 
overcompensation and unstable behavior. This generated a flurry of 
interest in the topic, during which Maxwell's classmate Edward John 
Routh generalized the results of Maxwell for the general class of linear 
systems. Independently, Adolf Hurwitz analyzed system stability 
using differential equations in 1877. This result is called the 



Routh-Hurwitz theorem 



[3] [4] 




A notable application of dynamic control was in the area of manned 
flight. The Wright Brothers made their first successful test flights on 
December 17, 1903 and were distinguished by their ability to control 
their flights for substantial periods (more so than the ability to produce 
lift from an airfoil, which was known). Control of the airplane was 
necessary for safe flight. 

By World War II, control theory was an important part of fire-control systems, guidance systems and electronics. 
The Space Race also depended on accurate spacecraft control. However, control theory also saw an increasing use in 
fields such as economics. 



Centrifugal governor in a Boulton & Watt engine 
of 1788 



People in systems and control 

Many active and historical figures made significant contribution to control theory, including, for example: 

Alexander Lyapunov (1857—1918) in the 1890s marks the beginning of stability theory. 

Harold S. Black (1898—1983), invented the concept of negative feedback amplifiers in 1927. He managed to 

develop stable negative feedback amplifiers in the 1930s. 

Harry Nyquist (1889—1976), developed the Nyquist stability criterion for feedback systems in the 1930s. 

Richard Bellman (1920—1984), developed dynamic programming since the 1940s. 

Andrey Kolmogorov (1903—1987) co-developed the Wiener-Kolmogorov filter (1941). 

Norbert Wiener (1894—1964) co-developed the Wiener-Kolmogorov filter and coined the term cybernetics in the 

1940s. 

John R. Ragazzini (1912—1988) introduced digital control and the z-transform in the 1950s. 

Lev Pontryagin (1908—1988) introduced the maximum principle and the bang-bang principle. 



Classical control theory 

To avoid the problems of the open-loop controller, control theory introduces feedback. A closed-loop controller uses 
feedback to control states or outputs of a dynamical system. Its name comes from the information path in the system: 
process inputs (e.g. voltage applied to an electric motor) have an effect on the process outputs (e.g. velocity or torque 
of the motor), which is measured with sensors and processed by the controller; the result (the control signal) is used 
as input to the process, closing the loop. 

Closed-loop controllers have the following advantages over open-loop controllers: 

• disturbance rejection (such as unmeasured friction in a motor) 



Control theory 



151 



• guaranteed performance even with model uncertainties, when the model structure does not match perfectly the 
real process and the model parameters are not exact 

• unstable processes can be stabilized 

• reduced sensitivity to parameter variations 

• improved reference tracking performance 

In some systems, closed-loop and open-loop control are used simultaneously. In such systems, the open-loop control 
is termed feedforward and serves to further improve reference tracking performance. 

A common closed-loop controller architecture is the PID controller. 



Closed-loop transfer function 

The output of the system y(t) is fed back through a sensor measurement F to the reference value r(t). The controller 
C then takes the error e (difference) between the reference and the output to change the inputs u to the system under 
control P. This is shown in the figure. This kind of controller is a closed-loop controller or feedback controller. 

This is called a single-input-single-output (SISO) control system; MIMO (i.e. Multi-Input-Multi-Output) systems, 
with more than one input/output, are common. In such cases variables are represented through vectors instead of 
simple scalar values. For some distributed parameter systems the vectors may be infinite-dimensional (typically 
functions). 



r + A e u 3 

— O — ► c — ► p — 1 

F 


► 







If we assume the controller C, the plant P, and the sensor F are linear and time-invariant (i.e.: elements of their 
transfer function C(s), P(s), and F(s) do not depend on time), the systems above can be analysed using the Laplace 
transform on the variables. This gives the following relations: 

Y{s) = P(s)U(s) 
U(s) = C(s)E(s) 
E(s) = R(s) - F(s)Y(s). 
Solving for Y(s) in terms of R(s) gives: 

P(s)C(s) 



Y(s) = 



l + F(s)P{s)C(s)J 

P{s)C(s 



R(s) = H(s)R(s). 

) 

-is referred to as the closed-loop transfer function of the system. 



The expression H(s) = ^ + F{s)p{s)c{s y 

The numerator is the forward (open-loop) gain from r to y, and the denominator is one plus the gain in going around 
the feedback loop, the so-called loop gain. If \P(s)C(s) I ^> 1, i-e. it has a large norm with each value of s, and if 
\F (s)\ ~ 1, then Y(s) is approximately equal to R(s). This simply means setting the reference to control the 
output. 



Control theory 152 

PID controller 

The PID controller is probably the most-used feedback control design. PID is an acronym for 
Proportional-Integral-Derivative, referring to the three terms operating on the error signal to produce a control 
signal. If u(t) is the control signal sent to the system, y(t) is the measured output and r(t) is the desired output, and 
tracking error e(t) = r(t) — y(t) > a PID controller has the general form 



u(t) = K P e(t) + Kj e(t)dt + K D —e(t). 



The desired closed loop dynamics is obtained by adjusting the three parameters Kp, i^and Kp> , often 
iteratively by "tuning" and without specific knowledge of a plant model. Stability can often be ensured using only 
the proportional term. The integral term permits the rejection of a step disturbance (often a striking specification in 
process control). The derivative term is used to provide damping or shaping of the response. PID controllers are the 
most well established class of control systems: however, they cannot be used in several more complicated cases, 
especially if MIMO systems are considered. 

Applying Laplace transformation results in the transformed PID controller equation 

1 



u 



(s) = Kpe(s) + ATj-e(s) + K D se(s) 



u{s) = {K P + K z - + K D s)e(s) 
s 

with the PID controller transfer function 

C( s ) = (Kp + K I - + K D s). 
s 

Modern control theory 

In contrast to the frequency domain analysis of the classical control theory, modern control theory utilizes the 
time-domain state space representation, a mathematical model of a physical system as a set of input, output and state 
variables related by first-order differential equations. To abstract from the number of inputs, outputs and states, the 
variables are expressed as vectors and the differential and algebraic equations are written in matrix form (the latter 
only being possible when the dynamical system is linear). The state space representation (also known as the 
"time-domain approach") provides a convenient and compact way to model and analyze systems with multiple inputs 
and outputs. With inputs and outputs, we would otherwise have to write down Laplace transforms to encode all the 
information about a system. Unlike the frequency domain approach, the use of the state space representation is not 
limited to systems with linear components and zero initial conditions. "State space" refers to the space whose axes 
are the state variables. The state of the system can be represented as a vector within that space. 

Topics in control theory 
Stability 

The stability of a general dynamical system with no input can be described with Lyapunov stability criteria. A linear 
system that takes an input is called bounded-input bounded-output (BIBO) stable if its output will stay bounded for 
any bounded input. Stability for nonlinear systems that take an input is input-to-state stability (ISS), which combines 
Lyapunov stability and a notion similar to BIBO stability. For simplicity, the following descriptions focus on 
continuous-time and discrete-time linear systems. 

Mathematically, this means that for a causal linear system to be stable all of the poles of its transfer function must 
satisfy some criteria depending on whether a continuous or discrete time analysis is used: 



Control theory 153 

• In continuous time, the Laplace transform is used to obtain the transfer function. A system is stable if the poles of 
this transfer function lie strictly in the closed left half of the complex plane (i.e. the real part of all the poles is less 
than zero). 

• In discrete time the Z-transform is used. A system is stable if the poles of this transfer function lie strictly inside 
the unit circle, i.e. the magnitude of the poles is less than one). 

When the appropriate conditions above are satisfied a system is said to be asymptotically stable: the variables of an 
asymptotically stable control system always decrease from their initial value and do not show permanent oscillations. 
Permanent oscillations occur when a pole has a real part exactly equal to zero (in the continuous time case) or a 
modulus equal to one (in the discrete time case). If a simply stable system response neither decays nor grows over 
time, and has no oscillations, it is marginally stable: in this case the system transfer function has non-repeated poles 
at complex plane origin (i.e. their real and complex component is zero in the continuous time case). Oscillations are 
present when poles with real part equal to zero have an imaginary part not equal to zero. 

Differences between the two cases are not a contradiction. The Laplace transform is in Cartesian coordinates and the 
Z-transform is in circular coordinates, and it can be shown that: 

• the negative-real part in the Laplace domain can map onto the interior of the unit circle 

• the positive-real part in the Laplace domain can map onto the exterior of the unit circle 

If a system in question has an impulse response of 

x[n] = 0.5 n u[n] 
then the Z-transform (see this example), is given by 

X{z) [ 



1-0.5Z- 1 

which has a pole in % = 0.5( zero imaginary part). This system is BIBO (asymptotically) stable since the pole is 
inside the unit circle. 

However, if the impulse response was 

x[n] — l.b n u[n] 
then the Z-transform is 

which has a pole at z = 1.5 an d is not BIBO stable since the pole has a modulus strictly greater than one. 

Numerous tools exist for the analysis of the poles of a system. These include graphical systems like the root locus, 
Bode plots or the Nyquist plots. 

Mechanical changes can make equipment (and control systems) more stable. Sailors add ballast to improve the 
stability of ships. Cruise ships use antiroll fins that extend transversely from the side of the ship for perhaps 30 feet 
(10 m) and are continuously rotated about their axes to develop forces that oppose the roll. 

Controllability and observability 

Controllability and observability are main issues in the analysis of a system before deciding the best control strategy 
to be applied, or whether it is even possible to control or stabilize the system. Controllability is related to the 
possibility of forcing the system into a particular state by using an appropriate control signal. If a state is not 
controllable, then no signal will ever be able to control the state. If a state is not controllable, but its dynamics are 
stable, then the state is termed Stabilizable. Observability instead is related to the possibility of "observing", through 
output measurements, the state of a system. If a state is not observable, the controller will never be able to determine 
the behaviour of an unobservable state and hence cannot use it to stabilize the system. However, similar to the 
stabilizability condition above, if a state cannot be observed it might still be detectable. 



Control theory 154 

From a geometrical point of view, looking at the states of each variable of the system to be controlled, every "bad" 
state of these variables must be controllable and observable to ensure a good behaviour in the closed-loop system. 
That is, if one of the eigenvalues of the system is not both controllable and observable, this part of the dynamics will 
remain untouched in the closed-loop system. If such an eigenvalue is not stable, the dynamics of this eigenvalue will 
be present in the closed-loop system which therefore will be unstable. Unobservable poles are not present in the 
transfer function realization of a state-space representation, which is why sometimes the latter is preferred in 
dynamical systems analysis. 

Solutions to problems of uncontrollable or unobservable system include adding actuators and sensors. 

Control specifications 

Several different control strategies have been devised in the past years. These vary from extremely general ones (PID 
controller), to others devoted to very particular classes of systems (especially robotics or aircraft cruise control). 

A control problem can have several specifications. Stability, of course, is always present: the controller must ensure 
that the closed-loop system is stable, regardless of the open-loop stability. A poor choice of controller can even 
worsen the stability of the open-loop system, which must normally be avoided. Sometimes it would be desired to 
obtain particular dynamics in the closed loop: i.e. that the poles have iJefAl < — A . where ^ is a fixed value 
strictly greater than zero, instead of simply ask that Re [A] < • 

Another typical specification is the rejection of a step disturbance; including an integrator in the open-loop chain (i.e. 
directly before the system under control) easily achieves this. Other classes of disturbances need different types of 
sub-systems to be included. 

Other "classical" control theory specifications regard the time-response of the closed-loop system: these include the 
rise time (the time needed by the control system to reach the desired value after a perturbation), peak overshoot (the 
highest value reached by the response before reaching the desired value) and others (settling time, quarter-decay). 
Frequency domain specifications are usually related to robustness (see after). 

Modern performance assessments use some variation of integrated tracking error (IAE,ISA,CQI). 

Model identification and robustness 

A control system must always have some robustness property. A robust controller is such that its properties do not 
change much if applied to a system slightly different from the mathematical one used for its synthesis. This 
specification is important: no real physical system truly behaves like the series of differential equations used to 
represent it mathematically. Typically a simpler mathematical model is chosen in order to simplify calculations, 
otherwise the true system dynamics can be so complicated that a complete model is impossible. 

System identification 

The process of determining the equations that govern the model's dynamics is called system identification. This can 
be done off-line: for example, executing a series of measures from which to calculate an approximated mathematical 
model, typically its transfer function or matrix. Such identification from the output, however, cannot take account of 
unobservable dynamics. Sometimes the model is built directly starting from known physical equations: for example, 
in the case of a mass-spring-damper system we know that rnxit) = —Kx(t) — Bi:(i) ■ Even assuming that a 
"complete" model is used in designing the controller, all the parameters included in these equations (called "nominal 
parameters") are never known with absolute precision; the control system will have to behave correctly even when 
connected to physical system with true parameter values away from nominal. 

Some advanced control techniques include an "on-line" identification process (see later). The parameters of the 
model are calculated ("identified") while the controller itself is running: in this way, if a drastic variation of the 
parameters ensues (for example, if the robot's arm releases a weight), the controller will adjust itself consequently in 
order to ensure the correct performance. 



Control theory 155 

Analysis 

Analysis of the robustness of a SISO control system can be performed in the frequency domain, considering the 
system's transfer function and using Nyquist and Bode diagrams. Topics include gain and phase margin and 
amplitude margin. For MIMO and, in general, more complicated control systems one must consider the theoretical 
results devised for each control technique (see next section): i.e., if particular robustness qualities are needed, the 
engineer must shift his attention to a control technique including them in its properties. 

Constraints 

A particular robustness issue is the requirement for a control system to perform properly in the presence of input and 
state constraints. In the physical world every signal is limited. It could happen that a controller will send control 
signals that cannot be followed by the physical system: for example, trying to rotate a valve at excessive speed. This 
can produce undesired behavior of the closed-loop system, or even break actuators or other subsystems. Specific 
control techniques are available to solve the problem: model predictive control (see later), and anti-wind up systems. 
The latter consists of an additional control block that ensures that the control signal never exceeds a given threshold. 

System classifications 
Linear control 

For MIMO systems, pole placement can be performed mathematically using a state space representation of the 
open-loop system and calculating a feedback matrix assigning poles in the desired positions. In complicated systems 
this can require computer-assisted calculation capabilities, and cannot always ensure robustness. Furthermore, all 
system states are not in general measured and so observers must be included and incorporated in pole placement 
design. 

Nonlinear control 

Processes in industries like robotics and the aerospace industry typically have strong nonlinear dynamics. In control 
theory it is sometimes possible to linearize such classes of systems and apply linear techniques, but in many cases it 
can be necessary to devise from scratch theories permitting control of nonlinear systems. These, e.g., feedback 
linearization, backstepping, sliding mode control, trajectory linearization control normally take advantage of results 
based on Lyapunov's theory. Differential geometry has been widely used as a tool for generalizing well-known linear 
control concepts to the non-linear case, as well as showing the subtleties that make it a more challenging problem. 

Main control strategies 

Every control system must guarantee first the stability of the closed-loop behavior. For linear systems, this can be 
obtained by directly placing the poles. Non-linear control systems use specific theories (normally based on 
Aleksandr Lyapunov's Theory) to ensure stability without regard to the inner dynamics of the system. The possibility 
to fulfill different specifications varies from the model considered and the control strategy chosen. Here a summary 
list of the main control techniques is shown: 

Adaptive control 

Adaptive control uses on-line identification of the process parameters, or modification of controller gains, 
thereby obtaining strong robustness properties. Adaptive controls were applied for the first time in the 
aerospace industry in the 1950s, and have found particular success in that field. 

Hierarchical control 

A Hierarchical control system is a type of Control System in which a set of devices and governing software is 
arranged in a hierarchical tree. When the links in the tree are implemented by a computer network, then that 
hierarchical control system is also a form of Networked control system. 



Control theory 



156 



Intelligent control 

Intelligent control uses various AI computing approaches like neural networks, Bayesian probability, fuzzy 
logic, machine learning, evolutionary computation and genetic algorithms to control a dynamic system. 

Optimal control 

Optimal control is a particular control technique in which the control signal optimizes a certain "cost index": 
for example, in the case of a satellite, the jet thrusts needed to bring it to desired trajectory that consume the 
least amount of fuel. Two optimal control design methods have been widely used in industrial applications, as 
it has been shown they can guarantee closed-loop stability. These are Model Predictive Control (MPC) and 
Linear-Quadratic-Gaussian control (LQG). The first can more explicitly take into account constraints on the 
signals in the system, which is an important feature in many industrial processes. However, the "optimal 
control" structure in MPC is only a means to achieve such a result, as it does not optimize a true performance 
index of the closed-loop control system. Together with PID controllers, MPC systems are the most widely 
used control technique in process control. 

Robust control 

Robust control deals explicitly with uncertainty in its approach to controller design. Controllers designed using 
robust control methods tend to be able to cope with small differences between the true system and the nominal 
model used for design. The early methods of Bode and others were fairly robust; the state-space methods 
invented in the 1960s and 1970s were sometimes found to lack robustness. A modern example of a robust 
control technique is H-infinity loop-shaping developed by Duncan McFarlane and Keith Glover of Cambridge 
University, United Kingdom. Robust methods aim to achieve robust performance and/or stability in the 
presence of small modeling errors. 

Stochastic control 

Stochastic control deals with control design with uncertainty in the model. In typical stochastic control 
problems, it is assumed that there exist random noise and disturbances in the model and the controller, and the 
control design must take into account these random deviations. 

See also 



Examples of control systems 

Automation 

Deadbeat Controller 

Distributed parameter 

systems 

Fractional order control 

H-infinity loop-shaping 

Hierarchical control system 

PID controller 

Model predictive control 

Process control 

Robust control 

Servomechanism 

State space (controls) 



Topics in control theory 

Coefficient diagram method 

Control reconfiguration 

Feedback 

H infinity 

Hankel singular value 

Krener's theorem 

Lead-lag compensator 

Radial basis function 

Robotic unicycle 

Root locus 

Signal-flow graphs 

Stable polynomial 

Underactuation 



Other related topics 

Automation and Remote Control 
Bond graph 
Control engineering 
Controller (control theory) 
Intelligent control 
Mathematical system theory 
Perceptual control theory 
Systems theory 
People in systems and control 
Time scale calculus 
Negative feedback amplifier 



Control theory 157 

Further reading 

Levine, William S., ed (1996). The Control Handbook. New York: CRC Press. ISBN 978-0-849-38570-4. 

Karl J. Astrom and Richard M. Murray (2008). Feedback Systems: An Introduction for Scientists and Engineers. 

[5] . Princeton University Press. ISBN 0691135762. 

Christopher Kilian (2005). Modern Control Technology. Thompson Delmar Learning. ISBN 1-4018-5806-6. 

Vannevar Bush (1929). Operational Circuit Analysis. John Wiley and Sons, Inc.. 

Robert F. Stengel (1994). Optimal Control and Estimation. Dover Publications. ISBN 0-486-68200-5, ISBN 

978-0-486-68200-6. 

Franklin et al. (2002). Feedback Control of Dynamic Systems (4 ed.). New Jersey: Prentice Hall. 

ISBN 0-13-032393-4. 

Joseph L. Hellerstein, Dawn M. Tilbury, and Sujay Parekh (2004). Feedback Control of Computing Systems. John 

Wiley and Sons. ISBN 0-47-126637-X, ISBN 978-0-471-26637-2. 

Diederich Hinrichsen and Anthony J. Pritchard (2005). Mathematical Systems Theory I - Modelling, State Space 

Analysis, Stability and Robustness. Springer. ISBN 0-978-3-540-44125-0. 

Andrei, Neculai (2005). Modern Control Theory - A historical Perspective . Retrieved 2007-10-10. 



Sontag, Eduardo (1998). Mathematical Control Theory: Deterministic Finite Dimensional Systems. Second 
Edition [7] . Springer. ISBN 0-387-984895. 

• Goodwin, Graham (2001). Control System Design. Prentice Hall. ISBN 0-13-958653-9. 

External links 

— rsi 

• Control Tutorials for Matlab - A set of worked through control examples solved by several different methods. 

References 

[1] Maxwell, J.C. (1867). "On Governors" (http://links.jstor.org/sici?sici=0370-1662(1867/1868)16<270:OG>2.0.CO;2-l). Proceedings of 

the Royal Society of London 16: 270-283. doi:10.1098/rspl.l867.0055. . Retrieved 2008-04-14. 
[2] Routh, E.J.; Fuller, A.T. (1975). Stability of motion. Taylor & Francis. 
[3] Routh, E.J. (1877). A Treatise on the Stability of a Given State of Motion, Particularly Steady Motion: Particularly Steady Motion. Macmillan 

and co.. 
[4] Hurwitz, A. (1964). "On The Conditions Under Which An Equation Has Only Roots With Negative Real Parts". Selected Papers on 

Mathematical Trends in Control Theory. 
[5] http://www.cds.caltech.edu/~murray/books/AM08/pdf/am08-complete_22Feb09.pdf 
[6] http://www.ici.ro/camo/neculai/history.pdf 

[7] http://www.math.rutgers.edu/~sontag/FTP_DIR/sontag_mathematical_control_theory_springer98.pdf 
[8] http://www.library.cmu.edu/ctms/ 



Genomics 158 



Genomics 



Genomics is the study of the genomes of organisms. The field includes intensive efforts to determine the entire DNA 
sequence of organisms and fine-scale genetic mapping efforts. The field also includes studies of intragenomic 
phenomena such as heterosis, epistasis, pleiotropy and other interactions between loci and alleles within the genome. 
In contrast, the investigation of the roles and functions of single genes is a primary focus of molecular biology or 
genetics and is a common topic of modern medical and biological research. Research of single genes does not fall 
into the definition of genomics unless the aim of this genetic, pathway, and functional information analysis is to 
elucidate its effect on, place in, and response to the entire genome's networks. 

For the United States Environmental Protection Agency, "the term "genomics" encompasses a broader scope of 
scientific inquiry associated technologies than when genomics was initially considered. A genome is the sum total of 
all an individual organism's genes. Thus, genomics is the study of all the genes of a cell, or tissue, at the DNA 
(genotype), mRNA (transcriptome), or protein (proteome) levels." 

History 

Genomics was established by Fred Sanger when he first sequenced the complete genomes of a virus and a 
mitochondrion. His group established techniques of sequencing, genome mapping, data storage, and bioinformatic 
analyses in the 1970-1980s. A major branch of genomics is still concerned with sequencing the genomes of various 
organisms, but the knowledge of full genomes has created the possibility for the field of functional genomics, mainly 
concerned with patterns of gene expression during various conditions. The most important tools here are microarrays 
and bioinformatics. Study of the full set of proteins in a cell type or tissue, and the changes during various 
conditions, is called proteomics. A related concept is materiomics, which is defined as the study of the material 
properties of biological materials (e.g. hierarchical protein structures and materials, mineralized biological tissues, 
etc.) and their effect on the macroscopic function and failure in their biological context, linking processes, structure 
and properties at multiple scales through a materials science approach. The actual term 'genomics' is thought to have 
been coined by Dr. Tom Roderick, a geneticist at the Jackson Laboratory (Bar Harbor, ME) over beer at a meeting 
held in Maryland on the mapping of the human genome in 1986. 

In 1972, Walter Fiers and his team at the Laboratory of Molecular Biology of the University of Ghent (Ghent, 
Belgium) were the first to determine the sequence of a gene: the gene for Bacteriophage MS2 coat protein. In 
1976, the team determined the complete nucleotide-sequence of bacteriophage MS2-RNA. The first DNA-based 
genome to be sequenced in its entirety was that of bacteriophage 0-X174; (5,368 bp), sequenced by Frederick 
Sanger in 1977. [4] 

The first free-living organism to be sequenced was that of Haemophilus influenzae (1.8 Mb) in 1995, and since then 
genomes are being sequenced at a rapid pace. 

As of September 2007, the complete sequence was known of about 1879 viruses , 577 bacterial species and 
roughly 23 eukaryote organisms, of which about half are fungi. Most of the bacteria whose genomes have been 
completely sequenced are problematic disease-causing agents, such as Haemophilus influenzae. Of the other 
sequenced species, most were chosen because they were well-studied model organisms or promised to become good 
models. Yeast {Saccharomyces cerevisiae) has long been an important model organism for the eukaryotic cell, while 
the fruit fly Drosophila melanogaster has been a very important tool (notably in early pre-molecular genetics). The 
worm Caenorhabditis elegans is an often used simple model for multicellular organisms. The zebrafish Brachydanio 
rerio is used for many developmental studies on the molecular level and the flower Arabidopsis thaliana is a model 
organism for flowering plants. The Japanese pufferfish (Takifugu rubripes) and the spotted green pufferfish 
(Tetraodon nigroviridis) are interesting because of their small and compact genomes, containing very little 
non-coding DNA compared to most species. The mammals dog (Canis familiaris), brown rat (Rattus 



Genomics 159 

norvegicus), mouse (Mus musculus), and chimpanzee {Pan troglodytes) are all important model animals in medical 
research. 

Human genomics 

A rough draft of the human genome was completed by the Human Genome Project in early 2001, creating much 
fanfare. By 2007 the human sequence was declared "finished" (less than one error in 20,000 bases and all 
chromosomes assembled. Display of the results of the project required significant bioinformatics resources. The 
sequence of the human reference assembly can be explored using the UCSC Genome Browser. 

Bacteriophage genomics 

Bacteriophages have played and continue to play a key role in bacterial genetics and molecular biology. Historically, 
they were used to define gene structure and gene regulation. Also the first genome to be sequenced was a 
bacteriophage. However, bacteriophage research did not lead the genomics revolution, which is clearly dominated by 
bacterial genomics. Only very recently has the study of bacteriophage genomes become prominent, thereby enabling 
researchers to understand the mechanisms underlying phage evolution. Bacteriophage genome sequences can be 
obtained through direct sequencing of isolated bacteriophages, but can also be derived as part of microbial genomes. 
Analysis of bacterial genomes has shown that a substantial amount of microbial DNA consists of prophage 
sequences and prophage-like elements. A detailed database mining of these sequences offers insights into the role of 
prophages in shaping the bacterial genome. 

Cyanobacteria genomics 

At present there are 24 cyanobacteria for which a total genome sequence is available. 15 of these cyanobacteria come 
from the marine environment. These are six Prochlorococcus strains, seven marine Synechococcus strains, 
Trichodesmium erythraeum IMS 101 and Crocosphaera watsonii WH8501. Several studies have demonstrated how 
these sequences could be used very successfully to infer important ecological and physiological characteristics of 
marine cyanobacteria. However, there are many more genome projects currently in progress, amongst those there are 
further Prochlorococcus and marine Synechococcus isolates, Acaryochloris and Prochloron, the N -fixing 
filamentous cyanobacteria Nodularia spumigena, Lyngbya aestuarii and Lyngbya majuscula, as well as 
bacteriophages infecting marine cyanobaceria. Thus, the growing body of genome information can also be tapped in 
a more general way to address global problems by applying a comparative approach. Some new and exciting 
examples of progress in this field are the identification of genes for regulatory RNAs, insights into the evolutionary 
origin of photosynthesis, or estimation of the contribution of horizontal gene transfer to the genomes that have been 
analyzed. 

See also 

Full Genome Sequencing 
Computational genomics 
Nitrogenomics 
Metagenomics 
Predictive Medicine 
Personal genomics 
Psychogenomics 



Genomics 160 

External links 

ri2i 

• Genomics Directory : A one-stop biotechnology resource center for bioentrepreneurs, scientists, and students 

• Annual Review of Genomics and Human Genetics 

ri4i 

• BMC Genomics : A BMC journal on Genomics 

• Genomics : UK companies and laboratories* Genomics journal 

ri7i 

• Genomics.org : An openfree wiki based Genomics portal 

n si 

• NHGRI : US government's genome institute 

ri9i 

• Pharmacogenomics in Drug Discovery and Development , a book on pharmacogenomics, diseases, 
personalized medicine, and therapeutics 

• Tishchenko P. D. Genomics: New Science in the New Cultural Situation 

T211 

• Undergraduate program on Genomic Sciences (Spanish) : One of the first undergraduate programs in the world 

[221 

• JCVI Comprehensive Microbial Resource 

T231 

• Pathema: A Clade Specific Bioinformatics Resource Center 

[241 

• KoreaGenome.org : The first Korean Genome published and the sequence is available freely. 

T251 

• GenomicsNetwork : Looks at the development and use of the science and technologies of genomics. 

• Institute for Genome Science : Genomics research. 

References 

[I] EPA Interim Genomics Policy (http://epa.gov/osa/spc/pdfs/genomics.pdf) 

[2] Min Jou W, Haegeman G, Ysebaert M, Fiers W (1972). "Nucleotide sequence of the gene coding for the bacteriophage MS2 coat protein". 

Nature lil (5350): 82-88. doi:10.1038/237082aO. PMID 4555447. 
[3] Fiers W, Contreras R, Duerinck F, Haegeman G, Iserentant D, Merregaert J, Min Jou W, Molemans F, Raeymaekers A, Van den Berghe A, 

Volckaert G, Ysebaert M (1976). "Complete nucleotide sequence of bacteriophage MS2 RNA: primary and secondary structure of the 

replicase gene". Nature 260 (5551): 500-507. doi:10.1038/260500a0. PMID 1264203. 
[4] Sanger F, Air GM, Barrell BG, Brown NL, Coulson AR, Fiddes CA, Hutchison CA, Slocombe PM, Smith M (1977). "Nucleotide sequence of 

bacteriophage phi X174 DNA". Nature 265 (5596): 687-695. doi:10.1038/265687a0. PMID 870828. 
[5] The Viral Genomes Resource, NCBI Friday, 14 September 2007 (http://www.ncbi.nlm.nih.gov/genomes/VIRUSES/virostat.html) 
[6] Genome Project Statistic, NCBI Friday, 14 September 2007 (http://www.ncbi.nlm.nih.gov/genomes/static/gpstat.html) 
[7] BBC article Human gene number slashed from Wednesday, 20 October 2004 (http://news.bbc.co.Uk/l/hi/sci/tech/3760766.stm) 
[8] CBSE News, Thursday, 16 October 2003 (http://www.cbse.ucsc.edu/news/2003/10/16/pufferfish_fruitfly/index.shtml) 
[9] NHGRI, pressrelease of the publishing of the dog genome (http://www.genome.gov/12511476) 
[10] McGrath S and van Sinderen D, ed (2007). Bacteriophage: Genetics and Molecular Biology (http://www.horizonpress.com/phage) (1st 

ed.). Caister Academic Press. ISBN 978-1-904455-14-1. . 

[II] Herrero A and Flores E, ed (2008). The Cyanobacteria: Molecular Biology, Genomics and Evolution (http://www.horizonpress.com/ 
cyan) (1st ed.). Caister Academic Press. ISBN 978-1-904455-15-8. . 

[12] http://www.genomicsdirectory.com 

[13] http://arjournals.annualreviews.org/loi/genom/ 

[14] http://www.biomedcentral.com/bmcgenomics/ 

[15] http://www.genomics.co.uk/companylist.php 

[16] http://www.elsevier.eom/wps/find/journaldescription.cws_home/622838/description#description 

[17] http://genomics.org 

[18] http://www.genome.gov/ 

[19] http://www.springer.com/humana+press/pharmacology+and+toxicology/book/978- 1-58829-887-4 

[20] http://www.zpu-journal.ru/ en/articles/detail. php?ID=342 

[21] http://www.lcg.unam.mx/ 

[22] http://cmr.jcvi.org/ 

[23] http://pathema.jcvi.org/ 

[24] http://koreagenome.org 

[25] http://genomicsnetwork.ac.uk 

[26] http://www.igs.umaryland.edu/research_topics.php 



Interactomics 161 



Interactomics 



Interactomics is a discipline at the intersection of bioinformatics and biology that deals with studying both the 
interactions and the consequences of those interactions between and among proteins, and other molecules within a 
cell . The network of all such interactions is called the Interactome. Interactomics thus aims to compare such 
networks of interactions (i.e., interactomes) between and within species in order to find how the traits of such 
networks are either preserved or varied. From a mathematical, or mathematical biology viewpoint an interactome 
network is a graph or a category representing the most important interactions pertinent to the normal physiological 
functions of a cell or organism. 

Interactomics is an example of "top-down" systems biology, which takes an overhead, as well as overall, view of a 
biosystem or organism. Large sets of genome-wide and proteomic data are collected, and correlations between 
different molecules are inferred. From the data new hypotheses are formulated about feedbacks between these 
molecules. These hypotheses can then be tested by new experiments . 

Through the study of the interaction of all of the molecules in a cell the field looks to gain a deeper understanding of 
genome function and evolution than just examining an individual genome in isolation . Interactomics goes beyond 
cellular proteomics in that it not only attempts to characterize the interaction between proteins, but between all 
molecules in the cell. 

Methods of interactomics 

The study of the interactome requires the collection of large amounts of data by way of high throughput experiments. 
Through these experiments a large number of data points are collected from a single organism under a small number 
of perturbations These experiments include: 

• Two-hybrid screening 

• Tandem Affinity Purification 

• X-ray tomography 

• Optical fluorescence microscopy 



Interactomics 



162 



Recent developments 

The field of interactomics is currently rapidly expanding and developing. While no biological interactomes have 
been fully characterized. Over 90% of proteins in Saccharomyces cerevisiae have been screened and their 
interactions characterized, making it the first interactome to be nearly fully specified 

Also there have been recent systematic attempts to explore the human interactome and 




Other species whose interactomes have been studied in some detail include Caenorhabditis elegans and Drosophila 
melanogaster. 



Criticisms and concerns 

Kiemer and Cesareni raise the following concerns with the current state of the field: 

• The experimental procedures associated with the field are error prone leading to "noisy results". This leads to 
30% of all reported interactions being artifacts. In fact, two groups using the same techniques on the same 
organism found less than 30% interactions in common. 

• Techniques may be biased, i.e. the technique determines which interactions are found. 

• Ineractomes are not nearly complete with perhaps the exception of S. cerivisiae. 

• While genomes are stable, interactomes may vary between tissues and developmental stages. 

• Genomics compares amino acids, and nucleotides which are in a sense unchangeable, but interactomics compares 
proteins and other molecules which are subject to mutation and evolution. 

• It is difficult to match evolutionary related proteins in distantly related species. 



Interactomics 163 

See also 

Interaction network 
Proteomics 
Metabolic network 
Metabolic network modelling 
Metabolic pathway 
Genomics 

Mathematical biology 
Systems biology 

External links 

• Interactomics.org . A dedicated interactomics web site operated under BioLicense. 

• Interactome.org . An interactome wiki site. 

171 

• PSIbase Structural Interactome Map of all Proteins. 

ro] 

• Omics.org . An omics portal site that is openfree (under BioLicense) 

• Genomics.org . A Genomics wiki site. 

• Comparative Interactomics analysis of protein family interaction networks using PSIMAP (protein structural 
ran 

interactome map) 

• Interaction interfaces in proteins via the Voronoi diagram of atoms 

• Using convex hulls to extract interaction interfaces from known structures. Panos Dafas, Dan Bolser, Jacek 
Gomoluch, Jong Park, and Michael Schroeder. Bioinformatics 2004 20: 1486-1490. 

• PSIbase: a database of Protein Structural Interactome map (PSIMAP). Sungsam Gong, Giseok Yoon, Insoo Jang 
Bioinformatics 2005. 

• Mapping Protein Family Interactions : Intramolecular and Intermolecular Protein Family Interaction Repertoires 
in the PDB and Yeast, Jong Park, Michael Lappe & Sarah A. TeichmannJ.M.B (2001). 

• Semantic Systems Biology 

References 

[1] Kiemer, L; G Cesareni (2007). "Comparative interactomics: comparing apples and pears?". TRENDS in Biochemistry 25: 448—454. 

doi:10.1016/j.tibtech.2007.08.002. 
[2] Bruggeman, F J; H V Westerhoff (2006). "The nature of systems biology". TRENDS in Microbiology 15: 45—50. 

doi:10.1016/j.tim.2006.11.003. 
[3] Krogan, NJ; et al. (2006). "Global landscape of protein complexes in the yeast Saccharomyeses Cerivisiae ". Nature 440: 637—643. 

doi: 10.1038/nature04670. 
[4] further citation needed 
[5] http://interactomics.org 
[6] http://interactome.org 
[7] http://psibase.kobic.re.kr 
[8] http://omics.org 

[9] http://bioinformatics.oxfordjournals.org/cgi/content/full/21/15/3234 
[10] http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TYR-4KXVD30-2&_user=10&_coverDate=ll%2F30%2F2006& 

_rdoc=l&_fmt=&_orig=search&_sort=d&view=c&_acct=C000050221&_version=l&_urlVersion=0&_userid=10& 

md5=8361bf3fe7834b4642cdda3b979de8bb 



164 



Chaotic Dynamics 



Butterfly effect 



The butterfly effect is a metaphor that 
encapsulates the concept of sensitive 
dependence on initial conditions in chaos 
theory; namely that small differences in the 
initial condition of a dynamical system may 
produce large variations in the long term 
behavior of the system. Although this may 
appear to be an esoteric and unusual 
behavior, it is exhibited by very simple 
systems: for example, a ball placed at the 
crest of a hill might roll into any of several 
valleys depending on slight differences in 
initial position. The butterfly effect is a 
common trope in fiction when presenting 
scenarios involving time travel and with 
"what if" scenarios where one storyline 
diverges at the moment of a seemingly 
minor event resulting in two significantly 
different outcomes. 



Sensitive dependency 
__ on initial conditions 

yt 



attractor D 




attractor B 

• 



Key: Blue squares represent initial states; 
black circles represent equilibria 

Point attractors in 2D phase space. 



►X 



Theory 

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on 
initial conditions are the two main ingredients for chaotic motion. They have the practical consequence of making 
complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case 
of weather), since it is impossible to measure the starting atmospheric conditions completely accurately. 



Origin of the concept and the term 

The term "butterfly effect" itself is related to the work of Edward Lorenz, and is based in chaos theory and sensitive 
dependence on initial conditions, already described in the literature in a particular case of the three-body problem by 
Henri Poincare in 1890 . He even later proposed that such phenomena could be common, say in meteorology. In 
1898 Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature, and Pierre 
Duhem discussed the possible general significance of this in 1908 . The idea that one butterfly could eventually 
have a far-reaching ripple effect on subsequent historic events seems first to have appeared in a 1952 short story by 
Ray Bradbury about time travel (see Literature and print here) although Lorenz made the term popular. In 1961, 
Lorenz was using a numerical computer model to rerun a weather prediction, when, as a shortcut on a number in the 
sequence, he entered the decimal .506 instead of entering the full .506127 the computer would hold. The result was a 
completely different weather scenario. Lorenz published his findings in a 1963 paper for the New York Academy 
of Sciences noting that "One meteorologist remarked that if the theory were correct, one flap of a seagull's wings 



Butterfly effect 



165 



could change the course of weather forever." Later speeches and papers by Lorenz used the more poetic butterfly. 
According to Lorenz, upon failing to provide a title for a talk he was to present at the 139th meeting of the American 
Association for the Advancement of Science in 1972, Philip Merilees concocted Does the flap of a butterfly's wings 
in Brazil set off a tornado in Texas? as a title. Although a butterfly flapping its wings has remained constant in the 
expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have 
varied widely. 

The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately 
alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in a certain location. The 
flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading 
to large-scale alterations of events (compare: domino effect). Had the butterfly not flapped its wings, the trajectory of 
the system might have been vastly different. While the butterfly does not "cause" the tornado in the sense of 
providing the energy for the tornado, it does "cause" it in the sense that the flap of its wings is an essential part of the 
initial conditions resulting in a tornado, and without that flap that particular tornado would not have existed. 



Illustration 



The butterfly effect in the Lorenz attractor 

time < t < 30 (larger) 



Z coordinate (larger) 





These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of 
time in the Lorenz attractor starting at two initial points that differ only by 10~ in the x-coordinate. Initially, the two trajectories seem coincident, as 
indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of 
the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at f=30. 



A Java animation of the Lorenz attractor shows the continuous evolution 



Butterfly effect 166 

Mathematical definition 

A dynamical system with evolution map f t displays sensitive dependence on initial conditions if points arbitrarily 

close become separate with increasing t. If M is the state space for the map f t , then f t displays sensitive 

dependence to initial conditions if there is a 6>0 such that for every point x€M and any neighborhood N containing x 
there exist a point y from that neighborhood N and a time t such that the distance 

d(r(x),r( y ))>6. 

The definition does not require that all points from a neighborhood separate from the base point x. 

Examples in semiclassical and quantum physics 

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of 
cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem. 
Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure 
quantum treatments; however, the sensitive dependence on initial conditions demonstrated in classical motion 

is included in the semiclassical treatments developed by Martin Gutzwiller and Delos and co-workers. 

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the 
time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of 

ri3i 

sensitivity of quantum systems to small changes in their given Hamiltonians. Poulin et al. present a quantum 
algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected 
to slightly different dynamics." They consider fidelity decay to be "the closest quantum analog to the (purely 
classical) butterfly effect." Whereas the classical butterfly effect considers the effect of a small change in the 
position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect 
of a small change in the Hamiltonian system with a given initial position and velocity. This quantum butterfly 

effect has been demonstrated experimentally. Quantum and semiclassical treatments of system sensitivity to 
initial conditions are known as quantum chaos. 

See also 

Avalanche effect 
Behavioral Cusp 
Black swan theory 
Cascading failure 
Causality 
Chain reaction 
Determinism 
Domino effect 
Dynamical systems 
Fractal 
Snowball effect 



Butterfly effect 167 

Further reading 

• Robert L. Devaney (2003). Introduction to Chaotic Dynamical Systems. Westview Press. ISBN 0-8133-4085-3. 

• Robert C. Hilborn (2004). "Sea gulls, butterflies, and grasshoppers: A brief history of the butterfly effect in 
nonlinear dynamics". American Journal of Physics 72: 425—427. doi: 10.1 1 19/1.1636492. 

External links 

ri9i 

• The meaning of the butterfly: Why pop culture loves the 'butterfly effect,' and gets it totally wrong , Peter 



Dizikes, Boston Globe, June 8, 2008 

(Cornell University) 

[21] 



From butterfly wings to single e-mail (Cornell University) 



New England Complex Systems Institute - Concepts: Butterfly Effect 

[221 
The Chaos Hypertextbook . An introductory primer on chaos 

Weisstein, Eric W., "Butterfly Effect [23] " from Math World. 



References 

[I] Some Historical Notes: History of Chaos Theory (http://www.wolframscience.com/reference/notes/971c) 
[2] Some Historical Notes: History of Chaos Theory (http://www.wolframscience.com/reference/notes/971c) 
[3] Some Historical Notes: History of Chaos Theory (http://www.wolframscience.com/reference/notes/971c) 

[4] Mathis, Nancy: "Storm Warning: The Story of a Killer Tornado", page x. Touchstone, 2007. ISBN 0-7432-8053-2 (http://books. google. 

com/books?id=tIY7o2g3aHkC&dq=isbn:0743280539) 
[5] "Butterfly Effects - Variations on a Meme" (http://clearnightsky.com/node/428). clearnightsky.com (http://www.clearnightsky.com). . 
[6] http://to-campos.planetaclix.pt/fractal/lorenz_eng.html 

[7] Postmodern Quantum Mechanics, EJ Heller, S Tomsovic, Physics Today, July 1993 

[8] Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (1990) Springer- Verlag, New York ISBN=0-387-97173-4. 
[9] What is... Quantum Chaos (http://www.ams.org/notices/200801/tx080100032p.pdf) by Ze'ev Rudnick (January 2008, Notices of the 

American Mathematical Society) 
[10] Quantum chaology, not quantum chaos, Michael Berry, 1989, Phys. Scr., 40, 335-336 doi: 10.1088/0031-8949/40/3/013. 

[II] Martin C. Gutzwiller (1971). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical Physics 12: 343. 
doi: 10.1063/1.1665596 

[12] Closed-orbit theory of oscillations in atomic photoabsorption cross sections in a strong electric field. II. Derivation of formulas, J Gao and 

JB Delos, Phys. Rev. A 46, 1455 - 1467 (1992) 
[13] Quantum Chaotic Environments, the Butterfly Effect, and Decoherence, Zbyszek P. Karkuszewski, Christopher Jarzynski, and Wojciech H. 

Zurek, Physical Review Letters VOLUME 89, NUMBER 17 (2002). 
[14] Exponential speed-up with a single bit of quantum information: Testing the quantum butterfly effect, David Poulin, Robin Blume-Kohout, 

Raymond Laflamme, and Harold Ollivier http://arxiv.org/PS_cache/quant-ph/pdf/0310/0310038vl.pdf 
[15] A Rough Guide to Quantum Chaos, David Poulin, http://www.iqc.ca/publications/tutorials/chaos.pdf 
[16] A. Peres, Quantum Theory: Concepts and Methods -Kluwer Academic, Dordrecht, 1995. 
[17] Quantum amplifier: Measurement with entangled spins, Jae-Seung Lee and A. K. Khitrin, JOURNAL OF CHEMICAL PHYSICS 

VOLUME 121, NUMBER 9 (2004) 
[18] A Rough Guide to Quantum Chaos, David Poulin, http://www.iqc.ca/publications/tutorials/chaos.pdf 
[19] http://www.boston.com/bostonglobe/ideas/articles/2008/06/08/the_meaning_of_the_butterfly/?page=full 
[20] http://www.news.cornell.edu/releases/Feb04/AAAS.Kleinberg.ws.html 
[21] http://necsi.org/guide/concepts/butterflyeffect.html 
[22] http://hypertextbook.com/chaos/ 
[23] http://mathworld.wolfram.com/ButterflyEffect.html 



Chaos theory 



168 



Chaos theory 



Chaos theory is a field of study in 
mathematics, physics, and philosophy 
studying the behavior of dynamical systems 
that are highly sensitive to initial conditions. 
This sensitivity is popularly referred to as 
the butterfly effect. Small differences in 
initial conditions (such as those due to 
rounding errors in numerical computation) 
yield widely diverging outcomes for chaotic 
systems, rendering long-term prediction 
impossible in general. This happens even 
though these systems are deterministic, 
meaning that their future behaviour is fully 

determined by their initial conditions, with 

121 
no random elements involved. In other 

words, the deterministic nature of these 

systems does not make them predictable. 

This behavior is known as deterministic 

chaos, or simply chaos. 

Chaotic behavior can be observed in many 

mi 

natural systems, such as the weather. Explanation of such behavior may be sought through analysis of a chaotic 
mathematical model, or through analytical techniques such as recurrence plots and Poincare maps. 




A plot of the Lorenz attractor for values r = 28, o 



Applications 

Chaos theory is applied in many scientific disciplines: mathematics, programming, microbiology, biology, computer 
science, economics, engineering, finance, philosophy, physics, politics, population dynamics, 

psychology, and robotics. 

Chaotic behavior has been observed in the laboratory in a variety of systems including electrical circuits, lasers, 

oscillating chemical reactions, fluid dynamics, and mechanical and magneto-mechanical devices, as well as 

141 
computer models of chaotic processes. Observations of chaotic behavior in nature include changes in weather, the 

dynamics of satellites in the solar system, the time evolution of the magnetic field of celestial bodies, population 

growth in ecology, the dynamics of the action potentials in neurons, and molecular vibrations. There is some 

controversy over the existence of chaotic dynamics in plate tectonics and in economics. 

One of the most successful applications of chaos theory has been in ecology, where dynamical systems such as the 
Ricker model have been used to show how population growth under density dependence can lead to chaotic 
dynamics. 

Chaos theory is also currently being applied to medical studies of epilepsy, specifically to the prediction of 
seemingly random seizures by observing initial conditions. 

A related field of physics called quantum chaos theory investigates the relationship between chaos and quantum 
mechanics. The correspondence principle states that classical mechanics is a special case of quantum mechanics, the 
classical limit. If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, it is 
unclear how exponential sensitivity to initial conditions can arise in practice in classical chaos. Recently, another 



Chaos theory 



169 



ri7i 

field, called relativistic chaos, has emerged to describe systems that follow the laws of general relativity. 

The initial conditions of three or more bodies interacting through gravitational attraction (see the n-body problem) 
can be arranged to produce chaotic motion. 



Chaotic dynamics 



„., [18] 




The map defined by x — > 4 x (1 — x) and y — > x + y if x 

+ y <l (x + y— I otherwise) displays sensitivity to 

initial conditions. Here two series of x and y values 

diverge markedly over time from a tiny initial 

difference. 



In common usage, "chaos" means "a state of disorder", but the 
adjective "chaotic" is defined more precisely in chaos theory. 
Although there is no universally accepted mathematical definition 
of chaos, a commonly-used definition says that, for a dynamical 
system to be classified as chaotic, it must have the following 

-*■ [19] 

properties: 

1 . it must be sensitive to initial conditions, 

2. it must be topologically mixing, and 

3. its periodic orbits must be dense. 

Sensitivity to initial conditions 

Sensitivity to initial conditions means that each point in such a 
system is arbitrarily closely approximated by other points with 
significantly different future trajectories. Thus, an arbitrarily small 
perturbation of the current trajectory may lead to significantly 
different future behaviour. However, it has been shown that the 
last two properties in the list above actually imply sensitivity to 
initial conditions and if attention is restricted to intervals, 

[22] 

the second property implies the other two (an alternative, and in general weaker, definition of chaos uses only the 

[23] 

first two properties in the above list ). It is interesting that the most practically significant condition, that of 
sensitivity to initial conditions, is actually redundant in the definition, being implied by two (or for intervals, one) 
purely topological conditions, which are therefore of greater interest to mathematicians. 

Sensitivity to initial conditions is popularly known as the "butterfly effect," so called because of the title of a paper 
given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C. 
entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas? The flapping wing 
represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale 
phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different 
(even the evolution of simple discrete systems, such as cellular automata, can heavily depend on initial conditions, 
and Stephen Wolfram has investigated a cellular automaton with this property, termed by him rule 30). 

A consequence of sensitivity to initial conditions is that if we start with only a finite amount of information about the 
system (as is usually the case in practice), then beyond a certain time the system will no longer be predictable. This 

[241 

is most familiar in the case of weather, which is generally predictable only about a week ahead. 

The Lyapunov exponent characterises the extent of the sensitivity to initial conditions. Quantitatively, two 
trajectories in phase space with initial separation 5Zo diverge 

|<5Z(i)| ^e Ai |<5Z | 
where X is the Lyapunov exponent. The rate of separation can be different for different orientations of the initial 
separation vector. Thus, there is a whole spectrum of Lyapunov exponents — the number of them is equal to the 
number of dimensions of the phase space. It is common to just refer to the largest one, i.e. to the Maximal Lyapunov 
exponent (MLE), because it determines the overall predictability of the system. A positive MLE is usually taken as 



Chaos theory 



170 



an indication that the system is chaotic. 




Topological mixing 

Topological mixing (or topological transitivity) means that the 
system will evolve over time so that any given region or open set 
of its phase space will eventually overlap with any other given 
region. This mathematical concept of "mixing" corresponds to the 
standard intuition, and the mixing of colored dyes or fluids is an 
example of a chaotic system. 

Topological mixing is often omitted from popular accounts of 
chaos, which equate chaos with sensitivity to initial conditions. 
However, sensitive dependence on initial conditions alone does 
not give chaos. For example, consider the simple dynamical 
system produced by repeatedly doubling an initial value. This 
system has sensitive dependence on initial conditions everywhere, 
since any pair of nearby points will eventually become widely 
separated. However, this example has no topological mixing, and 
therefore has no chaos. Indeed, it has extremely simple behaviour: 
all points except tend to infinity. 

Density of periodic orbits 

Density of periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits. 

Topologically mixing systems failing this condition may not display sensitivity to initial conditions, and hence may 

not be chaotic. For example, an irrational rotation of the circle is topologically transitive, but does not have dense 

[251 
periodic orbits, and hence does not have sensitive dependence on initial conditions. The one-dimensional logistic 

map defined by x — > 4 x (1 — x) is one of the simplest systems with density of periodic orbits. For example, 

0.3454915 -» 0.9045085 — > 0.3454915 is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, 

etc. (indeed, for all the periods specified by Sharkovskii's theorem). 

T271 
Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any one-dimensional system which 

exhibits a regular cycle of period three will also display regular cycles of every other length as well as completely 

chaotic orbits. 



The map defined by x — > 4 ;t (1 — x) and y — > x + y if X 
+ y < 1 (x + y — 1 otherwise) also displays topological 

mixing. Here the blue region is transformed by the 

dynamics first to the purple region, then to the pink and 

red regions, and eventually to a cloud of points 

scattered across the space. 



Chaos theory 



171 



Strange Attractors 

Some dynamical systems, like the 
one-dimensional logistic map defined by x 
— > 4 x (1 — x), are chaotic everywhere, but 
in many cases chaotic behaviour is found 
only in a subset of phase space. The cases of 
most interest arise when the chaotic 
behaviour takes place on an attractor, since 
then a large set of initial conditions will lead 
to orbits that converge to this chaotic region. 

An easy way to visualize a chaotic attractor 

is to start with a point in the basin of 

attraction of the attractor, and then simply 

plot its subsequent orbit. Because of the 

topological transitivity condition, this is 

likely to produce a picture of the entire final 

attractor, and indeed both orbits shown in 

the figure on the right give a picture of the 

general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz 

weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it 

was not only one of the first, but it is also one of the most complex and as such gives rise to a very interesting pattern 

which looks like the wings of a butterfly. 

Unlike fixed-point attractors and limit cycles, the attractors which arise from chaotic systems, known as strange 
attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as 
the Lorenz system) and in some discrete systems (such as the Henon map). Other discrete dynamical systems have a 
repelling structure called a Julia set which forms at the boundary between basins of attraction of fixed points — Julia 
sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and 
a fractal dimension can be calculated for them. 




The Lorenz attractor displays chaotic behavior. These two plots demonstrate 

sensitive dependence on initial conditions within the region of phase space 

occupied by the attractor. 



Chaos theory 



172 




Minimum complexity of a chaotic system 

Discrete chaotic systems, such as the 
logistic map, can exhibit strange attractors 
whatever their dimensionality. However, the 
Poincare-Bendixson theorem shows that a 
strange attractor can only arise in a 
continuous dynamical system (specified by 
differential equations) if it has three or more 
dimensions. Finite dimensional linear 
systems are never chaotic; for a dynamical 
system to display chaotic behaviour it has to 
be either nonlinear, or infinite-dimensional. 

The Poincare-Bendixson theorem states that 

a two dimensional differential equation has 

very regular behavior. The Lorenz attractor 

discussed above is generated by a system of 

three differential equations with a total of 

seven terms on the right hand side, five of 

which are linear terms and two of which are quadratic (and therefore nonlinear). Another well-known chaotic 

attractor is generated by the Rossler equations with seven terms on the right hand side, only one of which is 

ro on 

(quadratic) nonlinear. Sprott found a three dimensional system with just five terms on the right hand side, and 

[291 [30] 

with just one quadratic nonlinearity, which exhibits chaos for certain parameter values. Zhang and Heidel 
showed that, at least for dissipative and conservative quadratic systems, three dimensional quadratic systems with 
only three or four terms on the right hand side cannot exhibit chaotic behavior. The reason is, simply put, that 
solutions to such systems are asymptotic to a two dimensional surface and therefore solutions are well behaved. 

While the Poincare-Bendixson theorem means that a continuous dynamical system on the Euclidean plane cannot be 

chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behaviour. Perhaps 

T311 
surprisingly, chaos may occur also in linear systems, provided they are infinite-dimensional. A theory of linear 

chaos is being developed in the functional analysis, a branch of mathematical analysis. 



Bifurcation diagram of the logistic mapx — > r x (1 — x). Each vertical slice shows 

the attractor for a specific value of r. The diagram displays period-doubling as r 

increases, eventually producing chaos. 



History 



Chaos theory 



173 



The first discoverer of chaos was Henri Poincare. In the 1880s, while 

studying the three-body problem, he found that there can be orbits which are 

T321 
nonperiodic, and yet not forever increasing nor approaching a fixed point. 

[331 

In 1898 Jacques Hadamard published an influential study of the chaotic 

motion of a free particle gliding frictionlessly on a surface of constant 

negative curvature. In the system studied, "Hadamard's billiards," 

Hadamard was able to show that all trajectories are unstable in that all particle 

trajectories diverge exponentially from one another, with a positive Lyapunov 

exponent. 

Much of the earlier theory was developed almost entirely by mathematicians, 
under the name of ergodic theory. Later studies, also on the topic of nonlinear 
differential equations, were carried out by G.D. Birkhoff, A. N. 




[39] 



Barnsley fern created using the chaos 
game. Natural forms (ferns, clouds, 
mountains, etc.) may be recreated 
through an Iterated function system 
(IFS). 



Kolmogorov, L J L J L J M.L. Cartwright and J.E. Littlewood/ yi and 

Stephen Smale. Except for Smale, these studies were all directly inspired 

by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of 

Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had 

not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio 

circuits without the benefit of a theory to explain what they were seeing. 

Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after 
mid-century, when it first became evident for some scientists that linear theory, the prevailing system theory at that 
time, simply could not explain the observed behaviour of certain experiments like that of the logistic map. What had 
been beforehand excluded as measure imprecision and simple "noise" was considered by chaos theories as a full 
component of the studied systems. 

The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of 
chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by 
hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to 
visualize these systems. 

An early pioneer of the theory was Edward Lorenz whose interest in 

chaos came about accidentally through his work on weather prediction 

1411 
in 1961. Lorenz was using a simple digital computer, a Royal 

McBee LGP-30, to run his weather simulation. He wanted to see a 

sequence of data again and to save time he started the simulation in the 

middle of its course. He was able to do this by entering a printout of 

the data corresponding to conditions in the middle of his simulation 

which he had calculated last time. 




Turbulence in the tip vortex from an airplane 
wing. Studies of the critical point beyond which a 

system creates turbulence was important for 
Chaos theory, analyzed for example by the Soviet 

physicist Lev Landau who developed the 
Landau-Hopf theory of turbulence. David Ruelle 
and Floris Takens later predicted, against Landau, 

that fluid turbulence could develop through a 
strange attractor, a main concept of chaos theory. 



To his surprise the weather that the machine began to predict was 
completely different from the weather calculated before. Lorenz 
tracked this down to the computer printout. The computer worked with 
6-digit precision, but the printout rounded variables off to a 3-digit 
number, so a value like 0.506127 was printed as 0.506. This difference 
is tiny and the consensus at the time would have been that it should 
have had practically no effect. However Lorenz had discovered that 
small changes in initial conditions produced large changes in the 

1421 

long-term outcome. Lorenz's discovery, which gave its name to 



Lorenz attractors, proved that meteorology could not reasonably predict weather beyond a weekly period (at most). 



Chaos theory 174 

T431 
The year before, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. Beforehand, 

he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion 

of noise-containing periods to error-free periods was a constant — thus errors were inevitable and must be planned for 

T441 
by incorporating redundancy. Mandelbrot described both the "Noah effect" (in which sudden discontinuous 

changes can occur, e.g., in a stock's prices after bad news, thus challenging normal distribution theory in statistics, 

aka Bell Curve) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change 

afterwards). In 1967, he published "How long is the coast of Britain? Statistical self-similarity and fractional 

dimension," showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all 

T471 
scales, and is infinite in length for an infinitesimally small measuring device. Arguing that a ball of twine appears 

to be a point when viewed from far away (O-dimensional), a ball when viewed from fairly near (3-dimensional), or a 

curved strand (1 -dimensional), he argued that the dimensions of an object are relative to the observer and may be 

fractional. An object whose irregularity is constant over different scales ("self-similarity") is a fractal (for example, 

the Koch curve or "snowflake", which is infinitely long yet encloses a finite space and has fractal dimension equal to 

circa 1.2619, the Menger sponge and the Sierpihski gasket). In 1975 Mandelbrot published The Fractal Geometry of 

Nature, which became a classic of chaos theory. Biological systems such as the branching of the circulatory and 

bronchial systems proved to fit a fractal model. 

Chaos was observed by a number of experimenters before it was recognized; e.g., in 1927 by van der Pol and in 
1958 by R.L. Ives. However, as a graduate student in Chihiro Hayashi's laboratory at Kyoto University, 

Yoshisuke Ueda was experimenting with analog computers (that is, vacuum tubes) and noticed, on Nov. 27, 1961, 
what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, 
and did not allow him to report his findings until 1970. 

In December 1977 the New York Academy of Sciences organized the first symposium on Chaos, attended by David 
Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw (a physicist, 
part of the Eudaemons group with J. Doyne Farmer and Norman Packard who tried to find a mathematical method to 
beat roulette, and then created with them the Dynamical Systems Collective in Santa Cruz, California), and the 
meteorologist Edward Lorenz. 

The following year, Mitchell Feigenbaum published the noted article "Quantitative Universality for a Class of 

[531 
Nonlinear Transformations", where he described logistic maps. Feigenbaum had applied fractal geometry to the 

study of natural forms such as coastlines. Feigenbaum notably discovered the universality in chaos, permitting an 

application of chaos theory to many different phenomena. 

In 1979, Albert J. Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his 
experimental observation of the bifurcation cascade that leads to chaos and turbulence in convective 
Rayleigh— Benard systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum 
"for his brilliant experimental demonstration of the transition to turbulence and chaos in dynamical systems". 

Then in 1986 the New York Academy of Sciences co-organized with the National Institute of Mental Health and the 
Office of Naval Research the first important conference on Chaos in biology and medicine. There, Bernardo 
Huberman presented a mathematical model of the eye tracking disorder among schizophrenics. This led to a 
renewal of physiology in the 1980s through the application of chaos theory, for example in the study of pathological 
cardiac cycles. 

In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters describing for 
the first time self-organized criticality (SOC), considered to be one of the mechanisms by which complexity arises in 
nature. Alongside largely lab-based approaches such as the Bak— Tang— Wiesenfeld sandpile, many other 
investigations have centered around large-scale natural or social systems that are known (or suspected) to display 
scale-invariant behaviour. Although these approaches were not always welcomed (at least initially) by specialists in 
the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of 
natural phenomena, including: earthquakes (which, long before SOC was discovered, were known as a source of 



Chaos theory 175 

scale-invariant behaviour such as the Gutenberg— Richter law describing the statistical distribution of earthquake 

T571 
sizes, and the Omori law describing the frequency of aftershocks); solar flares; fluctuations in economic systems 

such as financial markets (references to SOC are common in econophysics); landscape formation; forest fires; 

landslides; epidemics; and biological evolution (where SOC has been invoked, for example, as the dynamical 

mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). 

Worryingly, given the implications of a scale-free distribution of event sizes, some researchers have suggested that 

another phenomenon that should be considered an example of SOC is the occurrence of wars. These "applied" 

investigations of SOC have included both attempts at modelling (either developing new models or adapting existing 

ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or 

characteristics of natural scaling laws. 

The same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced 
the general principles of chaos theory as well as its history to the broad public. At first the domain of work of a few, 
isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly 
under the name of nonlinear systems analysis. Alluding to Thomas Kuhn's concept of a paradigm shift exposed in 
The Structure of Scientific Revolutions (1962), many "chaologists" (as some self-nominated themselves) claimed that 
this new theory was an example of such a shift, a thesis upheld by J. Gleick. 

The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos 
theory continues to be a very active area of research, involving many different disciplines (mathematics, topology, 
physics, population biology, biology, meteorology, astrophysics, information theory, etc.). 

Chaos theory is employed in everyday life in some microwave ovens, to aid rapid and (most notably) evenly-spread 
defrosting using microwave energy; this function is known as Chaos Defrost and was first developed by Panasonic in 
2001. [58] 

Distinguishing random from chaotic data 

It can be difficult to tell from data whether a physical or other observed process is random or chaotic, because in 
practice no time series consists of pure 'signal.' There will always be some form of corrupting noise, even if it is 
present as round-off or truncation error. Thus any real time series, even if mostly deterministic, will contain some 
randomness. 

All methods for distinguishing deterministic and stochastic processes rely on the fact that a deterministic system 
always evolves in the same way from a given starting point. Thus, given a time series to test for determinism, 

one can: 

1. pick a test state; 

2. search the time series for a similar or 'nearby' state; and 

3. compare their respective time evolutions. 

Define the error as the difference between the time evolution of the 'test' state and the time evolution of the nearby 
state. A deterministic system will have an error that either remains small (stable, regular solution) or increases 
exponentially with time (chaos). A stochastic system will have a randomly distributed error. 

Essentially all measures of determinism taken from time series rely upon finding the closest states to a given 'test' 
state (i.e., correlation dimension, Lyapunov exponents, etc.). To define the state of a system one typically relies on 
phase space embedding methods. Typically one chooses an embedding dimension, and investigates the 
propagation of the error between two nearby states. If the error looks random, one increases the dimension. If you 
can increase the dimension to obtain a deterministic looking error, then you are done. Though it may sound simple it 
is not really. One complication is that as the dimension increases the search for a nearby state requires a lot more 
computation time and a lot of data (the amount of data required increases exponentially with embedding dimension) 
to find a suitably close candidate. If the embedding dimension (number of measures per state) is chosen too small 



Chaos theory 



176 



(less than the 'true' value) deterministic data can appear to be random but in theory there is no problem choosing the 
dimension too large — the method will work. 

When a non-linear deterministic system is attended by external fluctuations, its trajectories present serious and 
permanent distortions. Furthermore, the noise is amplified due to the inherent non-linearity and reveals totally new 
dynamical properties. Statistical tests attempting to separate noise from the deterministic skeleton or inversely isolate 
the deterministic part risk failure. Things become worse when the deterministic component is a non-linear feedback 
system. In presence of interactions between nonlinear deterministic components and noise, the resulting nonlinear 



series can display dynamics that traditional tests for nonlinearity are sometimes not able to capture 



[64] 



Cultural references 

Chaos theory has been mentioned in numerous novels and movies, such as Jurassic Park. 

See also 



Examples of chaotic systems 

Arnold's cat map 

Bouncing Ball Simulation 

System 

Chua's circuit 

Double pendulum 

Dynamical billiards 

Economic bubble 

Henon map 

Horseshoe map 

Logistic map 

Rossler attractor 

Standard map 

Swinging Atwood's machine 

Tilt A Whirl 

Coupled map lattice 

List of chaotic maps 



Other related topics 

Anosov diffeomorphism 

Bifurcation theory 

Butterfly effect 

Chaos theory in organizational development 

Complexity 

Control of chaos 

Edge of chaos 

Fractal 

• Julia set 

• Mandelbrot set 
Predictability 
Quantum chaos 
Santa Fe Institute 
Synchronization of chaos 



People 

• Mitchell Feigenbaum 

• Martin Gutzwiller 

• Michael Berry 

• Brosl Hasslacher 

• Michel Henon 

• Edward Lorenz 

• Ian Malcolm 

• Aleksandr Lyapunov 

• Benoit Mandelbrot 

• Henri Poincare 

• Otto Rossler 

• David Ruelle 

• Oleksandr Mikolaiovich 
Sharkovsky 

• Floris Takens 

• James A. Yorke 



Scientific literature 



Articles 

• A.N. Sharkovskii, "Co-existence of cycles of a continuous mapping of the line into itself", Ukrainian Math. J.. 
16:61-71 (1964) 

• Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." American Mathematical Monthly 82, 985-92, 1975. 



• Kolyada, S. F. "Li- Yorke sensitivity and other concepts of chaos 



[65],, 



Ukrainian Math. J. 56 (2004), 1242-1257. 



Textbooks 

• Alligood, K. T., Sauer, T., and Yorke, J. A. (1997). Chaos: an introduction to dynamical systems. Springer-Verlag 
New York, LLC. ISBN 0-387-94677-2. 

• Baker, G. L. (1996). Chaos, Scattering and Statistical Mechanics. Cambridge University Press. 
ISBN 0-521-39511-9. 



• Badii, R.; Politi A. (1997). "Complexity: hierarchical structures and scaling in physics 
University Press. ISBN 0521663857. 



„ [66] 



Cambridge 



Chaos theory 177 

• Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd ed,. Westview Press. 
ISBN 0-8133-4085-3. 

• Gollub, J. P.; Baker, G. L. (1996). Chaotic dynamics. Cambridge University Press. ISBN 0-521-47685-2. 

• Guckenheimer, J., and Holmes P. (1983). Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector 
Fields. Springer- Verlag New York, LLC. ISBN 0-387-90819-6. 

• Gutzwiller, Martin (1990). Chaos in Classical and Quantum Mechanics. Springer- Verlag New York, LLC. 
ISBN 0-387-97173-4. 

• Hoover, William Graham (1999,2001). Time Reversibility, Computer Simulation, and Chaos. World Scientific. 
ISBN 981-02-4073-2. 

• Kiel, L. Douglas; Elliott, Euel W. (1997). Chaos Theory in the Social Sciences. Perseus Publishing. 
ISBN 0-472-08472-0. 

• Moon, Francis (1990). Chaotic and Fractal Dynamics. Springer- Verlag New York, LLC. ISBN 0-471-54571-6. 

• Ott, Edward (2002). Chaos in Dynamical Systems. Cambridge University Press New, York. ISBN 0-521-01084-5. 

• Strogatz, Steven (2000). Nonlinear Dynamics and Chaos. Perseus Publishing. ISBN 0-7382-0453-6. 

• Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 0-19-850840-9. 

• Tel, Tamas; Gruiz, Marion (2006). Chaotic dynamics: An introduction based on classical mechanics. Cambridge 
University Press. ISBN 0-521-83912-2. 

• Tufillaro, Abbott, Reilly (1992). An experimental approach to nonlinear dynamics and chaos. Addison-Wesley 
New York. ISBN 0-201-55441-0. 

• Zaslavsky, George M. (2005). Hamiltonian Chaos and Fractional Dynamics. Oxford University Press. 
ISBN 0-198-52604-0. 

Semitechnical and popular works 

• Ralph H. Abraham and Yoshisuke Ueda (Ed.), The Chaos Avant-Garde: Memoirs of the Early Days of Chaos 
Theory, World Scientific Publishing Company, 2001, 232 pp. 
Michael Barnsley, Fractals Everywhere, Academic Press 1988, 394 pp. 

Richard J Bird, Chaos and Life: Complexity and Order in Evolution and Thought, Columbia University Press 
2003, 352 pp. 

John Briggs and David Peat, Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of 
Wholeness, Harper Perennial 1990, 224 pp. 

John Briggs and David Peat, Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change, Harper 
Perennial 2000, 224 pp. 

Lawrence A. Cunningham, From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient 
Capital Market Hypothesis, George Washington Law Review, Vol. 62, 1994, 546 pp. 

Leon Glass and Michael C. Mackey, From Clocks to Chaos: The Rhythms of Life, Princeton University Press 
1988, 272 pp. 

James Gleick, Chaos: Making a New Science, New York: Penguin, 1988. 368 pp. 
John Gribbin, Deep Simplicity, 

L Douglas Kiel, Euel W Elliott (ed.), Chaos Theory in the Social Sciences: Foundations and Applications , 
University of Michigan Press, 1997, 360 pp. 

Arvind Kumar, Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature , National 
Book Trust, 2003. 

Hans Lauwerier, Fractals, Princeton University Press, 1991. 
Edward Lorenz, The Essence of Chaos, University of Washington Press, 1996. 
Chapter 5 of Alan Marshall (2002) The Unity of nature, Imperial College Press: London 
Heinz-Otto Peitgen and Dietmar Saupe (Eds.), The Science of Fractal Images, Springer 1988, 312 pp. 



Chaos theory 178 

• Clifford A. Pickover, Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World , St Martins Pr 
1991. 

Ilya Prigogine and Isabelle Stengers, Order Out of Chaos, Bantam 1984. 

Heinz-Otto Peitgen and P. H. Richter, The Beauty of Fractals : Images of Complex Dynamical Systems, Springer 
1986,211pp. 

David Ruelle, Chance and Chaos, Princeton University Press 1993. 
Ivars Peterson, Newton's Clock: Chaos in the Solar System, Freeman, 1993. 
David Ruelle, Chaotic Evolution and Strange Attractors, Cambridge University Press, 1989. 
Peter Smith, Explaining Chaos, Cambridge University Press, 1998. 

Ian Stewart, Does God Play Dice?: The Mathematics of Chaos , Blackwell Publishers, 1990. 
Steven Strogatz, Sync: The emerging science of spontaneous order, Hyperion, 2003. 
Yoshisuke Ueda, The Road To Chaos, Aerial Pr, 1993. 

M. Mitchell Waldrop, Complexity : The Emerging Science at the Edge of Order and Chaos, Simon & Schuster, 
1992. 

External links 

Nonlinear Dynamics Research Group with Animations in Flash 
The Chaos group at the University of Maryland 

[221 

The Chaos Hypertextbook . An introductory primer on chaos and fractals. 
Society for Chaos Theory in Psychology & Life Sciences 

Nonlinear Dynamics Research Group at CSDC , Florence Italy 

[711 
Interactive live chaotic pendulum experiment , allows users to interact and sample data from a real working 

damped driven chaotic pendulum 

[721 
Nonlinear dynamics: how science comprehends chaos , talk presented by Sunny Auyang, 1998. 

[731 
Nonlinear Dynamics . Models of bifurcation and chaos by Elmer G Wiens 



[741 
Gleick's Chaos (excerpt) 



T751 

Systems Analysis, Modelling and Prediction Group at the University of Oxford. 



A page about the Mackey-Glass equation 

References 

[I] Stephen H. Kellert, In the Wake of Chaos: Unpredictable Order in Dynamical Systems, University of Chicago Press, 1993, p 32, ISBN 
0-226-42976-8. 

[2] Kellert, p. 56. 
[3] Kellert, p. 62. 
[4] Raymond Sneyers (1997) "Climate Chaotic Instability: Statistical Determination and Theoretical Background", Environmetrics, vol. 8, no. 5, 

pages 517—532. 
[5] Kyrtsou, C. and W. Labys, (2006). Evidence for chaotic dependence between US inflation and commodity prices, Journal of 

Macroeconomics, 28(1), pp. 256—266. 
[6] Kyrtsou, C. and W. Labys, (2007). Detecting positive feedback in multivariate time series: the case of metal prices and US inflation, Physica 

A, 377(1), pp. 227-229. 
[7] Kyrtsou, C, and Vorlow, C, (2005). Complex dynamics in macroeconomics: A novel approach, in New Trends in Macroeconomics, Diebolt, 

C, and Kyrtsou, C, (eds.), Springer Verlag. 
[8] Applying Chaos Theory to Embedded Applications (http://www.dspdesignline.com/ 

218101444 ;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=l) 
[9] Hristu-Varsakelis, D., and Kyrtsou, C, (2008): Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns, Discrete 

Dynamics in Nature and Society, Volume 2008, Article ID 138547, 7 pages, doi: 10. 1 155/2008/138547. 
[10] Kyrtsou, C. and M. Terraza, (2003). Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass 

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[II] Metaculture.net, metalinks: Applied Chaos (http://metalinks.metaculture.net/science/fractal/applications/default.asp), 2007. 



Chaos theory 179 

[12] Apostolos Serletis and Periklis Gogas, Purchasing Power Parity Nonlinearity and Chaos (http://www.informaworld.com/smpp/ 

content~content=a713761243~db=all~order=page), in: Applied Financial Economics, 10, 615—622, 2000. 
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v51yl997i4p359-385.html), in: Research in Economics, 51, 359-385, 1997. 
[15] Comdig.org, Complexity Digest 199.06 (http://www.comdig.org/index.php?id_issue=1999.06#194) 
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[17] A. E. Motter, Relativistic chaos is coordinate invariant (http://prola.aps.org/abstract/PRL/v91/i23/e231101), in: Phys. Rev. Lett. 91, 

231101 (2003). 
[18] Definition of chaos at Wiktionary. 
[19] Hasselblatt, Boris; Anatole Katok (2003). A First Course in Dynamics: With a Panorama of Recent Developments. Cambridge University 

Press. ISBN 0521587506. 
[20] Saber N. Elaydi, Discrete Chaos, Chapman & Hall/CRC, 1999, page 1 17, ISBN 1-58488-002-3. 
[21] William F. Basener, Topology and its applications, Wiley, 2006, page 42, ISBN 0-471-68755-3, 
[22] Michel Vellekoop; Raoul Berglund, "On Intervals, Transitivity = Chaos," The American Mathematical Monthly, Vol. 101, No. 4. (April, 

1994), pp. 353-355 (http://www.jstor.org/pss/2975629) 
[23] Alfredo Medio and Marji Lines, Nonlinear Dynamics: A Primer, Cambridge University Press, 2001, page 165, ISBN 0-521-55874-3. 
[24] Robert G. Watts, Global Warming and the Future of the Earth, Morgan & Claypool, 2007, page 17. 

[25] Devaney, Robert L. (2003). An Introduction to Chaotic Dynamical Systems, 2nd ed. Westview Press. ISBN 0-8133-4085-3. 
[26] Alligood, K. T., Sauer, T., and Yorke, J. A. (1997). Chaos: an introduction to dynamical systems. Springer- Verlag New York, LLC. 

ISBN 0-387-94677-2. 
[27] Li, T. Y. and Yorke, J. A. "Period Three Implies Chaos." American Mathematical Monthly 82, 985—92, 1975. (http://pb.math.univ.gda. 

pl/chaos/pdf/li- yorke.pdf) 
[28] Sprott.J.C. (1997). "Simplest dissipative chaotic flow". Physics Letters A 228: 271. doi:10.1016/S0375-9601(97)00088-l. 
[29] Fu, Z.; Heidel, J. (1997). "Non-chaotic behaviour in three-dimensional quadratic systems". Nonlinearity 10: 1289. 

doi:10.1088/0951-7715/10/5/014. 
[30] Heidel, J.; Fu, Z. (1999). "Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case". Nonlinearity 12: 617. 

doi:10.1088/0951-7715/12/3/012. 
[31] Bonet, J.; Marti'nez-Gimenez, F.; Peris, A. (2001). "A Banach space which admits no chaotic operator". Bulletin of the London Mathematical 

Society^: 196-198. doi:10.1112/blms/33.2.196. 
[32] Jules Henri Poincare (1890) "Sur le probleme des trois corps et les equations de la dynamique. Divergence des series de M. Lindstedt," Acta 

Mathematica, vol. 13, pages 1—270. 
[33] Florin Diacu and Philip Holmes (1996) Celestial Encounters: The Origins of Chaos and Stability, Princeton University Press. 
[34] Hadamard, Jacques (1898). "Les surfaces a courbures opposees et leurs lignes geodesiques". Journal de Mathematiques Pures et Appliquees 

4: pp. 27-73. 
[35] George D. Birkhoff, Dynamical Systems, vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: 

American Mathematical Society, 1927) 
[36] Kolmogorov, Andrey Nikolaevich (1941). "Local structure of turbulence in an incompressible fluid for very large Reynolds numbers". 

Doklady Akademii Nauk SSSR 30 (4): 301—305. Reprinted in: Proceedings of the Royal Society of London: Mathematical and Physical 

Sciences (Series A), vol. 434, pages 9-13 (1991). 
[37] Kolmogorov, A. N. (1941). "On degeneration of isotropic turbulence in an incompressible viscous liquid". Doklady Akademii Nauk SSSR 31 

(6): 538—540. Reprinted in: Proceedings of the Royal Society of London: Mathematical and Physical Sciences (Series A), vol. 434, pages 

15-17(1991). 
[38] Kolmogorov, A. N. (1954). "Preservation of conditionally periodic movements with small change in the Hamiltonian function". Doklady 

Akademii Nauk SSSR 98: 527—530. See also Kolmogorov— Arnold— Moser theorem 
[39] Mary L. Cartwright and John E. Littlewood (1945) "On non-linear differential equations of the second order, I: The equation y" + k(\—y )y' 

+ y = bkkcos(kt + a), k large," Journal of the London Mathematical Society, vol. 20, pages 180—189. See also: Van der Pol oscillator 
[40] Stephen Smale (January 1960) "Morse inequalities for a dynamical system," Bulletin of the American Mathematical Society, vol. 66, pages 

43-49. 
[41] Edward N. Lorenz, "Deterministic non-periodic flow," Journal of the Atmospheric Sciences, vol. 20, pages 130—141 (1963). 
[42] Gleick, James (1987). Chaos: Making a New Science. London: Cardinal, pp. 17. 

[43] Mandelbrot, Benoit (1963). "The variation of certain speculative prices". Journal of Business 36: pp. 394—419. 
[44] J.M. Berger and B. Mandelbrot (July 1963) "A new model for error clustering in telephone circuits," I.B.M. Journal of Research and 

Development, vol 7, pages 224—236. 
[45] B. Mandelbrot, The Fractal Geometry of Nature (N.Y., N.Y.: Freeman, 1977), page 248. 
[46] See also: Benoit B. Mandelbrot and Richard L. Hudson, The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward (N.Y., 

N.Y.: Basic Books, 2004), page 201. 



Chaos theory 180 

[47] Benoit Mandelbrot (5 May 1967) "How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension," Science, 

Vol. 156, No. 3775, pages 636-638. 
[48] B. van der Pol and J. van der Mark (1927) "Frequency demultiplication," Nature, vol. 120, pages 363—364. See also: Van der Pol oscillator 
[49] R.L. Ives (10 October 1958) "Neon oscillator rings," Electronics, vol. 31, pages 108—115. 
[50] See p. 83 of Lee W. Casperson, "Gas laser instabilities and their interpretation," pages 83—98 in: N. B. Abraham, F. T. Arecchi, and L. A. 

Lugiato, eds., Instabilities and Chaos in Quantum Optics II: Proceedings of the NATO Advanced Study Institute, II Ciocco, Italy, June 28— July 

7, 1987 (N.Y., N.Y.: Springer Verlag, 1988). 
[51] Ralph H. Abraham and Yoshisuke Ueda, eds., The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory (Singapore: World 

Scientific Publishing Co., 2001). See Chapters 3 and 4. 
[52] Sprott, J. Chaos and time-series analysis (http://books.google.com/books?id=SEDjdjPZ158C&pg=PA89&lpg=PA89&dq=ueda+ 

"Chihiro+Hayashi"&source=bl&ots=p9nZnB5MOD&sig=k2BrAnAyU84BHlMlrJ_-_K01mU0&hl=en& 

ei=mYLrSr_MEI_KNcXcgIMM&sa=X&oi=book_result&ct=result&resnum=4&ved=0CA8Q6AEwAw#v=onepage&q=ueda "Chihiro 

Hayashi"&f=false). Oxford. University Press, Oxford, UK, & New York, USA. 2003 
[53] Mitchell Feigenbaum (July 1978) "Quantitative universality for a class of nonlinear transformations," Journal of Statistical Physics, vol. 19, 

no. 1, pages 25—52. 
[54] "The Wolf Prize in Physics in 1986." (http://www. wolffund.org.il/cat.asp?id=25&cat_title=PHYSICS). . 
[55] Bernardo Huberman, "A Model for Dysfunctions in Smooth Pursuit Eye Movement" Annals of the New York Academy of Sciences, 

Vol. 504 Page 260 July 1987, Perspectives in Biological Dynamics and Theoretical Medicine 
[56] Per Bak, Chao Tang, and Kurt Wiesenfeld, "Self-organized criticality: An explanation of the 1/f noise," Physical Review Letters, vol. 59, 

no. 4, pages 381—384 (27 July 1987). However, the conclusions of this article have been subject to dispute. See: http://www.nslij-genetics. 

org/wli/lfnoise/lfnoise_square.html . See especially: Lasse Laurson, Mikko J. Alava, and Stefano Zapped, "Letter: Power spectra of 

self-organized critical sand piles," Journal of Statistical Mechanics: Theory and Experiment, 0511, L001 (15 September 2005). 
[57] F. Omori (1894) "On the aftershocks of earthquakes," Journal of the College of Science, Imperial University of Tokyo, vol. 7, pages 

111-200. 
[58] (http://www.independent.co.uk/news/science/chaos-theory-serves-up-solution-to-speedy-defrosting-in-microwaves-537426.html) The 

Independent, Fri 29 August 2003 
[59] Provenzale A. et al.: "Distinguishing between low-dimensional dynamics and randomness in measured time-series", in: Physica D, 

58:31^9, 1992 
[60] Sugihara G. and May R.: "Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series", in: Nature, 

344:734-41, 1990 
[61] Casdagli, Martin. "Chaos and Deterministic versus Stochastic Non-linear Modelling", in: Journal Royal Statistics Society: Series B, 54, nr. 2 

(1991), 303-28 
[62] Broomhead D. S. and King G. P.: "Extracting Qualitative Dynamics from Experimental Data", in: Physica 20D, 217—36, 1986 
[63] Kyrtsou, C, (2008). Re-examining the sources of heteroskedasticity: the paradigm of noisy chaotic models, Physica A, 387, pp. 6785—6789. 
[64] Kyrtsou, C, (2005). Evidence for neglected linearity in noisy chaotic models, International Journal of Bifurcation and Chaos, 15(10), pp. 

3391-3394. 
[65] http://www.springerlink.com/content/q00627510552020g/?p=93elf3daf93549dl850365a8800afb30&pi=3 
[66] http ://www. Cambridge. org/catalogue/catalogue. asp?isbn=052 1 663 857 
[67] http://lagrange.physics.drexel.edu 
[68] http://www.chaos.umd.edu 
[69] http://www. societyforchaostheory.org/ 
[70] http://www.csdc. unifi.it/mdswitch. html ?newlang=eng 
[71] http://physics.mercer.edu/pendulum/ 
[72] http://www.creatingtechnology.org/papers/chaos.htm 
[73] http://www.egwald.ca/nonlineardynamics/index.php 
[74] http://www.around.com/chaos.html 
[75] http://www.eng.ox.ac.uk/samp 
[76] http://www.mgix.com/snippets/7MackeyGlass 



Lorentz attractor 



181 



Lorentz attractor 



The Lorenz attractor, named for Edward N. Lorenz, is a fractal 
structure corresponding to the long-term behavior of the Lorenz 
oscillator. The Lorenz oscillator is a 3-dimensional dynamical 
system that exhibits chaotic flow, noted for its lemniscate shape. 
The map shows how the state of a dynamical system (the three 
variables of a three-dimensional system) evolves over time in a 
complex, non-repeating pattern. 




A plot of the trajectory Lorenz system for values p=28 



10, p = 8/3 



Overview 

The attractor itself, and the equations from which it is derived, 
were introduced by Edward Lorenz in 1963, who derived it from 
the simplified equations of convection rolls arising in the 
equations of the atmosphere. 

In addition to its interest to the field of non-linear mathematics, the 
Lorenz model has important implications for climate and weather 
prediction. The model is an explicit statement that planetary and 
stellar atmospheres may exhibit a variety of quasi-periodic 
regimes that are, although fully deterministic, subject to abrupt and 
seemingly random change. 

From a technical standpoint, the Lorenz oscillator is nonlinear, 

three-dimensional and deterministic. In 2001 it was proven by 

Warwick Tucker that for a certain set of parameters the system exhibits chaotic behavior and displays what is today 

called a strange attractor. The strange attractor in this case is a fractal of Hausdorff dimension between 2 and 3. 

Grassberger (1983) has estimated the Hausdorff dimension to be 2.06 ± 0.01 and the correlation dimension to be 

2.05 ±0.01. 




A trajectory of Lorenz's equations, rendered as a metal 
wire to show direction and 3D structure 



The system also arises in simplified models for lasers (Haken 1975) and dynamos (Knobloch 1981). 



Lorentz attractor 



182 



Equations 

The equations that govern the Lorenz oscillator are: 




Trajectory with scales added 



dx 

~dl 

dy 

dt 
dz 

~d~t 



■ a(y - x) 
x(p - z)-y 



xy — j3z 

where a is called the Prandtl number and pis called the Rayleigh number. All a , P, (3 > 0, but usually o = 
10, j3= 8/3 and pis varied. The system exhibits chaotic behavior for p= 28 but displays knotted periodic orbits 
for other values of p. For example, with p = 99.96it becomes a T{3,2) torus knot. 

Sensitive dependence on the initial condition 



Time t=l (Enlarge) 



Time t=2 (Enlarge) 






These figures — made using p=28, o = 10 and |3 = 8/3 — show three time segments of the 3-D evolution of 2 trajectories (one in blue, the other in 

yellow) in the Lorenz attractor starting at two initial points that differ only by 10 in the x-coordinate. Initially, the two trajectories seem coincident 

(only the yellow one can be seen, as it is drawn over the blue one) but, after some time, the divergence is obvious. 



Java animation of the Lorenz attractor shows the continuous evolution. 



[6] 



Lorentz attractor 



183 



Rayleigh number 



The Lorenz attractor for different values of p 




p=14, a=10, |5=8/3 (Enlarge) 




p=13, o=10, p=8/3 (Enlarge) 





p=15, o=10, (5=8/3 (Enlarge) 



p=28, o=10, (5=8/3 (Enlarge) 



For small values of p, the system is stable and evolves to one of two fixed point attractors. When p is larger than 24.28, the fixed points become 
repulsors and the trajectory is repelled by them in a very complex way, evolving without ever crossing itself. 



Java animation showing evolution for different values of p 



[6] 



Source code 

The source code to simulate the Lorenz attractor in GNU Octave follows. 



## 


Lorenz 


Attractor equations solved 


by ODE Solve 


## 


x ' = sigma* (y-x) 






## 


y ' = x 


* (rho - z ) - y 






## 


z ' = X 


*y - beta*z 






function 


dx = lorenzatt (X) 






rho = 


28; sigma = 10; beta = 8/3; 






dx = 


zeros (3,1); 








dx(l) 


= sigma* (X (2) 


- X(l)); 






dx(2) 


= X (1) * (rho - 


X(3)) - X(2); 






dx(3) 


= X(l) *X(2) - 


beta*X(3) ; 






return 






end 








## 


Using 


LSODE to solve 


the ODE syste 


m . 


clear all 








close all 








lsode_opt 


ions ( "absolute 


tolerance" , le 


-3) 



Lorentz attractor 184 

lsode_options ( "relative tolerance", le-4 ) 

t = linspace (0,25, le3) ; X0 = [0,1,1.05]; 

[X,T,MSG]=lsode (@lorenzatt,X0,t) ; 

T 

MSG 

plot3(X(:,l) ,X(:,2) ,X(:,3) ) 

view(45, 45) 

See also 

• List of chaotic maps 

• Takens' theorem 

• Mandelbrot set 

References 

Jonas Bergman, Knots in the Lorentz Equation , Undergraduate thesis, Uppsala University 2004. 

Fr0yland, J., Alfsen, K. H. (1984). "Lyapunov-exponent spectra for the Lorenz model". Phys. Rev. A 29: 

2928-2931. doi: 10. 1103/PhysRevA.29.2928. 

P. Grassberger and I. Procaccia (1983). "Measuring the strangeness of strange attractors" . Physica D 9: 

189-208. doi: 10. 1016/0167-2789(83)90298-1. 

Haken, H. (1975), "Analogy between higher instabilities in fluids and lasers", Physics Letters A 53 (1): 11—1%, 

doi: 10. 1016/0375-9601(75)90353-9. 

Lorenz, E. N. (1963). "Deterministic nonperiodic flow". /. Atmos. Sci. 20: 130—141. 

doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. 

Knobloch, Edgar (1981), "Chaos in the segmented disc dynamo", Physics Letters A 82 (9): 439—440, 

doi:10.1016/0375-9601(81)90274-7. 

Strogatz, Steven H. (1994). Nonlinear Systems and Chaos. Perseus publishing. 

Tucker, W. (2002). "A Rigorous ODE Solver and Smale's 14th Problem" [3] . Found. Comp. Math. 2: 53-117. 

External links 

Weisstein, Eric W., "Lorenz attractor from Math World. 

Lorenz attractor by Rob Morris, Wolfram Demonstrations Project. 

Lorenz equation on planetmath.org 

For drawing the Lorenz attractor, or coping with a similar situation using ANSI C and gnuplot. 

ro] 

Synchronized Chaos and Private Communications, with Kevin Cuomo . The implementation of Lorenz 



attractor in an electronic circuit 

nation (y 

[10] 



[91 
Lorenz attractor interactive animation (you need the Adobe Shockwave plugin) 



Levitated.net: computational art and design 

3D Attractors: Mac program to visualize and explore the Lorenz attractor in 3 dimensions 

3D VRML Lorenz attractor (you need a VRML viewer plugin) 

ri3i 

Essay on Lorenz attractors in J - see J programming language 

ri4i 

Applet for non-linear simulations (select "Lorenz attractor" preset), written by Viktor Bachraty in Jython 
Lorenz Attractor implemented in analog electronic 
Visualizing the Lorenz attractor in 3D with Python and VTK 



Lorentz attractor 



185 



References 

[I] http://www.teorfys.uu.se/en/node/467 

[2] http://adsabs. harvard.edu/cgi-bin/nph-bib_query ?bibcode=1983PhyD.... 9. .189G&db_key=PHY 

[3] http://www.math.uu.se/~warwick/main/rodes.html 

[4] http://mathworld.wolfram.com/LorenzAttractor.html 

[5] http://demonstrations.wolfram.com/LorenzAttractor/ 

[6] http://planetmath.org/encyclopedia/LorenzEquation.html 

[7] http : // w w w . mizuno . org/c/la/index. shtml 

[8] http://video.google.com/videoplay?docid=2875296564158834562&q=strogatz&ei=xr90SJ_SOpeG2wKB3Iy2DA&hl=en 

[9] http://toxi.co.uk/lorenz/ 

[10] http://www. levitated. net/daily/levLorenzAttractor. html 

[II] http://amath.colorado.edu/~juanga/3DAttractors.html 
[12] http://ibiblio.org/e-notes/VRML/Lorenz/Lorenz.htm 
[13] http://www.jsoftware.com/jwiki/Essays/Lorenz_Attractor 
[14] http://student.fiit.stuba.sk/~bachratv02/mes/applet.html 
[15] http://frank.harvard.edu/~paulh/misc/lorenz.htm 

[16] http://www.martinlaprise.info/2010/02/28/visualizing-the-lorentz-attractor-with-vtk/ 



Rossler attractor 



The Rossler attractor (pronounced 

/rosier/) is the attractor for the Rossler 

system, a system of three non-linear 

ordinary differential equations. These 

differential equations define a 

continuous -time dynamical system that 

exhibits chaotic dynamics associated with 

the fractal properties of the attractor. Some 

properties of the Rossler system can be 

deduced via linear methods such as 

eigenvectors, but the main features of the 

system require non-linear methods such as 

Poincare maps and bifurcation diagrams. 

The original Rossler paper says the Rossler 

attractor was intended to behave similarly to 

the Lorenz attractor, but also be easier to 

analyze qualitatively. An orbit within the 

attractor follows an outward spiral close to 

the X, y plane around an unstable fixed point. 

Once the graph spirals out enough, a second 

fixed point influences the graph, causing a 

rise and twist in the z -dimension. In the time domain, it becomes apparent that although each variable is oscillating 

within a fixed range of values, the oscillations are chaotic. This attractor has some similarities to the Lorenz attractor, 

but is simpler and 




Rossler attractor 



186 



has only one manifold. Otto Rossler designed the Rossler 
attractor in 1976, but the originally theoretical equations 
were later found to be useful in modeling equilibrium in 
chemical reactions. The defining equations are: 




Rossler attractor as a stereogram with q = 0.2' 6 — 0.2 

. c = 14 



dx 

~dl 

dy 

dt 
dz 

~dt 



-y 



x + ay 



= b + z(x — c) 



Rossler studied the chaotic attractor with a = 0.2' 6= 0.2- an d C 
5 = 0.1' an( ^ c = 14 nave been more commonly used since. 



5.7> though properties of a = Q.l, 



An analysis 



Rossler attractor 



187 



Some of the Rossler attractor's elegance is due to two of its equations 
being linear; setting % = fj, allows examination of the behavior on 
the X, y plane 




X^ y plane of Rossler attractor with 

a = 0.2. 6 = 0.2. c = 5.7 



dx 

~di 
dy 

dt 



= -y 

x + ay 
The stability in the X, y plane can then be found by calculating the eigenvalues of the Jacobian 



-1 



which 



are (a i s/a? — 4)/2- From this, we can see that when < a < 2> tne eigenvalues are complex and at least 
one has a positive real component, making the origin unstable with an outwards spiral on the X, y plane. Now 
consider the z plane behavior within the context of this range for a ■ So long as x is smaller than c , the c term 
will keep the orbit close to the X, y plane. As the orbit approaches x greater than c , the z -values begin to climb. 
As z climbs, though, the —z in the equation for dx/dt stops the growth in x ■ 

Fixed points 

In order to find the fixed points, the three Rossler equations are set to zero and the ( x , y , z) coordinates of each 
fixed point were determined by solving the resulting equations. This yields the general equations of each of the fixed 
point coordinates: 



X 



±v^ 



Aab 



±s/c 



4ab" 



2a 



±Vl 



Aab 



Which in turn can be used to show the actual fixed points for a given set of parameter values: 



+ v? 



Aab 



V^ 



Aab 



—c — \/c 2 - 


-Aab 


2a 


-c + Vc 2 - 


-Aab 



c 

J 



+ V^ 



Aab 



2a 



VC 2 



Aab 



\ 2 2a ' 2a J 

As shown in the general plots of the Rossler Attractor above, one of these fixed points resides in the center of the 
attractor loop and the other lies comparatively removed from the attractor. 



Rossler attractor 



188 



Eigenvalues and eigenvectors 

The stability of each of these fixed points can be analyzed by determining their respective eigenvalues and 
eigenvectors. Beginning with the Jacobian: 

'0 -1 -1 
1 a 

K z x — Cj 
the eigenvalues can be determined by solving the following cubic: 

—A + A (a + x — c) + A(ac — ax — 1 — z) + x — c + az = 

For the centrally located fixed point, Rossler's original parameter values of a=0.2, b=0.2, and c=5.7 yield eigenvalues 
of: 



Ai 
A 3 



0.0971028 + 0.9957862 
0.0971028 - 0.9957862 
-5.68718 



(Using Mathematica 7) 

The magnitude of a negative eigenvalue characterizes the level of attraction along the corresponding eigenvector. 
Similarly the magnitude of a positive eigenvalue characterizes the level of repulsion along the corresponding 
eigenvector. 

The eigenvectors corresponding to these eigenvalues are: 
0.7073 



vi 



f'2 



V3 



-0.07278 - 0.7032z 
0.0042 - 0.0007i 

0.7073 
0.07278 + 0.7032z 
0.0042 + 0.0007z 

0.1682 
-0.0286 

0.9853 

These eigenvectors have several 
interesting implications. First, the two 
eigenvalue/eigenvector pairs ( l>iand 
^2) are responsible for the steady 
outward slide that occurs in the main 
disk of the attractor. The last 
eigenvalue/eigenvector pair is 
attracting along an axis that runs 
through the center of the manifold and 
accounts for the z motion that occurs 
within the attractor. This effect is 
roughly demonstrated with the figure 
below. 

The figure examines the central fixed 
point eigenvectors. 



80 
70 
SO 
50- 
N 40 
30 
20- 
10 



Central Fixed Point Eigenvectors Examined 




10 




Examination of central fixed point eigenvectors: The blue line corresponds to the standard 
Rossler attractor generated with q = Q.2> 6 = 0.2' anc ' C = 5.7- 



The 



blue 



line 




Rossler attractor 189 

corresponds to the standard Rossler attractor generated with a = 0.2> 6 = 0.2> an d c 

The red dot in the center of this attractor is FPi ■ The red line intersecting 2 

that fixed point is an illustration of the repulsing plane generated by V\ >•' 

and l>2 • The green line is an illustration of the attracting Vs . The magenta 

line is generated by stepping backwards through time from a point on 

the attracting eigenvector which is slightly above FP\ — it illustrates 

the behavior of points that become completely dominated by that _.. , ... ~ , ~ 

r r J J Rossler attractor with q = (J. 2' = 0.2' 

vector. Note that the magenta line nearly touches the plane of the „ _ tc n 

attractor before being pulled upwards into the fixed point; this suggests 

that the general appearance and behavior of the Rossler attractor is largely a product of the interaction between the 

attracting ^and the repelling Uiand l>2plane. Specifically it implies that a sequence generated from the Rossler equations 

will begin to loop around FP\, start being pulled upwards into the V 3 vector, creating the upward arm of a curve that bends 

slightly inward toward the vector before being pushed outward again as it is pulled back towards the repelling plane. 

For the outlier fixed point, Rossler's original parameter values of q = fj.2> b = 0.2> an d c = 5.7yi e ld 
eigenvalues of: 

\ t = -0.0000046 + 5.4280259z 
A 2 = -0.0000046 - 5 .4280259i 
A 3 = 0.1929830 

The eigenvectors corresponding to these eigenvalues are: 

'0.0002422 + 0.1872055i N 
(| = [ 0.0344403 - 0.00 13136z 
0.9817159 

0.0002422 - 0.1872055?;\ 
, , = ( 0.0344403 + 0.0013136z 
0.9817159 J 

I 0.0049651 \ 
v 3 = -0.7075770 

\ 0.7066188 / 
Although these eigenvalues and eigenvectors exist in the Rossler attractor, their influence is confined to iterations of 
the Rossler system whose initial conditions are in the general vicinity of this outlier fixed point. Except in those 
cases where the initial conditions lie on the attracting plane generated by A]and A 2 > this influence effectively 
involves pushing the resulting system towards the general Rossler attractor. As the resulting sequence approaches the 
central fixed point and the attractor itself, the influence of this distant fixed point (and its eigenvectors) will wane. 



Rossler attractor 



190 



Poincare map 

The Poincare map is constructed by plotting the value of the function 

every time it passes through a set plane in a specific direction. An 

example would be plotting the y 7 lvalue every time it passes through 

the x = fjpl ane where x is changing from negative to positive, 

commonly done when studying the Lorenz attractor. In the case of the 

Rossler attractor, the x = 0pl ane i s uninteresting, as the map always 

crosses the % = fjplane at ^ = Qdue to the nature of the Rossler 

equations. In the a = O.lplane for a — 0.1, b = 0.1. C = 14 

the Poincare map shows the upswing in z values as x increases, as is 

to be expected due to the upswing and twist section of the Rossler plot. 

The number of points in this specific Poincare plot is infinite, but when 

a different c value is used, the number of points can vary. For 

example, with a c value of 4, there is only one point on the Poincare map, because the function yields a periodic 

orbit of period one, or if the c value is set to 12.8, there would be six points corresponding to a period six orbit. 



Poincare map for Rossler attractor with 

a = 0.1.6 = 0.1.c=14 



Mapping local maxima 

In the original paper on the Lorenz Attractor, Edward Lorenz analyzed 
the local maxima of z against the immediately preceding local 
maxima. When visualized, the plot resembled the tent map, implying 
that similar analysis can be used between the map and attractor. For the 
Rossler attractor, when the z n local maximum is plotted against the 
next local z maximum, Zn+1, the resulting plot (shown here for 
a = 0.2' b = 0.2' C = 5.7) i s unimodal, resembling a skewed 
Henon map. Knowing that the Rossler attractor can be used to create a 
pseudo 1-d map, it then follows to use similar analysis methods. The 
bifurcation diagram is specifically a useful analysis method. 

Variation of parameters 




Z n ™. Z n+ i 



Rossler attractor's behavior is largely a factor of the values of its constant parameters ( a, b > an d C ). In general 
varying each parameter has a comparable effect by causing the system to converge toward a periodic orbit, fixed 
point, or escape towards infinity, however the specific ranges and behaviors induced vary substantially for each 
parameter. Periodic orbits, or "unit cycles," of the Rossler system are defined by the number of loops around the 
central point that occur before the loops series begins to repeat itself. 

Bifurcation diagrams are a common tool for analyzing the behavior of chaotic systems. Bifurcation diagrams for the 
Rossler attractor are created by iterating through the Rossler ODEs holding two of the parameters constant while 
conducting a parameter sweep over a range of possible values for the third. The local x maxima for each varying 
parameter value is then plotted against that parameter value. These maxima are determined after the attractor has 
reached steady state and any initial transient behaviors have disappeared. This is useful in determining the 
relationship between periodicity and the selected parameter. Increasing numbers of points in a vertical line on a 
bifurcation diagram indicates the Rossler attractor behaves chaotically that value of the parameter being examined. 



Rossler attractor 



191 



Varying a 

In order to examine the behavior of the Rossler attractor for different values of a , b was fixed at 0.2, c was fixed 
at 5.7. Numerical examination of attractor's behavior over changing a suggests it has a disproportional influence 
over the attractor's behavior. Some examples of this relationship include: 

• a < : converges to the centrally located fixed point 

• a = 0.1 : urut cycle of period 1 

• a = 0.2 : standard parameter value selected by Rossler, chaotic 

• a = 0.3 : chaotic attractor, significantly more Mobius strip-like (folding over itself). 

• a = 0.35 : similar to .3, but increasingly chaotic 

• a = 0.38 : similar to .35, but increasingly chaotic 

If a gets even slightly larger than .38, it causes MATLAB to hang. Note this suggests that the practical range of a 
is very narrow. 



Varying fr 

The effect of £> on the Rossler 
attractor's behavior is best illustrated 
through a bifurcation diagram. This 
bifurcation diagram was created with 
a = 0.2. c = 5.7- As shown in the 
accompanying diagram, as 

approaches the attractor approaches 
infinity (note the upswing for very 
small values of £» . Comparative to the 
other parameters, varying \) seems to 
generate a greater range when period-3 
and period-6 orbits will occur. In 
contrast to a and c , higher values of 
I, systems that converge on a period- 1 
orbit instead of higher level orbits or 
chaotic attractors. 



ifmiotior Diiigrriiis f :,■ KrivA-r hJ/.m; ioi (Varying h) 




Bifurcation diagram for the Rossler attractor for varying £> 



Rossler attractor 



192 



Varying c 

The traditional bifurcation diagram for 
the Rossler attractor is created by 
varying c with q = 5 = 1 ■ This 
bifurcation diagram reveals that low 
values of c are periodic, but quickly 
become chaotic as c increases. This 
pattern repeats itself as c increases — 
there are sections of periodicity 
interspersed with periods of chaos, 
although the trend is towards higher 
order periodic orbits in the periodic 
sections as c increases. For example, 
the period one orbit only appears for 
values of c around 4 and is never 
found again in the bifurcation diagram. 
The same phenomena is seen with 
period three; until q = \% period 
three orbits can be found, but 
thereafter, they do not appear. 



/c 


Bifurcation d iagram 




M 






SO 






m 




raP?W 


lli 


j^ , ^^^ 3@Pw&"^' 




10 






ID 







10 



IS 10 35 30 35 40 



45 



Bifurcation diagram for the Rossler attractor for varying C 



A graphical illustration of the changing attractor over a range of c values illustrates the general behavior seen for all 
of these parameter analyses — the frequent transitions from ranges of relative stability and periodicity to completely 
chaotic and back again. 




(a) c=4, period 1 




1 








J 


I 










' 










1 










* t 










i \ 








"V ^ 






/ / 










' 


7 


4 


t 


in ia 



(b) c=6, period 2 









(e) c=9, chaotic 




11 








'0 








: 








1 








i 








.in 




^J^*" ~"^^ 




4 


* -ii 


-to -a o : ia >•: 


ro .~. 





(f) c=12, period 3 


■ 


...... 




, : ' : P^S . -"- ;>,.. 


in 


f :^P 


* I 


• « 


a 


^ifi^sL^zc^^' 


9J 


■?fl -ID • 10 3* M 



(g) c=12.6, period 6 



(h}c=13, chaotic 



(i)c=18, chaotic 



Rossler attractor 193 

The above set of images illustrates the variations in the post-transient Rossler system as c is varied over a range of 
values. These images were generated with q = fa = .1(a) q = 4, periodic orbit, (b) q = Q, period-2 orbit, (c) 
c = g 5, period-4 orbit, (d) q = g.7, period-8 orbit, (e) q = 9, sparse chaotic attractor. (f) q= \% period-3 
orbit, (g) c = 12.6, period-6 orbit, (h) c = 13, sparse chaotic attractor. (i) c = 18, filled-in chaotic attractor. 

Links to other topics 

The banding evident in the Rossler attractor is similar to a Cantor set rotated about its midpoint. Additionally, the 
half-twist in the Rossler attractor makes it similar to a Mobius strip. 

See also 

• Lorenz attractor 

• List of chaotic maps 

• Chaos theory 

• Dynamical system 

• Fractals 

• Otto Rossler 

References 

• E. N. Lorenz (1963). "Deterministic nonperiodic flow". /. Atmos. Sci. 20: 130—141. 
doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. 

• O. E. Rossler (1976). "An Equation for Continuous Chaos". Physics Letters 57A (5): 397—398. 

• O. E. Rossler (1979). "An Equation for Hyperchaos". Physics Letters 71A (2,3): 155-157. 

• Steven H. Strogatz (1994). Nonlinear Dynamics and Chaos. Perseus publishing. 

External links 

• Flash Animation using PovRay 

• Lorenz and Rossler attractors — Java animation 

• Java 3D interactive Rossler attractor 

• 3D Attractors: Mac program to visualize and explore the Rossler and Lorenz attractors in 3 dimensions 

• Rossler attractor in Scholarpedia 

References 

[1] http://lagrange.physics.drexel.edu/flash/rossray 
[2] http://mrmartin.net/code/RosslerAttractor.html 
[3] http://scholarpedia.org/article/Rossler_attractor 



List of chaotic maps 



194 



List of chaotic maps 



In mathematics, a chaotic map is a map that exhibits some sort of chaotic behavior. Maps may be parameterized by 
a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic 
maps often occur in the study of dynamical systems. 

Chaotic maps often generate fractals. Although a fractal may be constructed by an iterative procedure, some fractals 
are studied in and of themselves, as sets rather than in terms of the map that generates them. This is often because 
there are several different iterative procedures to generate the same fractal. 



List of chaotic maps 



Map 


Time 
domain 


Space 
domain 


Number of space 
dimensions 


Also known as 


Arnold's cat map 


discrete 


real 


2 




Baker's map 


discrete 


real 


2 




Bogdanov map 










Chossat-Goluhitsky symmetry map 










Circle map 


discrete 


real 


1 




Cob Web map 










Complex quadratic map 


discrete 


complex 


1 




Complex squaring map 


discrete 


complex 


1 




Complex Cubic map 










Degenerate Double Rotor map 










Double Rotor map 










Duffing map 


discrete 


real 


2 




Duffing equation 


continuous 


real 


1 




Dyadic transformation 


discrete 


real 


1 


2x mod 1 map, Bernoulli map, doubling map, 
sawtooth map 


Exponential map 


discrete 


complex 


2 




Gauss map 


discrete 


real 


1 


mouse map, Gaussian map 


Generalized Baker map 










Gingerbreadman map 


discrete 


real 


2 




Gumowski/Mira map 










Henon map 


discrete 


real 


2 




Henon with 5th order polynomial 










Hitzl-Zele map 










Horseshoe map 


discrete 


real 


2 




Ikeda map 


discrete 


real 


2 




Interval exchange map 


discrete 


real 


1 




Kaplan- Yorke map 


discrete 


real 


2 




Linear map on unit square 











List of chaotic maps 



195 



Logistic map 


discrete 


real 


1 




Lorenz attractor 


continuous 


real 


3 




Lorenz system's Poincare Return map 










Lozi map 










Nordmark truncated map 










Pomeau-Manneville maps for 
intermittent chaos 


discrete 


real 


land 2 


Normal-form maps for intermittency (Types I, 
and ffl) 


II 


Pulsed Rotor & standard map 










Quasiperiodicity map 










Rabinovich-Fabrikant equations 


continuous 


real 


3 




Random Rotate map 










Rossler map 


continuous 


real 


3 




Shobu-Ose-Mori piecewise-linear map 


discrete 


real 


1 


piecewise-linear approximation for 
Pomeau-Manneville Type I map 


Sinai map - See [1] 










Symplectic map 










Standard map 


discrete 


real 


2 


Chirikov standard map, Chirikov-Taylor map 


Tangent map 










Tent map 


discrete 


real 


1 




Tinkerbell map 


discrete 


real 


2 




Triangle map 










Van der Pol oscillator 


continuous 


real 


1 




Zaslavskii map 


discrete 


real 


2 




Zaslavskii rotation map 











List of fractals 



Cantor set 

Gravity set, or Mitchell-Green gravity set 

Julia set - derived from complex quadratic map 

Koch snowflake 

Lyapunov fractal 

Mandelbrot set - derived from complex quadratic map 

Menger sponge 

Sierpinski carpet 

Sierpinski triangle 



List of chaotic maps 196 

References 

[1] http://www.maths.ox.ac.uk/~mcsharry/papers/dynsysl8n3pl91y2003mcsharry.pdf 



197 



Other Applications 



Social network 



A social network is a social structure made of individuals (or organizations) called "nodes," which are tied 
(connected) by one or more specific types of interdependency, such as friendship, kinship, financial exchange, 
dislike, sexual relationships, or relationships of beliefs, knowledge or prestige. 

Social network analysis views social relationships in terms of network theory consisting of nodes and ties. Nodes 
are the individual actors within the networks, and ties are the relationships between the actors. The resulting 
graph-based structures are often very complex. There can be many kinds of ties between the nodes. Research in a 
number of academic fields has shown that social networks operate on many levels, from families up to the level of 
nations, and play a critical role in determining the way problems are solved, organizations are run, and the degree to 
which individuals succeed in achieving their goals. 

In its simplest form, a social network is a map of all of the relevant ties between all the nodes being studied. The 
network can also be used to measure social capital — the value that an individual gets from the social network. These 
concepts are often displayed in a social network diagram, where nodes are the points and ties are the lines. 



Social network analysis 

Social network analysis (related to network 
theory) has emerged as a key technique in 
modern sociology. It has also gained a 
significant following in anthropology, 
biology, communication studies, economics, 
geography, information science, 

organizational studies, social psychology, 
and sociolinguistics, and has become a 
popular topic of speculation and study. 

People have used the idea of "social 
network" loosely for over a century to 
connote complex sets of relationships 
between members of social systems at all 
scales, from interpersonal to international. 
In 1954, J. A. Barnes started using the term 
systematically to denote patterns of ties, 
encompassing concepts traditionally used by 
the public and those used by social 
scientists: bounded groups (e.g., tribes, 
families) and social categories (e.g., gender, 
ethnicity). Scholars such as S.D. Berkowitz, 
Stephen Borgatti, Ronald Burt, Kathleen 
Carley, Martin Everett, Katherine Faust, 




An example of a social network diagram. The node with the highest betweenness 
centrality is marked in yellow. 



Social network 198 

Linton Freeman, Mark Granovetter, David Knoke, David Krackhardt, Peter Marsden, Nicholas Mullins, Anatol 
Rapoport, Stanley Wasserman, Barry Wellman, Douglas R. White, and Harrison White expanded the use of 
systematic social network analysis. 

Social network analysis has now moved from being a suggestive metaphor to an analytic approach to a paradigm, 
with its own theoretical statements, methods, social network analysis software, and researchers. Analysts reason 
from whole to part; from structure to relation to individual; from behavior to attitude. They typically either study 
whole networks (also known as complete networks), all of the ties containing specified relations in a defined 
population, or personal networks (also known as egocentric networks), the ties that specified people have, such as 
their "personal communities". The distinction between whole/complete networks and personal/egocentric networks 
has depended largely on how analysts were able to gather data. That is, for groups such as companies, schools, or 
membership societies, the analyst was expected to have complete information about who was in the network, all 
participants being both potential egos and alters. Personal/egocentric studies were typically conducted when 
identities of egos were known, but not their alters. These studies rely on the egos to provide information about the 
identities of alters and there is no expectation that the various egos or sets of alters will be tied to each other. A 
snowball network refers to the idea that the alters identified in an egocentric survey then become egos themselves 
and are able in turn to nominate additional alters. While there are severe logistic limits to conducting snowball 
network studies, a method for examining hybrid networks has recently been developed in which egos in complete 
networks can nominate alters otherwise not listed who are then available for all subsequent egos to see. The 
hybrid network may be valuable for examining whole/complete networks that are expected to include important 
players beyond those who are formally identified. For example, employees of a company often work with 
non-company consultants who may be part of a network that cannot fully be defined prior to data collection. 

Several analytic tendencies distinguish social network analysis: 

There is no assumption that groups are the building blocks of society: the approach is open to studying 
less-bounded social systems, from nonlocal communities to links among websites. 

Rather than treating individuals (persons, organizations, states) as discrete units of analysis, it focuses on how 
the structure of ties affects individuals and their relationships. 

In contrast to analyses that assume that socialization into norms determines behavior, network analysis looks 
to see the extent to which the structure and composition of ties affect norms. 

The shape of a social network helps determine a network's usefulness to its individuals. Smaller, tighter networks can 
be less useful to their members than networks with lots of loose connections (weak ties) to individuals outside the 
main network. More open networks, with many weak ties and social connections, are more likely to introduce new 
ideas and opportunities to their members than closed networks with many redundant ties. In other words, a group of 
friends who only do things with each other already share the same knowledge and opportunities. A group of 
individuals with connections to other social worlds is likely to have access to a wider range of information. It is 
better for individual success to have connections to a variety of networks rather than many connections within a 
single network. Similarly, individuals can exercise influence or act as brokers within their social networks by 
bridging two networks that are not directly linked (called filling structural holes). 

The power of social network analysis stems from its difference from traditional social scientific studies, which 
assume that it is the attributes of individual actors — whether they are friendly or unfriendly, smart or dumb, 
etc. — that matter. Social network analysis produces an alternate view, where the attributes of individuals are less 
important than their relationships and ties with other actors within the network. This approach has turned out to be 
useful for explaining many real-world phenomena, but leaves less room for individual agency, the ability for 
individuals to influence their success, because so much of it rests within the structure of their network. 

Social networks have also been used to examine how organizations interact with each other, characterizing the many 
informal connections that link executives together, as well as associations and connections between individual 
employees at different organizations. For example, power within organizations often comes more from the degree to 



Social network 199 

which an individual within a network is at the center of many relationships than actual job title. Social networks also 
play a key role in hiring, in business success, and in job performance. Networks provide ways for companies to 
gather information, deter competition, and collude in setting prices or policies. 

History of social network analysis 

A summary of the progress of social networks and social network analysis has been written by Linton Freeman. 

Precursors of social networks in the late 1800s include Emile Durkheim and Ferdinand Tonnies. Tonnies argued that 
social groups can exist as personal and direct social ties that either link individuals who share values and belief 
{gemeinschaft) or impersonal, formal, and instrumental social links {gesellschaft). Durkheim gave a 
non-individualistic explanation of social facts arguing that social phenomena arise when interacting individuals 
constitute a reality that can no longer be accounted for in terms of the properties of individual actors. He 
distinguished between a traditional society — "mechanical solidarity" — which prevails if individual differences are 
minimized, and the modern society — "organic solidarity" — that develops out of cooperation between differentiated 
individuals with independent roles. 

Georg Simmel, writing at the turn of the twentieth century, was the first scholar to think directly in social network 
terms. His essays pointed to the nature of network size on interaction and to the likelihood of interaction in ramified, 
loosely -knit networks rather than groups (Simmel, 1908/1971). 

After a hiatus in the first decades of the twentieth century, three main traditions in social networks appeared. In the 
1930s, J.L. Moreno pioneered the systematic recording and analysis of social interaction in small groups, especially 
classrooms and work groups (sociometry), while a Harvard group led by W. Lloyd Warner and Elton Mayo explored 
interpersonal relations at work. In 1940, A.R. Radcliffe-Brown's presidential address to British anthropologists urged 

ro] 

the systematic study of networks. However, it took about 15 years before this call was followed-up systematically. 

Social network analysis developed with the kinship studies of Elizabeth Bott in England in the 1950s and the 
1950s- 1960s urbanization studies of the University of Manchester group of anthropologists (centered around Max 
Gluckman and later J. Clyde Mitchell) investigating community networks in southern Africa, India and the United 
Kingdom. Concomitantly, British anthropologist S.F. Nadel codified a theory of social structure that was influential 
in later network analysis. 

In the 1960s- 1970s, a growing number of scholars worked to combine the different tracks and traditions. One group 
was centered around Harrison White and his students at the Harvard University Department of Social Relations: Ivan 
Chase, Bonnie Erickson, Harriet Friedmann, Mark Granovetter, Nancy Howell, Joel Levine, Nicholas Mullins, John 
Padgett, Michael Schwartz and Barry Wellman. Also important in this early group were Charles Tilly, who focused 
on networks in political sociology and social movements, and Stanley Milgram, who developed the "six degrees of 
separation" thesis. Mark Granovetter and Barry Wellman are among the former students of White who have 
elaborated and popularized social network analysis. 

White's was not the only group. Significant independent work was done by scholars elsewhere: University of 
California Irvine social scientists interested in mathematical applications, centered around Linton Freeman, including 
John Boyd, Susan Freeman, Kathryn Faust, A. Kimball Romney and Douglas White; quantitative analysts at the 
University of Chicago, including Joseph Galaskiewicz, Wendy Griswold, Edward Laumann, Peter Marsden, Martina 
Morris, and John Padgett; and communication scholars at Michigan State University, including Nan Lin and Everett 
Rogers. A substantively-oriented University of Toronto sociology group developed in the 1970s, centered on former 
students of Harrison White: S.D. Berkowitz, Harriet Friedmann, Nancy Leslie Howard, Nancy Howell, Lome 
Tepperman and Barry Wellman, and also including noted modeler and game theorist Anatol Rapoport.In terms of 
theory, it critiqued methodological individualism and group-based analyses, arguing that seeing the world as social 

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networks offered more analytic leverage. 



Social network 200 

Research 

Social network analysis has been used in epidemiology to help understand how patterns of human contact aid or 
inhibit the spread of diseases such as HIV in a population. The evolution of social networks can sometimes be 
modeled by the use of agent based models, providing insight into the interplay between communication rules, rumor 
spreading and social structure. 

SNA may also be an effective tool for mass surveillance — for example the Total Information Awareness program 
was doing in-depth research on strategies to analyze social networks to determine whether or not U.S. citizens were 
political threats. 

Diffusion of innovations theory explores social networks and their role in influencing the spread of new ideas and 
practices. Change agents and opinion leaders often play major roles in spurring the adoption of innovations, although 
factors inherent to the innovations also play a role. 

Robin Dunbar has suggested that the typical size of a egocentric network is constrained to about 150 members due to 
possible limits in the capacity of the human communication channel. The rule arises from cross-cultural studies in 
sociology and especially anthropology of the maximum size of a village (in modern parlance most reasonably 
understood as an ecovillage). It is theorized in evolutionary psychology that the number may be some kind of limit 
of average human ability to recognize members and track emotional facts about all members of a group. However, it 
may be due to economics and the need to track "free riders", as it may be easier in larger groups to take advantage of 
the benefits of living in a community without contributing to those benefits. 

Mark Granovetter found in one study that more numerous weak ties can be important in seeking information and 
innovation. Cliques have a tendency to have more homogeneous opinions as well as share many common traits. This 
homophilic tendency was the reason for the members of the cliques to be attracted together in the first place. 
However, being similar, each member of the clique would also know more or less what the other members knew. To 
find new information or insights, members of the clique will have to look beyond the clique to its other friends and 
acquaintances. This is what Granovetter called "the strength of weak ties". 

Guanxi is a central concept in Chinese society (and other East Asian cultures) that can be summarized as the use of 

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personal influence. Guanxi can be studied from a social network approach. 

The small world phenomenon is the hypothesis that the chain of social acquaintances required to connect one 
arbitrary person to another arbitrary person anywhere in the world is generally short. The concept gave rise to the 
famous phrase six degrees of separation after a 1967 small world experiment by psychologist Stanley Milgram. In 
Milgram's experiment, a sample of US individuals were asked to reach a particular target person by passing a 
message along a chain of acquaintances. The average length of successful chains turned out to be about five 
intermediaries or six separation steps (the majority of chains in that study actually failed to complete). The methods 
(and ethics as well) of Milgram's experiment was later questioned by an American scholar, and some further research 

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to replicate Milgram's findings had found that the degrees of connection needed could be higher. Academic 
researchers continue to explore this phenomenon as Internet-based communication technology has supplemented the 
phone and postal systems available during the times of Milgram. A recent electronic small world experiment at 
Columbia University found that about five to seven degrees of separation are sufficient for connecting any two 
people through e-mail. 

Collaboration graphs can be used to illustrate good and bad relationships between humans. A positive edge between 
two nodes denotes a positive relationship (friendship, alliance, dating) and a negative edge between two nodes 
denotes a negative relationship (hatred, anger). Signed social network graphs can be used to predict the future 
evolution of the graph. In signed social networks, there is the concept of "balanced" and "unbalanced" cycles. A 
balanced cycle is defined as a cycle where the product of all the signs are positive. Balanced graphs represent a 
group of people who are unlikely to change their opinions of the other people in the group. Unbalanced graphs 
represent a group of people who are very likely to change their opinions of the people in their group. For example, a 



Social network 201 

group of 3 people (A, B, and C) where A and B have a positive relationship, B and C have a positive relationship, 
but C and A have a negative relationship is an unbalanced cycle. This group is very likely to morph into a balanced 
cycle, such as one where B only has a good relationship with A, and both A and B have a negative relationship with 
C. By using the concept of balances and unbalanced cycles, the evolution of signed social network graphs can be 
predicted. 

One study has found that happiness tends to be correlated in social networks. When a person is happy, nearby friends 
have a 25 percent higher chance of being happy themselves. Furthermore, people at the center of a social network 
tend to become happier in the future than those at the periphery. Clusters of happy and unhappy people were 
discerned within the studied networks, with a reach of three degrees of separation: a person's happiness was 
associated with the level of happiness of their friends' friends' friends. 

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Some researchers have suggested that human social networks may have a genetic basis. Using a sample of twins 

from the National Longitudinal Study of Adolescent Health, they found that in-degree (the number of times a person 

is named as a friend), transitivity (the probability that two friends are friends with one another), and betweenness 

centrality (the number of paths in the network that pass through a given person) are all significantly heritable. 

Existing models of network formation cannot account for this intrinsic node variation, so the researchers propose an 

alternative "Attract and Introduce" model that can explain heritability and many other features of human social 

networks. 

Application to Environmental Issues 

The 1984 book The IRG Solution argued that central media and government-type hierarchical organizations could 
not adequately understand the environmental crisis we were manufacturing, or how to initiate adequate solutions. It 
argued that the widespread introduction of Information Routing Groups was required to create a social network 
whose overall intelligence could collectively understand the issues and devise and implement correct workeable 
solutions and policies. 

Metrics (Measures) in social network analysis 

Betweenness 

The extent to which a node lies between other nodes in the network. This measure takes into account the 
connectivity of the node's neighbors, giving a higher value for nodes which bridge clusters. The measure 

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reflects the number of people who a person is connecting indirectly through their direct links. 
Bridge 

An edge is said to be a bridge if deleting it would cause its endpoints to lie in different components of a graph. 

Centrality 

This measure gives a rough indication of the social power of a node based on how well they "connect" the 
network. "Betweenness", "Closeness", and "Degree" are all measures of centrality. 

Centralization 

The difference between the number of links for each node divided by maximum possible sum of differences. A 
centralized network will have many of its links dispersed around one or a few nodes, while a decentralized 
network is one in which there is little variation between the number of links each node possesses. 

Closeness 

The degree an individual is near all other individuals in a network (directly or indirectly). It reflects the ability 
to access information through the "grapevine" of network members. Thus, closeness is the inverse of the sum 
of the shortest distances between each individual and every other person in the network. (See also: Proxemics) 
The shortest path may also be known as the "geodesic distance". 



Social network 202 

Clustering coefficient 

A measure of the likelihood that two associates of a node are associates themselves. A higher clustering 
coefficient indicates a greater 'cliquishness'. 

Cohesion 

The degree to which actors are connected directly to each other by cohesive bonds. Groups are identified as 
'cliques' if every individual is directly tied to every other individual, 'social circles' if there is less stringency of 
direct contact, which is imprecise, or as structurally cohesive blocks if precision is wanted. 

Degree 

The count of the number of ties to other actors in the network. See also degree (graph theory). 

(Individual-level) Density 

The degree a respondent's ties know one another/ proportion of ties among an individual's nominees. Network 
or global-level density is the proportion of ties in a network relative to the total number possible (sparse versus 
dense networks). 

Flow betweenness centrality 

The degree that a node contributes to sum of maximum flow between all pairs of nodes (not that node). 

Eigenvector centrality 

A measure of the importance of a node in a network. It assigns relative scores to all nodes in the network 
based on the principle that connections to nodes having a high score contribute more to the score of the node 
in question. 

Local Bridge 

An edge is a local bridge if its endpoints share no common neighbors. Unlike a bridge, a local bridge is 
contained in a cycle. 

Path Length 

The distances between pairs of nodes in the network. Average path-length is the average of these distances 
between all pairs of nodes. 

Prestige 

In a directed graph prestige is the term used to describe a node's centrality. "Degree Prestige", "Proximity 
Prestige", and "Status Prestige" are all measures of Prestige. See also degree (graph theory). 

Radiality 

Degree an individual's network reaches out into the network and provides novel information and influence. 
Reach 

The degree any member of a network can reach other members of the network. 
Structural cohesion 

The minimum number of members who, if removed from a group, would disconnect the group. 

Structural equivalence 

Refers to the extent to which nodes have a common set of linkages to other nodes in the system. The nodes 
don't need to have any ties to each other to be structurally equivalent. 

Structural hole 

Static holes that can be strategically filled by connecting one or more links to link together other points. 
Linked to ideas of social capital: if you link to two people who are not linked you can control their 
communication. 



Social network 



203 



Network analytic software 

Network analytic tools are used to represent the nodes (agents) and edges (relationships) in a network, and to analyze 
the network data. Like other software tools, the data can be saved in external files. Additional information comparing 
the various data input formats used by network analysis software packages is available at NetWiki. Network analysis 
tools allow researchers to investigate large networks like the Internet, disease transmission, etc. These tools provide 
mathematical functions that can be applied to the network model. 

Visual representation of social networks is important to understand the network data and convey the result of the 
analysis [22]. Network analysis tools are used to change the layout, colors, size and advanced properties of the 
network representation. 

Patents 



There has been rapid growth in the number of US patent applications 

that cover new technologies related to social networking. The number 

of published applications has been growing at about 250% per year 

over the past five years. There are now over 2000 published 

[231 
applications. Only about 100 of these applications have issued as 

patents, however, largely due to the multi-year backlog in examination 

of business method patents and ethical issues connected with this 



patent category 



[24] 



Growth in "Social Network" US Patent Applications 




2003 2004 2005 2006 2007 2008 2009 2010 

Year Published sourceiusno 



See also 



Clique 

Community of practice 

Dynamic network analysis 

Digital footprint 

FOAF (software) (Friend of a 

friend) 

Friendship paradox 

Knowledge management 

List of social networking websites 

Mathematical sociology 

Metcalfe's Law 

Network analysis 

Network of practice 

Network science 

Organizational patterns 

Small world phenomenon 

Social-circles network model 



Social networking service 
Social network aggregation 
Social network analysis software 
Social software 
Social unit 

Social web 
SocialRank 

Socio-technical systems 
Surveillance 
Triadic closure 
Value network 
Virtual community 
Virtual organization 
Weighted network 



Social network 204 



Further reading 



Barnes, J. A. "Class and Committees in a Norwegian Island Parish", Human Relations 7:39-58 

Berkowitz, Stephen D. 1982. An Introduction to Structural Analysis: The Network Approach to Social Research. 

Toronto: Butterworth. ISBN 0-409-81362-1 

T251 
Brandes, Ulrik, and Thomas Erlebach (Eds.)- 2005. Network Analysis: Methodological Foundations Berlin, 

Heidelberg: Springer- Verlag. 

Breiger, Ronald L. 2004. "The Analysis of Social Networks." Pp. 505—526 in Handbook of Data Analysis, edited 

by Melissa Hardy and Alan Bryman. London: Sage Publications. ISBN 0-7619-6652-8 Excerpts in pdf format 

Burt, Ronald S. (1992). Structural Holes: The Structure of Competition. Cambridge, MA: Harvard University 

Press. ISBN 0-674-84372-X 

(Italian) Casaleggio, Davide (2008). TU SEI RETE. La Rivoluzione del business, del marketing e delta politic a 

attraverso le reti sociali. ISBN 88-901826-5-2 

Carrington, Peter J., John Scott and Stanley Wasserman (Eds.). 2005. Models and Methods in Social Network 

Analysis. New York: Cambridge University Press. ISBN 978-0-521-80959-7 

Christakis, Nicholas and James H. Fowler "The Spread of Obesity in a Large Social Network Over 32 Years," 

New England Journal of Medicine 357 (4): 370-379 (26 July 2007) 

Doreian, Patrick, Vladimir Batagelj, and Anuska Ferligoj. (2005). Generalized Blockmodeling. Cambridge: 

Cambridge University Press. ISBN 0-521-84085-6 

Freeman, Linton C. (2004) The Development of Social Network Analysis: A Study in the Sociology of Science. 

Vancouver: Empirical Press. ISBN 1-59457-714-5 

Hill, R. and Dunbar, R. 2002. "Social Network Size in Humans." [27] Human Nature, Vol. 14, No. 1, pp. 53-72. 

Jackson, Matthew O. (2003). "A Strategic Model of Social and Economic Networks". Journal of Economic 

Theory 71: 44-74. doi:10.1006/jeth.l996.0108. pdf [28] 

Huisman, M. and Van Duijn, M. A. J. (2005). Software for Social Network Analysis. In P J. Carrington, J. Scott, 

& S. Wasserman (Editors), Models and Methods in Social Network Analysis (pp. 270—316). New York: 

Cambridge University Press. ISBN 978-0-521-80959-7 

Krebs, Valdis (2006) Social Network Analysis, A Brief Introduction. (Includes a list of recent SNA applications 

Web Reference .) 

Ligon, Ethan; Schechter, Laura, "The Value of Social Networks in rural Paraguay" , University of California, 

Berkeley, Seminar, March 25, 2009 , Department of Agricultural & Resource Economics, College of Natural 

Resources, University of California, Berkeley 

Lin, Nan, Ronald S. Burt and Karen Cook, eds. (2001). Social Capital: Theory and Research. New York: Aldine 

de Gruyter. ISBN 0-202-30643-7 

Mullins, Nicholas. 1973. Theories and Theory Groups in Contemporary American Sociology. New York: Harper 

and Row. ISBN 0-06-044649-8 

Muller-Prothmann, Tobias (2006): Leveraging Knowledge Communication for Innovation. Framework, Methods 

and Applications of Social Network Analysis in Research and Development, Frankfurt a. M. et al.: Peter Lang, 

ISBN 0-8204-9889-0. 

Manski, Charles F. (2000). "Economic Analysis of Social Interactions". Journal of Economic Perspectives 14: 

115-36. [31] viaJSTOR 

Moody, James, and Douglas R. White (2003). "Structural Cohesion and Embeddedness: A Hierarchical Concept 

of Social Groups." American Sociological Review 68(1): 103-127. [32] 

Newman, Mark (2003). "The Structure and Function of Complex Networks". SIAM Review 56: 167—256. 

doi: 10. 1137/S0036 14450342480. pdf [33] 

Nohria, Nitin and Robert Eccles (1992). Networks in Organizations, second ed. Boston: Harvard Business Press. 

ISBN 0-87584-324-7 



Social network 205 

Nooy, Wouter d., A. Mrvar and Vladimir Batagelj. (2005). Exploratory Social Network Analysis with Pajek. 

Cambridge: Cambridge University Press. ISBN 0-521-84173-9 

Scott, John. (2000). Social Network Analysis: A Handbook. 2nd Ed. Newberry Park, CA: Sage. ISBN 

0-7619-6338-3 

Sethi, Arjun. (2008). Valuation of Social Networking [34] 

Tilly, Charles. (2005). Identities, Boundaries, and Social Ties. Boulder, CO: Paradigm press. ISBN 

1-59451-131-4 

Valente, Thomas W. (1995). Network Models of the Diffusion of Innovations. Cresskill, NJ: Hampton Press. 

ISBN 1-881303-21-7 

Wasserman, Stanley, & Faust, Katherine. (1994). Social Network Analysis: Methods and Applications. 

Cambridge: Cambridge University Press. ISBN 0-521-38269-6 

Watkins, Susan Cott. (2003). "Social Networks." Pp. 909—910 in Encyclopedia of Population, rev. ed. Edited by 

Paul George Demeny and Geoffrey McNicoll. New York: Macmillan Reference. ISBN 0-02-865677-6 

Watts, Duncan J. (2003). Small Worlds: The Dynamics of Networks between Order and Randomness. Princeton: 

Princeton University Press. ISBN 0-691-1 1704-7 

Watts, Duncan J. (2004). Six Degrees: The Science of a Connected Age. W. W. Norton & Company. ISBN 

0-393-32542-3 

Wellman, Barry (1998). Networks in the Global Village: Life in Contemporary Communities. Boulder, CO: 

Westview Press. ISBN 0-8133-1150-0 

Wellman, Barry. 2001. "Physical Place and Cyber-Place: Changing Portals and the Rise of Networked 

Individualism." International Journal for Urban and Regional Research 25 (2): 227-52. 

Wellman, Barry and Berkowitz, Stephen D. (1988). Social Structures: A Network Approach. Cambridge: 

Cambridge University Press. ISBN 0-521-24441-2 

Weng, M. (2007). A Multimedia Social-Networking Community for Mobile Devices Interactive 

Telecommunications Program, Tisch School of the Arts/ New York University 

White, Harrison, Scott Boorman and Ronald Breiger. 1976. "Social Structure from Multiple Networks: I 

Blockmodels of Roles and Positions." American Journal of Sociology 81: 730-80. 



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The International Network for Social Network Analysis (INSNA) - professional society of social network 



External links 

• The International 1 
analysts, with more than 1,000 members 

• Annual International Workshop on Social Network Mining and Analysis SNAKDD - Annual computer 

science workshop for interdisciplinary studies on social network mining and analysis 
[371 

• VisualComplexity.com - a visual exploration on mapping complicated and complex networks 

no] 

• Center for Computational Analysis of Social and Organizational Systems (CASOS) at Carnegie Mellon 

• NetLab at the University of Toronto, studies the intersection of social, communication, information and 
computing networks 

• Netwiki (wiki page devoted to social networks; maintained at University of North Carolina at Chapel Hill) 

• Building networks for learning - A guide to on-line resources on strengthening social networking. 

T411 

• Program on Networked Governance - Program on Networked Governance, Harvard University 



Social network 206 

References 

[I] Linton Freeman, The Development of Social Network Analysis. Vancouver: Empirical Press, 2006. 

[2] Wellman, Barry and S.D. Berkowitz, eds., 1988. Social Structures: A Network Approach. Cambridge: Cambridge University Press. 
[3] Hansen, William B. and Reese, Eric L. 2009. Network Genie User Manual (https://secure.networkgenie.com/admin/documentation/ 

Network_Genie_Manual.pdf). Greensboro, NC: Tanglewood Research. 
[4] Freeman, Linton. 2006. The Development of Social Network Analysis. Vancouver: Empirical Pres, 2006; Wellman, Barry and S.D. Berkowitz, 

eds., 1988. Social Structures: A Network Approach. Cambridge: Cambridge University Press. 
[5] Scott, John. 1991. Social Network Analysis. London: Sage. 
[6] Wasserman, Stanley, and Faust, Katherine. 1994. Social Network Analysis: Methods and Applications. Cambridge: Cambridge University 

Press. 
[7] The Development of Social Network Analysis Vancouver: Empirical Press. 

[8] A.R. Radcliffe-Brown, "On Social Structure," Journal of the Royal Anthropological Institute: 70 (1940): 1-12. 
[9] [Nadel, SF. 1957. The Theory of Social Structure. London: Cohen and West. 
[10] The Networked Individual: A Profile of Barry Wellman. (http://www.semioticon.com/semiotix/semiotixl4/sem-14-05.html) 

[II] Mark Granovetter, "Introduction for the French Reader," Sociologica 2 (2007): 1-8; Wellman, Barry. 1988. "Structural Analysis: From 
Method and Metaphor to Theory and Substance." Pp. 19-61 in Social Structures: A Network Approach, edited by Barry Wellman and S.D. 
Berkowitz. Cambridge: Cambridge University Press. 

[12] Mark Granovetter, "Introduction for the French Reader," Sociologica 2 (2007): 1-8; Wellman, Barry. 1988. "Structural Analysis: From 

Method and Metaphor to Theory and Substance." Pp. 19-61 in Social Structures: A Network Approach, edited by Barry Wellman and S.D. 

Berkowitz. Cambridge: Cambridge University Press, (see also Scott, 2000 and Freeman, 2004). 
[13] Barry Wellman, Wenhong Chen and Dong Weizhen. "Networking Guanxi." Pp. 221-41 in Social Connections in China: Institutions, Culture 

and the Changing Nature of Guanxi, edited by Thomas Gold, Douglas Guthrie and David Wank. Cambridge University Press, 2002. 
[14] Could It Be A Big World After All? (http://www.judithkleinfeld.com/ar_bigworld.html): Judith Kleinfeld article. 
[15] Six Degrees: The Science of a Connected Age, Duncan Watts. 
[16] James H. Fowler and Nicholas A. Christakis. 2008. " Dynamic spread of happiness in a large social network: longitudinal analysis over 20 

years in the Framingham Heart Study, (http://www.bmj.com/cgi/content/full/337/dec04_2/a2338)" British Medical Journal. December 

4, 2008: doi:10.1136/bmj.a2338. Media account for those who cannot retrieve the original: Happiness: It Really is Contagious (http://www. 

npr.org/templates/story/story. php?storyId=) Retrieved December 5, 2008. 
[17] "Genes and the Friends You Make" (http://online.wsj.com/article/SB1233020408741 18079.html). Wall Street Journal. January 27, 

2009. . 
[18] Fowler, J. H. (10 February 2009). "Model of Genetic Variation in Human Social Networks" (http://jhfowler.ucsd.edu/ 

genes_and_social_networks.pdf) (PDF). Proceedings of the National Academy of Sciences 106 (6): 1720—1724. 

doi:10.1073/pnas.0806746106. . 
[19] The most comprehensive reference is: Wasserman, Stanley, & Faust, Katherine. (1994). Social Networks Analysis: Methods and 

Applications. Cambridge: Cambridge University Press. A short, clear basic summary is in Krebs, Valdis. (2000). "The Social Life of Routers." 

Internet Protocol Journal, 3 (December): 14-25. 
[20] Cohesive.blocking (http://intersci.ss.uci.edu/wiki/index.php/Cohesive_blocking) is the R program for computing structural cohesion 

according to the Moody-White (2003) algorithm. This wiki site provides numerous examples and a tutorial for use with R. 
[21] Moody, James, and Douglas R. White (2003). "Structural Cohesion and Embeddedness: A Hierarchical Concept of Social Groups." 

American Sociological Review 68(1):103-127. Online (http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf): (PDF file. 
[22] http://www.cmu.edu/joss/content/articles/volumel/Freeman.html 
[23] USPTO search on published patent applications mentioning "social network" (http://appft.uspto.gov/netacgi/nph-Parser?Sectl=PT02& 

Sect2=HITOFF&u=/netahtml/PTO/search-adv.html&r=0&p=l&f=S&l=50&Query=spec/"social+network"&d=PG01) 
[24] USPTO search on issued patents mentioning "social network" (http://patft.uspto.gov/netacgi/nph-Parser?Sectl=PT02& 

Sect2=HITOFF&u=/netahtml/PTO/search-adv.htm&r=0&p=l&f=S&l=50&Query=spec/"social+network"&d=PTXT) 
[25] http://www.springeronline.com/3-540-24979-6/ 
[26] http://www.u.arizona.edu/~breiger/NetworkAnalysis.pdf 
[27] http://www.liv.ac.uk/evolpsyc/Hill_Dunbar_networks.pdf 

[28] http://merlin.fae.ua.es/fvega/CourseNetworks-Alicante/Art%EDculos%20del%20curso/Jackson-Wolinsky-JET.pdf 
[29] http ://www. orgnet.com/sna. html 

[30] http://are.berkeley.edu/seminars/network%20value.pdf 
[31] http://links.jstor.org/sici?sici=0895-3309%28200022%2914%3A3%3C115%3AEAOSI%3E2.0.CO%3B2-I&size=LARGE& 

origin=JSTOR-enlargePage 
[32] http://www2.asanet.org/journals/ASRFeb03MoodyWhite.pdf 
[33] http://www.santafe.edu/files/gems/paleofoodwebs/Newman2003SIAM.pdf 
[34] http://fusion.dalmatech.com/%7Eadmin24/files/socialnetworkvaluation.pdf 
[35] http://www.insna.org 
[36] http://www.socialnetworkanalysis.info 



Social network 207 

[37] http://www.visualcomplexity.com 

[38] http://www.casos.cs.cmu.edu 

[39] http://www.chass.utoronto.ca/~wellman/netlab/ABOUT/index.html 

[40] http://learningforsustainability.net/social_learning/networks.php 

[41] http://www.ksg.harvard.edu/netgov 



Sociology and complexity science 



Sociology and complexity science (also referred to as sociology and complexity) is a new area of study within the 
larger field of complexity science — its acronym is SACS. SACS formally emerged around 1998, when 

sociologists and social scientists made what John Urry refers to as the complexity science turn; that is, the critical 
integration of the tools of complexity science (e.g., agent-based modeling, the new science of networks, etc) into the 
social sciences. 

Historically speaking, SACS is part of the systems tradition (systems thinking) within sociology. The systems 
tradition within sociology has three basic phases: (1) the classical era (late 1800s to 1920s), which included such 
scholars as Karl Marx, Max Weber, Vilfredo Pareto, Herbert Spencer and Emile Durkheim; (2) the prewar era 
(1940s to 1960s), which revolved around the work of Talcott Parsons and Robert Merton; and (3) the complexity 
turn era (1990s to present). 

Scholars involved include Duncan Watts, Albert-Laszlo Barabasi, Mark Newman, Immanuel Wallerstein, Manuel 
Castells, John Urry and Nigel Gilbert the creator and editor of the Journal of Artificial Societies and Social 
Simulation and a pioneer of computational sociology. 

While the substantive topics addresses by the scholars of SACS are numerous, there is a common focus. In one way 
or another, the overarching focus is on social complexity and social systems. Social systems are alternatively 
referred to as complex systems, complex adaptive systems or complex social systems. 

The SACS comprises five areas of research: (1) computational sociology, (2) the British-based School of 
Complexity (See John Urry (sociologist)) , (3) the Luhmann School of Complexity (see Niklas Luhmann), (4) 
sociocybernetics and, finally, (5) complex social network analysis (See complex network, social network and 
network society). Associated areas are complex organizations (see complexity theory and organizations), web 
science, e-social science (see e-science), computational economics, and computational political science. 



Historical background 



Sociology and complexity science 



208 




'A perspective on, and partial map of Complexity Science. The web version 
links to academics and research. 



lists 



An argument can be made that western 

sociology (including its various 

smaller, national sociologies) has been 

and continues to be a profession of 

complexity. The primary basis for 

this challenge is western society. To 

study society is, by definition, to study 

complexity. Starting with the industrial 

and "industrious" revolutions of the 

middle 1700s to early 1900s western 

society transitioned — teleology not 

implied — into a type of complexity 

that, in many ways, did not previously 

exist. Furthermore, as industrialization 

evolved into its later stages (i.e., 

Taylorism, Fordism, post-Fordism, etc), the complexity of western society evolved as well (See Arnold J. Toynbee). 

The latest developments in this complexity are post-industrialism and, most recently, across societies throughout the 

world, globalization. 

Classical era 

Of the numerous scholars writing during the middle 1800s to early 1900s, perhaps the best known systems thinkers 
were Auguste Comte, Herbert Spencer, Karl Marx, Max Weber, Emile Durkheim and Vilfredo Pareto. While not all 
of these scholars were sociologists, their systems thinking had a tremendous impact on organized sociology. Three 
characteristics identify these scholars as systems thinkers: (1) They conceptualized their work as a direct response to 
the increasing complexity of western society; (2) they conceptualized the changes taking place in western society in 
systems terms; that is, they treated western society (and its various substantive issues) as a system; and (3) their 
failure and successes show scholars today how best to think about social complexity in systems terms. Their failures 
include treating social systems in strictly biological terms— homeostasis, etc. Their successes include Pareto's 80/20 
rule and Durkheim's notion of system differentiation (sociology). 



the postwar era 

The postwar era in sociological systems thinking was influenced by Talcott Parsons action theory and, to a lesser 
extent by the work of Robert Merton. Parsons developed a theory of society and social evolution through a 
volunataristic methodology, is famous for his baroque theory of systems, known as structural functionalism. While 
Parsons theory developed an advance approach to the differentiational complexity of society, which forshadowed the 
development of complexity science and, more specifically, SACS, in two important ways. First, it foreshadowed 
SACS insomuch as it integrated sociological inquiry with systems science. While Parsons grounded his theory in a 
hierarchy of theoretical complexity in which classical socoiology was one, cybernetic and approaches within 
cognitive and biological science was others. In retrospect both system theory and cybernetics and be interpreted as 
precursors to complexity science. Second, through his development of the Department of Social Relations at 
Harvard, Parsons foreshadowed the trans-disciplinary, center-based orientation of complexity science— from the 
Santa Fe Institute to the Centre for Research in Social Simulation . Parsons sought to create an international, 
post-disciplinary, highly mobile community of scholars devoted to integrating sociology and systems thinking to 
enhance sociological inquiry. 



Sociology and complexity science 



209 



Complexity turn era 

The community of SACS is part of what John Urry (2005) calls the complexity turn in the social sciences. As Urry 
explains, most of the work being done within the SACS community got its start in the late 1990s, around the same 
time that complexity science was finally gaining international recognition; thanks, in large measure, to the growing 
prestige of the Santa Fe Institute (Santa Fe, New Mexico, USA), the birthplace of complexity science. During the 
late 1990s, the scholars of SACS were spread out across Western Europe and North America, working (for the most 
part) in intellectual and geographical isolation from one another, pursuing diverse areas of study that, at the time, 
seemed hardly related. In the late 1990s, these areas included: (1) computational sociology, (2) complex social 
network analysis, (3) sociocybernetics,(4) the Luhmann School of Complexity, and, (5) the British-based School of 
Complexity (BBC). The agenda basically was an exegetical restoration of the twin goals of Talcott Parsons: (a) 
thinking about the growing complexity of global society in systems terms and (b) integrating sociology with the 
latest advances in complexity science. 



Areas of research in SACS 



Walter JjuckLev 
[Complex Sana! 
Sly stain s) 



John Mi mere 
(.Msnaireria] Science) 



COMMUNITY OF SACS 

Niklasl.ulnnann 
( Society/A iitopoiesis) 



limn i I HI: .nil I illI 1 1 : 1 1 1 . 1 1 -. 

iSocial SvstemsTocltwx) 



Complex Social Network 
Analysis 

The goal of complex social network 

analysis (CSNA) is to study the 

dynamics of large, complex networks 

such as the internet (web science), 

global diseases, and corporate 

interactions. Through the usage of key 

concepts and methods in social 

network analysis, agent-based 

modeling, theoretical physics, and 

modern mathematics (particularly 

graph theory and fractal geometry), 

this field of inquiry has made some 

astonishing insights into the dynamics 

and structure of social systems (i.e., 

small-world phenomenon, scale-free 

networks, etc.). This area of research 

has two subclusters: the new science of networks and global network society. The former primarily emerges out of 

the work of Duncan Watts and colleagues, while the latter (which overlaps with the British-based School of 

Complexity) primarily emerges out of the work of John Urry and the sociological study of globalization. The latter 

also comes from the work of Manuel Castells and the later work of Immanuel Wallerstein which, since 1998, 




'Map 2: Map of Sociology and Complexity Science. *HERE FOR WEB VERSION OF 

MAP [15] 



increasingly makes use of complexity science, particularly the work of Ilya Prigogine 



[16] [17] [18] 



Computational Sociology 

The second area is computational sociology involving such scholars as Nigel Gilbert, Klaus Troitzsch, Scott Page, 
Joshua Epstein and Jiirgen Kliiver— see Map 2 for information on these scholars. The focus of researchers in this 
field, amount to two: social simulation and data-mining, both of which are subclusters within computational 
sociology. Social simulation uses the computer to create an artificial laboratory for the study of complex social 
systems, and data-mining uses machine intelligence to search for non-trivial patterns of relations in large, complex, 



Sociology and complexity science 210 

real-world databases. A variant of computational sociology is socionics. 

Luhmann School of Complexity 

The third field, and the one most different from the first two in terms of epistemology and method, is the Luhmann 
School of Complexity (LSC). Based primarily upon the work of Niklas Luhmann, the goal of this school of thought 
(which is very developed in Germany) is to reinvigorate the study of society as a complex social system. In this way, 

this perspective can be read as an attempt to succeed where Parsons failed, primarily by relying upon the latest 

T211 
advances in systems science and cybernetics, which are the same two fields Parsons drew upon to do his work. 

[22] [23] 

Sociocybernetics 

The fourth major area of research is sociocybernetics. The main goal of this field is to integrate sociology with 
second-order cybernetics and Niklas Luhmann, along with the latest advances in complexity science. In terms of 
scholarly work, the focus of sociocybernetics has been primarily conceptual and only slightly methodological or 
empirical. 

British-based School of Complexity 

The final area of research is the British-based School of Complexity (BBC), a small but growing school of 
thought. While the above four areas seek to make major advances in the discipline of sociology, the BBC is 

attempting to go further. They are not just interested in revising or advancing the current practice of sociology. They 
seek to reformulate the theories, concepts, methods and organizational arrangements of sociology through the 

[271 

employment of complexity science. A school of thought is a defined way of doing scholarly work, based on the 
teachings or instructions of a particular group of scholars. In the case of the BBC, the main scholars are David 
Byrne, John Urry, and Nigel Gilbert. The work being done through these scholars' writing include (1) establishing 
agent-based modeling as a legitimate form of sociological inquiry (which the BBC is doing through its work with 

T2R1 

computational sociology, specifically Nigel Gilbert); (2) problematizing the heavy numerical orientation of 

[291 [301 

mainstream complexity science by developing a qualitative-based, fuzzy-logic approach (including the 

[311 [321 

application of this work to the study of health and health care and urban environments); (3) constructing a 

[331 [341 

mobile sociology based on the latest advances in network and globalization theory; and (4) creating a number 

of innovative organizational arrangements, including the development of interdisciplinary centers, departments, 

[351 

conferences and associations devoted to the study of social complexity. When put together, all of this amounts to 
a new school of thinking in sociology, albeit not as developed yet as the Luhmann School of Complexity. 

See also 

• Complex adaptive 
system 
Complexity 
Complexity economics 
Computational sociology 
Generative sciences 
Multi-agent system 
Social network analysis 
Sociocybernetics 
Systems theory 



Sociology and complexity science 



211 



External links 

• Sociology and Complexity Science Website 

• Castellani and Hafferty (2009) Sociology and Complexity Science: A New Area of Inquiry 



[371 



SOCIOLOGY AND COMPLEXITY SCIENCE BLOG: An Educational Tool for Researchers and Students 



[38] 



On-line book "Simulation for the Social Scientist" by Nigel Gilbert and Klaus G Troitzsch, 1999, second edition 
2005 



[39] 



• Journal of Artificial Societies and Social Simulation 



[40] 



• From Factors to Actors: Computational Sociology and Agent-based Modeling - Review by Michael Macy and 



Robert Wilier 



[41] 



References 

[I] Byrne, David 1998. Complexity Theory and the Social Sciences. London: Routledge. 

[2] Castellani and Hafferty (2009) Sociology and Complexity Science: A New Area of Inquiry http://www.springer.com/physics/book/ 

978-3-540-88461-3] 
[3] Eve, Raymond, Sara Horsfall and Mary Lee 1997. Chaos, Complexity and Sociology: Myths, Models, and Theories. Thousand Oaks, CA: 

Sage Publications. 
[4] Urry, John 2005. "The Complexity Turn." Theory, Culture and Society, 22(5): 1-14. 
[5] Urry, John 2005. "The Complexity Turn." Theory, Culture and Society, 22(5): 1-14. 
[6] Castellani and Hafferty (2009) Sociology and Complexity Science: A New Area of Inquiry http://www.springer.com/physics/book/ 

978-3-540-88461-3] 
[7] http://www.personal.kent.edu/~bcastel3/complex_map.html 

[8] Collins, Randall 1994. Four Sociological Traditions. New York, NY: Oxford University Press. 
[9] Luhmann, Niklas 1995. Social Systems. Stanford CA: Stanford University Press. 
[10] Gerhardt U (2002) Talcott Parsons: An Intellectual Biography. Cambridge, UK: Cambridge University Press. 

[II] Capra F (1996) The Web of Life. New York, NY: Anchor Books Doubleday. 
[12] http://cress.soc.surrey.ac.uk/ 

[13] Waldrop M (1992) Complexity: The Emerging Science at the Edge of Order and Chaos. New York, NY: Simon & Schuster. 

[14] Castellani and Hafferty (2009) Sociology and Complexity Science: A New Area of Inquiry http://www.springer.com/physics/book/ 

978-3-540-88461-3]</ 
[15] http://www. personal. kent.edu/~bcastel3/soc%26complex_map. html'"CLICK 
[16] Barabasi AL (2003) Linked: The New Science of Networks. Cambridge, MA: Perseus Publishing. 

[17] Freeman L (2004) The Development of Social Network Analysis: A Study in the Sociology of Science. Vancouver Canada: Empirical Press. 
[18] Watts D (2004) The New Science of Networks. Annual Review of Sociology 30: 243-270. 

[19] Gilbert N, Troitzsch K (2005) Simulation for Social Scientists, 2nd Edition. New York, NY: Open University Press 
[20] Epstein J (2007) Generative Social Science: Studies in Agent-Based Computational Modeling. Princeton, NJ: Princeton University Press. 
[21] Knodt E (1995) Forward. In Luhmann N Social Systems: Outline of a General Theory, Translated by Eva Knodt. Stanford, CA: Stanford 

University Press. 
[22] Luhmann N (1982) The Differentiation of Society. New York, NY: Columbia University Press. 
[23] Moeller HG (2006) Luhmann Explained: From Souls to Systems. Chicago, IL: Open Court. 

[24] Geyer F, van der Zouwen J (1992) Sociocybernetics. In Negoita CV Handbook of Cybernetics. New York, NY: Marcel Dekker, pp. 95—124. 
[25] Castellani and Hafferty (2009) Sociology and Complexity Science: A New Area of Inquiry http://www.springer.com/physics/book/ 

978-3-540-88461-3] 
[26] McLennan G (2003) Sociology's Complexity. Sociology 37(3): 547-564. 
[27] McLennan G (2003) Sociology's Complexity. Sociology 37(3): 547-564. 

[28] Gilbert N, Troitzsch K (2005) Simulation for Social Scientists, 2nd Edition. New York, NY: Open University Press 
[29] Byrne, D.S. 2005. Complexity, Configuration and Cases. Theory, Culture & Society 22(5): 95-111. 
[30] Byrne D (2001) What is Complexity Science? Thinking as a Realist About Measurement and Cities and Arguing for Natural History. 

Emergence 3(1): 61—76. 
[31] Tim Blackman (2006). Placing Health: Neighbourhood Renewal, Health Improvement and Complexity. Bristol: Policy Press. 
[32] Curtis and Riva (2009) Health Geographies II: Complexity and Health Care Systems and Policy, Progress in Human Geography pp, 1-8 
[33] Urry J (2003) Global Complexity. Oxford, UK: Blackwell Publishing 
[34] Urry J (2000) Sociology Beyond Societies. London, UK: Routledge. 
[35] Castellani and Hafferty (2009) Sociology and Complexity Science: A New Area of Inquiry http://www.springer.com/physics/book/ 

978-3-540-88461-3] 
[36] http://www.personal.kent.edu/~bcastel3/ 



Sociology and complexity science 212 

[37] http://www.springer.com/physics/book/978-3-540-88461-3 

[38] http://sacswebsite.blogspot.com/ 

[39] http://cress.soc.surrey.ac.uk/s4ss/ 

[40] http://jasss.soc.surrey.ac.uk/JASSS.html 

[41] http://www.casos.cs.cmu.edu/education/phd/classpapers/Macy_Factors_2001.pdf 



Sociocybernetics 



Sociocybernetics is an independent chapter of science in sociology based upon the General Systems Theory and 
cybernetics. 

It also has a basis in Organizational Development (OD) consultancy practice and in Theories of Communication, 
theories of psychotherapies and computer sciences. The International Sociological Association has a specialist 
research committee in the area — RC51 — which publishes the (electronic) Journal of Sociocybernetics. 

The term "socio" in the name of sociocybernetics refers to any social system (as defined, among others, by Talcott 
Parsons and Niklas Luhmann). 

The idea to study society as a system can be traced back to the origin of sociology when the emergent idea of 
functional differentiation has been applied for the first time to society by Auguste Comte. 

The basic goal for which sociocybernetics was created, is the production of a theoretical framework as well as 
information technology tools for responding to the basic challenges individuals, couples, families, groups, 
companies, organizations, countries, international affairs are facing today. 

Sociocybernetics analyzes social 'forces' 

One of the tasks of sociocybernetics is to map, measure, harness, and find ways of intervening in the parallel 
network of social forces that influence human behavior. Sociocyberneticists' task is to understand the guidance and 
control mechanisms that govern the operation of society (and the behavior of individuals more generally) in practice 
and then to devise better ways of harnessing and intervening in them — that is to say to devise more effective ways to 
operate these mechanisms, or to modify them according to the opinions of the cyberneticist. 

Sociocybernetics aims to generate a general theoretical framework for 
understanding cooperative behavior. 

It claims to give a deep understanding of the General Theory of Evolution. The outlook that Sociocybernetics uses 
when analyzing any living system lies in a Basic Law of SocioCybernetics. It says: All living systems go through 
five levels of interrelations (social contracts) of its subsystems: 

• A. Aggression: survive or die 

• B. Bureaucracy: follow the norms and rules 

• C. Competition: my gain is your loss 

• D. Decision: disclosing individual feelings, intentions 

• E. Empathy: cooperation in one unified interest 

Going through these five phases of relationship theoretically gives the framework for the sociocybernetic study of 
any evolutionary system. It serves as an "equation for life." 



Sociocybernetics 



213 



Issues and challenges 

Recent research from the Santa Fe Institute presents the idea that social systems like cities don't behave like 
organisms as has been proposed by some in sociocybernetics. 

Perhaps the most basic challenges faced by sociocyberneticians are those that stem from Bookchin's work "The 
Ecology of Freedom and the emergence and decline of Hierarchy". 

Bookchin's argument is that what have often been described as "primitive" societies are best thought of as "organic" 
societies. People within them have differentiated roles as do the cells of a body, but this differentiation is largely 
reversible. Coordination between the cells is not organized by some "center" but through a network of feedback 
(cybernetic) processes. Particularly important are organisms' ability to evolve as well as reproduce. But simply 
saying that the process is "autopoietic" is to evade the task of identifying the multiple and mutually reinforcing 
cybernetic processes that are at work. 

Yet Bookchin's claim, which appears to be thoroughly documented, is that the evolution of organic societies into our 
current, vastly destructive, hierarchical societies - over millennia - has also taken place through some ... (almost 
cancerous?) ... unstoppable autopoietic process. If we are to halt this process ... which is about to destroy us as a 
species, probably carrying the planet as we know it with us, it will be necessary to map and find ways of intervening 
in the sociocybernetic processes involved. No centralised system-wide, command-and-control oriented, change will 
suffice. Systems intervention requires complex systems-oriented intervention targeted at nodes in the system, not 
system-wide change based on "common sense". 

See also 



Anthropology 
Cliodynamics 
Complex systems 
Dynastic cycle 
General systems theory 
List of cycles 
Psychology 
Social cycle theory 
Sociology 



Superorganisms 
Systems philosophy 
Systems thinking 
War cycles 
World-systems theory 



Further reading 



[3]., 



• Felix Geyer and Johannes van der Zouwen (1992). "Sociocybernetics in: Handbook of Cybernetics (C.V. 
Negoita, ed.). New York: Marcel Dekker, 1992 , pp. 95-124. 

• Felix Geyer (1994). "The Challenge of Sociocybernetics [4] ". In: Kybernetes. 24(4):6-32, 1995. Copyright MCB 
University Press 1995 

• Felix Geyer (2001). "Sociocybernetics [5] " In: Kybernetes, Vol. 31 No. 7/8, 2002, pp. 1021-1042. 

• Raven, J. (1994). Managing Education for Effective Schooling: The Most Important Problem Is to Come to Terms 
with Values. Unionville, New York: Trillium Press. (OCLC 34483891) 

• Raven, J. (1995). The New Wealth of Nations: A New Enquiry into the Nature and Origins of the Wealth of 
Nations and the Societal Learning Arrangements Needed for a Sustainable Society. Unionville, New York: Royal 
Fireworks Press; Sudbury, Suffolk: Bloomfield Books. (ISBN 0-89824-232-0) 



Sociocybernetics 



214 



External links 

• Center for Sociocybernetics Studies Bonn 



[6] 



References 

[1] http://www.unizar.es/sociocybernetics 

[2] Luis M. A. Bettencourt, Jose Lobo, Dirk Helbing, Christian Kuhnert, and Geoffrey B. West. Growth, innovation, scaling and the pace of life 

in cities, http://www.pnas.org/cgi/content/abstract/0610172104vl 

[3] http://www.unizar.es/sociocybernetics/chen/felix/pfge8.html 

[4] http://www.unizar.es/sociocybernetics/chen/felix/pfge2.html 

[5] http://www.unizar.es/sociocybernetics/chen/felix/pfgel6.pdf 

[6] http://www.sociocybernetics.eu 



Systems engineering 



Systems engineering is an interdisciplinary field 
of engineering that focuses on how complex 
engineering projects should be designed and 
managed. Issues such as logistics, the 
coordination of different teams, and automatic 
control of machinery become more difficult 
when dealing with large, complex projects. 
Systems engineering deals with work-processes 
and tools to handle such projects, and it overlaps 
with both technical and human-centered 
disciplines such as control engineering and 
project management. 




Systems engineering techniques are used in complex projects: spacecraft 

design, computer chip design, robotics, software integration, and bridge 

building. Systems engineering uses a host of tools that include modeling and 

simulation, requirements analysis and scheduling to manage complexity. 



Systems engineering 



215 



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QFD House of Quality for Enterprise Product 
Development Processes 



The term systems engineering can be traced back to Bell Telephone 
Laboratories in the 1940s. The need to identify and manipulate the 
properties of a system as a whole, which in complex engineering 
projects may greatly differ from the sum of the parts' properties, 
motivated the Department of Defense, NASA, and other industries to 



apply the discipline 



[2] 



When it was no longer possible to rely on design evolution to improve 
upon a system and the existing tools were not sufficient to meet 
growing demands, new methods began to be developed that addressed 
the complexity directly. The evolution of systems engineering, which 
continues to this day, comprises the development and identification of 
new methods and modeling techniques. These methods aid in better 
comprehension of engineering systems as they grow more complex. 
Popular tools that are often used in the Systems Engineering context 
were developed during these times, including USL, UML, QFD, and 
IDEFO. 



In 1990, a professional society for systems engineering, the National 
Council on Systems Engineering (NCOSE), was founded by representatives from a number of US corporations and 
organizations. NCOSE was created to address the need for improvements in systems engineering practices and 
education. As a result of growing involvement from systems engineers outside of the U.S., the name of the 
organization was changed to the International Council on Systems Engineering (INCOSE) in 1995. Schools in 
several countries offer graduate programs in systems engineering, and continuing education options are also 



available for practicing engineers 



[5] 



Concept 

Some definitions 

"An interdisciplinary approach and means to enable the realization of successful systems" — INCOSE handbook, 2004. 

"System engineering is a robust approach to the design, creation, and operation of systems. In simple terms, the approach consists of 
identification and quantification of system goals, creation of alternative system design concepts, performance of design trades, selection 

and implementation of the best design, verification that the design is properly built and integrated, and post-implementation assessment of 

[7] 
how well the system meets (or met) the goals." — NASA Systems engineering handbook, 1995. 



"The Art and Science of creating effective systems, using whole system, whole life principles" OR "The Art and Science of creating 

„[8] 



optimal solution systems to complex issues and problems" 
INCOSE (UK), 2007. 



Derek Hitchins, Prof, of Systems Engineering, former president of 



"The concept from the engineering standpoint is the evolution of the engineering scientist, i.e., the scientific generalist who maintains a 

broad outlook. The method is that of the team approach. On large-scale-system problems, teams of scientists and engineers, generalists as 

well as specialists, exert their joint efforts to find a solution and physically realize it... The technique has been variously called the systems 

„[9] 



approach or the team development method.' 



■ Harry H. Goode & Robert E. Machol, 1957. 



"The Systems Engineering method recognizes each system is an integrated whole even though composed of diverse, specialized 
structures and sub-functions. It further recognizes that any system has a number of objectives and that the balance between them may 
differ widely from system to system. The methods seek to optimize the overall system functions according to the weighted objectives and 



Systems engineering 216 

to achieve maximum compatibility of its parts." — Systems Engineering Tools by Harold Chestnut, 1965. 

Systems Engineering signifies both an approach and, more recently, as a discipline in engineering. The aim of 
education in Systems Engineering is to simply formalize the approach and in doing so, identify new methods and 
research opportunities similar to the way it occurs in other fields of engineering. As an approach, Systems 
Engineering is holistic and interdisciplinary in flavor. 

Origins and traditional scope 

The traditional scope of engineering embraces the design, development, production and operation of physical 
systems, and systems engineering, as originally conceived, falls within this scope. "Systems engineering", in this 
sense of the term, refers to the distinctive set of concepts, methodologies, organizational structures (and so on) that 
have been developed to meet the challenges of engineering functional physical systems of unprecedented 
complexity. The Apollo program is a leading example of a systems engineering project. 

The use of the term "systems engineering" has evolved over time to embrace a wider, more holistic concept of 
"systems" and of engineering processes. This evolution of the definition has been a subject of ongoing controversy 
[11], and the term continues to be applied to both the narrower and broader scope. 

Holistic view 

Systems Engineering focuses on analyzing and eliciting customer needs and required functionality early in the 
development cycle, documenting requirements, then proceeding with design synthesis and system validation while 
considering the complete problem, the system lifecycle. Oliver et al. claim that the systems engineering process can 
be decomposed into 

• a Systems Engineering Technical Process, and 

• a Systems Engineering Management Process. 

Within Oliver's model, the goal of the Management Process is to organize the technical effort in the lifecycle, while 

the Technical Process includes assessing available information, defining effectiveness measures, to create a behavior 

ri2i 
model, create a structure model, perform trade-off analysis, and create sequential build & test plan. 

Depending on their application, although there are several models that are used in the industry, all of them aim to 
identify the relation between the various stages mentioned above and incorporate feedback. Examples of such 

ri3i 

models include the Waterfall model and the VEE model. 

Interdisciplinary field 

ri4i 

System development often requires contribution from diverse technical disciplines. By providing a systems 
(holistic) view of the development effort, systems engineering helps meld all the technical contributors into a unified 
team effort, forming a structured development process that proceeds from concept to production to operation and, in 
some cases, to termination and disposal. 

This perspective is often replicated in educational programs in that Systems Engineering courses are taught by 
faculty from other engineering departments which, in effect, helps create an interdisciplinary environment. 

Managing complexity 

The need for systems engineering arose with the increase in complexity of systems and projects. When speaking in 
this context, complexity incorporates not only engineering systems, but also the logical human organization of data. 
At the same time, a system can become more complex due to an increase in size as well as with an increase in the 
amount of data, variables, or the number of fields that are involved in the design. The International Space Station is 
an example of such a system. 



Systems engineering 



217 



The development of smarter control algorithms, microprocessor design, and analysis of environmental systems also 
come within the purview of systems engineering. Systems engineering encourages the use of tools and methods to 
better comprehend and manage complexity in systems. Some examples of these tools can be seen here: 

Modeling and Simulation, 
Optimization, 
System dynamics, 
Systems analysis, 
Statistical analysis, 
Reliability analysis, and 
Decision making 

Taking an interdisciplinary approach to engineering systems is inherently complex since the behavior of and 
interaction among system components is not always immediately well defined or understood. Defining and 
characterizing such systems and subsystems and the interactions among them is one of the goals of systems 
engineering. In doing so, the gap that exists between informal requirements from users, operators, marketing 
organizations, and technical specifications is successfully bridged. 



Scope 




The scope of Systems Engineering activities 



[18] 



One way to understand the motivation 
behind systems engineering is to see it 
as a method, or practice, to identify 
and improve common rules that exist 
within a wide variety of systems. 
Keeping this in mind, the principles of 
Systems Engineering — holism, 
emergent behavior, boundary, et al. — 
can be applied to any system, complex 
or otherwise, provided systems 

ri9i 

thinking is employed at all levels. 
Besides defense and aerospace, many 
information and technology based 
companies, software development 
firms, and industries in the field of 
electronics & communications require 
Systems engineers as part of their 



team 



[20] 



An analysis by the INCOSE Systems Engineering center of excellence (SECOE) indicates that optimal effort spent 

r2ii 
on Systems Engineering is about 15-20% of the total project effort. At the same time, studies have shown that 

[21] 

Systems Engineering essentially leads to reduction in costs among other benefits. However, no quantitative 
survey at a larger scale encompassing a wide variety of industries has been conducted until recently. Such studies are 

[221 [231 

underway to determine the effectiveness and quantify the benefits of Systems engineering. 

Systems engineering encourages the use of modeling and simulation to validate assumptions or theories on systems 
and the interactions within them. 

Use of methods that allow early detection of possible failures, in Safety engineering, are integrated into the design 
process. At the same time, decisions made at the beginning of a project whose consequences are not clearly 
understood can have enormous implications later in the life of a system, and it is the task of the modern systems 



Systems engineering 218 

engineer to explore these issues and make critical decisions. There is no method which guarantees that decisions 
made today will still be valid when a system goes into service years or decades after it is first conceived but there are 
techniques to support the process of systems engineering. Examples include the use of soft systems methodology, 
Jay Wright Forrester's System dynamics method and the Unified Modeling Language (UML), each of which are 
currently being explored, evaluated and developed to support the engineering decision making process. 

Education 

Education in systems engineering is often seen as an extension to the regular engineering courses, reflecting the 
industry attitude that engineering students need a foundational background in one of the traditional engineering 
disciplines (e.g. mechanical engineering, industrial engineering, computer science, electrical engineering) plus 
practical, real-world experience in order to be effective as systems engineers. Undergraduate university programs in 
systems engineering are rare. 

INCOSE maintains a continuously updated Directory of Systems Engineering Academic Programs worldwide. As 
of 2006, there are about 75 institutions in United States that offer 130 undergraduate and graduate programs in 
systems engineering. Education in systems engineering can be taken as SE-centric or Domain-centric. 

• SE-centric programs treat systems engineering as a separate discipline and all the courses are taught focusing on 
systems engineering practice and techniques. 

• Domain-centric programs offer systems engineering as an option that can be exercised with another major field in 
engineering. 

Both these patterns cater to educate the systems engineer who is able to oversee interdisciplinary projects with the 
depth required of a core-engineer. 

Systems engineering topics 

Systems engineering tools are strategies, procedures, and techniques that aid in performing systems engineering on a 
project or product. The purpose of these tools vary from database management, graphical browsing, simulation, and 

rofil 

reasoning, to document production, neutral import/export and more. 

System 

There are many definitions of what a system is in the field of systems engineering. Below are a few authoritative 
definitions: 

• ANSI/EIA-632-1999: 'An aggregation of end products and enabling products to achieve a given purpose." 

• IEEE Std 1220-1998: "A set or arrangement of elements and processes that are related and whose behavior 
satisfies customer/operational needs and provides for life cycle sustainment of the products." 

• ISO/TEC 15288:2008: "A combination of interacting elements organized to achieve one or more stated 

..[31] 

purposes. 

• NASA Systems Engineering Handbook: "(1) The combination of elements that function together to produce the 
capability to meet a need. The elements include all hardware, software, equipment, facilities, personnel, 
processes, and procedures needed for this purpose. (2) The end product (which performs operational functions) 

and enabling products (which provide life-cycle support services to the operational end products) that make up a 

..[32] 
system. 

• INCOSE Systems Engineering Handbook: "homogeneous entity that exhibits predefined behavior in the real 
world and is composed of heterogeneous parts that do not individually exhibit that behavior and an integrated 

T331 

configuration of components and/or subsystems." 

• INCOSE: "A system is a construct or collection of different elements that together produce results not obtainable 
by the elements alone. The elements, or parts, can include people, hardware, software, facilities, policies, and 



Systems engineering 219 

documents; that is, all things required to produce systems-level results. The results include system level qualities, 
properties, characteristics, functions, behavior and performance. The value added by the system as a whole, 
beyond that contributed independently by the parts, is primarily created by the relationship among the parts; that 
is, how they are interconnected." 

The systems engineering process 

n si 
Depending on their application, tools are used for various stages of the systems engineering process: 

Using models 

Models play important and diverse roles in systems engineering. A model can be defined in several ways, 
including: 

• An abstraction of reality designed to answer specific questions about the real world 

• An imitation, analogue, or representation of a real world process or structure; or 

• A conceptual, mathematical, or physical tool to assist a decision maker. 

Together, these definitions are broad enough to encompass physical engineering models used in the verification of a 
system design, as well as schematic models like a functional flow block diagram and mathematical (i.e., quantitative) 
models used in the trade study process. This section focuses on the last. 

The main reason for using mathematical models and diagrams in trade studies is to provide estimates of system 
effectiveness, performance or technical attributes, and cost from a set of known or estimable quantities. Typically, a 
collection of separate models is needed to provide all of these outcome variables. The heart of any mathematical 
model is a set of meaningful quantitative relationships among its inputs and outputs. These relationships can be as 
simple as adding up constituent quantities to obtain a total, or as complex as a set of differential equations describing 
the trajectory of a spacecraft in a gravitational field. Ideally, the relationships express causality, not just 
correlation. 

Tools for graphic representations 

Initially, when the primary purpose of a systems engineer is to comprehend a complex problem, graphic 
representations of a system are used to communicate a system's functional and data requirements. Common 
graphical representations include: 

Functional Flow Block Diagram (FFBD) 

Data Flow Diagram (DFD) 

N2 (N-Squared) Chart 

IDEFO Diagram 

UML Use case diagram 

UML Sequence diagram 

USL Function Maps and Type Maps. 

Enterprise Architecture frameworks, like TOGAF, MODAF, Zachman Frameworks etc. 

A graphical representation relates the various subsystems or parts of a system through functions, data, or interfaces. 
Any or each of the above methods are used in an industry based on its requirements. For instance, the N2 chart may 
be used where interfaces between systems is important. Part of the design phase is to create structural and behavioral 
models of the system. 

Once the requirements are understood, it is now the responsibility of a Systems engineer to refine them, and to 
determine, along with other engineers, the best technology for a job. At this point starting with a trade study, systems 
engineering encourages the use of weighted choices to determine the best option. A decision matrix, or Pugh 
method, is one way (QFD is another) to make this choice while considering all criteria that are important. The trade 



Systems engineering 220 

study in turn informs the design which again affects the graphic representations of the system (without changing the 
requirements). In an SE process, this stage represents the iterative step that is carried out until a feasible solution is 
found. A decision matrix is often populated using techniques such as statistical analysis, reliability analysis, system 
dynamics (feedback control), and optimization methods. 

At times a systems engineer must assess the existence of feasible solutions, and rarely will customer inputs arrive at 
only one. Some customer requirements will produce no feasible solution. Constraints must be traded to find one or 
more feasible solutions. The customers' wants become the most valuable input to such a trade and cannot be 
assumed. Those wants/desires may only be discovered by the customer once the customer finds that he has 
overconstrained the problem. Most commonly, many feasible solutions can be found, and a sufficient set of 
constraints must be defined to produce an optimal solution. This situation is at times advantageous because one can 
present an opportunity to improve the design towards one or many ends, such as cost or schedule. Various modeling 
methods can be used to solve the problem including constraints and a cost function. 

Systems Modeling Language (SysML), a modeling language used for systems engineering applications, supports the 

[371 
specification, analysis, design, verification and validation of a broad range of complex systems. 

Universal Systems Language (USL) is a systems oriented object modeling language with executable (computer 
independent) semantics for defining complex systems, including software. 

Closely related fields 

Many related fields may be considered tightly coupled to systems engineering. These areas have contributed to the 
development of systems engineering as a distinct entity. 

Cognitive systems engineering 

Cognitive systems engineering (CSE) is a specific approach to the description and analysis of human-machine 

[39] 
systems or sociotechnical systems. The three main themes of CSE are how humans cope with complexity, 

how work is accomplished by the use of artefacts, and how human-machine systems and socio-technical 

systems can be described as joint cognitive systems. CSE has since its beginning become a recognised 

scientific discipline, sometimes also referred to as Cognitive Engineering. The concept of a Joint Cognitive 

System (JCS) has in particular become widely used as a way of understanding how complex socio-technical 

systems can be described with varying degrees of resolution. The experience with CSE has been described in 

two books that summarises the field after more than 20 years of work, namely and 

Configuration Management 

Like Systems Engineering, Configuration Management as practiced in the defence and aerospace industry is a 
broad systems-level practice. The field parallels the taskings of Systems Engineering; where Systems 
Engineering deals with requirements development, allocation to development items and verification, 
Configuration Management deals with requirements capture, traceability to the development item, and audit of 
development item to ensure that it has achieved the desired functionality that Systems Engineering and/or Test 
and Verification Engineering have proven out through objective testing. 

Control engineering 

Control engineering and its design and implementation of control systems, used extensively in nearly every 
industry, is a large sub-field of Systems Engineering. The cruise control on an automobile and the guidance 
system for a ballistic missile are two examples. Control systems theory is an active field of applied 
mathematics involving the investigation of solution spaces and the development of new methods for the 
analysis of the control process. 

Industrial engineering 

Industrial engineering is a branch of engineering that concerns the development, improvement, 
implementation and evaluation of integrated systems of people, money, knowledge, information, equipment, 



Systems engineering 221 

energy, material and process. Industrial engineering draws upon the principles and methods of engineering 
analysis and synthesis, as well as mathematical, physical and social sciences together with the principles and 
methods of engineering analysis and design to specify, predict and evaluate the results to be obtained from 
such systems. 

Interface design 

Interface design and its specification are concerned with assuring that the pieces of a system connect and 
inter-operate with other parts of the system and with external systems as necessary. Interface design also 
includes assuring that system interfaces be able to accept new features, including mechanical, electrical and 
logical interfaces, including reserved wires, plug-space, command codes and bits in communication protocols. 
This is known as extensibility. Human-Computer Interaction (HCI) or Human-Machine Interface (HMI) is 
another aspect of interface design, and is a critical aspect of modern Systems Engineering. Systems 
engineering principles are applied in the design of network protocols for local-area networks and wide-area 
networks. 

Operations research 

Operations research supports systems engineering. The tools of operations research are used in systems 
analysis, decision making, and trade studies. Several schools teach SE courses within the operations research 

or industrial engineering department, highlighting the role systems engineering plays in complex projects. 

T421 
operations research, briefly, is concerned with the optimization of a process under multiple constraints. 

Reliability engineering 

Reliability engineering is the discipline of ensuring a system will meet the customer's expectations for 
reliability throughout its life; i.e. it will not fail more frequently than expected. Reliability engineering applies 
to all aspects of the system. It is closely associated with maintainability, availability and logistics engineering. 
Reliability engineering is always a critical component of safety engineering, as in failure modes and effects 
analysis (FMEA) and hazard fault tree analysis, and of security engineering. Reliability engineering relies 
heavily on statistics, probability theory and reliability theory for its tools and processes. 

Performance engineering 

Performance engineering is the discipline of ensuring a system will meet the customer's expectations for 
performance throughout its life. Performance is usually defined as the speed with which a certain operation is 
executed or the capability of executing a number of such operations in a unit of time. Performance may be 
degraded when an operations queue to be executed is throttled when the capacity is of the system is limited. 
For example, the performance of a packet-switched network would be characterised by the end-to-end packet 
transit delay or the number of packets switched within an hour. The design of high-performance systems 
makes use of analytical or simulation modeling, whereas the delivery of high-performance implementation 
involves thorough performance testing. Performance engineering relies heavily on statistics, queuing theory 
and probability theory for its tools and processes. 

Program management and project management. 

Program management (or programme management) has many similarities with systems engineering, but has 
broader-based origins than the engineering ones of systems engineering. Project management is also closely 
related to both program management and systems engineering. 

Safety engineering 

The techniques of safety engineering may be applied by non-specialist engineers in designing complex 
systems to minimize the probability of safety-critical failures. The "System Safety Engineering" function helps 
to identify "safety hazards" in emerging designs, and may assist with techniques to "mitigate" the effects of 
(potentially) hazardous conditions that cannot be designed out of systems. 

Security engineering 



Systems engineering 222 

Security engineering can be viewed as an interdisciplinary field that integrates the community of practice for 
control systems design, reliability, safety and systems engineering. It may involve such sub-specialties as 
authentication of system users, system targets and others: people, objects and processes. 

Software engineering 

From its beginnings Software engineering has helped shape modern Systems Engineering practice. The 
techniques used in the handling of complexes of large software-intensive systems has had a major effect on the 
shaping and reshaping of the tools, methods and processes of SE. 

See also 

Lists Topics 

• List of production topics • Management cybernetics 

• List of systems engineers • Enterprise systems engineering 

• List of types of systems engineering • System of systems engineering (SoSE) 

• List of systems engineering at universities 



Further reading 

• Harold Chestnut, Systems Engineering Methods. Wiley, 1967. 

• Harry H. Goode, Robert E. Machol System Engineering: An Introduction to the Design of Large-scale Systems, 
McGraw-Hill, 1957. 

• David W. Oliver, Timothy P. Kelliher & James G. Keegan, Jr. Engineering Complex Systems with Models and 
Objects. McGraw-Hill, 1997. 

• Simon Ramo, Robin K. St.Clair, The Systems Approach: Fresh Solutions to Complex Problems Through 
Combining Science and Practical Common Sense, Anaheim, CA: KNI, Inc, 1998. 

• Andrew P. Sage, Systems Engineering. Wiley IEEE, 1992. 

• Andrew P. Sage, Stephen R. Olson, Modeling and Simulation in Systems Engineering, 2001. 

T431 

• Dale Shermon, Systems Cost Engineering , Gower publishing, 2009 

External links 

INCOSE [44] homepage. 

[45] 

Systems Engineering Fundamentals. Defense Acquisition University Press, 2001 

Shishko, Robert et al. NASA Systems Engineering Handbook. NASA Center for AeroSpace Information, 2005. 

Systems Engineering Handbook [47] NASA/SP-2007-6105 Revl, December 2007. 

Derek Hitchins, World Class Systems Engineering , 1997 '. 

[49] 
Parallel product alternatives and verification & validation activities 



Systems engineering 223 

References 

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[2] Arthur D. Hall (1962). A Methodology for Systems Engineering. Van Nostrand Reinhold. ISBN 0442030460. 
[3] Andrew Patrick Sage (1992). Systems Engineering. Wiley IEEE. ISBN 0471536393. 

[4] INCOSE Resp Group (11 June 2004). "Genesis of INCOSE" (http://www.incose.org/about/genesis.aspx). . Retrieved 2006-07-11. 

[5] INCOSE Education & Research Technical Committee. "Directory of Systems Engineering Academic Programs" (http://www.incose.org/ 

educationcareers/academicprogramdirectory.aspx). . Retrieved 2006-07-11. 
[6] Systems Engineering Handbook, version 2a. INCOSE. 2004. 
[7] NASA Systems Engineering Handbook. NASA. 1995. SP-610S. 

[8] "Derek Hitchins" (http://incose.org.uk/people-dkh.htm). INCOSE UK. . Retrieved 2007-06-02. 
[9] Goode, Harry H.; Robert E. Machol (1957). System Engineering: An Introduction to the Design of Large-scale Systems. McGraw-Hill. p. 8. 

LCCN 56-11714. 
[10] Chestnut, Harold (1965). Systems Engineering Tools. Wiley. ISBN 0471154482. 

[II] http://citeseerx.ist.psu. edu/viewdoc/download?doi=10. 1.1. 86. 7496&rep=repl&type=pdf 

[12] Oliver, David W.; Timothy P. Kelliher, James G. Keegan, Jr. (1997). Engineering Complex Systems with Models and Objects. McGraw-Hill. 

pp. 85-94. ISBN 0070481881. 
[13] "The SE VEE" (http://www.gmu.edu/departments/seor/insert/robot/robot2.html). SEOR, George Mason University. . Retrieved 

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and Practical Common Sense (http://www.incose.org/ProductsPubs/DOC/SystemsApproach.pdf). Anaheim, CA: KNI, Inc.. . 
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2007-05-25. 
[16] "ESD Faculty and Teaching Staff" (http://esd.mit.edu/people/faculty.html). Engineering Systems Division, MIT. . Retrieved 

2007-05-25. 
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Press, 2001 
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SE.pdf) (PDF). INCOSE, UK. . Retrieved 2007-06-07. 
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introsal.html). George Mason University. . Retrieved 2007-06-07. 
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2007-06-07. 
[22] "Surveying Systems Engineering Effectiveness" (http://www.splc.net/programs/acquisition-support/presentations/surveying.pdf) 

(PDF). . Retrieved 2007-06-07. 
[23] "Systems Engineering Cost Estimation by Consensus" (http://www.valerdi.com/cosysmo/rvalerdi.doc). . Retrieved 2007-06-07. 
[24] Andrew P. Sage, Stephen R. Olson (2001). Modeling and Simulation in Systems Engineering (http://intl-sim.sagepub.com/cgi/content/ 

abstract/76/2/90). SAGE Publications. . Retrieved 2007-06-02. 
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Research. . Retrieved 2007-06-02. 
[26] "Didactic Recommendations for Education in Systems Engineering" (http://www.gaudisite.nl/DidacticRecommendationsSESlides.pdf) 

(PDF). . Retrieved 2007-06-07. 
[27] "Perspectives of Systems Engineering Accreditation" (http://sistemas.unmsm.edu.pe/occa/material/INCOSE-ABET-SE-SF-21Mar06. 

pdf) (PDF). INCOSE. . Retrieved 2007-06-07. 
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[29] "Processes for Engineering a System", ANSI/EIA-632-1999, ANSI/EIA, 1999 (http://webstore.ansi.org/RecordDetail.aspx?sku=ANSI/ 

EIA-632-1999) 
[30] "Standard for Application and Management of the Systems Engineering Process -Description", IEEE Std 1220- 1998, IEEE, 1998 (http:// 

standards.ieee.org/reading/ieee/std_public/description/se/1220-1998_desc.html) 
[31] "Systems and software engineering - System life cycle processes", ISO/IEC 15288:2008, ISO/IEC, 2008 (http://www. 15288.com/) 
[32] "NASA Systems Engineering Handbook", Revision 1, NASA/SP-2007-6105, NASA, 2007 (http://education.ksc.nasa.gov/ 

esmdspacegrant/Documents/NASA SP-2007-6105 Rev 1 Final 31Dec2007.pdf) 
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[34] "A Consensus of the INCOSE Fellows", INCOSE, 2006 (http://www.incose.org/practice/fellowsconsensus.aspx) 
[35] NASA (1995). "System Analysis and Modeling Issues". In: NASA Systems Engineering Handbook (http://human.space.edu/old/docs/ 

Systems_Eng_Handbook.pdf) June 1995. p.85. 



Systems engineering 224 

[36] Long, Jim (2002) (PDF). Relationships between Common Graphical Representations in System Engineering (http://www.vitechcorp.com/ 

whitepapers/files/20070103 1634430.CommonGraphicalRepresentations_2002.pdf). Vitech Corporation. . 
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Specification Project, pp. pp 23. . Retrieved 2007-07-03. 
[38] Hamilton, M. Hackler, W.R., "A Formal Universal Systems Semantics for SysML, 17th Annual International Symposium, INCOSE 2007, 

San Diego, CA, June 2007. 
[39] Hollnagel E. & Woods D. D. (1983). Cognitive systems engineering: New wine in new bottles. International Journal of Man-Machine 

Studies, 18, 583-600. 
[40] Hollnagel, E. & Woods, D. D. (2005) Joint cognitive systems: The foundations of cognitive systems engineering. Taylor & Francis 
[41] Woods, D. D. & Hollnagel, E. (2006). Joint cognitive systems: Patterns in cognitive systems engineering. Taylor & Francis. 
[42] (see articles for discussion: (http://www.boston.com/globe/search/stories/reprints/operationeverything062704.html) and (http://www. 

sas.com/news/sascom/2004q4/feature_tech.html)) 
[43] http://www.gowerpublishing.com/isbn/978056688612 
[44] http://www.incose.org 

[45] http://www.dau.mil/pubs/pdf/SEFGuide%2001-01.pdf 

[46] http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19960002194_1996102194.pdf 

[47] http://education.ksc.nasa.gov/esmdspacegrant/Documents/NASA%20SP-2007-6105%20Rev%201%20Final%2031Dec2007.pdf 
[48] http://www.hitchins.net/WCSE.html 
[49] http://www.inderscience. com/search/index. php?action=record&rec_id=25267 



Sociobiology 



Sociobiology is a synthesis of scientific disciplines which attempts to explain social behavior in animal species by 
considering the Darwinian advantages specific behaviors may have. It is often considered a branch of biology and 
sociology, but also draws from ethology, anthropology, evolution, zoology, archaeology, population genetics and 
other disciplines. Within the study of human societies, sociobiology is closely related to the fields of human 
behavioral ecology and evolutionary psychology. 

Sociobiology investigates social behaviors, such as mating patterns, territorial fights, pack hunting, and the hive 
society of social insects. Just as selection pressure led to animals evolving useful ways of interacting with the natural 
environment, it led to the genetic evolution of advantageous social behavior. 

Sociobiology has become one of the greatest scientific controversies of the late 20th and early 21st centuries, 
especially in the context of explaining human behavior. Applied to non-humans, sociobiology is uncontroversial. 
Criticism, most notably made by Richard Lewontin and Stephen Jay Gould, centers on sociobiology's contention that 
genes play an ultimate role in human behavior and that traits such as aggressiveness can be explained by biology 
rather than a person's social environment. Many sociobiologists, however, cite a complex relationship between 
nature and nurture. In response to the controversy, anthropologist John Tooby and psychologist Leda Cosmides 
launched evolutionary psychology as a branch of sociobiology made less controversial by avoiding questions of 
human biodiversity. 

Definition 

E.O. Wilson defines sociobiology as: "The extension of population biology and evolutionary theory to social 

• t - ..[l] 
organisation 

Sociobiology is based on the premise that some behaviors (both social and individual) are at least partly inherited 
and can be affected by natural selection. It begins with the idea that behaviors have evolved over time, similar to the 
way that physical traits are thought to have evolved. It predicts therefore that animals will act in ways that have 
proven to be evolutionarily successful over time, which can among other things result in the formation of complex 
social processes conducive to evolutionary fitness. 

The discipline seeks to explain behavior as a product of natural selection. Behavior is therefore seen as an effort to 
preserve one's genes in the population. Inherent in sociobiological reasoning is the idea that certain genes or gene 



Sociobiology 



225 



combinations that influence particular behavioral traits can be inherited from generation to generation. 



Introductory examples 

For example, newly dominant male lions often will kill cubs in the pride that were not sired by them. This behaviour 
is adaptive in evolutionary terms because killing the cubs eliminates competition for their own offspring and causes 
the nursing females to come into heat faster, thus allowing more of his genes to enter into the population. 
Sociobiologists would view this instinctual cub-killing behavior as being inherited through the genes of successfully 
reproducing male lions, whereas non-killing behaviour may have "died out" as those lions were less successful in 
reproducing. 

Genetic mouse mutants have now been harnessed to illustrate the power that genes exert on behaviour. For example, 
the transcription factor FEV (aka Petl) has been shown, through its role in maintaining the serotonergic system in 
the brain, to be required for normal aggressive and anxiety-like behavior . Thus, when FEV is genetically deleted 
from the mouse genome, male mice will instantly attack other males, whereas their wild-type counterparts take 
significantly longer to initiate violent behaviour. In addition, FEV has been shown to be required for correct maternal 
behaviour in mice, such that their offspring do not survive unless cross-fostered to other wild-type female mice 

A genetic basis for instinctive behavioural traits among non-human species, such as in the above example, is 
commonly accepted among many biologists; however, attempting to use a genetic basis to explain complex 
behaviours in human societies has remained extremely controversial. 



History 

According to the OED, John Paul Scott coined the word "sociobiology" 
at a 1946 conference on genetics and social behaviour, and became 
widely used after it was popularized by Edward O. Wilson in his 1975 
book, Sociobiology: The New Synthesis. However, the influence of 
evolution on behavior has been of interest to biologists and philosophers 
since soon after the discovery of the evolution itself. Peter Kropotkin's 
Mutual Aid: A Factor of Evolution, written in the early 1890s, is a 
popular example. Antecedents of modern sociobiological thinking can be 
traced to the 1960s and the work of such biologists as Robert Trivers and 
William D. Hamilton. 




E. O. Wilson, a central figure in the history of 
sociobiology. 



Nonetheless, it was Wilson's book that pioneered and popularized the 

attempt to explain the evolutionary mechanics behind social behaviors 

such as altruism, aggression, and nurturence, primarily in ants (Wilson's own research specialty) but also in other 

animals. The final chapter of the book is devoted to sociobiological explanations of human behavior, and Wilson 

later wrote a Pulitzer Prize winning book, On Human Nature, that addressed human behavior specifically. 



Sociobiology 226 

Sociobiological theory 

Sociobiologists believe that human behavior, as well as nonhuman animal behavior, can be partly explained as the 
outcome of natural selection. They contend that in order fully to understand behavior, it must be analyzed in terms of 
evolutionary considerations. 

Natural selection is fundamental to evolutionary theory. Variants of hereditary traits which increase an organism's 
ability to survive and reproduce will be more greatly represented in subsequent generations, i.e., they will be 
"selected for". Thus, inherited behavioral mechanisms that allowed an organism a greater chance of surviving and/or 
reproducing in the past are more likely to survive in present organisms. That inherited adaptive behaviors are present 
in nonhuman animal species has been multiply demonstrated by biologists, and it has become a foundation of 
evolutionary biology. However, there is continued resistance by some researchers over the application of 
evolutionary models to humans, particularly from within the social sciences, where culture has long been assumed to 
be the predominant driver of behavior. 

Sociobiology is based upon two fundamental premises: 

• Certain behavioral traits are inherited, 

• Inherited behavioral traits have been honed by natural selection. Therefore, these traits were probably "adaptive" 
in the species" evolutionarily evolved environment. 

Sociobiology uses Nikolaas Tinbergen's four categories of questions and explanations of animal behavior. Two 
categories are at the species level; two, at the individual level. The species-level categories (often called "ultimate 
explanations") are 

• the function (i.e., adaptation) that a behavior serves and 

• the evolutionary process (i.e., phylogeny) that resulted in this functionality. 

The individual-level categories (often called "proximate explanations") are 

• the development of the individual (i.e., ontogeny) and 

• the proximate mechanism (e.g., brain anatomy and hormones). 

Sociobiologists are interested in how behavior can be explained logically as a result of selective pressures in the 
history of a species. Thus, they are often interested in instinctive, or intuitive behavior, and in explaining the 
similarities, rather than the differences, between cultures. For example, mothers within many species of mammals — 
including humans — are very protective of their offspring. Sociobiologists reason that this protective behavior likely 
evolved over time because it helped those individuals which had the characteristic to survive and reproduce. Over 
time, individuals who exhibited such protective behaviours would have had more surviving offspring than did those 
who did not display such behaviours, such that this parental protection would increase in frequency in the 
population. In this way, the social behavior is believed to have evolved in a fashion similar to other types of 
nonbehavioral adaptations, such as (for example) fur or the sense of smell. 

Individual genetic advantage often fails to explain certain social behaviors as a result of gene-centred selection, and 
evolution may also act upon groups. The mechanisms responsible for group selection employ paradigms and 
population statistics borrowed from game theory. E.O. Wilson argued that altruistic individuals must reproduce their 
own altruistic genetic traits for altruism to survive. When altruists lavish their resources on non-altruists at the 
expense of their own kind, the altruists tend to die out and the others tend to grow. In other words, altruism is more 
likely to survive if altruists practice the ethic that "charity begins at home." 

Within sociobiology, a social behavior is first explained as a sociobiological hypothesis by finding an evolutionarily 
stable strategy that matches the observed behavior. Stability of a strategy can be difficult to prove, but usually, a 
well-formed strategy will predict gene frequencies. The hypothesis can be supported by establishing a correlation 
between the gene frequencies predicted by the strategy, and those expressed in a population. Measurement of genes 
and gene-frequencies can be problematic, however, because a simple statistical correlation can be open to charges of 
circularity (Circularity can occur if the measurement of gene frequency indirectly uses the same measurements that 



Sociobiology 227 

describe the strategy). 

Altruism between social insects and littermates has been explained in such a way. Altruistic behavior in some 
animals has been correlated to the degree of genome shared between altruistic individuals. A quantitative description 
of infanticide by male harem-mating animals when the alpha male is displaced as well as rodent female infanticide 
and fetal resorption are active areas of study. In general, females with more bearing opportunities may value 
offspring less, and may also arrange bearing opportunities to maximize the food and protection from mates. 

An important concept in sociobiology is that temperamental traits within a gene pool and between gene pools exist in 
an ecological balance. Just as an expansion of a sheep population might encourage the expansion of a wolf 
population, an expansion of altruistic traits within a gene pool may also encourage the expansion of individuals with 
dependent traits. 

Sociobiology is sometimes associated with arguments over the "genetic" basis of intelligence. While sociobiology is 
predicated on the observation that genes do affect behavior, it is perfectly consistent to be a sociobiologist while 
arguing that measured IQ variations between individuals reflect mainly cultural or economic rather than genetic 
factors. However, many critics point out that the usefulness of sociobiology as an explanatory tool breaks down once 
a trait is so variable as to no longer be exposed to selective pressures. In order to explain aspects of human 
intelligence as the outcome of selective pressures, it must be demonstrated that those aspects are inherited, or 
genetic, but this does not necessarily imply differences among individuals: a common genetic inheritance could be 
shared by all humans, just as the genes responsible for number of limbs are shared by all individuals. An even more 
sensitive subject is race and intelligence. 

Researchers performing twin studies have argued that differences between people on behavioral traits such as 

creativity, extroversion and aggressiveness are between 45% to 75% due to genetic differences, and intelligence is 

said by some to be about 80% genetic after one matures (discussed at Intelligence quotient#Genetics vs 

environment). However, critics (such as the evolutionary geneticist R. C Lewontin) have highlighted serious flaws in 

mi 
twin studies, such as the inability of researchers to separate environmental, genetic, and dialectic effects on twins. 

Criminality is actively under study, but extremely controversial. There are arguments that in some environments 
criminal behavior might be adaptive. 

Criticism 

Many critics draw an intellectual link between sociobiology and biological determinism, the belief that most human 
differences can be traced to specific genes rather than differences in culture or social environments. Critics also draw 
parallels between biological determinism as an underlying philosophy to the social Darwinian and eugenics 
movements of the early 20th century, and controversies in the history of intelligence testing. Steven Pinker argues 
that critics have been overly swayed by politics and a "fear" of biological determinism. However, all these critics 
have claimed that sociobiology fails on scientific grounds, independent of their political critiques. In particular, 
Lewontin, Rose & Kamin drew a detailed distinction between the politics and history of an idea and its scientific 
validity, as has Stephen Jay Gould. 

Wilson and his supporters counter the intellectual link by denying that Wilson had a political agenda, still less a 
right-wing one. They pointed out that Wilson had personally adopted a number of liberal political stances and had 
attracted progressive sympathy for his outspoken environmentalism. They argued that as scientists they had a duty to 
uncover the truth whether that was politically correct or not. They argued that sociobiology does not necessarily lead 
to any particular political ideology as many critics implied. Many subsequent sociobiologists, including Robert 
Wright, Anne Campbell, Frans de Waal and Sarah Blaffer Hrdy, have used sociobiology to argue quite separate 
points. Noam Chomsky came to the defense of sociobiology's methodology, noting that it was the same methodology 
he used in his work on linguistics. However, he roundly criticized the sociobiologists' actual conclusions about 
humans as lacking substance. He also noted that the anarchist Peter Kropotkin had made similar arguments in his 
book Mutual Aid: A Factor of Evolution, although focusing more on altruism than aggression, suggesting that 



Sociobiology 



228 



anarchist societies were feasible because of an innate human tendency to cooperate. 

Wilson's claims that he had never meant to imply what ought to be, only what is the case are supported by his 
writings, which are descriptive, not prescriptive. However, many critics have pointed out that the language of 
sociobiology often slips from "is" to "ought", leading sociobiologists to make arguments against social reform on 
the basis that socially progressive societies are at odds with our innermost nature. For example, some groups have 
supported positions of ethnic nepotism. Views such as this, however, are often criticized as examples of the 
naturalistic fallacy, when reasoning jumps from descriptions about what is to prescriptions about what ought to be. 
(A common example is the justification of militarism if scientific evidence showed warfare was part of human 
nature.) It has also been argued that opposition to stances considered anti-social, such as ethnic nepotism, are based 
on moral assumptions, not bioscientific assumptions, meaning that it is not vulnerable to being disproved by 
bioscientific advances. ' The history of this debate, and others related to it, are covered in detail by Cronin 
(1992), Segerstrale (2000) and Alcock (2001). Adaptationists such as Steven Pinker have also suggested that the 
debate has a strong ad hominem component. 



See also 



Concepts 

Biocultural anthropology 

Biosocial theory 

Cultural selection theory 

Dual inheritance theory 

Ethics and evolutionary psychology 

Evolutionary psychology 

Evolutionary developmental 

psychology 

Human behavioral ecology 

Iterated prisoner's dilemma 

Kin selection 

Prisoner's dilemma 

Social evolution 

Sociophysiology 

Evolutionary ethics 



Well-known sociobiologists 

Richard Dawkins 
Edward O. Wilson 
W. D. Hamilton 
Robert Trivers 
George C. Williams 
Sarah Blaffer Hrdy 
Richard Machalek 
Steven Pinker 
Francois Nielsen 



Books 



Sociobiology: The New Synthesis by E. O. Wilson, 1975 

The Blank Slate: The Modern Denial of Human Nature by Steven Pinker 

The Selfish Gene by Richard Dawkins 

Biology, Ideology and Human Nature: Not In Our Genes by Richard Lewontin, Steven Rose & Leon Kamin 



Sociobiology 229 

References 

Bibliography 

• Alcock, John (2001). The Triumph of Sociobiology. Oxford: Oxford University Press. Directly rebuts several of 
the above criticisms and misconceptions listed above. 

• Barkow, Jerome (Ed.). (2006) Missing the Revolution: Darwinism for Social Scientists. Oxford: Oxford 
University Press. 

• Cronin, H. (1992). The Ant and the Peacock: Altruism and Sexual Selection from Darwin to Today. Cambridge: 
Cambridge University Press. 

• Nancy Etcoff (1999). Survival of the Prettiest: The Science of Beauty. Anchor Books. ISBN 0-385-47942-5. 

• Haugan, G0rill (2006) Nursing home patients' spirituality. Interaction of the spiritual, physical, emotional and 
social dimensions (Faculty of Nursing, S0r-Tr0ndelag University College Norwegian University of Science and 
Technology) 

• Richard M. Lerner (1992). Final Solutions: Biology, Prejudice, and Genocide. Pennsylvania State University 
Press. ISBN 0-271-00793-1. 

• Richards, Janet Radcliffe (2000). Human Nature After Darwin: A Philosophical Introduction. London: Routledge. 

• Segerstrale, Ullica (2000). Defenders of the Truth: The Battle for Science in the Sociobiology Debate and Beyond. 
Oxford: Oxford University Press. 

• Gisela Kaplan, Lesley J Rogers (2003). Gene Worship: Moving Beyond the Nature/Nurture Debate over Genes, 
Brain, and Gender. Other Press. ISBN 1-59051-034-8. 

External links 

• Sociobiology (Stanford Encyclopedia of Philosophy) - Harmon Holcomb & Jason Byron 

ri3i 

• The Sociobiology of Sociopathy, Mealey, 1995 

ri4i 

• Speak, Darwinists! Interviews with leading sociobiologists. 



Genetic Similarity and Ethnic Nationalism - An Attempted Sociobiological Explanation of the scientific basis 



Race and Creation - Richard Dawkins 
Genetic Similarity and Ethnic I 
for Political Group Formation. 

ri7i 

• A brief history on sociobiology 

References 

[I] Wilson, E.O. (1978) On Human Nature Page x, Cambridge, Ma: Harvard 

[2] Hendricks TJ, Fyodorov DV, Wegman LJ, Lelutiu NB, Pehek EA, Yamamoto B, Silver J, Weeber EJ, Sweatt JD, Deneris ES. Pet-1 ETS 

gene plays a critical role in 5-HT neuron development and is required for normal anxiety-like and aggressive behaviour. Neuron. 2003 Jan 

23;37(2):233-47 
[3] Lerch-Haner JK, Frierson D, Crawford LK, Beck SG, Deneris ES. Serotonergic transcriptional programming determines maternal behavior 

and offspring survival. Nat Neurosci. 2008 Sep;ll(9):1001-3. 
[4] Richard Lewontin, Leon Kamin, Steven Rose (1984). Not in Our Genes: Biology, Ideology, and Human Nature. Pantheon Books. 

ISBN 0-394-50817-3. 
[5] The Sociobiology Of Sociopathy: An Integrated (http://www.bbsonline.org/Preprints/01dArchive/bbs.mealey.html) 
[6] Pinker, Steven (2002). The Blank Slate: The Modern Denial of Human Nature. New York: Viking. 
[7] Gould, S.J. (1996) "The Mismeasure of Man", Introduction to the Revised Edition 

[8] Chomsky, Noam (1995). "Rollback, Part II." (http://www.chomsky.mfo/articles/199505--.htm#TXT2.23) ZMagazine 8 (Feb.): 20-31. 
[9] Salter, Frank (2007). "Ethnic nepotism as heuristic." (http://books. google. com/books?id=usZR5aXFmiUC&pg=PA541) In R. Dunbar and 

L. Barrett Oxford handbook of evolutionary psychology. Oxford: Oxford University Press, pp. 541-551. 
[10] http://plato.stanford.edu/entries/sociobiology/ 

[II] http://www.uky.edu/AS/Philosophy/HarmonHolcomb.htm 
[12] http://www.pitt.edu/~jmbl65 

[13] http://www.bbsonline.org/Preprints/01dArchive/bbs.mealey.html 
[14] http://www.froes.dds.nl 



Sociobiology 230 

[15] http://www.prospect-magazine.co. uk/article_details.php?id=6467 

[16] http://www. psychology. uwo.ca/faculty/rushtonpdfs/N&N%202005-l.pdf 

[17] http://www.nytimes.com/2008/07/15/science/15wils.html?pagewanted=l 



Theoretical biology 



Mathematical and theoretical biology is an interdisciplinary academic research field with a range of applications in 
biology, medicine and biotechnology. The field may be referred to as mathematical biology or biomathematics 

[2] 

to stress the mathematical side, or as theoretical biology to stress the biological side. It includes at least four 
major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), 
bioinformatics and computational biomodelinglbiocomputing. 

Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, 
using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in 
biological, biomedical and biotechnology research. For example, in cell biology, protein interactions are often 
represented as "cartoon" models, which, although easy to visualize, do not accurately describe the systems studied. 
In order to do this, precise mathematical models are required. By describing the systems in a quantitative manner, 
their behavior can be better simulated, and hence properties can be predicted that might not be evident to the 
experimenter. 



Importance 



Applying mathematics to biology has a long history, but only recently has there been an explosion of interest in the 
field. Some reasons for this include: 

• the explosion of data-rich information sets, due to the genomics revolution, which are difficult to understand 
without the use of analytical tools, 

• recent development of mathematical tools such as chaos theory to help understand complex, nonlinear 
mechanisms in biology, 

• an increase in computing power which enables calculations and simulations to be performed that were not 
previously possible, and 

• an increasing interest in in silico experimentation due to ethical considerations, risk, unreliability and other 
complications involved in human and animal research. 

Areas of research 

Several areas of specialized research in mathematical and theoretical biology as well as external links 

to related projects in various universities are concisely presented in the following subsections, including also a large 
number of appropriate validating references from a list of several thousands of published authors contributing to this 
field. Many of the included examples are characterised by highly complex, nonlinear, and supercomplex 
mechanisms, as it is being increasingly recognised that the result of such interactions may only be understood 
through a combination of mathematical, logical, physical/chemical, molecular and computational models. Due to the 
wide diversity of specific knowledge involved, biomathematical research is often done in collaboration between 
mathematicians, biomathematicians, theoretical biologists, physicists, biophysicists, biochemists, bioengineers, 
engineers, biologists, physiologists, research physicians, biomedical researchers, oncologists, molecular biologists, 
geneticists, embryologists, zoologists, chemists, etc. 



Theoretical biology 23 1 

Computer models and automata theory 

A monograph on this topic summarizes an extensive amount of published research in this area up to 1987, 
including subsections in the following areas: computer modeling in biology and medicine, arterial system models, 
neuron models, biochemical and oscillation networks, quantum automata , quantum computers in molecular 
biology and genetics, cancer modelling, neural nets, genetic networks, abstract relational biology, 
metabolic-replication systems, category theory applications in biology and medicine, automata theory, cellular 
automata, tessallation models and complete self-reproduction , chaotic systems in organisms, relational 

biology and organismic theories. This published report also includes 390 references to peer-reviewed articles 

by a large number of authors. 

Modeling cell and molecular biology 

This area has received a boost due to the growing importance of molecular biology. 

ri7i 
Mechanics of biological tissues 

Theoretical enzymology and enzyme kinetics 

Cancer modelling and simulation 

Modelling the movement of interacting cell populations 

Mathematical modelling of scar tissue formation 

["221 

Mathematical modelling of intracellular dynamics 

T231 
Mathematical modelling of the cell cycle 

Modelling physiological systems 

T241 

• Modelling of arterial disease 

T251 

• Multi-scale modelling of the heart 

Molecular set theory 

Molecular set theory was introduced by Anthony Bartholomay, and its applications were developed in mathematical 
biology and especially in Mathematical Medicine. Molecular set theory (MST) is a mathematical formulation of 
the wide-sense chemical kinetics of biomolecular reactions in terms of sets of molecules and their chemical 
transformations represented by set-theoretical mappings between molecular sets. In a more general sense, MST is the 
theory of molecular categories defined as categories of molecular sets and their chemical transformations represented 
as set-theoretical mappings of molecular sets. The theory has also contributed to biostatistics and the formulation of 
clinical biochemistry problems in mathematical formulations of pathological, biochemical changes of interest to 
Physiology, Clinical Biochemistry and Medicine. 

Population dynamics 

Population dynamics has traditionally been the dominant field of mathematical biology. Work in this area dates back 
to the 19th century. The Lotka— Volterra predator-prey equations are a famous example. In the past 30 years, 
population dynamics has been complemented by evolutionary game theory, developed first by John Maynard Smith. 
Under these dynamics, evolutionary biology concepts may take a deterministic mathematical form. Population 
dynamics overlap with another active area of research in mathematical biology: mathematical epidemiology, the 
study of infectious disease affecting populations. Various models of the spread of infections have been proposed and 
analyzed, and provide important results that may be applied to health policy decisions. 



Theoretical biology 232 

Mathematical methods 

A model of a biological system is converted into a system of equations, although the word 'model' is often used 
synonymously with the system of corresponding equations. The solution of the equations, by either analytical or 
numerical means, describes how the biological system behaves either over time or at equilibrium. There are many 
different types of equations and the type of behavior that can occur is dependent on both the model and the equations 
used. The model often makes assumptions about the system. The equations may also make assumptions about the 
nature of what may occur. 

Mathematical biophysics 

The earlier stages of mathematical biology were dominated by mathematical biophysics, described as the application 
of mathematics in biophysics, often involving specific physical/mathematical models of biosystems and their 
components or compartments. 

The following is a list of mathematical descriptions and their assumptions. 

Deterministic processes (dynamical systems) 

A fixed mapping between an initial state and a final state. Starting from an initial condition and moving forward in 
time, a deterministic process will always generate the same trajectory and no two trajectories cross in state space. 

• Difference equations/Maps — discrete time, continuous state space. 

• Ordinary differential equations — continuous time, continuous state space, no spatial derivatives. See also: 
Numerical ordinary differential equations. 

• Partial differential equations — continuous time, continuous state space, spatial derivatives. See also: Numerical 
partial differential equations. 

Stochastic processes (random dynamical systems) 

A random mapping between an initial state and a final state, making the state of the system a random variable with a 
corresponding probability distribution. 

• Non-Markovian processes — generalized master equation — continuous time with memory of past events, discrete 
state space, waiting times of events (or transitions between states) discretely occur and have a generalized 
probability distribution. 

• Jump Markov process — master equation — continuous time with no memory of past events, discrete state space, 
waiting times between events discretely occur and are exponentially distributed. See also: Monte Carlo method 
for numerical simulation methods, specifically continuous -time Monte Carlo which is also called kinetic Monte 
Carlo or the stochastic simulation algorithm. 

• Continuous Markov process — stochastic differential equations or a Fokker-Planck equation — continuous time, 
continuous state space, events occur continuously according to a random Wiener process. 

Spatial modelling 

One classic work in this area is Alan Turing's paper on morphogenesis entitled The Chemical Basis of 
Morphogenesis, published in 1952 in the Philosophical Transactions of the Royal Society. 

rofil 

• Travelling waves in a wound-healing assay 

• Swarming behaviour 

• A mechanochemical theory of morphogenesis 

T311 

• Biological pattern formation 

[32] 

• Spatial distribution modeling using plot samples 



Theoretical biology 



233 



Phylogenetics 

Phylogenetics is an area that deals with the reconstruction and analysis of phylogenetic (evolutionary) trees and 

[331 
networks based on inherited characteristics 



Model example: the cell cycle 

The eukaryotic cell cycle is very complex and is one of the most studied topics, since its misregulation leads to 
cancers. It is possibly a good example of a mathematical model as it deals with simple calculus but gives valid 
results. Two research groups have produced several models of the cell cycle simulating several organisms. 

They have recently produced a generic eukaryotic cell cycle model which can represent a particular eukaryote 
depending on the values of the parameters, demonstrating that the idiosyncrasies of the individual cell cycles are due 
to different protein concentrations and affinities, while the underlying mechanisms are conserved (Csikasz-Nagy et 
al., 2006). 

By means of a system of ordinary differential equations these models show the change in time (dynamical system) of 
the protein inside a single typical cell; this type of model is called a deterministic process (whereas a model 
describing a statistical distribution of protein concentrations in a population of cells is called a stochastic process). 
To obtain these equations an iterative series of steps must be done: first the several models and observations are 
combined to form a consensus diagram and the appropriate kinetic laws are chosen to write the differential 
equations, such as rate kinetics for stoichiometric reactions, Michaelis-Menten kinetics for enzyme substrate 
reactions and Goldbeter— Koshland kinetics for ultrasensitive transcription factors, afterwards the parameters of the 
equations (rate constants, enzyme efficiency coefficients and Michealis constants) must be fitted to match 
observations; when they cannot be fitted the kinetic equation is revised and when that is not possible the wiring 
diagram is modified. The parameters are fitted and validated using observations of both wild type and mutants, such 
as protein half-life and cell size. 

In order to fit the parameters the differential equations need to be studied. This can be done either by simulation or 
by analysis. 

In a simulation, given a starting vector (list of the values of the variables), the progression of the system is calculated 
by solving the equations at each time-frame in small increments. 

In analysis, the proprieties of the 

equations are used to investigate the 

behavior of the system depending of 

the values of the parameters and 

variables. A system of differential 

equations can be represented as a 

vector field, where each vector 

described the change (in concentration 

of two or more protein) determining 

where and how fast the trajectory 

L 
(simulation) is heading. Vector fields 

can have several special points: a 

stable point, called a sink, that attracts in all directions (forcing the concentrations to be at a certain value), an 

unstable point, either a source or a saddle point which repels (forcing the concentrations to change away from a 

certain value), and a limit cycle, a closed trajectory towards which several trajectories spiral towards (making the 

concentrations oscillate). 

A better representation which can handle the large number of variables and parameters is called a bifurcation 



BIFURCATION DIAGRAM 

Fixed Points 




stable steady -state- 
Mass dictates tiic ;ici ve cydiiii levels because stable steady- 
states attract {rc:;s\ ■jo cisc-rvaUes) keeping [MPF] consiant 



• o Stable/Unstable limit cycle max/min: 

Tlhe system is in a loop so at that mass tho [f.lPFf will oscill: 

Singularities 

Saddle Node; 

5W' A ;,:,,(.'!■:> 2-r.u an :i;isuiblo stead y-slatos annihilate-, boyosid 
SN2 ivtiich Ihcis are no Btpafcrium points: those bifurcation 
events v.ilt trigger the exit framGf and G2 i e spec t!v sly 



we 



cell mass- (au.) 

dMass*di = S^o^-Mass [exponential growth) 
d[C3n2JM = (k s1 + k^ |SBF])mass - ka- [Gin 2] 

The parameter mass directly conlrolE cyelin levels, osjjfc-sslr j 

implicitly i(£ yet UnKhOwn mass dependant, pjntM mechanism 



Hopf Bifurcation 

A siable and an unslable steady-states annihilate .-esuhmg ^n 
an uiiii;ia:t- ihiit cycle [eigisiwalijiss have no Real part) 



SNIPER SNIPER Bifurcation 



diagram(Bifurcation theory): the presence of these special steady-state points at certain values of a parameter (e.g. 
mass) is represented by a point and once the parameter passes a certain value, a qualitative change occurs, called a 



Theoretical biology 234 

bifurcation, in which the nature of the space changes, with profound consequences for the protein concentrations: the 
cell cycle has phases (partially corresponding to Gl and G2) in which mass, via a stable point, controls cyclin levels, 
and phases (S and M phases) in which the concentrations change independently, but once the phase has changed at a 
bifurcation event (Cell cycle checkpoint), the system cannot go back to the previous levels since at the current mass 
the vector field is profoundly different and the mass cannot be reversed back through the bifurcation event, making a 
checkpoint irreversible. In particular the S and M checkpoints are regulated by means of special bifurcations called a 
Hopf bifurcation and an infinite period bifurcation. 

See also 

Abstract relational biology 

Artificial life 

Biocybernetics 

Bioinformatics 

Biologically inspired computing 

Biosemiotics 

Biostatistics 

T31 
Cellular automata 

Coalescent theory 

Complex systems biology 

Computational biology 

Digital morphogenesis 

t~> • i • u- i [3] [39] [40] [41] [42] [43] 

Dynamical systems in biology 

Epidemiology 

Evolution theories and Population Genetics 

• Population genetics models 

• Molecular evolution theories 
Ewens's sampling formula 
Excitable medium 
Journal of Theoretical Biology 
Mathematical models 

Molecular modelling 
Molecular modelling on GPU 

Software for molecular modeling 

[441 
Metabolic -replication systems 

Models of Growth and Form 

Neighbour-sensing model 

Morphometries 

/-. • • ♦ /A c\ [48] [45] 

Orgamsmic systems (OS) 

r> t [48] [39] [46] 

Orgamsmic supercategones 

Population dynamics of fisheries 

Protein folding, also blue Gene and folding© home 

Quantum computers 

Quantum genetics 

[471 

Relational biology 

Self-reproduction (also called self-replication in a more general context). 

Computational gene models 



Theoretical biology 235 

• Systems biology 

• Theoretical biology 

• Theoretical ecology 

• Topological models of morphogenesis 

• DNA topology 

• DNA sequencing theory 

For use of basic arithmetics in biology, see relevant topic, such as Serial dilution. 

• Biographies 

Charles Darwin 
D'Arcy Thompson 
Joseph Fourier 
Charles S. Peskin 
Nicolas Rashevsky 
Robert Rosen 
Rosalind Franklin 
Francis Crick 
Rene Thom 
Vito Volterra 

Societies and Institutes 

• Division of Mathematical Biology at NIMR 

• Society for Mathematical Biology 

• European Society for Mathematical and Theoretical Biology 

References 

• Nicolas Rashevsky. (1938)., Mathematical Biophysics. Chicago: University of Chicago Press. 

• Robert Rosen, Dynamical system theory in biology. New York, Wiley-Interscience (1970) ISBN 0-471-73550-7 

• Israel, G., 2005, "Book on mathematical biology" in Grattan-Guinness, I., ed., Landmark Writings in Western 
Mathematics. Elsevier: 936-44. 

• Israel G (1988). "On the contribution of Volterra and Lotka to the development of modern biomathematics". 
History and Philosophy of the Life Sciences 10 (1): 37-49. PMID 3045853. 

• Scudo FM (March 1971). "Vito Volterra and theoretical ecology". Theoretical Population Biology 2 (1): 1—23. 
doi:10.1016/0040-5809(71)90002-5. PMID 4950157. 

• S.H. Strogatz, Nonlinear dynamics and Chaos: Applications to Physics, Biology, Chemistry, and Engineering. 
Perseus, 2001, ISBN 0-7382-0453-6 

• N.G van Kampen, Stochastic Processes in Physics and Chemistry, North Holland., 3rd ed. 2001, ISBN 
0-444-89349-0 

• I. C. Baianu., Computer Models and Automata Theory in Biology and Medicine., Monograph, Ch.l 1 in M. 
Witten (Editor), Mathematical Models in Medicine, vol. 7., Vol. 7: 1513-1577 (1987),Pergamon Press:New York, 
(updated by Hsiao Chen Lin in 2004 ISBN 0-08-036377-6 
P.G Drazin, Nonlinear systems. C.U.P., 1992. ISBN 0-521-40668-4 
L. Edelstein-Keshet, Mathematical Models in Biology. SIAM, 2004. ISBN 0-07-554950-6 

G Forgacs and S. A. Newman, Biological Physics of the Developing Embryo. C.U.P., 2005. ISBN 0-521-78337-2 
A. Goldbeter, Biochemical oscillations and cellular rhythms. C.U.P., 1996. ISBN 0-521-59946-6 
L.G Harrison, Kinetic theory of living pattern. C.U.P., 1993. ISBN 0-521-30691-4 



Theoretical biology 236 

• F. Hoppensteadt, Mathematical theories of populations: demographics, genetics and epidemics. SIAM, 
Philadelphia, 1975 (reprinted 1993). ISBN 0-89871-017-0 

• D.W. Jordan and P. Smith, Nonlinear ordinary differential equations, 2nd ed. O.U.P., 1987. ISBN 0-19-856562-3 

• J.D. Murray, Mathematical Biology. Springer- Verlag, 3rd ed. in 2 vols.: Mathematical Biology: I. An 
Introduction, 2002 ISBN 0-387-95223-3; Mathematical Biology: II. Spatial Models and Biomedical Applications, 
2003 ISBN 0-387-95228-4. 

• E. Renshaw, Modelling biological populations in space and time. C.U.P., 1991. ISBN 0-521-44855-7 

• S.I. Rubinow, Introduction to mathematical biology. John Wiley, 1975. ISBN 0-471-74446-8 

• L.A. Segel, Modeling dynamic phenomena in molecular and cellular biology. C.U.P., 1984. ISBN 0-521-27477-X 

• L. Preziosi, Cancer Modelling and Simulation. Chapman Hall/CRC Press, 2003. ISBN 1-58488-361-8. 

Theoretical biology 

• Bonner, J. T. 1988. The Evolution of Complexity by Means of Natural Selection. Princeton: Princeton University 
Press. 

• Hertel, H. 1963. Structure, Form, Movement. New York: Reinhold Publishing Corp. 

• Mangel, M. 1990. Special Issue, Classics of Theoretical Biology (part 1). Bull. Math. Biol. 52(1/2): 1-318. 

• Mangel, M. 2006. The Theoretical Biologist's Toolbox. Quantitative Methods for Ecology and Evolutionary 
Biology. Cambridge University Press. 

• Prusinkiewicz, P. & Lindenmeyer, A. 1990. The Algorithmic Beauty of Plants. Berlin: Springer-Verlag. 

• Reinke, J. 1901. Einleitung in die theoretische Biologic Berlin: Verlag von Gebriider Paetel. 

• Thompson, D.W. 1942. On Growth and Form. 2nd ed. Cambridge: Cambridge University Press: 2. vols. 

• Uexkiill, J. v. 1920. Theoretische Biologic Berlin: Gebr. Paetel. 

• Vogel, S. 1988. Life's Devices: The Physical World of Animals and Plants. Princeton: Princeton University Press. 

• Waddington, C.H. 1968-1972. Towards a Theoretical Biology. 4 vols. Edinburg: Edinburg University Press. 



Hoppensteadt, F. (September 1995), "Getting Started in Mathematical Biology" , Notices of American 



Further reading 

• Hoppensteadt, F. (Sej 

Mathematical Society. 

[57] 

• Reed, M. C. (March 2004), "Why Is Mathematical Biology So Hard?" , Notices of American Mathematical 
Society. 

• May, R. M. (2004), "Uses and Abuses of Mathematics in Biology", Science 303 (5659): 790-793, 
doi: 10. 1 126/science. 1094442. 

• Murray, J. D. (1988), "How the leopard gets its spots?" [58] , Scientific American 258 (3): 80-87. 

• Schnell, S.; Grima, R.; Maini, P. K. (2007), "Multiscale Modeling in Biology" [59] , American Scientist 95: 
134-142. 

• Chen, Katherine C; Calzone, Laurence; Csikasz-Nagy, Attila (2004), "Integrative analysis of cell cycle control in 
budding yeast", Mol Biol Cell 15 (8): 3841-3862, doi:10.1091/mbc.E03-ll-0794. 

• Csikasz-Nagy, Attila; Battogtokh, Dorjsuren; Chen, Katherine C; Novak, Bela; Tyson, John J. (2006), "Analysis 
of a generic model of eukaryotic cell-cycle regulation", Biophys J. 90 (12): 4361—4379, 

doi: 10. 1529/biophysj. 106.08 1240. 

• Fuss, H; Dubitzky, Werner; Downes, C. Stephen; Kurth, Mary Jo (2005), "Mathematical models of cell cycle 
regulation", Brief Bioinform. 6 (2): 163-177, doi:10.1093/bib/6.2.163. 

• Lovrics, Anna; Csikasz-Nagy, Attila; Zselyl, Istvan Gy; Zador, Judit; Turanyi, Tamas; Novak, Bela (2006), 
"Time scale and dimension analysis of a budding yeast cell cycle model", BMC Bioinform. 9 (7): 494, 
doi:10.1186/1471-2105-7-494. 



Theoretical biology 237 

External links 

The Society for Mathematical Biology 

Theoretical and mathematical biology website 

Complexity Discussion Group 

UCLA Biocybernetics Laboratory 

TUCS Computational Biomodelling Laboratory 

Nagoya University Division of Biomodeling 

Technische Universiteit Biomodeling and Informatics 

BioCybernetics Wiki, a vertical wiki on biomedical cybernetics and systems biology 

Bulletin of Mathematical Biology 

European Society for Mathematical and Theoretical Biology 

Journal of Mathematical Biology 

Biomathematics Research Centre at University of Canterbury 

T721 
Centre for Mathematical Biology at Oxford University 

T731 
Mathematical Biology at the National Institute for Medical Research 

T741 
Institute for Medical BioMathematics 

[751 
Mathematical Biology Systems of Differential Equations from EqWorld: The World of Mathematical 

Equations 

Systems Biology Workbench - a set of tools for modelling biochemical networks 

T771 
The Collection of Biostatistics Research Archive 

T781 

Statistical Applications in Genetics and Molecular Biology 

[791 

The International Journal of Biostatistics 

Theoretical Modeling of Cellular Physiology at Ecole Normale Superieure, Paris 

Lists of references 

A general list of Theoretical biology/Mathematical biology references, including an updated list of actively 
contributing authors 

roil 

A list of references for applications of category theory in relational biology 

[89] 

An updated list of publications of theoretical biologist Robert Rosen 

~ [OQ] 

Theory of Biological Anthropology (Documents No. 9 and 10 in English) 

Drawing the Line Between Theoretical and Basic Biology (a forum article by Isidro T. Savillo) 

Related journals 

roc] 

Acta Biotheoretica 

Bioinformatics 

Biological Theory 

BioSystems 

Bulletin of Mathematical Biology [68] 

Ecological Modelling 

Journal of Mathematical Biology 

Journal of Theoretical Biology 

Journal of the Royal Society Interface 

[92] 
Mathematical Biosciences 

Medical Hypotheses 

[94] 
Rivista di Biologia-Biology Forum 

Theoretical and Applied Genetics 

Theoretical Biology and Medical Modelling 



Theoretical biology 238 

[97] 

• Theoretical Population Biology 

• Theory in Biosciences (formerly: Biologisches Zentralblatt) 

Related societies 

• ESMTB: European Society for Mathematical and Theoretical Biology 

[99] 

• The Israeli Society for Theoretical and Mathematical Biology 

• Societe Francophone de Biologie Theorique 

• International Society for Biosemiotic Studies 

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[3] Baianu, I. C; Brown, R.; Georgescu, G.; Glazebrook, J. F. (2006). "Complex Non-linear Biodynamics in Categories, Higher Dimensional 

Algebra and Lukasiewicz— Moisil Topos: Transformations of Neuronal, Genetic and Neoplastic Networks". Axiomathes 16: 65. 

doi:10.1007/sl0516-005-3973-8. 
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2008-09-10. 
[7] J. R. Junck. Ten Equations that Changed Biology: Mathematics in Problem-Solving Biology Curricula, Bioscene, (1997), 1-36 (http://acube. 

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[9] "bibliography for category theory/algebraic topology applications in physics" (http://planetphysics.org/encyclopedia/ 

BibliographyForCategoryTheoryAndAlgebraicTopologyApplicationsInTheoreticalPhysics.html). PlanetPhysics. . Retrieved 2010-03-17. 
[10] "bibliography for mathematical biophysics and mathematical medicine" (http://planetphysics.org/encyclopedia/ 

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[II] Modem Cellular Automata by Kendall Preston and M. J. B. Duff http://books.google.co. uk/books?id=10_0q_e-u_UC&dq=cellular+ 
automata+and+tessalation&pg=PPl&ots=ciXYCF3AYm&source=citation&sig=CtaUDhisM7MalS7rZfXvp689y-8&hl=en&sa=X& 
oi=book_result&resnum=12&ct=result 

[12] "Dual Tessellation - from Wolfram Math World" (http://mathworld.wolfram.com/DualTessellation.html). Mathworld.wolfram.com. 

2010-03-03. . Retrieved 2010-03-17. 
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Medicine, vol. 7., Ch.ll Pergamon Press, New York, 1513-1577. http://cogprints.org/3687/ 
[14] "Computer models and automata theory in biology and medicine I KLI Theory Lab" (http://theorylab.org/node/56690). Theorylab.org. 

2009-05-26. . Retrieved 2010-03-17. 
[15] Currently available for download as an updated PDF: http://cogprints.ecs.soton.ac.uk/archive/00003718/01/ 

COMPUTER_SIMULATIONCOMPUTABILITYBIOSYSTEMSrefnew.pdf 
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2010-03-17. 
[18] Oprisan, Sorinel A.; Oprisan, Ana (2006). "A Computational Model of Oncogenesis using the Systemic Approach". Axiomathes 16: 155. 

doi:10.1007/sl0516-005-4943-x. 
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2010-03-17. 
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Ma.hw.ac.uk. . Retrieved 2010-03-17. 
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[23] (http://mpf.biol.vt.edu/Research.html) 
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html). Maths.gla.ac.uk. . Retrieved 2010-03-17. 



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org/?op=getobj&from=objects&id=10770D 
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[33] Charles Semple (2003), Phylogenetics (http://books. google. co.uk/books?id=uR8i2qetjSAC), Oxford University Press, ISBN 

978-0-19-850942-4 
[34] "The JJ Tyson Lab" (http://mpf.biol.vt.edu/Tyson Lab. html). Virginia Tech. . Retrieved 2008-09-10. 
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2010-03-17. 
[38] Baianu, I. C; Brown, R.; Glazebrook, J. F. (2007). "Categorical Ontology of Complex Spacetime Structures: the Emergence of Life and 

Human Consciousness". Axiomathes 17: 223. doi: 10. 1007/sl05 16-007-901 1-2. 
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doi: 10.1007/BF02476770. 
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oclc/101642 
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539-61. doi:10.1007/BF02476770. PMID 4327361. 
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[50] "KLI Theory Lab" (http://www.kli.ac.at/theorylab/index.html). Kli.ac.at. . Retrieved 2010-03-17. 



Theoretical genetics 240 



Theoretical genetics 



Population genetics is the study of allele frequency distribution and change under the influence of the four main 
evolutionary processes: natural selection, genetic drift, mutation and gene flow. It also takes into account the factors 
of population subdivision and population structure. It attempts to explain such phenomena as adaptation and 
speciation. 

Population genetics was a vital ingredient in the emergence of the modern evolutionary synthesis. Its primary 
founders were Sewall Wright, J. B. S. Haldane and R. A. Fisher, who also laid the foundations for the related 
discipline of quantitative genetics. 

Fundamentals 

Population genetics concerns the genetic constitution of populations and how this constitution changes with time. A 
population is a set of organisms in which any pair of members can breed together. This implies that all members 
belong to the same species and live near each other. 

For example, all of the moths of the same species living in an isolated forest are a population. A gene in this 
population may have several alternate forms, which account for variations between the phenotypes of the organisms. 
An example might be a gene for coloration in moths that has two alleles: black and white. A gene pool is the 
complete set of alleles for a gene in a single population; the allele frequency for an allele is the fraction of the genes 
in the pool that is composed of that allele (for example, what fraction of moth coloration genes are the black allele). 
Evolution occurs when there are changes in the frequencies of alleles within a population of interbreeding 
organisms; for example, the allele for black color in a population of moths becoming more common. 

To understand the mechanisms that cause a population to evolve, it is useful to consider what conditions are required 
for a population not to evolve. The Hardy-Weinberg principle states that the frequencies of alleles (variations in a 
gene) in a sufficiently large population will remain constant if the only forces acting on that population are the 
random reshuffling of alleles during the formation of the sperm or egg, and the random combination of the alleles in 
these sex cells during fertilization. Such a population is said to be in Hardy-Weinberg equilibrium as it is not 



evolving. 



Theoretical genetics 



241 



Hardy- Weinberg principle 

The Hardy— Weinberg principle states 
that both allele and genotype 
frequencies in a population remain 
constant — that is, they are in 
equilibrium — from generation to 
generation unless specific disturbing 
influences are introduced. Outside the 
lab, one or more of these "disturbing 
influences" are always in effect. Hardy 
Weinberg equilibrium is impossible in 
nature. Genetic equilibrium is an ideal 
state that provides a baseline to measure 
genetic change against. 






p-0 
g-1 



0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 

Hardy— Weinberg principle for two alleles: the horizontal axis shows the two allele 

frequencies p and q and the vertical axis shows the genotype frequencies. Each graph 

shows one of the three possible genotypes. 



Allele frequencies in a population 

remain static across generations, 

provided the following conditions are at 

hand: random mating, no mutation (the 

alleles don't change), no migration or 

emigration (no exchange of alleles between populations), infinitely large population size, and no selective pressure 

for or against any traits. 

In the simplest case of a single locus with two alleles: the dominant allele is denoted A and the recessive a and their 
frequencies are denoted by p and q\ freq(A) = p; freq(a) = q; p + q = 1 . If the population is in equilibrium, then we 

2 2 

will have freq(AA) = p for the AA homozygotes in the population, freq(aa) = q for the aa homozygotes, and 
freq(Aa) = 2pq for the heterozygotes. 

Based on these equations, useful but difficult-to-measure facts about a population can be determined. For example, a 
patient's child is a carrier of a recessive mutation that causes cystic fibrosis in homozygous recessive children. The 
parent wants to know the probability of her grandchildren inheriting the disease. In order to answer this question, the 
genetic counselor must know the chance that the child will reproduce with a carrier of the recessive mutation. This 
fact may not be known, but disease frequency is known. We know that the disease is caused by the homozygous 
recessive genotype; we can use the Hardy— Weinberg principle to work backward from disease occurrence to the 
frequency of heterozygous recessive individuals. 

Scope and theoretical considerations 

The mathematics of population genetics were originally developed as part of the modern evolutionary synthesis. 
According to Beatty (1986), it defines the core of the modern synthesis. 

According to Lewontin (1974), the theoretical task for population genetics is a process in two spaces: a "genotypic 
space" and a "phenotypic space". The challenge of a complete theory of population genetics is to provide a set of 
laws that predictably map a population of genotypes (G ) to a phenotype space (P ), where selection takes place, and 
another set of laws that map the resulting population (P ) back to genotype space (G ) where Mendelian genetics can 
predict the next generation of genotypes, thus completing the cycle. Even leaving aside for the moment the 
non-Mendelian aspects of molecular genetics, this is clearly a gargantuan task. Visualizing this transformation 
schematically: 



G 1 ^ P 1 ^ p 2 ^ G2 Z4 G\ 



Theoretical genetics 242 

(adapted from Lewontin 1974, p. 12). XD 

T represents the genetic and epigenetic laws, the aspects of functional biology, or development, that transform a 
genotype into phenotype. We will refer to this as the "genotype-phenotype map". T is the transformation due to 
natural selection, T are epigenetic relations that predict genotypes based on the selected phenotypes and finally T 
the rules of Mendelian genetics. 

In practice, there are two bodies of evolutionary theory that exist in parallel, traditional population genetics operating 
in the genotype space and the biometric theory used in plant and animal breeding, operating in phenotype space. The 
missing part is the mapping between the genotype and phenotype space. This leads to a "sleight of hand" (as 
Lewontin terms it) whereby variables in the equations of one domain, are considered parameters or constants, where, 
in a full-treatment they would be transformed themselves by the evolutionary process and are in reality functions of 
the state variables in the other domain. The "sleight of hand" is assuming that we know this mapping. Proceeding as 
if we do understand it is enough to analyze many cases of interest. For example, if the phenotype is almost 
one-to-one with genotype (sickle-cell disease) or the time-scale is sufficiently short, the "constants" can be treated as 
such; however, there are many situations where it is inaccurate. 

The four processes 
Natural selection 

Natural selection is the process by which heritable traits that make it more likely for an organism to survive and 
successfully reproduce become more common in a population over successive generations. 

The natural genetic variation within a population of organisms means that some individuals will survive more 
successfully than others in their current environment. Factors which affect reproductive success are also important, 
an issue which Charles Darwin developed in his ideas on sexual selection. 

Natural selection acts on the phenotype, or the observable characteristics of an organism, but the genetic (heritable) 
basis of any phenotype which gives a reproductive advantage will become more common in a population (see allele 
frequency). Over time, this process can result in adaptations that specialize organisms for particular ecological niches 
and may eventually result in the emergence of new species. 

Natural selection is one of the cornerstones of modern biology. The term was introduced by Darwin in his 
groundbreaking 1859 book On the Origin of Species, in which natural selection was described by analogy to 
artificial selection, a process by which animals and plants with traits considered desirable by human breeders are 
systematically favored for reproduction. The concept of natural selection was originally developed in the absence of 
a valid theory of heredity; at the time of Darwin's writing, nothing was known of modern genetics. The union of 
traditional Darwinian evolution with subsequent discoveries in classical and molecular genetics is termed the modern 
evolutionary synthesis. Natural selection remains the primary explanation for adaptive evolution. 

Genetic drift 

Genetic drift is the change in the relative frequency in which a gene variant (allele) occurs in a population due to 
random sampling and chance. That is, the alleles in the offspring in the population are a random sample of those in 
the parents. And chance has a role in determining whether a given individual survives and reproduces. A population's 
allele frequency is the fraction or percentage of its gene copies compared to the total number of gene alleles that 
share a particular form. 

Genetic drift is an important evolutionary process which leads to changes in allele frequencies over time. It may 
cause gene variants to disappear completely, and thereby reduce genetic variability. In contrast to natural selection, 
which makes gene variants more common or less common depending on their reproductive success, the changes 
due to genetic drift are not driven by environmental or adaptive pressures, and may be beneficial, neutral, or 



Theoretical genetics 243 

detrimental to reproductive success. 

The effect of genetic drift is larger in small populations, and smaller in large populations. Vigorous debates wage 
among scientists over the relative importance of genetic drift compared with natural selection. Ronald Fisher held the 
view that genetic drift plays at the most a minor role in evolution, and this remained the dominant view for several 
decades. In 1968 Motoo Kimura rekindled the debate with his neutral theory of molecular evolution which claims 
that most of the changes in the genetic material are caused by genetic drift. 

Mutation 

Mutations are changes in the DNA sequence of a cell's genome and are caused by radiation, viruses, transposons and 
mutagenic chemicals, as well as errors that occur during meiosis or DNA replication. Errors are introduced 

particularly often in the process of DNA replication, in the polymerization of the second strand. These errors can 
also be induced by the organism itself, by cellular processes such as hypermutation. 

Mutations can have an impact on the phenotype of an organism, especially if they occur within the protein coding 
sequence of a gene. Error rates are usually very low (1 error in every 10 million— 100 million bases) due to the 
"proofreading" ability of DNA polymerases. Without proofreading, error rates are a thousand-fold higher. 

Chemical damage to DNA occurs naturally as well, and cells use DNA repair mechanisms to repair mismatches and 
breaks in DNA. Nevertheless, the repair sometimes fails to return the DNA to its original sequence. 

In organisms that use chromosomal crossover to exchange DNA and recombine genes, errors in alignment during 

ri3i 

meiosis can also cause mutations. Errors in crossover are especially likely when similar sequences cause partner 
chromosomes to adopt a mistaken alignment; this makes some regions in genomes more prone to mutating in this 
way. These errors create large structural changes in DNA sequence — duplications, inversions or deletions of entire 
regions, or the accidental exchanging of whole parts between different chromosomes (called translocation). 

Mutation can result in several different types of change in DNA sequences; these can either have no effect, alter the 
product of a gene, or prevent the gene from functioning. Studies in the fly Drosophila melanogaster suggest that if a 
mutation changes a protein produced by a gene, this will probably be harmful, with about 70 percent of these 

ri4i 

mutations having damaging effects, and the remainder being either neutral or weakly beneficial. Due to the 
damaging effects that mutations can have on cells, organisms have evolved mechanisms such as DNA repair to 

ro] 

remove mutations. Therefore, the optimal mutation rate for a species is a trade-off between costs of a high 
mutation rate, such as deleterious mutations, and the metabolic costs of maintaining systems to reduce the mutation 
rate, such as DNA repair enzymes. Viruses that use RNA as their genetic material have rapid mutation rates, 
which can be an advantage since these viruses will evolve constantly and rapidly, and thus evade the defensive 
responses of e.g. the human immune system. 

no] 

Mutations can involve large sections of DNA becoming duplicated, usually through genetic recombination. These 
duplications are a major source of raw material for evolving new genes, with tens to hundreds of genes duplicated in 
animal genomes every million years. Most genes belong to larger families of genes of shared ancestry. Novel 
genes are produced by several methods, commonly through the duplication and mutation of an ancestral gene, or by 

[21] [221 

recombining parts of different genes to form new combinations with new functions. 

Here, domains act as modules, each with a particular and independent function, that can be mixed together to 

[23] 
produce genes encoding new proteins with novel properties. For example, the human eye uses four genes to make 

structures that sense light: three for color vision and one for night vision; all four arose from a single ancestral 

T241 
gene. Another advantage of duplicating a gene (or even an entire genome) is that this increases redundancy; this 

allows one gene in the pair to acquire a new function while the other copy performs the original function. 

[27] [2S] 

Other types of mutation occasionally create new genes from previously noncoding DNA. 



Theoretical genetics 244 

Gene flow 

Gene flow is the exchange of genes between populations, which are usually of the same species. Examples of 
gene flow within a species include the migration and then breeding of organisms, or the exchange of pollen. Gene 
transfer between species includes the formation of hybrid organisms and horizontal gene transfer. 

Migration into or out of a population can change allele frequencies, as well as introducing genetic variation into a 
population. Immigration may add new genetic material to the established gene pool of a population. Conversely, 
emigration may remove genetic material. As barriers to reproduction between two diverging populations are required 
for the populations to become new species, gene flow may slow this process by spreading genetic differences 
between the populations. Gene flow is hindered by mountain ranges, oceans and deserts or even man-made 
structures such as the Great Wall of China, which has hindered the flow of plant genes. 

Depending on how far two species have diverged since their most recent common ancestor, it may still be possible 

T311 
for them to produce offspring, as with horses and donkeys mating to produce mules. Such hybrids are generally 

infertile, due to the two different sets of chromosomes being unable to pair up during meiosis. In this case, closely 

related species may regularly interbreed, but hybrids will be selected against and the species will remain distinct. 

However, viable hybrids are occasionally formed and these new species can either have properties intermediate 

T321 

between their parent species, or possess a totally new phenotype. The importance of hybridization in creating new 

[331 
species of animals is unclear, although cases have been seen in many types of animals, with the gray tree frog 

being a particularly well-studied example. 

Hybridization is, however, an important means of speciation in plants, since polyploidy (having more than two 
copies of each chromosome) is tolerated in plants more readily than in animals. Polyploidy is important in 

hybrids as it allows reproduction, with the two different sets of chromosomes each being able to pair with an 

T371 
identical partner during meiosis. Polyploids also have more genetic diversity, which allows them to avoid 

T3S1 

inbreeding depression in small populations. 

Horizontal gene transfer is the transfer of genetic material from one organism to another organism that is not its 

offspring; this is most common among bacteria. In medicine, this contributes to the spread of antibiotic resistance, 

as when one bacteria acquires resistance genes it can rapidly transfer them to other species. Horizontal transfer of 

genes from bacteria to eukaryotes such as the yeast Saccharomyces cerevisiae and the adzuki bean beetle 

Callosobruchus chinensis may also have occurred. An example of larger-scale transfers are the eukaryotic 

[43] 
bdelloid rotifers, which appear to have received a range of genes from bacteria, fungi, and plants. Viruses can 

also carry DNA between organisms, allowing transfer of genes even across biological domains. Large-scale gene 

transfer has also occurred between the ancestors of eukaryotic cells and prokaryotes, during the acquisition of 

chloroplasts and mitochondria. 

Gene flow is the transfer of alleles from one population to another. 

Migration into or out of a population may be responsible for a marked change in allele frequencies. Immigration may 
also result in the addition of new genetic variants to the established gene pool of a particular species or population. 

There are a number of factors that affect the rate of gene flow between different populations. One of the most 
significant factors is mobility, as greater mobility of an individual tends to give it greater migratory potential. 
Animals tend to be more mobile than plants, although pollen and seeds may be carried great distances by animals or 
wind. 

Maintained gene flow between two populations can also lead to a combination of the two gene pools, reducing the 
genetic variation between the two groups. It is for this reason that gene flow strongly acts against speciation, by 
recombining the gene pools of the groups, and thus, repairing the developing differences in genetic variation that 
would have led to full speciation and creation of daughter species. 

For example, if a species of grass grows on both sides of a highway, pollen is likely to be transported from one side 
to the other and vice versa. If this pollen is able to fertilise the plant where it ends up and produce viable offspring, 



Theoretical genetics 245 

then the alleles in the pollen have effectively been able to move from the population on one side of the highway to 
the other. 

Genetic structure 

Because of physical barriers to migration, along with limited vagility, and natal philopatry, natural populations are 
rarely panmictic (Buston et ah, 2007). There is usually a geographic range within which individuals are more closely 
related to one another than those randomly selected from the general population. This is described as the extent to 
which a population is genetically structured (Repaci et al, 2007). 

Microbial population genetics 

Microbial population genetics is a rapidly advancing field of investigation with relevance to many other theoretical 
and applied areas of scientific investigations. The population genetics of microorganisms lays the foundations for 
tracking the origin and evolution of antibiotic resistance and deadly infectious pathogens. Population genetics of 
microorganisms is also an essential factor for devising strategies for the conservation and better utilization of 
beneficial microbes (Xu, 2010). 

History 




...... 

Bistort betulariaf. typica is the white-bodied form of the peppered moth. 




Bistort betulariaf. carbonaria is the black-bodied form of the peppered moth. 

Population genetics 

The Mendelian and biometrician models were eventually reconciled, when population genetics was developed. A 
key step was the work of the British biologist and statistician R.A. Fisher. In a series of papers starting in 1918 and 
culminating in his 1930 book The Genetical Theory of Natural Selection, Fisher showed that the continuous variation 
measured by the biometricians could be produced by the combined action of many discrete genes, and that natural 
selection could change gene frequencies in a population, resulting in evolution. In a series of papers beginning in 
1924, another British geneticist, J.B.S. Haldane, applied statistical analysis to real-world examples of natural 
selection, such as the evolution of industrial melanism in peppered moths, and showed that natural selection worked 
at an even faster rate than Fisher assumed. 

The American biologist Sewall Wright, who had a background in animal breeding experiments, focused on 
combinations of interacting genes, and the effects of inbreeding on small, relatively isolated populations that 
exhibited genetic drift. In 1932, Wright introduced the concept of an adaptive landscape and argued that genetic drift 
and inbreeding could drive a small, isolated sub-population away from an adaptive peak, allowing natural selection 
to drive it towards different adaptive peaks. Fisher and Wright had some fundamental disagreements and a 



Theoretical genetics 246 

controversy about the relative roles of selection and drift continued for much of the century between the Americans 
and the British. The Frenchman Gustave Malecot was also important early in the development of the discipline. 

The work of Fisher, Haldane and Wright founded the discipline of population genetics. This integrated natural 
selection with Mendelian genetics, which was the critical first step in developing a unified theory of how evolution 
worked. 

John Maynard Smith was Haldane's pupil, whilst W.D. Hamilton was heavily influenced by the writings of Fisher. 
The American George R. Price worked with both Hamilton and Maynard Smith. American Richard Lewontin and 
Japanese Motoo Kimura were heavily influenced by Wright. 

Modern evolutionary synthesis 

In the first few decades of the 20th century, most field naturalists continued to believe that Lamarckian and 
orthogenic mechanisms of evolution provided the best explanation for the complexity they observed in the living 
world. However, as the field of genetics continued to develop, those views became less tenable. Theodosius 
Dobzhansky, a postdoctoral worker in T. H. Morgan's lab, had been influenced by the work on genetic diversity by 
Russian geneticists such as Sergei Chetverikov. He helped to bridge the divide between the foundations of 
microevolution developed by the population geneticists and the patterns of macroevolution observed by field 
biologists, with his 1937 book Genetics and the Origin of Species. 

Dobzhansky examined the genetic diversity of wild populations and showed that, contrary to the assumptions of the 
population geneticists, these populations had large amounts of genetic diversity, with marked differences between 
sub-populations. The book also took the highly mathematical work of the population geneticists and put it into a 
more accessible form. In Great Britain E.B. Ford, the pioneer of ecological genetics, continued throughout the 1930s 
and 1940s to demonstrate the power of selection due to ecological factors including the ability to maintain genetic 
diversity through genetic polymorphisms such as human blood types. Ford's work would contribute to a shift in 
emphasis during the course of the modern synthesis towards natural selection over genetic drift. 

See also 

Coalescent theory 

Dual inheritance theory 

Ecological genetics 

Evolutionarily Significant Unit 

Ewens's sampling formula 

Fitness landscape 

Founder effect 

Genetic diversity 

Genetic drift 

Genetic erosion 

Genetic pollution 

Gene pool 

Genotype-phenotype distinction 

Habitat fragmentation 

Hardy-Weinberg principle 

Microevolution 

Molecular evolution 

Muller's ratchet 

Mutational meltdown 

Neutral theory of molecular evolution 



Theoretical genetics 247 

• Population bottleneck 

• Quantitative genetics 

• Reproductive compensation 

• Selection 

• Small population size 

• Viral quasispecies 

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Theoretical genetics 249 

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External links 

• Yale University (http://alfred.med.yale.edu/alfred/) 

• EHSTRAFD.org - Earth Human STR Allele Frequencies Database (http://www.ehstrafd.org) 

• History of population genetics (http://www.esp.org/books/sturt/history/contents/sturt-history-ch-17.pdf) 

• National Geographic: Atlas of the Human Journey (https://www5.nationalgeographic.com/genographic/atlas. 
html) (Haplogroup-based human migration maps) 

• Monash Virtual Laboratory (http://vlab.infotech.monash.edu.au/simulations/cellular-automata/ 
population-genetics/) - Simulations of habitat fragmentation and population genetics online at Monash 
University's Virtual Laboratory. 



Theoretical ecology 250 



Theoretical ecology 



Theoretical ecology refers to several intellectual traditions. The tradition pursued in universities and scientific 
journals under the rubric of theoretical ecology addresses the equations and probability distributions that govern the 
demography and biogeography of species. Common topics of theoretical ecology include population dynamics and 
especially the mathematics of food webs, and competition. 

To a large extent theoretical ecology draws on the work of G. Evelyn Hutchinson and his students. Brothers H.T. 
Odum and E.P. Odum are seen as the true founders of modern theoretical ecology (sometimes described as 
ecosystem ecology). Robert MacArthur brought theory to community ecology. Daniel Simberloff was the student of 
E.O. Wilson, with whom MacArthur collaborated on The Theory of Island Biogeography, a seminal work in the 
development of theoretical ecology. Simberloff went on to add rigour to experimental ecology and was one of the 
stalwarts in the SLOSS Debate (whether it is preferable to protect a Single Large or Several Small reserves) and 
forced supporters of Jared Diamond's community assembly rules to defend their ideas through Neutral Model 
Analysis. Simberloff also played a key role in the (ongoing) debate on the utility of corridors for connecting isolated 
reserves (with Reed Noss taking the lead on the opposing side). 

MacArthur's students Stephen Hubbell and Michael Rosenzweig combined theoretical and practical elements into 
works that extended MacArthur and Wilson's Island Biogeography Theory - Hubbell with his Unified Neutral 
Theory of Biodiversity and Biogeography and Rosenzweig with is Species Diversity in Space and Time. 

Other key theoretical ecologists include Robert May, Robert Rosen, author of "Life Itself", G. David Tilman, and 
Robert Ulanowicz, author of Ecology: The Ascendant Perspective. 



Theoretical Ecologists 

G. Evelyn Hutchinson 
Robert MacArtur 
Edward O. Wilson 
Simon Levin 
Richard Levins 
Robert May 
George Sugihara 
Joel Cohen 
Robert V. ONeill 
Howard T. Odum 
Donald DeAngelis 
G David Tilman 
Robert Ulanowicz 



Theoretical ecology 25 1 

See also 

• Ecosystem model 

• Mathematical biology 

• Population dynamics 

• Population modeling 

• Theoretical biology 



Population dynamics 



Population dynamics is the branch of life sciences that studies short- and long-term changes in the size and age 
composition of populations, and the biological and environmental processes influencing those changes. Population 
dynamics deals with the way populations are affected by birth and death rates, and by immigration and emigration, 
and studies topics such as aging populations or population decline. 



History 



Population dynamics has traditionally been the dominant branch of mathematical biology, which has a history of 
more than 210 years, although more recently the scope of mathematical biology has greatly expanded. The first 
principle of population dynamics is widely regarded as the exponential law of Malthus, as modelled by the 
Malthusian growth model. The early period was dominated by demographic studies such as the work of Benjamin 
Gompertz and Pierre Francois Verhulst in the early 19th century, who refined and adjusted the Malthusian 
demographic model. 

A more general model formulation was proposed by F.J. Richards in 1959, further expanded by Simon Hopkins, in 
which the models of Gompertz, Verhulst and also Ludwig von Bertalanffy are covered as special cases of the general 
formulation. The Lotka— Volterra predator-prey equations are another famous example. The computer game SimCity 
and the MMORPG Ultima Online, among others, tried to simulate some of these population dynamics. 

In the past 30 years, population dynamics has been complemented by evolutionary game theory, developed first by 
John Maynard Smith. Under these dynamics, evolutionary biology concepts may take a deterministic mathematical 
form. Population dynamics overlap with another active area of research in mathematical biology: mathematical 
epidemiology, the study of infectious disease affecting populations. Various models of viral spread have been 
proposed and analysed, and provide important results that may be applied to health policy decisions. 

Fisheries and wildlife management 

In fisheries and wildlife management, population is affected by three dynamic rate functions. 

• Natality or birth rate, often recruitment, which means reaching a certain size or reproductive stage. Usually refers 
to the age a fish can be caught and counted in nets 

• Population growth rate, which measures the growth of individuals in size and length. More important in fisheries, 
where population is often measured in biomass. 

• Mortality, which includes harvest mortality and natural mortality. Natural mortality includes non-human 
predation, disease and old age. 

If N is the number of individuals at time 1 then 

N 1 = N +B -D + I-E 

where N is the number of individuals at time 0, B is the number of individuals born, D the number that died, I the 
number that immigrated, and E the number that emigrated between time and time 1 . 



Population dynamics 252 

If we measures these rates over many time intervals, we can determine how a population's density changes over time. 
Immigration and emigration are present, but are usually not measured. 

All of these are measured to determine the harvestable surplus, which is the number of individuals that can be 
harvested from a population without affecting long term stability, or average population size. The harvest within the 
harvestable surplus is considered compensatory mortality, where the harvest deaths are substituting for the deaths 
that would occur naturally. It started in Europe. Harvest beyond that is additive mortality, harvest in addition to all 
the animals that would have died naturally. These terms are not the universal good and evil of population 
management, for example, in deer, the DNR are trying to reduce deer population size overall to an extent, since 
hunters have reduced buck competition and increased deer population unnaturally. 

Intrinsic rate of increase 

The rate at which a population increases in size, i.e. the change in population size over a particular period of time is 
known as the intrinsic rate of increase. The concept is commonly used in insect population biology to determine 
how environmental factors affect the rate at which pest populations increase. 

See also 

Lotka— Volterra equation 

Minimum viable population 

Maximum sustainable yield 

Nicholson-Bailey model 

Nurgaliev's law 

Population cycle 

Population ecology 

Population genetics 

Population modeling 

Ricker model 

Societal collapse 

System dynamics 

Population dynamics of fisheries 

References 

[1] Jahn, GC, LP Almazan, and J Pacia. 2005. Effect of nitrogen fertilizer on the intrinsic rate of increase of the rusty plum aphid, Hysteroneura 
setariae (Thomas) (Homoptera: Aphididae) on rice (Oryza sativa L.). Environmental Entomology 34 (4): 938-943. (http://docserver.esa. 
catchword.org/deliver/cw/pdf/esa/freepdfs/0046225x/v34n4s26.pdf) 

• Introduction to Social Macrodynamics: Compact Macromodels of the World System Growth by Andrey 
Korotayev, Artemy Malkov, and Daria Khaltourina. ISBN 5-484-00414-4 

• Turchin, P. 2003. Complex Population Dynamics: a Theoretical/Empirical Synthesis. Princeton, NJ: Princeton 
University Press. 

• Weiss, V. 2007. The population cycle drives human history - from a eugenic phase into a dysgenic phase and 
eventual collapse. The Journal of Social, Political and Economic Studies 32: 327-358 (http://www.jspes.org/ 
fall2007_weiss.html) 



Population dynamics 



253 



External links 

• GreenBoxes code sharing network (http://iugo-cafe.org/greenboxes). Greenboxes (Beta) is a repository for 
open-source population modelling and PVA code. Greenboxes allows users an easy way to share their code and to 
search for others shared code. 

• The Virtual Handbook on Population Dynamics (http://www.thomas-brey.de/science/virtualhandbook). An 
online compilation of state-ot-the-art basic tools for the analysis of population dynamics with emphasis on benthic 
invertebrates. 

• Creatures! (http://wwwZ.futureskill.com) High School interactive simulation program that implements an agent 
based simulation of grass, rabbits and foxes. 



Ecology 




wtmxuL 

The scientific discipline of ecology encompasses areas from global processes (above), to the study of marine and 
terrestrial habitats (middle) to interspecific interactions such as predation and pollination (below). 

Ecology (from Greek: olkoq, "house" or "living relations"; -XoyLa, "study of") is the interdisciplinary scientific study 
of the distributions, abundance and relations of organisms and their interactions with the environment. Ecology is 
also the study of ecosystems. Ecosystems describe the web or network of relations among organisms at different 
scales of organization. Since ecology refers to any form of biodiversity, ecologists research everything from tiny 
bacteria's role in nutrient recycling to the effects of tropical rain forest on the Earth's atmosphere. The discipline of 
ecology emerged from the natural sciences in the late 19th century. Ecology is not synonymous with environment, 
environmentalism, or environmental science. Ecology is closely related to the disciplines of physiology, 

Ml 

evolution, genetics and behavior. 

Like many of the natural sciences, a conceptual understanding of ecology is found in the broader details of study, 
including: 

• life processes explaining adaptations 

• distribution and abundance of organisms 

• the movement of materials and energy through living communities 

• the successional development of ecosystems, and 



the abundance and distribution of biodiversity in context of the environment 



[1] [2] [3] 



Ecology 



254 



Ecology is distinguished from natural history, which deals primarily with the descriptive study of organisms. It is a 
sub-discipline of biology, which is the study of life. 

There are many practical applications of ecology in conservation biology, wetland management, natural resource 
management (agriculture, forestry , fisheries), city planning (urban ecology), community health, economics, basic & 
applied science and it provides a conceptual framework for understanding and researching human social interaction 
(human ecology). 




Levels of organization and study 

Because ecology deals with ever-changing 
ecosystems, both time and space must be 

considered when describing ecological 

191 
phenomena. In regards to time, it can take 

thousands of years for ecological processes 

to mature. The life-span of a tree, for 

example, can include different successional 

or serai stages leading to mature old-growth 

forests. The ecological process is extended 

even further through time as trees topple over and decay. 

Ecosystems are also classified at different spatial scales: the area of an ecosystem can vary greatly from tiny to vast. 
For instance, several generations of an aphid population and their predators might exist on a single leaf. Inside each 
of those aphids exist diverse communities of bacteria. The scale of study must at times be quite large, when 
studying the life of the tree in the forest where bacteria and aphids live. To understand tree growth, for example, 
soil type, moisture content, slope of the land, forest canopy closure, and other local site variables must all be 
examined; to understand the ecology of the forest, complex global factors such as climate must be considered 



Ecosystems regenerate after a disturbance such as fire, forming mosaics of 

different age groups structured across a landscape. Pictured are different serai 

stages in forested ecosystems starting from pioneers colonizing a disturbed site and 

maturing in successional stages leading to old-growth forests. 



[121 



Long-term ecological studies provide important track records to better understand ecosystems over space and time. 

1131 
The International Long Term Ecological Network manages and exchanges scientific information among research 

sites. The longest experiment in existence is the Park Grass Experiment that was initiated in 1856. Another 

example includes the Hubbard Brook study in operation since 1960. Ecology is also complicated by the fact that 

small scale patterns do not necessarily explain large scale phenomena, otherwise captured in the expression 'the sum 

is greater than the parts'. These emergent phenomena operate at different environmental scales of influence, 

ranging from molecular to planetary scales, and require different sets of scientific explanation. 

To structure the study of ecology into a manageable framework of understanding, the biological world is 
conceptually organized as a nested hierarchy of organization, ranging in scale from genes, to cells, to tissues, to 
organs, to organisms, to species and up to the level of the biosphere. Ecosystems are primarily researched at (but 
not restricted to) three key levels of organization, including organisms, populations, and communities. Ecologists 
study ecosystems by sampling a certain number of individuals that are representative of a population. Ecosystems 
consist of communities interacting with each other and the environment. In ecology, communities are created by the 
interaction of the populations of different species in an area 



[21] [22] 



Biodiversity is an attribute of a site or area that consists of the variety within and among biotic communities, whether influenced by 
humans or not, at any spatial scale from microsites and habitat patches to the entire biosphere. 

Biodiversity (an amalgamation of the words biological diversity) describes all varieties of life from genes to 
ecosystems and spans every level of biological organization. There are many ways to index, measure, and represent 

[24] 

biodiversity. Biodiversity includes species diversity, ecosystem diversity, genetic diversity and the complex 
processes operating at and among these respective levels. Biodiversity plays an important role in 

1271 T2R1 

ecological health as much as it does for human health. Preventing or prioritizing species extinctions is one 



Ecology 



255 



way to preserve biodiversity, but populations, the genetic diversity within them and ecological processes, such as 
migration, are being threatened on global scales and disappearing rapidly as well. Conservation priorities and 
management techniques require different approaches and considerations to address the full ecological scope of 
biodiversity. Populations and species migration, for example, are more sensitive indicators of ecosystem services that 



sustain and contribute natural capital toward the well-being of humanity 



[29] [30] [31] [32] 



An understanding of 



biodiversity has practical application for ecosystem-based conservation planners as they make ecologically 



responsible decisions in management recommendations to consultant firms, governments and industry 



[33] 



Ecological niche 

The ecological niche is a central concept in the ecology of organisms. There are 
many definitions of the niche dating back to 1917, but George Evelyn 

T371 T381 

Hutchinson made conceptual advances in 1957 and introduced the most 

widely accepted definition: "The niche is the set of biotic and abiotic conditions 

in which a species is able to persist and maintain stable population sizes." 

'519 

The ecological niche is divided into the fundamental and the realized niche. 

The fundamental niche is the set of environmental conditions under which a 

species is able to persist. The realized niche is the set of environmental plus 

ecological conditions under which a species is able to persist. 

Organisms have functional traits that are uniquely adapted to the ecological 

niche. A trait is a measurable property of an organism that strongly influences 

[39] 
its performance. Biogeographical patterns and range distributions are 

explained or predicted through knowledge and understanding of a species niche 

requirements. For example, the uniquely adapted nature of each species to 

their ecological niche means that they are able to competitively exclude other 

similarly adapted species from having an overlapping geographic range. This is 

[41] 

called the competitive exclusion principle. Important to the concept of niche 
is habitat. The habitat describes the environment over which a species is known 

[42] 

to occur and the type of community that is formed as a result. For example, 
habitat might refer to an aquatic or terrestrial environment that can be further 
categorized as montane or alpine ecosystems. 




Termite mounds with varied heights of 

chimneys regulate gas exchange, 
temperature and other environmental 
parameters that are needed to sustain 

the internal physiology of the entire 

, [34ff35] 
colony. 



Ecology 



256 



Organisms are subject to environmental pressures, but they are also modifiers 
of their habitats. The regulatory feedback between organisms and their 
environment can modify conditions from local (e.g., a pond) to global scales 
(e.g., Gaia) and over time and even after death, such as decaying logs or silica 
skeleton deposits from marine organisms. This process of ecosystem 
engineering has also been called niche construction. Ecosystem engineers are 
defined as: "...organisms that directly or indirectly modulate the availability of 
resources to other species, by causing physical state changes in biotic or abiotic 

[45] -373 

materials. In so doing they modify, maintain and create habitats." 

The ecological engineering concept has stimulated a new appreciation for the 
degree of influence that organisms have on the ecosystem and evolutionary 
process. The niche construction concept highlights a previously 
underappreciated feedback mechanism of natural selection imparting forces on 
the abiotic niche. An example of natural selection through ecosystem 

engineering occurs in the nests of social insects, including ants, bees, wasps, 
and termites. There is an emergent homeostasis in the structure of the nest that 
regulates, maintains and defends the physiology of the entire colony. Termite 
mounds, for example, maintain a constant internal temperature through the 
design of air-conditioning chimneys. The structure of the nests themselves are subject to the forces of natural 
selection. Moreover, the nest can survive over successive generations, which means that ancestors inherit both 



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adapt and modify their environment by 

forming calcium carbonate skeletons 

that provide growing conditions for 

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genetic material and a legacy niche that was constructed before their time 



[34] [35] [47] 



Population ecology 

The population is the unit of analysis in population ecology. A population consists of individuals of the same species 

that live, interact and migrate through the same niche and habitat. A primary law of population ecology is the 

[49] 
Malthusian growth model. This law states that: 

"...a population will grow (or decline) exponentially as long as the environment experienced by all individuals in the 
population remains constant." 

This Malthusian premise provides the basis for formulating predictive theories and tests that follow. Simplified 
population models usually start with four variables including death, birth, immigration, and emigration. 
Mathematical models are used to calculate changes in population demographics using a null model. A null model is 
used as a null hypothesis for statistical testing. The null hypothesis states that random processes create observed 
patterns. Alternatively the patterns differ significantly from the random model and require further explanation. 
Models can be mathematically complex where "...several competing hypotheses are simultaneously confronted with 
the data." An example of an introductory population model describes a closed population, such as on an island, 
where immigration and emigration does not take place. In these island models the per capita rates of change are 
described as: 

dN/dT = B - D = bN - dN = (b - d)N = rN , 

where N is the total number of individuals in the population, B is the number of births, D is the number of deaths, b 

and d are the per capita rates of birth and death respectively, and r is the per capita rate of population change. This 

[491 
formula can be read out as the rate of change in the population (dN/dT) is equal to births minus deaths (B — D). 

[51] 



Using these modelling techniques, Malthus' population principle of growth was later transformed into a model 
known as the logistic equation: 



dN/dT = aN(l-N/K), 



Ecology 257 

where N is the number of individuals measured as biomass density, a is the maximum per-capita rate of change, and 
K is the carrying capacity of the population. The formula can be read as follows: the rate of change in the population 
(dN/dT) is equal to growth (aN) that is limited by carrying capacity (1 —N/K). The discipline of population ecology 
builds upon these introductory models to further understand demographic processes in real study populations and 
conduct statistical tests. The field of population ecology often uses data on life history and matrix algebra to develop 
projection matrices on fecundity and survivorship. This information is used for managing wildlife stocks and setting 
harvest quotas. 

T531 
A list of terms that define various types of natural groupings of individuals that are used in population studies 



Term Definition 

Species All individuals of a species. 

population 



Metapopulation A set of spatially disjunct populations, among which there is some immigration. 

Population A group of conspecific individuals that is demographically, genetically, or spatially disjunct from other 

groups of individuals. 



Aggregation A spatially clustered group of individuals. 

Deme A group of individuals more genetically similar to each other than to other individuals, usually with some 

degree of spatial isolation as well. 



Local population A group of individuals within an investigator-delimited area smaller than the geographic range of the species 
and often within a population (as defined above). A local population could be a disjunct population as well. 

Subpopulation An arbitrary spatially-delimited subset of individuals from within a population (as defined above). 



r/A-Selection theory 

A important concept in population ecology is r/K-selection theory. The concept was introduced in 1967 in a book 
entitled The Theory of Island Biogeography and was one of the first predictive models to explain life-history 
evolution. The premise behind this model is that forces of natural selection change according to the density of the 
population. When an island is first colonized, the density of individuals is low and the population size increases with 
reduced levels of competition and an abundance of available resources. Under such circumstances a population 
experiences density independent forces of natural selection, which is called r-selection. When the population 
becomes crowded, it reaches the island's carrying capacity, and individuals compete more heavily for limited 
resources. Under crowded conditions the population experiences density-dependent forces of natural selection, called 
^-selection. 

In the r/K-selection model, the first variable r is the intrinsic rate of natural increase in population size and the 
second variable K is the carrying capacity of a population. Different species evolve different life-history strategies 
spanning a continuum between these two selective forces. An r-selected species is one that has high birth rates, low 
levels of parental investment, and high rates of mortality before individuals reach maturity. Evolution favors high 
rates of fecundity in r-selected species. Many kinds of insects and invasive species exhibit r-selected characteristics. 
In contrast, a ^-selected species has low rates of fecundity, high levels of parental investment in the young, and low 
rates of mortality as individuals mature. Humans and elephants are examples of species exhibiting ^-selected 
characteristics, including longevity and efficiency in the conversion of more resources into fewer offspring. 



Ecology 258 

Metapopulation ecology 

Populations are also studied and conceptualized through the metapopulation concept. The metapopulation concept 

was introduced in 1969 :"as a population of populations which go extinct locally and recolonize. 

[59] 
Metapopulation ecology is another statistical approach that is often used in conservation research. 

Metapopulation research simplifies the landscape into patches of varying levels of quality. Like the r/K-selection 

model, metapopulation models have also been used to explain life-history evolution, such as the ecological stability 

of amphibian metamorphosis shifting life stages out of aquatic patches and into terrestrial patches. In 

metapopulation terminology there are emigrants (individuals that leave a patch), immigrants (individuals that move 

into a patch) and sites are classed either as sources or sinks. A site is a generic term that refers to places where 

ecologists sample populations, such as ponds or defined sampling areas in a forest. Source patches are productive 

sites that generate a seasonal supply of juveniles that migrate to other patch locations. Sink patches are unproductive 

sites that only receive migrants and will go extinct unless rescued by an adjacent source patch or environmental 

conditions become more favorable. Metapopulation models examine patch dynamics over time to answer questions 

about spatial and demographic ecology. The ecology of metapopulations is a dynamic process of extinction and 

colonization. Small patches of lower quality (i.e., sinks) are maintained or rescued by a seasonal influx of new 

immigrants. A dynamic metapopulation structure evolves from year to year, where some patches are sinks in dry 

years and become sources when conditions are more favorable. Ecologists use a mixture of computer models and 

field studies to explain metapopulation structure. 

Community ecology 

Community ecology examines how interactions among species and their environment affect the abundance, distribution and 
diversity of species within communities. 

Johnson & Stinchcomb 

Community ecology is a subdiscipline of ecology which studies the distribution, abundance, demography, and 
interactions between coexisting populations. An example of a study in community ecology might measure primary 
production in a wetland in relation to decomposition and consumption rates. This requires an understanding of the 
community connections between plants (i.e., primary producers) and the decomposers (e.g., fungi and bacteria). 
or the analysis of predator-prey dynamics affecting amphibian biomass. Food webs and trophic levels are two 
widely employed conceptual models used to explain the linkages among species. 



Ecology 



259 



Food webs 

Food webs are a type of concept map that illustrate ecological pathways, usually starting with solar energy being 
used by plants during photosynthesis. As plants grow, they accumulate carbohydrates and are eaten by grazing 
herbivores. Step by step lines or relations are drawn until a web of life is illustrated. 

There are different ecological dimensions that can be 
mapped to create more complicated food webs, 
including: species composition (type of species), 
richness (number of species), biomass (the dry weight 
of plants and animals), productivity (rates of 
conversion of energy and nutrients into growth), and 
stability (food webs over time). A food web diagram 
illustrating species composition shows how change in a 
single species can directly and indirectly influence 
many others. Microcosm studies are used to simplify 
food web research into semi-isolated units such as 
small springs, decaying logs, and laboratory 
experiments using organisms that reproduce quickly, 
such as daphnia feeding on algae grown under 
controlled environments in jars of water 



[73] [74] 




Principles gleaned from food web microcosm studies 

are used to extrapolate smaller dynamic concepts to 

larger systems. Food webs are limited because they 

are generally restricted to a specific habitat, such as a 

cave or a pond. The food web illustration (right) only shows a small part of the complexity connecting the aquatic 

system to the adjacent terrestrial land. Many of these species migrate into other habitats to distribute their effects on 

a larger scale. In other words, food webs are incomplete, but are nonetheless a valuable tool in understanding 

community ecosystems. 

Food chain length is another way of describing food webs as a measure of the number of species encountered as 
energy or nutrients move from the plants to top predators. ' There are different ways of calculating food chain 
length depending on what parameters of the food web dynamic are being considered: connectance, energy, or 
interaction. In a simple predator-prey example, a deer is one step removed from the plants it eats (chain length = 
1) and a wolf that eats the deer is two steps removed (chain length = 2). The relative amount or strength of influence 
that these parameters have on the food web address questions about: 

• the identity or existence of a few dominant species (called strong interactors or keystone species) 

• the total number of species and food-chain length (including many weak interactors) and 

[741 

• how community structure, function and stability is determined. 



Ecology 



260 



Trophic dynamics 




The Greek root of the word troph, tpoepr], trophe, 
means food or feeding. Links in food-webs 
primarily connect feeding relations or trophism 
among species. Biodiversity within ecosystems 
can be organized into vertical and horizontal 
dimensions. The vertical dimension represents 
feeding relations that become further removed 
from the base of the food chain up toward top 

predators. The horizontal dimension represents 

T771 
the abundance or biomass at each level. When 

the relative abundance or biomass of each 

functional feeding group is stacked into their 

respective trophic levels they naturally sort into a 

T7R1 

'pyramid of numbers'. Functional groups are 

broadly categorized as autotrophs (or primary 

producers), heterotrophs (or consumers), and 

detrivores (or decomposers). Heterotrophs can be further sub-divided into different functional groups, including: 

primary consumers (strict herbivores), secondary consumers (predators that feed exclusively on herbivores) and 

[79] 
tertiary consumers (predators that feed on a mix of herbivores and predators). Omnivores do not fit neatly into a 

functional category because they eat both plant and animal tissues. It has been suggested, however, that omnivores 

have a greater functional influence as predators because relative to herbivores they are comparatively inefficient at 

[80] 

grazing. 



Soils / Decomposers 
A stylized image of a trophic pyramid with soil illustrated at the base 



Ecologist collect data on trophic levels and food webs to statistically model and mathematically calculate 
parameters, such as those used in other kinds of network analysis (e.g., graph theory), to study emergent patterns and 
properties shared among ecosystems. The emergent pyramidal arrangement of trophic levels with amounts of energy 
transfer decreasing as species become further removed from the source of production is one of several patterns that is 

1721 TRll TR21 

repeated amongst the planets ecosystems. The size of each level in the pyramid generally represents 

biomass, which can be measured as the dry weight of an organism. Autotrophs may have the highest global 
proportion of biomass, but they are closely rivaled or surpassed by microbes. 

The decomposition of dead organic matter, such as leaves falling on the forest floor, turns into soils that feed plant 
production. The total sum of the planet's soil ecosystems is called the pedosphere where a very large proportion of 
the Earth's biodiversity sorts into other trophic levels. Invertebrates that feed and shred larger leaves, for example, 
create smaller bits for smaller organisms in the feeding chain. Collectively, these are the detrivores that regulate soil 
formation. Tree roots, fungi, bacteria, worms, ants, beetles, centipedes, spiders, mammals, birds, reptiles, 

amphibians and other less familiar creatures all work to create the trophic web of life in soil ecosystems. As 
organisms feed and migrate through soils they physically displace materials, which is an important ecological 
process called bioturbation. Biomass of soil microorganisms are influenced by and feed back into the trophic 
dynamics of the exposed solar surface ecology. Paleoecological studies of soils places the origin for bioturbation to a 
time before the Cambrian period. Other events, such as the evolution of trees and amphibians moving into land in the 
Devonian period played a significant role in the development of soils and ecological trophism. 



Ecology 261. 

List of ecological functional groups, definitions and examples 



Functional Group 


Definition and Examples 


Producers or 
Autotrophs 


Usually plants or cyanobacteria that are capable of photosynthesis but could be other organisms such as 
the bacteria near ocean vents that are capable of chemosynthesis. 


Consumers or 
Heterotrophs 


Animals, which can be primary consumers (herbivorous), or secondary or tertiary consumers 
(carnivorous and omnivores). 


Decomposers or 
Detritivores 


Bacteria, fungi, and insects which degrade organic matter of all types and restore nutrients to the 
environment. The producers will then consume the nutrients, completing the cycle. 



Functional trophic groups sort out hierarchically into pyramidic trophic levels because it requires specialized 
adaptations to become a photosynthesizer or a predator, so few organisms have the adaptations needed to combine 
both abilities. This explains why functional adaptations to trophism (feeding) organizes different species into 
emergent functional groups. Trophic levels are part of the holistic or complex systems view of ecosystems. 
Each trophic level contains unrelated species that grouped together because they share common ecological functions. 
Grouping functionally similar species into a trophic system gives a macroscopic image of the larger functional 
design. 

Links in a food-web illustrate direct trophic relations among species, but there are also indirect effects that can alter 

the abundance, distribution, or biomass in the trophic levels. For example, predators eating herbivores indirectly 

influence the control and regulation of primary production in plants. Although the predators do not eat the plants 

directly, they regulate the population of herbibores that are directly linked to plant trophism. The net effect of direct 

and indirect relations is called trophic cascades. Trophic cascades are separated into species-level cascades, where 

only a subset of the food-web dynamic is impacted by a change in population numbers, and community-level 

cascades, where a change in population numbers has a dramatic effect on the entire food-web, such as the 

[92] 
distribution of plant biomass. 

Keystone species 

A keystone species is a species that is disproportionately connected to more species in the food-web. Keystone 

species have lower levels of biomass in the trophic pyramid relative to the importance of their role. The many 

connections that a keystone species holds means that it maintains the organization and structure of entire 

communities. The loss of a keystone species results in a range of dramatic cascading effects that alters trophic 

[93] [94] 
dynamics, other food-web connections and can cause the extinction of other species in the community. 

Sea otters {Enhydra lutris) are commonly cited as an example of a keystone species because they limit the density of 

sea urchins that feed on kelp. If sea otters are removed from the system, the urchins graze until the kelp beds 

[95] 
disappear and this has a dramatic effect on community structure. Hunting of sea otters, for example, is thought to 

have indirectly lead to the extinction of the Steller's Sea Cow (Hydrodamalis gigas). While the keystone species 

concept has been used extensively as a conservation tool, it has been criticized for being poorly defined from an 

operational stance. It is very difficult to experimentally determine in each different ecosystem what species may hold 

a keystone role. Furthermore, food-web theory suggests that keystone species may not be all that common. It is 

[92] [95] 
therefore unclear how generally the keystone species model can be applied. 

Biome and biosphere 

Ecological units of organization are defined through reference to any magnitude of space and time on the planet. 
Communities of organisms, for example, are somewhat arbitrarily defined, but the processes of life integrate at 
different levels and organize into more complex wholes. Biomes, for example, are a larger unit of organization that 
categorize regions of the Earth's ecosystems mainly according to the structure and composition of vegetation. 
Different researchers have applied different methods to define continental boundaries of biomes dominated by 



Ecology 262 

different functional types of vegetative communities that are limited in distribution by climate, precipitation, weather 

and other environmental variables. Examples of biome names include: tropical rainforest, temperate broadleaf and 

mixed forests, temperate deciduous forest, taiga, tundra, hot desert, and polar desert. Other researchers have 

recently started to categorize other types of biomes, such as the human and oceanic microbiomes. To a microbe, the 

[99] 
human body is a habitat and a landscape. The microbiome has been largely discovered through advances in 

molecular genetics that have revealed a hidden richness of microbial diversity on the planet. The oceanic 

microbiome plays a significant role in the ecological biogeochemistry of the planet's oceans. 

Ecological theory has been used to explain self-emergent regulatory phenomena at the planetary scale. The largest 
scale of ecological organization is the biosphere: the total sum of ecosystems on the planet. Ecological relations 
regulate the flux of energy, nutrients, and climate all the way up to the planetary scale. For example, the dynamic 
history of the planetary CO and O composition of the atmosphere has been largely determined by the biogenic flux 
of gases coming from respiration and photosynthesis, with levels fluctuating over time and in relation to the ecology 
and evolution of plants and animals. When sub-component parts are organized into a whole there are oftentimes 

ri9i 

emergent properties that describe the nature of the system. This the Gaia hypothesis, and is an example of holism 
applied in ecological theory. The ecology of the planet acts as a single regulatory or holistic unit called Gaia. 
The Gaia hypothesis states that there is an emergent feedback loop generated by the metabolism of living organisms 
that maintains the temperature of the Earth and atmospheric conditions within a narrow self-regulating range of 
tolerance. 

Ecology and evolution 

Ecology and evolution are considered sister disciplines of the life sciences. Natural selection, life history, 
development, adaptation, populations, and inheritance are examples of concepts that thread equally into ecological 
and evolutionary theory. Morphological, behavioural and/or genetic traits, for example, can be mapped onto 
evolutionary trees to study the historical development of a species in relation to their functions and roles in different 
ecological circumstances. In this framework, the analytical tools of ecologists and evolutionists overlap as they 
organize, classify and investigate life through common systematic principals, such as phylogenetics or the Linnaean 
system of taxonomy. The two disciplines often appear together, such as in the title of the journal Trends in 
Ecology and Evolution} There is no sharp boundary separating ecology from evolution and they differ more in 
their areas of applied focus. Both disciplines discover and explain emergent and unique properties and processes 
operating across different spatial or temporal scales of organization. While the boundary between 

ecology and evolution is not always clear, it is understood that ecologists study the abiotic and biotic factors that 
influence the evolutionary process. 



Ecology 



263 



Behavioral ecology 



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All organisms are motile to some 
extent. Behavioural ecology is the study 
of ethology and its ecological and 
evolutionary implications. Ethology is 
the study of observable movement or 
behaviour in nature. This could include 
investigations of motile sperm of plants 
and zooplankton swimming toward the 
female egg, the cultivation of fungi by 
weevils, the mating dance of a 
salamander, or social gatherings of 
amoeba. [108][109][110][111][112] 

Adaptation is the central unifying 
concept in behavioral 

ecology. "International Society for 
Behavioral Ecology" . Behaviors can be recorded as traits and inherited in much the same way that eye and hair 

color can. Behaviours evolve and become adapted to the ecosystem because they are subject to the forces of natural 

T251 
selection. Hence, behaviors can be adaptive, meaning that they evolve functional utilities that increases 

reproductive success for the individuals that inherit such traits. This is also the technical definition for fitness in 

1251 
biology, which is a measure of reproductive success over successive generations. 



300 500 700 

wavelength 

Social display and color variation in differently adapted species of chameleons 

(Bradypodion spp.). Chameleons change their skin color to match their background as a 

behavioral defense mechanism and also use color to communicate with other members 

of their species, such as dominant (left) versus submissive (right) patterns shown in the 

three species (A-C) above. 



[115] 



Predator-prey interactions are an introductory concept into food-web studies as well as behavioural ecology. 
Prey species can exhibit different kinds of behavioural adaptations to predators, such as avoid, flee or defend. Many 
prey species are faced with multiple predators that differ in the degree of danger posed. To be adapted to their 
environment and face predatory threats, organisms must balance their energy budgets as they invest in different 
aspects of their life history, such as growth, feeding, mating, socializing, or modifying their habitat. Hypotheses 
posited in behavioural ecology are generally based on adaptive principals of conservation, optimization or 
efficiency. For example, 

"The threat-sensitive predator avoidance hypothesis predicts that prey should assess the degree of threat posed by 
different predators and match their behavior according to current levels of risk." 

"The optimal flight initiation distance occurs where expected postencounter fitness is maximized, which depends on 
the prey's initial fitness, benefits obtainable by not fleeing, energetic escape costs, and expected fitness loss due to 



predation risk. 



,[118 



The behaviour of long-toed salamanders {Ambystoma macrodactylum) presents an example in this context. When 
threatened, the long-toed salamander defends itself by waving its tail and secreting a white milky fluid. The 

excreted fluid is distasteful, toxic and adhesive, but it is also used for nutrient and energy storage during hibernation. 
Hence, salamanders subjected to frequent predatory attack will be energetically compromised as they use up their 

[121] [122] 

energy stores. 



Ecology 



264 



Ecological interactions can be divided into host and associate 
relationships. A host is any entity that harbors another that is 

M241 

called the associate. Host and associate relationships among 
species that are mutually or reciprocally beneficial are called 
mutualisms. If the host and associate are physically connected, the 
relationship is called symbiosis. Approximately 60% of all plants, 
for example, have a symbiotic relationship with arbuscular 
mycorrhizal fungi. Symbiotic plants and fungi exchange 
carbohydrates for mineral nutrients. Symbiosis differs from 

indirect mutualisms where the organisms live apart. For example, 
tropical rainforests regulate the Earth's atmosphere. Trees living in 
the equatorial regions of the planet supply oxygen into the 
atmosphere that sustains species living in distant polar regions of 
the planet. This relationship is called commensalism because many 
other host species receive the benefits of clean air at no cost or 
harm to the associate tree species supplying the oxygen. The 
host and associate relationship is called parasitism if one species 
benefits while the other suffers. Competition among species or 
among members of the same species is defined as reciprocal 
antagonism, such as grasses competing for growth space. 




Symbiosis: Leafhoppers (Eurymela fenestrata) are 

protected by ants (Iridomyrmex purpureus) in a 

symbiotic relationship. The ants protect the leafhoppers 

from predators and in return the leafhoppers feeding on 

plants exude honeydew from their anus that provides 

[123] 
energy and nutrients to tending ants. 



Ecology 



265 




Parasites: A harvestman spider is parasiticized by mites. This is parasitism 

because the spider suffers as its juices are slowly sucked out and the mites gain all 

the benefits of a host to travel on and feed off. 



Popular ecological study systems for 
mutualism include, fungus-growing ants 
employing agricultural symbiosis, bacteria 
living in the guts of insects and other 
organisms, the fig wasp and yucca moth 
pollination complex, lichens with fungi and 
photosynthetic algae, and corals with 
photosynthetic algae 



[128] [129] 



Intraspecific behaviours are notable in the 
social insects, slime moulds, social spiders, 
human society, and naked mole rats where 
eusocialism has evolved. Social behaviours 
include reciprocally beneficial behaviours 

i ■ A * [25] [HO] [130] 

among kin and nest mates. 
Social behaviours evolve from kin and 
group selection. Kin selection explains 
altruism through genetic relationships, 
whereby an altruistic behaviour leading to 
death is rewarded by the survival of genetic 
copies distributed among surviving 
relatives. The social insects, including ants, 
bees and wasps are most famously studied 
for this type of relationship because the 
male drones are clones that share the same 



[25] 



genetic make-up as every other male in the colony. In contrast, group selectionists find examples of altruism 
among non-genetic relatives and explain this through selection acting on the group, whereby it becomes selectively 
advantageous for groups if their members express altruistic behaviours to one another. Groups that are 
predominantely altruists beat groups that are predominantely selfish. 



A often quoted behavioural ecology hypothesis is known as Lack's brood reduction hypothesis (named after David 
Lack). Lack's hypothesis posits an evolutionary and ecological explanation as to why birds lay a series of eggs with 
an asynchronous delay leading to nestlings of mixed age and weights. According to Lack, this brood behaviour is an 
ecological insurance that allows the larger birds to survive in poor years and all birds to survive when food is 
plentiful. [132] [133] 

Elaborate sexual displays and posturing are encountered in the behavioural ecology of animals. The birds of 
paradise, for example, display elaborate ornaments and song during courtship. These displays serve a dual purpose of 
signalling healthy or well-adapted individuals and good genes. The elaborate displays are driven by sexual selection 



as an advertisement of quality of traits among male suitors 



[134] 



Biogeography 

The word biogeography is an amalgamation of biology and geography. Biogeography is the comparative study of the 
geographic distribution of organisms and the corresponding evolution of their traits in space and time. " The 
Journal of Biogeography was established in 1974. [136] Biogeography and ecology share many of their disciplinary 
roots. For example, the theory of island biogeography, published by the mathematician Robert MacArthur and 
ecologist Edward O. Wilson in 1967 is considered one of the fundamentals of ecological theory. 



Ecology 266 

Biogeography has a long history in the natural sciences where questions arise concerning the spatial distribution of 
plants and animals. Ecology and evolution provide the explanatory context for biogeographical studies. 
Biogeographical patterns result from ecological processes that influence range distributions, such as migration and 
dispersal. and from historical processes that split populations or species into different areas. The 

biogeographic processes that result in the natural splitting of species explains much of the modern distribution of the 
Earth's biota. The splitting of lineages in a species is called vicariance biogeography and it is a sub-discipline of 
biogeography. There are also practical applications in the field of biogeography concerning ecological 

systems and processes. For example, the range and distribution of biodiversity and invasive species responding to 
climate change is a serious concern and active area of research in context of global warming. 

Molecular Ecology 

The important relationship between ecology and genetic inheritance predates modern techniques for molecular 
analysis. Molecular ecological research became more feasible with the development of genetic technologies, such as 
the polymerase chain reaction (PCR). The rise of molecular technologies and influx of research questions into this 

M421 

new ecological field resulted in the publication Molecular Ecology in 1992. Molecular ecology uses various 

analytical techniques to study genes in an evolutionary and ecological context. In 1994, professor John Avise played 
a leading role in this area of science with the publication of his book, Molecular Markers, Natural History and 
Evolution. Newer technologies opened a wave of genetic analysis into organisms once difficult to study from an 
ecological or evolutionary standpoint, such as bacteria, fungi and nematodes. Molecular ecology engendered a new 
research paradigm to investigate ecological questions considered otherwise intractable. Molecular ecology revealed 
previously obscured details in the intricacies of nature and improved resolution into probing questions about 
behavioral and biogeographical ecology. For example, molecular ecology revealed promiscuous sexual behavior and 

[1441 

multiple male partners in tree swallows previously thought to be socially monogamous. In a biogeographical 
context, the marriage between genetics, ecology and evolution resulted in a new sub-discipline called 
phylogeography. 

Ecology and the environment 

"The environment of any organism is the class composed of the sum of those phenomena that enter a reaction system of the 

organism or otherwise directly impinge upon it to affect its mode of life at any time throughout its life cycle as ordered by the 

demands of the ontogeny of the organism or as ordered by any other condition of the organism that alters its environmental 

demands." 

„ t , [146] :332 

Mason et al. 

The environment is dynamically interlinked with ecology. Like the term ecology, environment has different 
conceptual meanings and to many these terms also overlap with the concept of nature. Environment "...includes the 
physical world, the social world of human relations and the built world of human creation." ' This section 

describes the physical environmental attributes or parameters that are external to the level of biological organization 
under investigation, including abiotic factors such as temperature, radiation, light, chemistry, climate and geology, 
and biotic factors, including genes, cells, organisms, members of the same species (conspecifics) and other species 
that share a habitat. The physical environmental connection means that the laws of thermodynamics applies to 

ecology. Armed with an understanding of metabolic and thermodynamic principles a complete accounting of energy 

[1491 
and material flow can be traced through an ecosystem. 

Environmental and ecological relations are studied through reference to conceptually manageable and isolated parts. 
However, once the effective environmental components are understood they conceptually link back together as a 
holocoenotic[150] system. In other words, the organism and the environment form a dynamic whole (or 

[1511 "252 

umwelt). ' Change in one ecological or environmental factor can concurrently affect the dynamic state of an 

[152] [153] 

entire ecosystem. 



Ecology 



267 



Ecological studies are necessarily holistic as opposed to reductionistic. Holism has three scientific meanings 

or uses that identify with: 1) the mechanistic complexity of ecosystems, 2) the practical description of patterns in 
quantitative reductionist terms where correlations may be identified but nothing is understood about the causal 
relations without reference to the whole system, which leads to 3) a metaphysical hierarchy whereby the causal 
relations of larger systems are understood without reference to the smaller parts. An example of the metaphysical 
aspect to holism is the trend of increased exterior thickness in shells of different species. The reason for a thickness 
increase can be understood through reference to principals of natural selection via predation without any reference to 
the biomolecular properties of the exterior shells. 



Metabolism and the early atmosphere 

Metabolism - the rate at which energy and material resources are taken up from the environment, transformed within an organism, 

and allocated to maintenance, growth and reproduction - is a fundamental physiological trait. 

c t t , [156] :991 
Ernst et al. 

The Earth formed approximately 4.5 billion years ago and environmental conditions were too extreme for life to 
form for the first 500 million years. During this early Hadean period, the Earth started to cool allowing time for a 
crust and oceans to form. Environmental conditions were unsuitable for the origins of life for the first billion years 
after the Earth formed. The Earth's atmosphere transformed from hydrogen dominant, to one composed mostly of 
methane and ammonia. Over the next billion years the metabolic activity of life transformed the atmosphere to 
higher concentrations of carbon dioxide, nitrogen, and water vapor. These gases changed the way that light from the 
sun hit the Earth's surface and greenhouse effects trapped in heat. There were untapped sources of free energy within 
the mixture of reducing and oxidizing gasses that set the stages for primitive ecosystems to evolve and, in turn, the 
atmosphere also evolved. 

Throughout history, the Earth's atmosphere and biogeochemical 
cycles has been in a dynamic equilibrium with planetary 
ecosystems. The history is characterized by periods of significant 
transformation followed by millions of years of stability. The 
evolution of the earliest organisms, likely anaerobic methanogen 
microbes, started the process by converting atmospheric hydrogen 




The leaf is the primary site of photosynthesis in most 
plants. 



into methane (4H + 



CO, 



CH + 2H O). Anoxygenic 



photosynthesis converting hydrogen sulfide into other sulfur 

hv H> CH O -> HO -> + 



compounds or water (2H S + CO 



CH 2 



H 2 



2S or 2H 2 + C0 2 + hv 



CH O + HO), as occurs in deep sea 



hydrothermal vents today, reduced hydrogen concentrations and 

increased atmospheric methane. Early forms of fermentation also 

increased levels of atmospheric methane. The climatic history of 

Earth is characterized by a series of major transitions separated by long periods of relative stability. The transition to 

an oxygen dominant atmosphere (the Great Oxidation) did not begin until approximately 2.4-2.3 billion years ago, 



but photosynthetic processes started 0.3 to 1 billion years prior. 



[159] [160] 



Radiation: heat, temperature and light 

The biology of life operates within a certain range of temperatures. Heat is a form of energy that regulates 
temperature. Heat affects growth rates, activity, behavior and primary production. Temperature is largely dependent 
on the incidence of solar radiation. The latitudinal and longitudinal spatial variation of temperature greatly affects 
climates and consequently the distribution of biodiversity and levels of primary production in different ecosystems or 
biomes across the planet. Heat and temperature relate importantly to metabolic activity. Poikilotherms, for example, 
have a body temperature that is largely regulated and dependent on the temperature of the external environment. In 



Ecology 268 

contrast, homeotherms regulate their internal body temperature by expending metabolic energy. 

There is a relationship between light, primary production, and ecological energy budgets. Sunlight is the primary 
input of energy into the planet's ecosystems. Light is composed of electromagnetic energy of different wavelengths. 
Radiant energy from the sun generates heat, provides photons of light measured as active energy in the chemical 
reactions of life, and also acts as a catalyst for genetic mutation. Plants, algae, and some bacteria absorb 

light and assimilate the energy through photosynthesis. Organisms capable of assimilating energy by photosynthesis 
or through inorganic fixation of H S are autotrophs. Autotrophs — responsible for primary production — assimilate 

light energy that becomes metabolically stored as potential energy in the form of biochemical enthalpic bonds. 

[149] 

Physical environments 

Water 

Wetland conditions such as shallow water, high plant productivity, and anaerobic substrates provide a suitable environment for 
important physical, biological, and chemical processes. Because of these processes, wetlands play a vital role in global nutrient and 
element cycles.' 

The rate of diffusion of carbon dioxide and oxygen is approximately 10,000 times slower in water than it is in air. 
When soils become flooded, they quickly lose oxygen from low-concentration (hypoxic) to an (anoxic) environment 
where anaerobic bacteria thrive among the roots. Water also influences the spectral properties of light that becomes 
more diffuse as it is reflected off the water surface and submerged particles. Aquatic plants exhibit a wide 
variety of morphological and physiological adaptations that allow them to survive, compete and diversify these 
environments. For example, the roots and stems develop large cellular air spaces to allow for the efficient 
transportation gases (for example, CO and O ) used in respiration and photosynthesis. In drained soil, 
microorganisms use oxygen during respiration. In aquatic environments, anaerobic soil microorganisms use nitrate, 
manganic ions, ferric ions, sulfate, carbon dioxide and some organic compounds. The activity of soil microorganisms 
and the chemistry of the water reduces the oxidation-reduction potentials of the water. Carbon dioxide, for example, 
is reduced to methane (CH ) by methanogenic bacteria. Salt water also requires special physiological adaptations to 
deal with water loss. Salt water plants (or halophytes) are able to osmo-regulate their internal salt (NaCl) 
concentrations or develop special organs for shedding salt away. The physiology offish is also specially adapted 
to deal with high levels of salt through osmoregulation. Their gills form electrochemical gradients that mediate salt 
excrusion in salt water and uptake in fresh water. 

Gravity 

The shape and energy of the land is affected to a large degree by gravitational forces. On a larger scale, the 
distribution of gravitational forces on the earth are uneven and influence the shape and movement of tectonic plates 
as well as having an influence on geomorphic processes such as orogeny and erosion. These forces govern many of 
the geophysical properties and distributions of ecological biomes across the Earth. On a organism scale, gravitational 
forces provide directional cues for plant and fungal growth (gravitropism), orientation cues for animal migrations, 
and influence the biomechanics and size of animals. Ecological traits, such as allocation of biomass in trees during 
growth are subject to mechanical failure as gravitational forces influence the position and structure of branches and 
leaves. The cardiovascular systems of all animals are functionally adapted to overcome pressure and 

gravitational forces that change according to the features of organisms (e.g., height, size, shape), their behavior (e.g., 
diving, running, flying), and the habitat occupied (e.g., water, hot deserts, cold tundra). 



Ecology 



269 



Pressure 

Climatic and osmotic pressure places physiological constraints on organisms, such as flight and respiration at high 
altitudes, or diving to deep ocean depths. These constraints influence vertical limits of ecosystems in the biosphere as 
organisms are physiologically sensitive and adapted to atmospheric and osmotic water pressure differences. 
Oxygen levels, for example, decrease with increasing pressure and are a limiting factor for life at higher 
altitudes. Water transportation through trees is another important ecophysiological parameter dependent upon 
pressure. Water pressure in the depths of oceans requires adaptations to deal with the different living 

conditions. Mammals, such as whales, dolphins and seals are adapted to deal with changes in sound due to water 
pressure differences 



[168] 



Wind and turbulence 

Turbulent forces in air and water have significant effects on the 
environment and ecosystem distribution, form and dynamics. On a 
planetary scale, ecosystems are affected by circulation patterns in 
the global trade winds. Wind power and the turbulent forces it 

creates can influence heat, nutrient, and biochemical profiles of 

[21 
ecosystems. For example, wind running over the surface of a 

lake creates turbulence, mixing the water column and influencing 

the environmental profile to create thermally layered zones, 

partially governing how fish, algae, and other parts of the aquatic 

ecology are structured. Wind speed and turbulence also 

exert influence on rates of evapotranspiration rates and energy 

budgets in plants and animals. Wind speed, temperature 

and moisture content can vary as winds travel across different 

landfeatures and elevations. The westerlies, for example, come 

into contact with the coastal and interior mountains of western 

i 

North America to produce a rain shadow on the leeward side of 

the mountain. The air expands and moisture condenses as the winds move up in elevation which can cause 

precipitation; this is called orographic lift. This environmental process produces spatial divisions in biodiversity, as 

species adapted to wetter conditions are range-restricted to the coastal mountain valleys and unable to migrate across 

the xeric ecosystems of the Columbia Basin to intermix with sister lineages that are segregated to the interior 




The architecture of inflorescence in grasses is subject 

to the physical pressures of wind and shaped by the 

forces of natural selection facilitating wind-pollination 
. ... . [169] [170] 

(or anemophily). 



mountain systems 



[174] [175] 



Fire 




Forest fires modify the land by leaving behind an environmental mosaic that diversifies the landscape into different serai stages 
and habitats of varied quality (left). Some species are adapted to forest fires, such as pine trees that open their cones only after 
fire exposure (right). 

Plants convert carbon dioxide into biomass and emit oxygen into the atmosphere. Approximately 350 million 
years ago (near the Devonian period) the photosynthetic process brought the concentration of atmospheric oxygen 

[1771 

above 17%, which allowed combustion to occur. Fire releases CO and converts fuel into ash and tar. Fire is a 

n 7si 
significant ecological parameter that raises many issues pertaining to its control and suppression in management. 

[1791 
While the issue of fire in relation to ecology and plants has been recognized for a long time, Charles Cooper 



Ecology 270 

brought attention to the issue of forest fires in relation to the ecology of forest fire suppression and management in 
the 1960s. [180] [181] 

Fire creates environmental mosaics and a patchiness to ecosystem age and canopy structure. Native North Americans 
were among the first to influence fire regimes by controlling their spread near their homes or by lighting fires to 
stimulate the production of herbaceous foods and basketry materials. The altered state of soil nutrient supply and 
cleared canopy structure also opens new ecological niches for seedling establishment. Most ecosystem are 

adapted to natural fire cycles. Plants, for example, are equipped with a variety of adaptations to deal with forest fires. 
Some species (e.g., Pinus halepensis) cannot germinate until after their seeds have lived through a fire. This 
environmental trigger for seedlings is called serotiny. Some compounds from smoke also promote seed 

• „■ [186] 

germination. 

Biogeochemistry 

Ecologists study and measure nutrient budgets to understand how these materials are regulated and flow through the 
environment. This research has led to an understanding that there is a global feedback between 

ecosystems and the physical parameters of this planet including minerals, soil, pH, ions, water and atmospheric 
gases. There are six major elements, including H (hydrogen), C (carbon), N (nitrogen), O (oxygen), S (sulfur), and P 
(phosphorus) that form the constitution of all biological macromolecules and feed into the Earth's geochemical 
processes. From the smallest scale of biology the combined effect of billions upon billions of ecological processes 
amplify and ultimately regulate the biogeochemical cycles of the Earth. Understanding the relations and cycles 
mediated between these elements and their ecological pathways has significant bearing toward understanding global 
biogeochemistry. 

The ecology of global carbon budgets gives one example of the linkage between biodiversity and biogeochemistry. 

For starters, the Earth's oceans are estimated to hold 40,000 gigatonnes (Gt) carbon, vegetation and soil is estimated 

n ssi 
to hold 2070 Gt carbon, and fossil fuel emissions are estimated to emit an annual flux of 6.3 Gt carbon. At 

different times in the Earth's history there has been major restructuring in these global carbon budgets that was 

regulated to a large extent by the ecology of the land. For example, through the early-mid Eocene volcanic 

outgassing, the oxidation of methane stored in wetlands, and seafloor gases increased atmospheric CO 

concentrations to levels as high as 3500 ppm. In the Oligocene, from 25 to 32 million years ago, there was 

another significant restructuring in the global carbon cycle as grasses evolved a special type of C4 photosynthesis 

and expanded their ranges. This new photosynthetic pathway evolved in response to the drop in atmospheric CO 

concentrations below 550 ppm. Ecosystem functions such as these feed back significantly into global 

atmospheric models for carbon cycling. Loss in the abundance and distribution of biodiversity causes global carbon 

cycle feedbacks that are expected to increase rates of global warming in the next century. The effect of global 

warming melting large sections of permafrost creates a new mosaic of flooded areas where decomposition results in 

the emission of methane (CH ). Hence, there is a relationship between global warming, decomposition and 

respiration in soils and wetlands producing significant climate feedbacks and altered global biogeochemical 

cycles. There is concern over increases in atmospheric methane in the context of the global carbon cycle, 

because methane is also a greenhouse gas that is 23 times more effective at absorbing long-wave radiation on a 100 

year time scale. 

Historical roots of ecology 

Unlike many of the scientific disciplines, ecology has a complex and winding origin due in large part to its 
interdisciplinary nature. Several published books provide extensive coverage of the classics. In the early 

20th century, ecology was an analytical form of natural history. The descriptive nature of natural history 

included examination of the interaction of organisms with both their environment and their community. Such 
examinations were conducted by important natural historians including James Hutton and Jean-Baptiste Lamarck 



Ecology 



271 



contributed to the development of ecology. The term "ecology" (German: Oekologie) is a more recent scientific 
development and was first coined by the German biologist Ernst Haeckel in his book Generelle Morpologie der 
Organismen (1866). 

By ecology we mean the body of knowledge concerning the economy of nature-the investigation of the total relations of the animal 
both to its inorganic and its organic environment; including, above all, its friendly and inimical relations with those animals and 
plants with which it comes directly or indirectly into contact-in a word, ecology is the study of all those complex interrelations 
referred to by Darwin as the conditions of the struggle of existence. 

Haeckel's definition quoted in Esbjorn-Hargens 




Ernst Haeckel (left) and Eugenius Warming (right), two early founders of ecology 

Opinions differ on who was the founder of modern ecological theory. Some mark Haeckel's definition as the 



beginning 



[201] 



others say it was Eugen Warming with the writing of Oecology of Plants: An Introduction to the 



[2021 

Study of Plant Communities (1895). Ecology may also be thought to have begun with Carl Linnaeus' research 
principals on the economy of nature that matured in the early 18th century. He founded an early branch of 

ecological study he called the economy of nature. The works of Linnaeus influenced Darwin in The Origin of 

[204] 

Species where he adopted the usage of Linnaeus' phrase on the economy or polity of nature. Linnaeus made the 
first to attempt to define the balance of nature, which had previously been held as an assumption rather than 
formulated as a testable hypothesis. Haeckel, who admired Darwin's work, defined ecology in reference to the 
economy of nature which has lead some to question if ecology is synonymous with Linnaeus' concepts for the 

[2031 

economy of nature. Biogeographer Alexander von Humbolt was also foundational and was among the first to 
recognize ecological gradients and alluded to the modern ecological law of species to area relationships. 

The modern synthesis of ecology is a young science, which first attracted substantial formal attention at the end of 

the 19th century (around the same time as evolutionary studies) and become even more popular during the 1960s 

11991 
environmental movement. However, many observations, interpretations and discoveries relating to ecology 

extend back to much earlier studies in natural history. For example, the concept on the balance or regulation of 

nature can be traced back to Herodotos (died c. 425 BC) who described an early account of mutualism along the Nile 

river where crocodiles open their mouths to beneficially allow sandpipers safe access to remove leeches. In the 

broader contributions to the historical development of the ecological sciences, Aristotle is considered one of the 

earliest naturalists who had an influential role in the philosophical development of ecological sciences. One of 

Aristotle's students, Theophrastus, made astute ecological observations about plants and posited a philosophical 

stance about the autonomous relations between plants and their environment that is more in line with modern 

ecological thought. Both Aristotle and Theophrastus made extensive observations on plant and animal migrations, 

[2071 

biogeography, physiology, and their habits in what might be considered a modern analog of the ecological niche. 

[208] 



Ecology 



272 



From Aristotle to Darwin the natural world was 
predominantly considered static and unchanged since its 
original creation. Prior to The Origin of Species there was 
little appreciation or understanding of the dynamic and 
reciprocal relations between organisms, their adaptations and 
their modifications to the environment. While 

Charles Darwin is most notable for his treatise on 

[2121 
evolution, he is also one of the founders of soil 

ecology. In The Origin of Species Darwin also made 

note of the first ecological experiment that was published in 

[209] 

1816. In the science leading up to Darwin the notion of 
evolving species was gaining popular support. This scientific 
paradigm changed the way that researchers approached the ecological sciences 




The layout of the first ecological experiment, noted by 
Charles Darwin in The Origin of Species, was studied in a 
grass garden at Woburn Abbey in 1817. The experiment 

studied the performance of different mixtures of species 

■ " "1] 



planted in different kinds of soils 
[214] 



[209] [210] 



Nowhere can one see more clearly illustrated what may be called the sensibility of such an organic complex,— expressed by the fact 
that whatever affects any species belonging to it, must speedily have its influence of some sort upon the whole assemblage. He will 
thus be made to see the impossibility of studying any form completely, out of relation to the other forms,— the necessity for taking a 
comprehensive survey of the whole as a condition to a satisfactory understanding of any part. 

Stephen Forbes (1887) [215] 



[216] 



Ecology after the turn of 20th century 

The first American ecology book was published in 1905 by Frederic Clements. L ^ 1UJ In his book, Clements forwarded 
the idea of plant communities as a superorganism. This publication launched a debate between ecological holism and 
individualism that lasted until the 1970s. The Clements superorganism concept proposed that ecosystems progress 
through regular and determined stages of serai development that are analogous to developmental stages of an 
organism whose parts function to maintain the integrity of the whole. The Clementsian paradigm was challenged by 



Henry Gleason 



[217] 



According to Gleason, ecological communities develop from the unique and coincidental 



association of individual organisms. This perceptual shift placed the focus back onto the life histories of individual 

T21 81 

organisms and how this relates to the development of community associations. 

The Clementsian superorganism concept has not been completely rejected, but it was an overextended application of 
holism, which remains a significant theme in contemporary ecological studies. Holism was first introduced 
in 1926 by a polarizing historical figure, a South African General named Jan Christian Smuts. Smuts was inspired by 
Clement's superorganism theory when he developed and published on the unifying concept of holism, which runs in 
stark contrast to his racial views as the father of apartheid. Around the same time, Charles Elton pioneered the 

T7R1 r"7Qi 

concept of food chains in his classical book "Animal Ecology". Elton defined ecological relations using 
concepts of food-chains, food-cycles, food-size, and described numerical relations among different functional groups 
and their relative abundance. Elton's term 'food-cycle' was replaced by 'food-web' in a subsequent ecological 

[2211 

text. Elton's book broke conceptual ground by illustrating complex ecological relations through simpler 

food- web diagrams. 

[2221 

The number of authors publishing on the topic of ecology has grown considerably since the turn of 20th century. 
The explosion of information available to the modern researcher of ecology makes it an impossible task for one 
individual to sift through the entire history. Hence, the identification of classics in the history of ecology is a difficult 

[223] 

designation to make. 




Ecology 273 

Parallel development 

Ecology has developers in many nations, including Russia's Vladimir Vernadsky and his founding of the biosphere 

T2241 
concept in the 1920s or Japan's Kinji Imanishi and his concepts of harmony in nature and habitat segregation in 

["2251 

the 1950s. The scientific recognition or importance of contributions to ecology from other cultures is hampered 
by language and translation barriers. The history of ecology remains an active area of study, often published in 
the Journal of the History of Biology [226]. 

Ecosystem services and the biodiversity crisis 

Increasing globalization of human activities and rapid movements of people as well as their goods and services suggest that mankind 
is now in an era of novel coevolution of ecological and socioeconomic systems at regional and global scales. 

The ecosystems of planet Earth are coupled to human environments. 
Ecosystems regulate the global geophysical cycles of energy, climate, 
soil nutrients, and water that in turn support and grow natural capital 
(including the environmental, physiological, cognitive, cultural, and 

spiritual dimensions of life). Ultimately, every manufactured product 

12191 
in human environments comes from natural systems. Ecosystems 

are considered common-pool resources because ecosystems do not 

[2271 

exclude beneficiaries and they can be depleted or degraded. For 
example, green space within communities provides common-pool 
health services. Research shows that people who are more engaged 
with regular access to natural areas have lower rates of diabetes, heart 
,,,,,..., , disease and psychological disorders. These ecological health 

A bumblebee pollinating a flower, one example r J " " 

of an ecosystem service services are regularly depleted through urban development projects 

[2291 [2301 

that do not factor in the common-pool value of ecosystems. 
The ecological commons delivers a diverse supply of community services that sustains the well-being of human 

[2311 [2321 

society. The Millennium Ecosystem Assessment, an international UN initiative involving more than 1,360 

experts worldwide, identifies four main ecosystem service types having 30 sub-categories stemming from natural 
capital. The ecological commons includes provisioning (e.g., food, raw materials, medicine, water supplies), 
regulating (e.g., climate, water, soil retention, flood retention), cultural (e.g., science and education, artistic, 

[2331 

spiritual), and supporting (e.g., soil formation, nutrient cycling, water cycling) services. 

Policy and human institutions should rarely assume that human enterprise is benign. A safer assumption holds that human enterprise 

[2341 '95 
almost always exacts an ecological toll - a debit taken from the ecological commons. 

[2351 

Ecology is an economic science that uses many of the same terms and methods that are used in accounting. 
Natural capital is the stock of materials or information stored in biodiversity that generates services that can enhance 
the welfare of communities. Population losses are the more sensitive indicator of natural capital than are species 
extinction in the accounting of ecosystem services. The prospect for recovery in the economic crisis of nature is 
grim. Populations, such as local ponds and patches of forest are being cleared away and lost at rates that exceed 
species extinctions. 

While we are used to thinking of cities as geographically discrete places, most of the land "occupied" by their residents lies far 
beyond their borders. The total area of land required to sustain an urban region (its "ecological footprint") is typically at least an 
order of magnitude greater than that contained within municipal boundaries or the associated built-up area. 

The WWF 2008 living planet report and other researchers report that human civilization has exceeded the 

r?3Rl T2391 

bio-regenerative capacity of the planet. This means that human consumption is extracting more natural 

resources than can be replenished by ecosystems around the world. In 1992, professor William Rees developed the 
concept of our ecological footprint. The ecological footprint is a way of accounting the level of impact that human 



Ecology 274 

[2371 

development is having on the Earth's ecosystems. All indications are that the human enterprise is unsustainable 

[2391 

as the ecological footprint of society is placing too much stress on the ecology of the planet. The mainstream 
growth-based economic system adopted by governments worldwide does not include a price or markets for natural 

[240] 

capital. This type of economic system places further ecological debt onto future generations. 

Human societies are increasingly being placed under stress as the ecological commons is diminished through an 

[241] -44 

accounting system that has incorrectly assumed "... that nature is a fixed, indestructible capital asset.' ' While 

nature is resilient and it does regenerate, there are limits to what can be extracted, but conventional monetary 

[242] [2431 

analyses are unable to detect the problem. Evidence of the limits in natural capital are found in the global 

[244] 

assessments of biodiversity, which indicate that the current epoch, the Anthropocene is a sixth mass extinction. 

[245] [2461 

Species loss is accelerating at 100—1000 times faster than average background rates in the fossil record. The 

ecology of the planet has been radically transformed by human society and development causing massive loss of 
ecosystem services that otherwise deliver and freely sustain equitable benefits to human society through the 
ecological commons. The ecology of the planet is further threatened by global warming, but investments in nature 
conservation can provide a regulatory feedback to store and regulate carbon and other greenhouse gases. 
The field of conservation biology involves ecologists that are researching the nature of the biodiversity threat and 

[2491 

searching for solutions to sustain the planet's ecosystems for future generations. 

Many human-nature interactions occur indirectly due to the production and use of human-made (manufactured and synthesized) 
products, such as electronic appliances, furniture, plastics, airplanes, and automobiles. These products insulate humans from the 
natural environment, leading them to perceive less dependence on natural systems than is the case, but all manufactured products 
ultimately come from natural systems. 

"Human activities are associated directly or indirectly with nearly every aspect of the current extinction 

,,[245] : 11472 

spasm. 

The current wave of threats, including massive extinction rates and concurrent loss of natural capital to the detriment 
of human society, is happening rapidly. This is called a biodiversity crisis, because 50% of the worlds species are 
predicted to go extinct within the next 50 years. The world's fisheries are facing dire challenges as the threat 

[2521 

of global collapse appears imminent, with serious ramifications for the well-being of humanity. Governments of 
the G8 met in 2007 and set forth The Economics of Ecosystems and Biodiversity' (TEEB) initiative [253]: 

In a global study we will initiate the process of analyzing the global economic benefit of biological 
diversity, the costs of the loss of biodiversity and the failure to take protective measures versus the costs 
of effective conservation. 

Ecologists are teaming up with economists to measure the wealth of ecosystems and to express their value as a way 
of finding solutions to the biodiversity crisis. Some researchers have attempted to place a dollar figure 

on ecosystem services, such as the value that the Canadian boreal forest is contributing to global ecosystem services. 
If ecologically intact, the boreal forest has an estimated value of US$3.7 trillion. The boreal forest ecosystem is one 

[25R1 

of the planet's great atmospheric regulators and it stores more carbon than any other biome on the planet. The 
annual value for ecological services of the Boreal Forest is estimated at US$93.2 billion, or 2.5 greater than the 
annual value of resource extraction. The economic value of 17 ecosystem services for the entire biosphere 

1 o ro qa"i 

(calculated in 1997) has an estimated average value of US$33 trillion (10 ) per year. These ecological 

economic values are not currently included in calculations of national income accounts, the GDP and they have no 
price attributes because they exist mostly outside of the global markets. The loss of natural capital continues 

to accelerate and goes undetected by mainstream monetary analysis. 




Bachalpsee in the Swiss Alps; generally 

mountainous areas are less affected by human 

activity. 



Ecology 275 

See also 

Acoustic ecology 

Agroecology 

Aquatic ecosystems 

Biodiversity 

Biotope 

Biogeography 

Climate 

Conservation movement 

Earth science 

Ecoacoustics 

Ecohydrology 

Ecological economics 

Ecological Forecasting 

Ecology movement 

Ecology of contexts 

Ecosystem 

Ecosystem model 

Ecological psychology 

Ecological Relationships 

Ecosystem services 

Ecotope 

ELDIS, database ecological aspects of economical development. 

Environment 

Forest farming 

Forest gardening 

Habitat conservation 

Human ecology 

Industrial ecology 

Insect ecology 

Knowledge ecology 

Landscape ecology 

Landscape limnology 

Molecular ecology 

Natural capital 

Natural landscape 

Natural resource 

Natural resource management 

Nature 

Palaeoecology 

Social ecology 

Soil 

Sustainability 

Sustainable development 



Ecology 276 

Lists 

• Index of biology articles 

• Glossary of ecology 

• List of ecologists 

• List of important publications in biology#Ecology 

• Outline of biology 

Further reading 

Allee, W. C. (1932). Animal life and social growth. Baltimore: The Williams & Wilkins Company and 

Associates. 

Allee, W.; Emerson, A. E., Park, O., Park, T., and Schmidt, K. P. (1949). Principles of Animal Ecology. W. B. 

Saunders Company. ISBN 0721611206. 

Begon, M.; Townsend, C. R., Harper, J. L. (2006). Ecology: From individuals to ecosystems. (4th ed.). Blackwell. 

ISBN 1405111178. 

Brinson, M. M.; Lugo, A. E.; Brown, S. (1984). "Primary Productivity, Decomposition and Consumer Activity in 

Freshwater Wetlands." . Annual Review of Ecology and Systematics 12: 123—161. 

Clements, F. E. (1905). Research Methods in Ecology. Lincoln, Nebraska: University Publ.. 

Costanza, R.; dArge, R.; de Groot, R.; Farberk, S.; Grasso, M.; Hannon, B.; et al. (1997). "The value of the 

world's ecosystem services and natural capital." . Nature 387: 253—260. 

Davie, R. D.; Welsh, H. H. (2004). "On the Ecological Role of Salamanders". Annual Review of Ecology and 

Systematics 35: 405-434. 

Elton, C. S. (1927). Animal Ecology. London, UK.: Sidgwick and Jackson. 

Forbes, S. (1887). "The lake as a microcosm" . Bull, of the Scientific Association (Peoria, IL : .): 77—87. 

Gleason, H. A. (1926). "The Individualistic Concept of the Plant Association" . Bulletin of the Torrey 

Botanical Club 53 (1): 7—26. 

Hanski, I. (2008). "The world that became ruined." [266] . EMBO reports 9: S34-S36. 

Hastings, A. B.; Crooks, J. E.; Cuddington, J. A.; Jones, K.; Lambrinos, C. J.; Talley, J. G; et al. (2007). 

"Ecosystem engineering in space and time". Ecology Letters 10 (2): 153—164. 

doi:10.1111/j.l461-0248.2006.00997.x. PMID 17257103. 

Kormondy, E. (1995). Concepts of ecology. (4th ed.). Benjamin Cummings. ISBN 0134781163. 

Liu, J.; Dietz, T.; Carpenter, S. R.; Folke, C; Alberti, M.; Redman, C. L.; et al. (2009). "Coupled Human and 

Natural Systems" . AMBIO: A Journal of the Human Environment 36 (8): 639—649. 



Lovelock, J. (2003). "The living Earth". Nature 426 (6968): 769-770. doi:10.1038/426769a. PMID 14685210. 

Mcintosh, R. (1985). The Background of Ecology: Concept and Theory. . New York: Cambridge University 

Press. ISBN 0-521-24935. 

Mcintosh, R. P. (1989). "Citation Classics of Ecology" [269] . The Quarterly Review of Biology 64 (1): 31-49. 

Odum, E. P. (1977). "The emergence of ecology as a new integrative discipline". Science 195: 1289—1293. 

Odum, E. P.; Brewer, R. W.; Barrett, G W. (2004). Fundamentals of Ecology (Fifth Edition ed.). Brooks Cole. 

ISBN 978-0534420666. 

Omerod, S.J.; Pienkowski, M.W.; Watkinson, A.R. (1999). "Communicating the value of ecology". Journal of 

Applied Ecology 36: 847-855. 

Rickleffs, Robert, E. (1996). The Economy of Nature. University of Chicago Press, pp. 678. ISBN 0716738473. 

Smith, R.; Smith, R. M. (2000). Ecology and Field Biology. (6th ed.). Prentice Hall. ISBN 0321042905. 

[270] 

Whittaker, R. H; Levin, S. A.; Root, R. B. (1973). "Niche, Habitat, and Ecotope" . The American Naturalist 
107 (955): 321-338. 



Ecology 277 

T2711 

• Wiens, J. J.; Graham, C. H. (2005), "Integrating Evolution, Ecology, and Conservation Biology" , Annual 

Review of Ecology, Evolution, and Systematics 36: 519—539 

External links 

["2721 

• stanford.edu/entries/ecology/ Ecology (Stanford Encyclopedia of Philosophy) 

• Science Aid: Ecology [273] High School (GCSE, Alevel) Ecology. 



Ecology Journals List of scientific journals related to Ecology 

T2751 
Ecology Dictionary - Explanation of Ecological Terms 

[113] International Society for Behavioral Ecology 



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Ecology 



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[263] http://www.uvm.edu/giee/publications/Nature_Paper.pdf 

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client=firefox-a&cd=l#v=onepage&q=&f=false 

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[271] http://life.bio.sunysb.edu/ee/grahamlab/pdf/Wiens_Graham_AnnRev2005.pdf 

[272] http://plato. 

[273] http://scienceaid.co.uk/biology/ecology/index.html 

[274] http://ekolojinet.com/journals.html 

[275] http://ecologydictionary.org 



Systems ecology 



Systems ecology is an interdisciplinary field 
of ecology, taking a holistic approach to the 
study of ecological systems, especially 
ecosystems. Systems ecology can be seen as 
an application of general systems theory to 
ecology. Central to the systems ecology 
approach is the idea that an ecosystem is a 
complex system exhibiting emergent 
properties. Systems ecology focuses on 
interactions and transactions within and 
between biological and ecological systems, 
and is especially concerned with the way the 
functioning of ecosystems can be influenced 
by human interventions. It uses and extends 
concepts from thermodynamics and 
develops other macroscopic descriptions of 
complex systems. 




Ecological analysis of CO in an ecosystem 



Systems ecology 288 

Overview 

Systems ecology seeks a holistic view of the interactions and transactions within and between biological and 
ecological systems. Systems ecologists realise that the function of any ecosystem can be influenced by human 
economics in fundamental ways. They have therefore taken an additional transdisciplinary step by including 
economics in the consideration of ecological-economic systems. In the words of R.L. Kitching: 

• Systems ecology can be defined as the approach to the study of ecology of organisms using the techniques and 
philosophy of systems analysis: that is, the methods and tools developed, largely in engineering, for studying, 
characteriszing and making predictions about complex entities, that is, systems.. 

• In any study of an ecological system, an essential early procedure is to draw a diagram of the system of interest ... 
diagrams indicate the system's boundaries by a solid line. Within these boundaries, series of components are 
isolated which have been chosen to represent that portion of the world in which the systems analyst is interested 
... If there are no connections across the systems' boundaries with the surrounding systems environments, the 
systems are described as closed. Ecological work, however, deals almost exclusively with open systems. 

As a mode of scientific enquiry, a central feature of Systems Ecology is the general application of the principles of 
energetics to all systems at any scale. Perhaps the most notable proponent of this view was Howard T. Odum - 
sometimes considered the father of ecosystems ecology. In this approach the principles of energetics constitute 
ecosystem principles. Reasoning by formal analogy from one system to another enables the Systems Ecologist to see 
principles functioning in an analogous manner across system-scale boundaries. H.T. Odum commonly used the 
Energy Systems Language as a tool for making systems diagrams and flow charts. 

The fourth of these principles, the principle of maximum power efficiency, takes central place in the analysis and 
synthesis of ecological systems. The fourth principle suggests that the most evolutionarily advantageous system 
function occurs when the environmental load matches the internal resistance of the system. The further the 
environmental load is from matching the internal resistance, the further the system is away from its sustainable 
steady state. Therefore the systems ecologist engages in a task of resistance and impedance matching in ecological 
engineering, just as the electronic engineer would do. 

Summary of relationships in systems ecology 

The image to the right is a summary of 
relationships between the storage quantity 
Q, the forces X, N, and the outflows /, 
resistance R, conductivity L, time constants 
T, and transfer coefficients k of ecosystem 
metabolism. The transfer coefficient "k", is 
also known as the metabolic constant. 

"All these relationships are 
automatically implied by the energy 
circuit symbol ". 





J 


= LX 






J = L' N 


L = l 

R 


J 


= 1X 

R 


"■*. 


>' 


J = 1 N 

R 


XorN=Q 
C 




•'• 


J = _J_Q 
RC 






k = _A 
RC 






J = kQ 






RC = T 




summary 


J = _Q 

T 
of relationships 







Systems ecology 289 

Closely related fields 
Deep Ecology 

Deep Ecology is a school of philosophy pioneered by the Norwegian Philosopher, Gandhian scholar and 
environmental activist Arne Naess. Created in 1973 at an environmental conference in Budapest, it argues that the 
school of environmental management is anthropocentric, that the natural environment is not only "more complex 
than we imagine, it is more complex than we can imagine" . Concerned with the development of an "ecological 
self", which views the human ego as a part of a living system, rather than apart from such systems, "Experiential 
Deep Ecology" of Joanna Macy, John Seed and others, seeks to transcend altruism with a deeper self-interest, based 
upon biospherical equality beyond human chauvinism. 

Earth systems engineering and management 

Earth systems engineering and management (ESEM) is a discipline used to analyze, design, engineer and manage 
complex environmental systems. It entails a wide range of subject areas including anthroplogy, engineering, 
environmental science, ethics and philosophy. At its core, ESEM looks to "rationally design and manage coupled 
human-natural systems in a highly integrated and ethical fashion" 

Ecological economics 

Ecological economics is a transdisciplinary field of academic research that addresses the dynamic and spatial 
interdependence between human economies and natural ecosystems. Ecological economics brings together and 
connects different disciplines, within the natural and social sciences but especially between these broad areas. As the 
name suggests, the field is made up of researchers with a background in economics and ecology. An important 
motivation for the emergence of ecological economics has been criticism on the assumptions and approaches of 
traditional (mainstream) environmental and resource economics. 

Ecological energetics 

Ecological energetics is the quantitative study of the flow of energy through ecological systems. It aims to uncover 
the principles which describe the propensity of such energy flows through the trophic, or 'energy availing' levels of 
ecological networks. In systems ecology the principles of ecosystem energy flows or "ecosystem laws" (i.e. 
principles of ecological energetics) are considered formally analogous to the principles of energetics. 

Ecological humanities 

Ecological humanities aims to bridge the divides between the sciences and the humanities, and between Western, 
Eastern and Indigenous ways of knowing nature. Like ecocentric political theory, the ecological humanities are 
characterised by a connectivity ontology and a commitment to two fundamental axioms relating to the need to 
submit to ecological laws and to see humanity as part of a larger living system. 




Systems ecology 290 

Ecosystem ecology 

Ecosystem ecology is the integrated study of biotic and abiotic 
components of ecosystems and their interactions within an ecosystem 
framework. This science examines how ecosystems work and relates 
this to their components such as chemicals, bedrock, soil, plants, and 
animals. Ecosystem ecology examines physical and biological 
structure and examines how these ecosystem characteristics interact. 

The relationship between systems ecology and ecosystem ecology is 

complex. Much of systems ecology can be considered a subset of , . . . .. , . .. . 

r J CJ A riparian forest in the White Mountains, New 

ecosystem ecology. Ecosystem ecology also utilizes methods that have Hampshire (USA) 

little to do with the holistic approach of systems ecology. However, 

systems ecology more actively considers external influences such as economics that usually fall outside the bounds 

of ecosystem ecology. Whereas ecosystem ecology can be defined as the scientific study of ecosystems, systems 

ecology is more of a particular approach to the study of ecological systems and phenomena that interact with these 

systems. 

Industrial ecology 

Industrial ecology is the study of industrial processes as linear (open loop) systems, in which resource and capital 
investments move through the system to become waste, to a closed loop system where wastes become inputs for new 
processes. 

See also 

Agroecology 

Ecological literacy 

Economics and energy 

Emergy 

Energetics 

Energy Systems Language 

Holism in science 

Holistic management 

Landscape ecology 

Antireductionism 

Literature 

• Gregory Bateson, Steps to an Ecology of Mind, 2000. 

• Kenneth Edmund Ferguson, Systems Analysis in Ecology, WATT, 1966, 276 pp. 

• Efraim Halfon, Theoretical Systems Ecology: Advances and Case Studies, 1979. 

• J. W. Haefner, Modeling Biological Systems: Principles and Applications, London., UK, Chapman and Hall 1996, 
473 pp. 

• Richard F Johnston, Peter W Frank, Charles Duncan Michener, Annual Review of Ecology and Systematics, 1976, 
307 pp. 

• RL. Kitching, Systems ecology, University of Queensland Press, 1983. 

• Howard T. Odum, Systems Ecology: An Introduction, Wiley-Interscience, 1983. 

• Howard T. Odum, Ecological and General Systems: An Introduction to Systems Ecology. University Press of 
Colorado, Niwot, CO, 1994. 



Systems ecology 291 

• Friedrich Recknagel, Applied Systems Ecology: Approach and Case Studies in Aquatic Ecology, 1989. 

• James. Sanderson & Larry D. Harris, Landscape Ecology: A Top-down Approach, 2000, 246 pp. 

• Sheldon Smith, Human Systems Ecology: Studies in the Integration of Political Economy, 1989. 

External links 

Organisations 

• Systems Ecology Department at the Stockholm University. 

• Systems Ecology Department at the University of Amsterdam. 

• Systems ecology Lab [7] at SUNY-ESF. 

ro] 

• Systems Ecology program at the University of Florida 

• Terrestrial Systems Ecology of ETH Zurich. 

References 

[1] R.L. Kitching 1983, p.9. 

[2] (Kitching 1983, p. 11) 

[3] H.T.Odum 1994, p. 26. 

[4] A statement attributed to British biologist J.B.S. Haldane 

[5] http://www.ecology.su.se/ 

[6] http://www.falw.vu.nl/Onderzoeksinstituten/index.cfm/home_subsection.cfm/subsectionid/ 

55B99586-E22B-4B5C-89C65764F35DDB54 

[7] http://www.esf.edu/efb/hall/se/ 

[8] http://www.ees.ufl.edu/homepp/brown/syseco/ 

[9] http://www.sysecol.ethz.ch/ 



Ecological genetics 



Ecological genetics is the study of genetics in the context of the interactions among organisms and between the 
organisms and their environment. While molecular genetics studies the structure and function of genes at a molecular 
level, ecological genetics (and the related field of population genetics) studies phenotypic evolution in natural 
populations of organisms. Research in this field is of traits of ecological significance — that is, traits related to 
fitness, which affect an organism's survival and reproduction (e.g., flowering time, drought tolerance, sex ratio). 

Studies are often done on insects and other organisms that have short generation times, and thus evolve at high rates. 

History 

Although work on natural populations had been done previously, it is acknowledged that the field was founded by 
the English biologist E.B. Ford (1901-1988) in the early 20th century. Ford was taught genetics at Oxford University 
by Julian Huxley, and started research on the genetics of natural populations in 1924. Ecological Genetics is the title 
of his 1964 'magnum opus' on the subject (4th ed 1975). Other notable ecological geneticists would include 
Theodosius Dobzhansky who worked on chromosome polymorphism in fruit flies. As a young researcher in Russia, 
Dobzhansky had been influenced by Sergei Chetverikov, who also deserves to be remembered as a founder of 
genetics in the field, though his significance was not appreciated until much later. 

Philip Sheppard, Cyril Clarke, Bernard Kettlewell and A.J. Cain were all strongly influenced by Ford; their careers 
date from the post WWII era. Collectively, their work on lepidopterans, and on human blood groups, established the 
field, and threw light on selection in natural populations where its role had been once doubted. 

Work of this kind needs long-term funding, as well as grounding in both ecology and genetics. These are both 
difficult requirements. Research projects can last longer than a researcher's career; for instance, research into 



Ecological genetics 292 

mimicry started 150 years ago, and is still going strongly. Funding of this type of research is still rather erratic, but at 
least the value of working with natural populations in the field cannot now be doubted. 

See also 

• antibiotic resistance 

• peppered moth, Bistort betularia, 

• pesticide resistance 

• polymorphism (biology) 

• scarlet tiger moth, Calimorpha dominula, 

References 

• Ford E.B. (1964). Ecological Genetics 

• Cain A.J. and W.B. Provine (1992). Genes and ecology in history. In: R.J. Berry, T.J. Crawford and G.M. Hewitt 
(eds). Genes in Ecology. Blackwell Scientific: Oxford. (Provides a good historical background) 

• Conner, J.K. and Hard, D. L. "A Primer of Ecological Genetics". Sinauer Associates, Inc.; Sunderland, Mass. 
(2004) Provides basic and intermediate level processes and methods. 



Molecular evolution 



Molecular evolution is the process of evolution at the scale of DNA, RNA, and proteins. Molecular evolution 
emerged as a scientific field in the 1960s as researchers from molecular biology, evolutionary biology and 
population genetics sought to understand recent discoveries on the structure and function of nucleic acids and 
protein. Some of the key topics that spurred development of the field have been the evolution of enzyme function, 
the use of nucleic acid divergence as a "molecular clock" to study species divergence, and the origin of 
non-functional or junk DNA. Recent advances in genomics, including whole-genome sequencing, high-throughput 
protein characterization, and bioinformatics have led to a dramatic increase in studies on the topic. In the 2000s, 
some of the active topics have been the role of gene duplication in the emergence of novel gene function, the extent 
of adaptive molecular evolution versus neutral drift, and the identification of molecular changes responsible for 
various human characteristics especially those pertaining to infection, disease, and cognition. 

Principles of molecular evolution 
Mutations 

Mutations are permanent, transmissible changes to the genetic material (usually DNA or RNA) of a cell. Mutations 
can be caused by copying errors in the genetic material during cell division and by exposure to radiation, chemicals, 
or viruses, or can occur deliberately under cellular control during the processes such as meiosis or hypermutation. 
Mutations are considered the driving force of evolution, where less favorable (or deleterious) mutations are removed 
from the gene pool by natural selection, while more favorable (or beneficial) ones tend to accumulate. Neutral 
mutations do not affect the organism's chances of survival in its natural environment and can accumulate over time, 
which might result in what is known as punctuated equilibrium; the modern interpretation of classic evolutionary 
theory. 



Molecular evolution 293 

Causes of change in allele frequency 

There are three known processes that affect the survival of a characteristic; or, more specifically, the frequency of an 
allele (variant of a gene): 

• Genetic drift describes changes in gene frequency that cannot be ascribed to selective pressures, but are due 
instead to events that are unrelated to inherited traits. This is especially important in small mating populations, 
which simply cannot have enough offspring to maintain the same gene distribution as the parental generation. 

• Gene flow or Migration: or gene admixture is the only one of the agents that makes populations closer genetically 
while building larger gene pools. 

• Selection, in particular natural selection produced by differential mortality and fertility. Differential mortality is 
the survival rate of individuals before their reproductive age. If they survive, they are then selected further by 
differential fertility — that is, their total genetic contribution to the next generation. In this way, the alleles that 
these surviving individuals contribute to the gene pool will increase the frequency of those alleles. Sexual 
selection, the attraction between mates that results from two genes, one for a feature and the other determining a 
preference for that feature, is also very important. 

Molecular study of phylogeny 

Molecular systematics is a product of the traditional field of systematics and molecular genetics. It is the process of 
using data on the molecular constitution of biological organisms' DNA, RNA, or both, in order to resolve questions 
in systematics, i.e. about their correct scientific classification or taxonomy from the point of view of evolutionary 
biology. 

Molecular systematics has been made possible by the availability of techniques for DNA sequencing, which allow 
the determination of the exact sequence of nucleotides or bases in either DNA or RNA. At present it is still a long 
and expensive process to sequence the entire genome of an organism, and this has been done for only a few species. 
However, it is quite feasible to determine the sequence of a defined area of a particular chromosome. Typical 
molecular systematic analyses require the sequencing of around 1000 base pairs. 

The driving forces of evolution 

Depending on the relative importance assigned to the various forces of evolution, three perspectives provide 

evolutionary explanations for molecular evolution. 

Hi 
While recognizing the importance of random drift for silent mutations, selectionists hypotheses argue that 

balancing and positive selection are the driving forces of molecular evolution. Those hypotheses are often based on 

the broader view called panselectionism, the idea that selection is the only force strong enough to explain evolution, 

relaying random drift and mutations to minor roles. 

Neutralists hypotheses emphasize the importance of mutation, purifying selection and random genetic drift. The 
introduction of the neutral theory by Kimura, quickly followed by King and Jukes' own findings, lead to a fierce 
debate about the relevance of neodarwinism at the molecular level. The Neutral theory of molecular evolution states 
that most mutations are deleterious and quickly removed by natural selection, but of the remaining ones, the vast 
majority are neutral with respect to fitness while the amount of advantageous mutations is vanishingly small. The 
fate of neutral mutations are governed by genetic drift, and contribute to both nucleotide polymorphism and fixed 
differences between species. 

Mutationists hypotheses emphasize random drift and biases in mutation patterns. Sueoka was the first to propose 
a modern mutationist view. He proposed that the variation in GC content was not the result of positive selection, but 
a consequence of the GC mutational pressure. 



Molecular evolution 



294 



Related fields 

An important area within the study of molecular evolution is the use of molecular data to determine the correct 
biological classification of organisms. This is called molecular systematics or molecular phylogenetics. 

Tools and concepts developed in the study of molecular evolution are now commonly used for comparative 
genomics and molecular genetics, while the influx of new data from these fields has been spurring advancement in 
molecular evolution. 



Key researchers in molecular evolution 

Some researchers who have made key contributions to the development of the field: 

Motoo Kimura — Neutral theory 

Masatoshi Nei — Adaptive evolution 

Walter M. Fitch — Phylogenetic reconstruction 

Walter Gilbert — RNA world 

Joe Felsenstein — Phylogenetic methods 

Susumu Ohno — Gene duplication 

John H. Gillespie — Mathematics of adaptation 



Journals and societies 

Journals dedicated to molecular evolution include Molecular Biology and Evolution, Journal of Molecular 
Evolution, and Molecular Phylogenetics and Evolution. Research in molecular evolution is also published in journals 
of genetics, molecular biology, genomics, systematics, or evolutionary biology. The Society for Molecular Biology 
and Evolution publishes the journal "Molecular Biology and Evolution" and holds an annual international 

meeting. 

See also 



History of molecular evolution 
Chemical evolution 
Evolution 
Genetic drift 

E. coli long-term evolution 
experiment 
Evolutionary physiology 



Genomic organization 
Horizontal gene transfer 
Human evolution 
Evolution of dietary 
antioxidants 
Molecular clock 

Comparative phylogenetics 



Neutral theory of molecular evolution 
Nucleotide diversity 
Parsimony 
Population genetics 

Selection 



Molecular evolution 295 

Further reading 

• Li, W.-H. (2006). Molecular Evolution. Sinauer. ISBN 0878934804. 

• Lynch, M. (2007). The Origins of Genome Architecture. Sinauer. ISBN 0878934847. 

References 

[I] Graur, D. and Li, W.-H. (2000). Fundamentals of molecular evolution. Sinauer. 

[2] Gillespie, J. H (1991). The Causes of Molecular Evolution. Oxford University Press, New York. ISBN 0-19-506883-1. 

[3] Kimura, M. (1983). The Neutral Theory of Molecular Evolution. Cambridge University Press, Cambridge. ISBN 0-521-23109-4. 

[4] Kimura, Motoo (1968). "Evolutionary rate at the molecular level" (http://www2.hawaii.edu/~khayes/Journal_Club/fall2006/ 

Kimura_1968_Nature.pdf). Nature 217: 624-626. doi:10.1038/217624a0. . 
[5] King, J.L. and Jukes, T.H. (1969). "Non-Darwinian Evolution" (http://www.blackwellpublishing.com/ridley/classictexts/king.pdf). 

Science 164: 788-798. doi:10.1126/science.l64.3881.788. PMID 5767777. . 
[6] Nachman M. (2006). "Detecting selection at the molecular level" in: Evolutionary Genetics: concepts and case studies, pp. 103—118. 
[7] The nearly neutral theory expanded the neutralist perspective, suggesting that several mutations are nearly neutral, which means both random 

drift and natural selection is relevant to their dynamics. 
[8] Ohta, T (1992). "The nearly neutral theory of molecular evolution". Annual Review of Ecology and Systematics 23: 263—286. 

doi:10.1146/annurev.es.23.110192.001403. 
[9] Nei, M. (2005). "Selectionism and Neutralism in Molecular Evolution". Molecular Biology and Evolution 22(12): 2318—2342. 

doi:10.1093/molbev/msi242. PMID 16120807. 
[10] Sueoka, N. (1964). "On the evolution of informational macromolecules". in In: Bryson, V. and Vogel, H.J.. Evolving genes and proteins. 

Academic Press, New- York. pp. 479—496. 

[II] http://www.smbe.org 



Evolutionary history of life 



The evolutionary history of life on Earth traces the processes by which living and fossil organisms evolved. It 
stretches from the origin of life on Earth, thought to be over 3500 million years ago, to the present day. The 

similarities between all present day organisms indicate the presence of a common ancestor from which all known 

T21 
species have diverged through the process of evolution. 

Microbial mats of coexisting bacteria and archaea were the dominant form of life in the early Archean and many of 

[31 
the major steps in early evolution are thought to have taken place within them. The evolution of oxygenic 

photosynthesis, around 3500 million years ago, eventually led to the oxygenation of the atmosphere, beginning 

around 2400 million years ago. While eukaryotic cells may have been present earlier, their evolution 

accelerated when they began to use oxygen in their metabolism. The earliest evidence of complex eukaryotes with 

organelles, dates from 1850.0 million years ago. Later, around 1700 million years ago, multicellular organisms 

181 

began to appear, with differentiated cells performing specialised functions. 

The earliest land plants date back to around 450.0 million years ago, though evidence suggests that algal scum 

formed on the land as early as 1200 million years ago. Land plants were so successful that they are thought to 

ri2i ri3i 

have contributed to the late Devonian extinction event. Invertebrate animals appear during the Vendian period, 

while vertebrates originated about 525 million years ago during the Cambrian explosion. 

During the Permian period, synapsids, including the ancestors of mammals, dominated the land, but the 

11 71 11 81 

Permian— Triassic extinction event 251.0 million years ago came close to wiping out all complex life. During 
the recovery from this catastrophe, archosaurs became the most abundant land vertebrates, displacing therapsids in 
the mid-Triassic. One archosaur group, the dinosaurs, dominated the Jurassic and Cretaceous periods, while 

[21] 

the ancestors of mammals survived only as small insectivores. After the Cretaceous— Tertiary extinction event 65 

[221 [231 [24] 

million years ago killed off the non-avian dinosaurs mammals increased rapidly in size and diversity. Such 

mass extinctions may have accelerated evolution by providing opportunities for new groups of organisms to 

diversify. 



Evolutionary history of life 296 

Fossil evidence indicates that flowering plants appeared and rapidly diversified in the Early Cretaceous, between 130 
million years ago and 90 million years ago, probably helped by coevolution with pollinating insects. 

Flowering plants and marine phytoplankton are still the dominant producers of organic matter. Social insects 
appeared around the same time as flowering plants. Although they occupy only small parts of the insect "family 
tree", they now form over half the total mass of insects. Humans evolved from a lineage of upright-walking apes 

no] 

whose earliest fossils date from over 6 million years ago. Although early members of this lineage had 

[291 
chimp-sized brains, there are signs of a steady increase in brain size after about 3 million years ago. 

Earliest history of Earth 

History of Earth and its life 



Hadean 

Archean 

Protero 
-zoic 

Phanero 
-zoic 

Eo 
Paleo 
Meso 

Neo 
Paleo 



Evolutionary history of life 297 

Meso 

Neo 

Paleo 

Meso 

Ceno 

Scale: 
Millions of years 

The oldest meteorite fragments found on Earth are about 4540 million years old, and this has convinced scientists 

nil 
that the whole Solar system, including Earth, formed around that time. About 40 million years later a planetoid 

T321 
struck the Earth, throwing into orbit the material that formed the Moon. 

[331 [3n 

Until recently the oldest rocks found on Earth were about 3800 million years old, and this led scientists to 

believe for decades that Earth's surface was molten until then. Hence they named this part of Earth's history the 

[34] [35] 

Hadean eon, whose name means "hellish". However analysis of zircons formed 4400 to 4000 million years 

ago indicates that Earth's crust solidified about 100 million years after the planet's formation and that Earth quickly 

acquired oceans and an atmosphere, which may have been capable of supporting life. 

T371 
Evidence from the Moon indicates that from 4000 to 3800 million years ago it suffered a Late Heavy 

Bombardment by debris that was left over from the formation of the Solar system, and Earth, having stronger 

gravity, should have experienced an even heavier bombardment. While there is no direct evidence of 

T371 
conditions on Earth 4000 to 3800 million years ago, there is no reason to think that the Earth was not also 

[39] 
affected by this late heavy bombardment. This event may well have stripped away any previous atmosphere and 

oceans; in this case gases and water from comet impacts may have contributed to their replacement, although 

volcanic outgassing on Earth would have contributed at least half. 

Earliest evidence for life on Earth 

The earliest identified organisms were minute and relatively featureless, so their fossils look like small rods, which 
are very difficult to tell apart from structures which form through physical processes. The oldest undisputed evidence 
of life on Earth, interpreted as fossilized bacteria, dates to 3000 million years ago. Other finds in rocks dated 
to about 3500 million years ago have been interpreted as bacteria, and geochemical evidence seemed to show 

the presence of life 3800 million years ago. However these analyses were closely scrutinized, and 

T451 
non-biological processes were found which could produce all of the "signatures of life" that had been reported. 

While this does not prove that the structures found had a non-biological origin, they cannot be taken as clear 

evidence for the presence of life. Currently, the oldest unchallenged evidence for life is geochemical signatures from 

rocks deposited 3400 million years ago, although there has been little time for these recent reports (2006) 

to be examined by critics. 



Evolutionary history of life 



298 



Origins of life on Earth 

Biochemists reason that all living 
organisms on Earth must share a single 
last universal ancestor, because it 
would be virtually impossible that two 
or more separate lineages could have 
independently developed the many 
complex biochemical mechanisms 
shared by all living organisms. 
However the earliest organisms for 
which fossil evidence is available are 
bacteria, which are far too complex to 
have arisen directly from non-living 
materials. The lack of fossil or 
geochemical evidence for earlier types 
of organism has left plenty of scope for 
hypotheses, which fall into two main 
groups: that life arose spontaneously 
on Earth, and that it was "seeded" from 



Animals Fungi Gram-positives 
„.. . . i / Chlamydiae 

Slime moulds \ \ / / 

Plants \ \^ If/ ® reen nonsulfur bacteria 
Algae s ^Actinobacteria 

Planctomycetes 

Spirochaetes 

Fusobacteria 

Cyanobacteria 
(blue-green algae) 

Thermophilic 
sulfate-reducers 



Protozoa 



Crenarchaeota 
Nanoarchaeota 
Euryarchaeota 




Acidobacteria 



Proteobacteria 



Evolutionary tree showing the divergence of modern species from their common ancestor 

in the center. Ciccarelli, F.D., Doerks, T., von Mering, C, Creevey, C.J., et al (2006). 

"Toward automatic reconstruction of a highly resolved tree of life". Science 311 (5765): 

1283-7. doi:10.1126/science.H23061. PMID 16513982. The three domains are colored, 

with bacteria blue, archaea green, and eukaryotes red. 



elsewhere in the universe 



[52] 



Life "seeded" from elsewhere 



[53] 



The idea that life Earth was "seeded" from elsewhere in the universe dates back at least to the fifth century BC. In 
the twentieth century it was proposed by the physical chemist Svante Arrhenius, by the astronomers Fred Hoyle 
and Chandra Wickramasinghe, and by molecular biologist Francis Crick and chemist Leslie Orgel. There are 

three main versions of the "seeded from elsewhere" hypothesis: from elsewhere in our Solar system via fragments 

T571 
knocked into space by a large meteor impact, in which case the only credible source is Mars; by alien visitors, 

possibly as a result of accidental contamination by micro-organisms that they brought with them; and from 

outside the Solar system but by natural means. Experiments suggest that some micro-organisms can survive 

the shock of being catapulted into space and some can survive exposure to radiation for several days, but there is no 

T571 
proof that they can survive in space for much longer periods. Scientists are divided over the likelihood of life 

T581 1571 

arising independently on Mars, or on other planets in our galaxy. 



Independent emergence on Earth 

Life on earth is based on carbon and water. Carbon provides stable frameworks for complex chemicals and can be 
easily extracted from the environment, especially from carbon dioxide. The only other element with similar chemical 
properties, silicon, forms much less stable structures and, because most of its compounds are solids, would be more 
difficult for organisms to extract. Water is an excellent solvent and has two other useful properties: the fact that ice 
floats enables aquatic organisms to survive beneath it in winter; and its molecules have electrically negative and 
positive ends, which enables it to form a wider range of compounds than other solvents can. Other good solvents, 

such as ammonia, are liquid only at such low temperatures that chemical reactions may be too slow to sustain life, 

[591 
and lack water's other advantages. Organisms based on alternative biochemistry may however be possible on 

other planets. 

Research on how life might have emerged unaided from non-living chemicals focuses on three possible starting 
points: self-replication, an organism's ability to produce offspring that are very similar to itself; metabolism, its 



Evolutionary history of life 299 

ability to feed and repair itself; and external cell membranes, which allow food to enter and waste products to leave, 
but exclude unwanted substances. Research on abiogenesis still has a long way to go, since theoretical and 
empirical approaches are only beginning to make contact with each other. 

Replication first: RNA world 

The replicator in virtually all known life is deoxyribonucleic acid. DNA's structure and replication systems are far more complex 
than those of the original replicator. 

Even the simplest members of the three modern domains of life use DNA to record their "recipes" and a complex 
array of RNA and protein molecules to "read" these instructions and use them for growth, maintenance and 
self-replication. This system is far too complex to have emerged directly from non-living materials. The 
discovery that some RNA molecules can catalyze both their own replication and the construction of proteins led to 
the hypothesis of earlier life-forms based entirely on RNA. These ribozymes could have formed an RNA world in 
which there were individuals but no species, as mutations and horizontal gene transfers would have meant that the 
offspring in each generation were quite likely to have different genomes from those that their parents started with. 
RNA would later have been replaced by DNA, which is more stable and therefore can build longer genomes, 
expanding the range of capabilities a single organism can have. Ribozymes remain as the main 

components of ribosomes, modern cells' "protein factories". 

Although short self-replicating RNA molecules have been artificially produced in laboratories, doubts have been 
raised about where natural non-biological synthesis of RNA is possible. The earliest "ribozymes" may have been 

[71] [72] 

formed of simpler nucleic acids such as PNA, TNA or GNA, which would have been replaced later by RNA. 

In 2003 it was proposed that porous metal sulfide precipitates would assist RNA synthesis at about 100 °C (212 °F) 

and ocean-bottom pressures near hydrothermal vents. In this hypothesis lipid membranes would be the last major cell 

[73] 
components to appear and until then the proto-cells would be confined to the pores. 

Metabolism first: Iron-sulfur world 

A series of experiments starting in 1997 showed that early stages in the formation of proteins from inorganic 
materials including carbon monoxide and hydrogen sulfide could be achieved by using iron sulfide and nickel sulfide 
as catalysts. Most of the steps required temperatures of about 100 °C (212 °F) and moderate pressures, although one 

stage required 250 °C (482 °F) and a pressure equivalent to that found under 7 kilometres (4.3 mi) of rock. Hence it 

T741 
was suggested that self-sustaining synthesis of proteins could have occurred near hydrothermal vents. 

Membranes first: Lipid world 




= water-attracting heads of lipid molecules 
= water-repellent tails 
Cross-section through a liposome. 

It has been suggested that double-walled "bubbles" of lipids like those that form the external membranes of cells may 

T751 
have been an essential first step. Experiments that simulated the conditions of the early Earth have reported the 

formation of lipids, and these can spontaneously form liposomes, double-walled "bubbles", and then reproduce 

themselves. Although they are not intrinsically information-carriers as nucleic acids are, they would be subject to 



Evolutionary history of life 300 

natural selection for longevity and reproduction. Nucleic acids such as RNA might then have formed more easily 
within the liposomes than they would have outside. 

The clay theory 

RNA is complex and there are doubts about whether it can be produced non-biologically in the wild. Some clays, 

notably montmorillonite, have properties that make them plausible accelerators for the emergence of an RNA world: 

they grow by self-replication of their crystalline pattern; they are subject to an analog of natural selection, as the clay 

"species" that grows fastest in a particular environment rapidly becomes dominant; and they can catalyze the 

[771 
formation of RNA molecules. Although this idea has not become the scientific consensus, it still has active 

[78] 

supporters. 

Research in 2003 reported that montmorillonite could also accelerate the conversion of fatty acids into "bubbles", 
and that the "bubbles" could encapsulate RNA attached to the clay. These "bubbles" can then grow by absorbing 
additional lipids and then divide. The formation of the earliest cells may have been aided by similar processes. 

A similar hypothesis presents self-replicating iron-rich clays as the progenitors of nucleotides, lipids and amino 
acids. 

Environmental and evolutionary impact of microbial mats 

Microbial mats are multi-layered, multi-species colonies of 
bacteria and other organisms that are generally only a few 
millimeters thick, but still contain a wide range of chemical 
environments, each of which favors a different set of 

roi ] 

micro-organisms. To some extent each mat forms its own food 




chain, as the by-products of each group of micro-organisms 

rs2i 
generally serve as "food" for adjacent groups. 

Stromatolites are stubby pillars built as microbes in mats slowly 
migrate upwards to avoid being smothered by sediment deposited 

roi ] 

on them by water/ There has been vigorous debate about the Modem stromatolites in Shark Bay, Western Australia. 

1411 

validity of alleged fossils from before 3000 million years 

ago, with critics arguing that so-called stromatolites could have been formed by non-biological processes. In 
2006 another find of stromatolites was reported from the same part of Australia as previous ones, in rocks dated to 
3500 million years ago. 

In modern underwater mats the top layer often consists of photosynthesizing cyanobacteria which create an 
oxygen-rich environment, while the bottom layer is oxygen-free and often dominated by hydrogen sulfide emitted by 
the organisms living there. It is estimated that the appearance of oxygenic photosynthesis by bacteria in mats 
increased biological productivity by a factor of between 100 and 1,000. The reducing agent used by oxygenic 
photosynthesis is water, which is much more plentiful than the geologically-produced reducing agents required by 

roc] 

the earlier non-oxygenic photosynthesis. From this point onwards life itself produced significantly more of the 
resources it needed than did geochemical processes. Oxygen is toxic to organisms that are not adapted to it, but 
greatly increases the metabolic efficiency of oxygen-adapted organisms. 

141 T891 

Oxygen became a significant component of Earth's atmosphere about 2400 million years ago. Although 

eukaryotes may have been present much earlier, the oxygenation of the atmosphere was a prerequisite for the 

evolution of the most complex eukaryotic cells, from which all multicellular organisms are built. The boundary 

between oxygen-rich and oxygen-free layers in microbial mats would have moved upwards when photosynthesis 

shut down overnight, and then downwards as it resumed on the next day. This would have created selection pressure 

for organisms in this intermediate zone to acquire the ability to tolerate and then to use oxygen, possibly via 



Evolutionary history of life 301 

131 
endosymbiosis, where one organism lives inside another and both of them benefit from their association. 

Cyanobacteria have the most complete biochemical "toolkits" of all the mat-forming organisms. Hence they are the 

most self-sufficient of the mat organisms and were well-adapted to strike out on their own both as floating mats and 

131 
as the first of the phytoplankton, providing the basis of most marine food chains. 

Diversification of eukaryotes 



Eukaryotes 


Bikonta 
Unikonta 


Apusozoa 

Archaeplastida (Land plants, green algae, red algae, and glaucophytes) 

Chromalveolata 

Rhizaria 

Excavata 

Amoebozoa 

Opisthokonta 

Metazoa 
(Animals) 

Choanozoa 

Eumycota (Fungi) 



One possible family tree of eukaryotes 

Eukaryotes may have been present long before the oxygenation of the atmosphere, but most modern eukaryotes 

require oxygen, which their mitochondria use to fuel the production of ATP, the internal energy supply of all known 

1921 
cells. In the 1970s it was proposed and, after much debate, widely accepted that eukaryotes emerged as a result of 

a sequence of endosymbioses between "procaryotes". For example: a predatory micro-organism invaded a large 

procaryote, probably an archaean, but the attack was neutralized, and the attacker took up residence and evolved into 

the first of the mitochondria; one of these chimeras later tried to swallow a photosynthesizing cyanobacterium, but 

the victim survived inside the attacker and the new combination became the ancestor of plants; and so on. After each 

endosymbiosis began, the partners would have eliminated unproductive duplication of genetic functions by 

re-arranging their genomes, a process which sometimes involved transfer of genes between them. Another 

hypothesis proposes that mitochondria were originally sulfur- or hydrogen-metabolising endosymbionts, and became 

nsun 
[99] 



oxygen-consumers later. On the other hand mitochondria might have been part of eukaryotes' original 



equipment. 

There is a debate about when eukaryotes first appeared: the presence of steranes in Australian shales may indicate 
that eukaryotes were present 2700 million years ago; however an analysis in 2008 concluded that these 

chemicals infiltrated the rocks less than 2200 million years ago and prove nothing about the origins of 

eukaryotes. Fossils of the alga Grypania have been reported in 1850.0 million-year-old rocks (originally 
dated to 2100 million years ago but later revised ), and indicates that eukaryotes with organelles had already 

evolved. A diverse collection of fossil algae were found in rocks dated between 1500 million years ago and 

1400 million years ago. The earliest known fossils of fungi date from 1430 million years ago. 



Evolutionary history of life 



302 



Multicellular organisms and sexual reproduction 



Multicellularity 

The simplest definitions of "multicellular", for example "having 
multiple cells", could include colonial cyanobacteria like Nostoc. 
Even a professional biologist's definition such as "having the same 
genome but different types of cell" would still include some 
genera of the green alga Volvox, which have cells that specialize 
in reproduction. Multicellularity evolved independently in 

organisms as diverse as sponges and other animals, fungi, plants, 
brown algae, cyanobacteria, slime moulds and myxobacteria. 

For the sake of brevity this article focuses on the organisms 
that show the greatest specialization of cells and variety of cell 
types, although this approach to the evolution of complexity could 
be regarded as "rather anthropocentric 




..[114] 



A slime mold solves a maze. The mold (yellow) 

explored and filled the maze (left). When the 

researchers placed sugar (red) at two separate points, 

the mold concentrated most of its mass there and left 

only the most efficient connection between the two 

points (right). 



The initial advantages of multicellularity may have included: increased resistance to predators, many of which 
attacked by engulfing; the ability to resist currents by attaching to a firm surface; the ability to reach upwards to 
filter-feed or to obtain sunlight for photosynthesis; the ability to create an internal environment that gives 

protection against the external one; and even the opportunity for a group of cells to behave "intelligently" by 
sharing information. These features would also have provided opportunities for other organisms to diversify, by 
creating more varied environments than flat microbial mats could 



[115] 



Multicellularity with differentiated cells is beneficial to the organism as a whole but disadvantageous from the point 
of view of individual cells, most of which lose the opportunity to reproduce themselves. In an asexual multicellular 
organism, rogue cells which retain the ability to reproduce may take over and reduce the organism to a mass of 
undifferentiated cells. Sexual reproduction eliminates such rogue cells from the next generation and therefore 
appears to be a prerequisite for complex multicellularity. 

The available evidence indicates that eukaryotes evolved much earlier but remained inconspicuous until a rapid 
diversification around 1000 million years ago. The only respect in which eukaryotes clearly surpass bacteria and 

archaea is their capacity for variety of forms, and sexual reproduction enabled eukaryotes to exploit that advantage 
by producing organisms with multiple cells that differed in form and function. 



Evolution of sexual reproduction 

The defining characteristic of sexual reproduction is recombination, in which each of the offspring receives 50% of 
its genetic inheritance from each of the parents. Bacteria also exchange DNA by bacterial conjugation, the 

benefits of which include resistance to antibiotics and other toxins, and the ability to utilize new metabolites. 
However conjugation is not a means of reproduction, and is not limited to members of the same species — there are 
cases where bacteria transfer DNA to plants and animals. 

The disadvantages of sexual reproduction are well-known: the genetic reshuffle of recombination may break up 
favorable combinations of genes; and since males do not directly increase the number of offspring in the next 
generation, an asexual population can out-breed and displace in as little as 50 generations a sexual population that is 
equal in every other respect. Nevertheless the great majority of animals, plants, fungi and protists reproduce 
sexually. There is strong evidence that sexual reproduction arose early in the history of eukaryotes and that the genes 



controlling it have changed very little since then 
puzzle. 



[120] 



How sexual reproduction evolved and survived is an unsolved 



Evolutionary history of life 303 

The Red Queen Hypothesis suggests that sexual reproduction provides protection against parasites, because it is 
easier for parasites to evolve means of overcoming the defenses of genetically identical clones than those of sexual 
species that present moving targets, and there is some experimental evidence for this. However there is still doubt 
about whether it would explain the survival of sexual species if multiple similar clone species were present, as one of 
the clones may survive the attacks of parasites for long enough to out-breed the sexual species. 

The Mutation Deterministic Hypothesis assumes that each organism has more than one harmful mutation and the 
combined effects of these mutations are more harmful than the sum of the harm done by each individual mutation. If 
so, sexual recombination of genes will reduce the harm done that bad mutations do to offspring and at the same time 
eliminate some bad mutations from the gene pool by isolating them in individuals that perish quickly because they 
have an above-average number of bad mutations. However the evidence suggests that the MDH's assumptions are 
shaky, because many species have on average less than one harmful mutation per individual and no species that has 
been investigated shows evidence of synergy between harmful mutations. 

The random nature of recombination causes the relative abundance of alternative traits to vary from one generation 
to another. This genetic drift is insufficient on its own to make sexual reproduction advantageous, but a combination 
of genetic drift and natural selection may be sufficient. When chance produces combinations of good traits, natural 
selection gives a large advantage to lineages in which these traits become genetically linked. On the other hand the 
benefits of good traits are neutralized if they appear along with bad traits. Sexual recombination gives good traits the 
opportunities to become linked with other good traits, and mathematical models suggest this may be more than 
enough to offset the disadvantages of sexual reproduction. Other combinations of hypotheses that are inadequate 
on their own are also being examined. 

Fossil evidence for multicellularity and sexual reproduction 

Horodyskia may have been an early metazoan, or a colonial foraminiferan 

M231 T71 

The earliest known fossil organism that is clearly multicellular, Qingshania, dated to 1700 million years ago, 

appears to consist of virtually identical cells. A red alga called Bangiomorpha, dated at 1200 million years ago, is 

the earliest known organism which has differentiated, specialized cells, and is also the oldest known 

sexually-reproducing organism. The 1430 million-year-old fossils interpreted as fungi appear to have been 

multicellular with differentiated cells. The "string of beads" organism Horodyskia, found in rocks dated from 

1500 million years ago to 900.0 million years ago, may have been an early metazoan; however it has 

r 1221 

also been interpreted as a colonial foraminiferan. 



Evolutionary history of life 304 

Emergence of animals 



Bilaterians 

Deuterostomes (chordates, hemichordates, echinoderms) 

Protostomes 

Ecdysozoa (anthropods, nematodes, tardigrades, etc.) 

Lophotrochozoa (molluscs, annelids, brachiopods, etc.) 

Acoelomorpha 

Cnidaria (jellyfish, sea anemones, hydras) 
Ctenophora (comb jellies) 

Placozoa 

Porifera (sponges): Calcarea 

Porifera: Hexactinellida & Demospongiae 

Choanoflagellata 
Mesomycetozoea 

A family tree of the animals. 

Animals are multicellular eukaryotes, and are distinguished from plants, algae, and fungi by lacking cell 

[1071 ri2si 

walls. All animals are motile, if only at certain life stages. All animals except sponges have bodies 

differentiated into separate tissues, including muscles, which move parts of the animal by contracting, and nerve 

tissue, which transmits and processes signals. 

The earliest widely-accepted animal fossils are rather modern-looking cnidarians (the group that includes jellyfish, 
sea anemones and hydras), possibly from around 580 million years ago, although fossils from the Doushantuo 

Formation can only be dated approximately. Their presence implies that the cnidarian and bilaterian lineages had 
already diverged. 

The Ediacara biota, which flourished for the last 40 million years before the start of the Cambrian, were the first 
animals more than a very few centimeters long. Many were flat and had a "quilted" appearance, and seemed so 

[1331 

strange that there was a proposal to classify them as a separate kingdom, Vendozoa. Others, however, been 

interpreted as early molluscs (Kimberella ), echinoderms (Arkarua ), and arthropods (Spriggina, 

n 3ri 
Parvancorina ). There is still debate about the classification of these specimens, mainly because the diagnostic 

features which allow taxonomists to classify more recent organisms, such as similarities to living organisms, are 

generally absent in the Ediacarans. However there seems little doubt that Kimberella was at least a triploblastic 

[1391 
bilaterian animal, in other words significantly more complex than cnidarians. 

The small shelly fauna are a very mixed collection of fossils found between the Late Ediacaran and Mid Cambrian 
periods. The earliest, Cloudina, shows signs of successful defense against predation and may indicate the start of an 
evolutionary arms race. Some tiny Early Cambrian shells almost certainly belonged to molluscs, while the owners of 
some "armor plates", Halkieria and Microdictyon, were eventually identified when more complete specimens were 



Evolutionary history of life 



305 




Opabinia made the largest single 
contribution to modern interest in the 
Cambrian explosion. 



found in Cambrian lagerstatten that preserved soft-bodied animals. 

In the 1970s there was already a debate about whether the emergence of the 
modern phyla was "explosive" or gradual but hidden by the shortage of 
Pre-Cambrian animal fossils. A re-analysis of fossils from the Burgess 
Shale lagerstatte increased interest in the issue when it revealed animals, such 
as Opabinia, which did not fit into any known phylum. At the time these were 
interpreted as evidence that the modern phyla had evolved very rapidly in the 
"Cambrian explosion" and that the Burgess Shale's "weird wonders" showed 
that the Early Cambrian was a uniquely experimental period of animal 

ri42i 

evolution. Later discoveries of similar animals and the development of 
new theoretical approaches led to the conclusion that many of the "weird 
wonders" were evolutionary "aunts" or "cousins" of modern groups — for 

example that Opabinia was a member of the lobopods, a group which includes the ancestors of the arthropods, and 

11441 
that it may have been closely related to the modern tardigrades. Nevertheless there is still much debate about 

whether the Cambrian explosion was really explosive and, if so, how and why it happened and why it appears unique 

in the history of animals. 

Most of the animals at the heart of the Cambrian explosion debate 

are protostomes, one of the two main groups of complex animals. 

One deuterostome group, the echinoderms, many of which have 

hard calcite "shells", are fairly common from the Early Cambrian 

small shelly fauna onwards. Other deuterostome groups are 

soft-bodied, and most of the significant Cambrian deuterostome 

fossils come from the Chengjiang fauna, a lagerstatte in 

China. The Chengjiang fossils Haikouichthys and 

Myllokunmingia appear to be true vertebrates, and Haikouichthys had distinct vertebrae, which may have been 

slightly mineralized. Vertebrates with jaws, such as the Acanthodians, first appeared in the Late Ordovician. 




Acanthodians were among the earliest vertebrates with 
. [146] 
jaws 



Colonization of land 

Adaptation to life on land is a major challenge: all land organisms need to avoid drying-out and all those above 
microscopic size have to resist gravity; respiration and gas exchange systems have to change; reproductive systems 
cannot depend on water to carry eggs and sperm towards each other. Although the earliest good evidence of 



land plants and animals dates back to the Ordovician period (488 to 444 



[153] 



million years ago), modern land 



ecosystems only appeared in the late Devonian, about 385 to 359 million years ago. 



Evolutionary history of life 



306 



Evolution of soil 

Before the colonization of land, soil, a combination of mineral particles and decomposed organic matter, did not 
exist. Land surfaces would have been either bare rock or unstable sand produced by weathering. Water and any 
nutrients in it would have drained away very quickly. 

Films of cyanobacteria, which are not plants but use the same 
photosynthesis mechanisms, have been found in modern deserts, 
and only in areas that are unsuitable for vascular plants. This 
suggests that microbial mats may have been the first organisms to 
colonize dry land, possibly in the Precambrian. Mat-forming 
cyanobacteria could have gradually evolved resistance to 
desiccation as they spread from the seas to tidal zones and then to 



land 



[155] 



Lichens, which are symbiotic combinations of a fungus 




Lichens growing on concrete 



(almost always an ascomycete) and one or more photosynthesizers 

(green algae or cyanobacteria), are also important colonizers 

of lifeless environments, and their ability to break down rocks contributes to soil formation in situations where 

plants cannot 

Silurian. 



plants cannot survive. The earliest known ascomycete fossils date from 423 to 419 million years ago in the 



Soil formation would have been very slow until the appearance of burrowing animals, which mix the mineral and 
organic components of soil and whose feces are a major source of the organic components. Burrows have been 
found in Ordovician sediments, and are attributed to annelids ("worms") or arthropods. 



Plants and the Late Devonian wood crisis 

In aquatic algae, almost all cells are capable of photosynthesies 
and are nearly independent. Life on land required plants to become 
internally more complex and specialized: photosynthesis was most 
efficient at the top; roots were required in order to extract water 
from the ground; the parts in between became supports and 



transport systems for water and nutrients 



[151] [159] 



Spores of land plants, possibly rather like liverworts, have been 
found in Mid Ordovician rocks dated to about 476 million 

years ago. In Mid Silurian rocks 430 million years ago there 

are fossils of actual plants including clubmosses such as 
Baragwanathia; most were under 10 centimetres (3.9 in) high, and 
some appear closely related to vascular plants, the group that 
includes trees. 




By the Late Devonian 370 million years ago, trees such as 

Archaeopteris were so abundant that they changed river systems 
from mostly braided to mostly meandering, because their roots bound the soil firmly 
Devonian wood crisis", because: 



Reconstruction of Cooksonia, a vascular plant from the 
Silurian. 



[163] 



In fact they caused a "Late 



They removed more carbon dioxide from the atmosphere, reducing the greenhouse effect and thus causing an ice 
age in the Carboniferous period. In later ecosystems the carbon 



Evolutionary history of life 



307 



dioxide "locked up" in wood is returned to the atmosphere by 
decomposition of dead wood. However the earliest fossil 
evidence of fungi that can decompose wood also comes from 
the Late Devonian. 

The increasing depth of plants' roots led to more washing of 
nutrients into rivers and seas by rain. This caused algal blooms 
whose high consumption of oxygen caused anoxic events in 
deeper waters, increasing the extinction rate among deep-water 



animals 



[165] 




Land invertebrates 



Fossilized trees from the Mid-Devonian Gilboa fossil 
forest. 



Animals had to change their feeding and excretory systems, and most land animals developed internal fertilization of 
their eggs. The difference in refractive index between water and air required changes in their eyes. On the other hand 
in some ways movement and breathing became easier, and the better transmission of high-frequency sounds in air 
encouraged the development of hearing. 

Some trace fossils from the Cambrian-Ordovician boundary about 490.0 million years ago are interpreted as the 

tracks of large amphibious arthropods on coastal sand dunes, and may have been made by euthycarcinoids, 
which are thought to be evolutionary "aunts" of myriapods. Other trace fossils from the Late Ordovician a little 
million years ago probably represent land invertebrates, and there is clear evidence of numerous 



over 445 



[170] 



arthropods on coasts and alluvial plains shortly before the Silurian-Devonian boundary, about 415 million years 

[172] 

ago, including signs that some arthropods ate plants. Arthropods were well pre-adapted to colonise land, because 
their existing jointed exoskeletons provided protection against desiccation, support against gravity and a means of 

[173] 

locomotion that was not dependent on water. 

The fossil record of other major invertebrate groups on land is poor: none at all for non-parasitic flatworms, 
nematodes or nemerteans; some parasitic nematodes have been fossilized in amber; annelid worm fossils are known 
from the Carboniferous, but they may still have been aquatic animals; the earliest fossils of gastropods on land date 
from the Late Carboniferous, and this group may have had to wait until leaf litter became abundant enough to 
provide the moist conditions they need. 

The earliest confirmed fossils of flying insects date from the Late Carboniferous, but it is thought that insects 
developed the ability to fly in the Early Carboniferous or even Late Devonian. This gave them a wider range of 
ecological niches for feeding and breeding, and a means of escape from predators and from unfavorable changes in 
the environment. About 99% of modern insect species fly or are descendants of flying species. 



Land vertebrates 




Acanthostega changed views about the early evolution 
of tetrapods 



Evolutionary history of life 



308 



"Fish" 



Osteolepiformes ("fish") 

Panderichthyidae 

Obruchevichthidae 

Acanthostega 

Ichthyostega 

Tulerpeton 

Early amphibians 

Anthracosauria 
Amniotes 



ri771 

Family tree of tetrapods 

Tetrapods, vertebrates with four limbs, evolved from other rhipidistians over a relatively short timespan during the 
Late Devonian, between 370 million years ago and 360 million years ago. From the 1950s to the early 

1980s it was thought that tetrapods evolved from fish that had already acquired the ability to crawl on land, possibly 
in order to go from a pool that was drying out to one that was deeper. However in 1987 nearly-complete fossils of 
Acanthostega from about 363 million years ago showed that this Late Devonian transitional animal had legs and 

both lungs and gills, but could never have survived on land: its limbs and its wrist and ankle joints were too weak to 
bear its weight; its ribs were too short to prevent its lungs from being squeezed flat by its weight; its fish-like tail fin 
would have been damaged by dragging on the ground. The current hypothesis is that Acanthostega, which was about 
1 metre (3.3 ft) long, was a wholly aquatic predator that hunted in shallow water. Its skeleton differed from that of 
most fish, in ways that enabled it to raise its head to breathe air while its body remained submerged, including: its 
jaws show modifications that would have enabled it to gulp air; the bones at the back of its skull are locked together, 
providing strong attachment points for muscles that raised its head; the head is not joined to the shoulder girdle and it 



has a distinct neck. 



[176] 



The Devonian proliferation of land plants may help to explain why air-breathing would have been an advantage: 
leaves falling into streams and rivers would have encouraged the growth of aquatic vegetation; this would have 
attracted grazing invertebrates and small fish that preyed on them; they would have been attractive prey but the 
environment was unsuitable for the big marine predatory fish; air-breathing would have been necessary because 
these waters would have been short of oxygen, since warm water holds less dissolved oxygen than cooler marine 
water and since the decomposition of vegetation would have used some of the oxygen. 

Later discoveries revealed earlier transitional forms between Acanthostega and completely fish-like animals. 
Unfortunately there is then a gap of about 30 million years between the fossils of ancestral tetrapods and Mid 
Carboniferous fossils of vertebrates that look well-adapted for life on land. Some of these look like early relatives of 
modern amphibians, most of which need to keep their skins moist and to lay their eggs in water, while others are 



Evolutionary history of life 309 

accepted as early relatives of the amniotes, whose water-proof skins and eggs enable them to live and breed far from 

. [177] 

water. 

Dinosaurs, birds and mammals 

Amniotes 

Synapsids 

Early synapsids (extinct) 

Pelycosaurs 

Extinct pelycosaurs 

Therapsids 

Extinct therapsids 

Mammaliformes 

Extinct 
mammaliformes 

Mammals 



Sauropsids 

n R2i 

Anapsids; whether turtles belong here is debated 

Captorhinidae and Protorothyrididae 

Diapsids 

Araeoscelidia (extinct) 

Squamata (lizards and snakes) 

Archosaurs 

Extinct archosaurs 

Crocodilians 

Pterosaurs (extinct) 
Dinosaurs 

Theropods 

Extinct 
theropods 

Birds 

Sauropods 
(extinct) 

Ornithischians (extinct) 



Evolutionary history of life 310 

Possible family tree of dinosaurs, birds and mammals 

Amniotes, whose eggs can survive in dry environments, probably evolved in the Late Carboniferous period, between 
330 million years ago and 314 million years ago. The earliest fossils of the two surviving amniote groups, 

synapsids and sauropsids, date from around 313 million years ago. The synapsid pelycosaurs and their 

descendants the therapsids are the most common land vertebrates in the best-known Permian fossil beds, between 

n R8i n 71 

229.0 million years ago and 251.0 million years ago. However at the time these were all in temperate zones 

at middle latitudes, and there is evidence that hotter, drier environments nearer the Equator were dominated by 

sauropsids and amphibians. 

The Permian-Triassic extinction wiped out almost all land vertebrates, as well as the great majority of other 

ri9ii ri92i 

life. During the slow recovery from this catastrophe, estimated to be 30M years, a previously obscure 



sauropsid group became the most abundant and diverse terrestrial vertebrates: a few fossils of archosauriformes 
("shaped like archosaurs") have been found in Late Permian rocks, but by the Mid Triassic archosaurs were the 
dominant land vertebrates. Dinosaurs distinguished themselves from other archosaurs in the Late Triassic, and 

ri94i 

became the dominant land vertebrates of the Jurassic and Cretaceous periods, between 199 million years ago 

and 65 million years ago. 

During the Late Jurassic, birds evolved from small, predatory theropod dinosaurs. The first birds inherited teeth 
and long, bony tails from their dinosaur ancestors, but some developed horny, toothless beaks by the very Late 
Jurassic and short pygostyle tails by the Early Cretaceous. 

While the archosaurs and dinosaurs were becoming more dominant in the Triassic, the mammaliform successors of 
the therapsids could only survive as small, mainly nocturnal insectivores. This apparent set-back may actually have 
promoted the evolution of mammals, for example nocturnal life may have accelerated the development of 
endothermy ("warm-bloodedness") and hair or fur. By 195 million years ago in the Early Jurassic there 

were animals that were very nearly mammals. Unfortunately there is a gap in the fossil record throughout the 

[2021 

Mid Jurassic. However fossil teeth discovered in Madagascar indicate that true mammals existed at least 167 

[2031 r2041 

million years ago. After dominating land vertebrate niches for about 150 million years, the dinosaurs 

T221 
perished 65 million years ago in the Cretaceous— Tertiary extinction along with many other groups of 

organisms. Mammals throughout the time of the dinosaurs had been restricted to a narrow range of taxa, sizes 

and shapes, but increased rapidly in size and diversity after the extinction, with bats taking to the air within 

13 million years, and cetaceans to the sea within 15 million years. 



Evolutionary history of life 



311 



Flowering plants 



Gymnosperms 



Gymnosperms 



Gnetales 
(gymnosperm) 

Welwitschia 
(gymnosperm) 

Ephedra 
(gymnosperm) 

Bennettitales 

Angiosperms 
(flowering plants) 



One possible family tree of flowering 
plants. 



Angiosperms 
(flowering plants) 



Cycads 
(gymnosperm) 

Bennettitales 



Gingko 



Gnetales 
(gymnosperm) 

Conifers 
(gymnosperm) 



Another possible family tree. 



[211] 



The 250,000 to 400,000 species of flowering plants outnumber all other ground plants combined, and are the 
dominant vegetation in most terrestrial ecosystems. There is fossil evidence that flowering plants diversified rapidly 
in the Early Cretaceous, between 130 million years ago and 90 million years ago, and that their rise 

was associated with that of pollinating insects. Among modern flowering plants Magnolias are thought to be 
close to the common ancestor of the group. However paleontologists have not succeeded in identifying the 
earliest stages in the evolution of flowering plants 



[210] [211] 



Social insects 

The social insects are remarkable because the great majority of individuals in each colony are sterile. This appears 
contrary to basic concepts of evolution such as natural selection and the selfish gene. In fact there are very few 
eusocial insect species: only 15 out of approximately 2,600 living families of insects contain eusocial species, and it 
seems that eusociality has evolved independently only 12 times among arthropods, although some eusocial lineages 
have diversified into several families. Nevertheless social insects have been spectacularly successful; for example 
although ants and termites account for only about 2% of known insect species, they form over 50% of the total mass 
of insects. Their ability to control a territory appears to be the foundation of their success 



[212] 



Evolutionary history of life 



312 



The sacrifice of breeding opportunities by most individuals has 
long been explained as a consequence of these species' unusual 
haplodiploid method of sex determination, which has the 
paradoxical consequence that two sterile worker daughters of the 
same queen share more genes with each other than they would 
with their offspring if they could breed. However Wilson and 
Holldobler argue that this explanation is faulty: for example, it is 
based on kin selection, but there is no evidence of nepotism in 
colonies that have multiple queens. Instead, they write, eusociality 
evolves only in species that are under strong pressure from 
predators and competitors, but in environments where it is possible 
to build "fortresses"; after colonies have established this security, 
they gain other advantages though co-operative foraging. In 
support of this explanation they cite the appearance of eusociality 
in bathyergid mole rats, which are not haplodiploid. 

The earliest fossils of insects have been found in Early Devonian 

rocks from about 400 million years ago, which preserve only 

a few varieties of flightless insect. The Mazon Creek lagerstatten 

from the Late Carboniferous, about 300 million years ago, 

include about 200 species, some gigantic by modern standards, 

and indicate that insects had occupied their main modern ecological niches as herbivores, detritivores and 

insectivores. Social termites and ants first appear in the Early Cretaceous, and advanced social bees have been found 

T2171 

in Late Cretaceous rocks but did not become abundant until the Mid Cenozoic. 




These termite mounds have survived a bush fire. 



Humans 



[28] 



Modern humans evolved from a lineage of upright-walking apes that has been traced back over 6 million years 

nioi [9191 

ago to Sahelanthropus. The first known stone tools were made about 2.5 million years ago, apparently by 

[2201 

Australopithecus garhi, and were found near animal bones that bear scratches made by these tools. The earliest 
hominines had chimp-sized brains, but there has been a fourfold increase in the last 3 million years; a statistical 

analysis suggests that hominine brain sizes depend almost completely on the date of the fossils, while the species to 

r22ii 
which they are assigned has only slight influence. There is a long -running debate about whether modern humans 

evolved all over the world simultaneously from existing advanced hominines or are descendants of a single small 

population in Africa, which then migrated all over the world less than 200,000 years ago and replaced previous 

[222] 

hominine species. There is also debate about whether anatomically-modern humans had an intellectual, cultural 
and technological "Great Leap Forward" under 100,000 years ago and, if so, whether this was due to neurological 

[2231 

changes that are not visible in fossils. 



Evolutionary history of life 



313 



Mass extinctions 





Cm | O | S | 


D 


c 


P 


|Tr 


1 


K 


Pg N 


50- 




















40- 




















30- 






















20- 






















io-|U 




















0- 





















542 500 450 400 350 300 250 200 150 100 50 
K-T 







Tr-J 
P-Tr 
LateD 
O-S 

Millions of years ago 

Apparent extinction intensity, i.e. the fraction of genera going extinct at any given time, as reconstructed from the fossil record. 

(Graph not meant to include recent epoch of Holocene extinction event) 

[224] 

Life on earth has suffered occasional mass extinctions at least since 542 million years ago. Although they are 

disasters at the time, mass extinctions have sometimes accelerated the evolution of life on earth. When dominance of 
particular ecological niches passes from one group of organisms to another, it is rarely because the new dominant 
group is "superior" to the old and usually because an extinction event eliminates the old dominant group and makes 

t rt. [225] [226] 

way for the new one. 

The fossil record appears to show that the gaps between mass extinctions are becoming longer and the average and 
background rates of extinction are decreasing. Both of these phenomena could be explained in one or more 



ways 



.[227] 



The oceans may have become more hospitable to life over the last 500 million years and less vulnerable to mass 
extinctions: dissolved oxygen became more widespread and penetrated to greater depths; the development of life 
on land reduced the run-off of nutrients and hence the risk of eutrophication and anoxic events; and marine 

T?2R] T22Q] 

ecosystems became more diversified so that food chains were less likely to be disrupted. 
Reasonably complete fossils are very rare, most extinct organisms are represented only by partial fossils, and 
complete fossils are rarest in the oldest rocks. So paleontologists have mistakenly assigned parts of the same 
organism to different genera which were often defined solely to accommodate these finds — the story of 
Anomalocaris is an example of this. The risk of this mistake is higher for older fossils because these are often 
unlike parts of any living organism. Many of the "superfluous" genera are represented by fragments which are not 



found again and the "superfluous" genera appear to become extinct very quickly 



[227] 



Evolutionary history of life 



314 




542 500 450 400 350 300 250 200 150 100 

► 

All genera 

"Well-defined" genera 

Trend line 

"Big Five" mass extinctions 

Other mass extinctions 

Million years ago 

Thousands of genera 

Phanerozoic biodiversity as shown by the fossil record 

Biodiversity in the fossil record, which is 

"the number of distinct genera alive at any given time; that is, those whose first occurrence predates and 
whose last occurrence postdates that time" 

T2311 
shows a different trend: a fairly swift rise from 542 to 400 million years ago; a slight decline from 400 to 200 

[2321 

million years ago, in which the devastating Permian— Triassic extinction event is an important factor; and a swift 
rise from 200 " million years ago to the present. 



The present 

Oxygenic photosynthesis accounts for virtually all of the production of organic matter from non-organic ingredients. 

Production is split about evenly between land and marine plants, and phytoplankton are the dominant marine 

a [234] 

producers. 

The processes that drive evolution are still operating. Well-known examples include the changes in coloration of the 
peppered moth over the last 200 years and the more recent appearance of pathogens that are resistant to 
antibiotics. There is even evidence that humans are still evolving, and possibly at an accelerating rate over 

the last 40,000 years. [237] 



Evolutionary history of life 315 

See also 

• Evolution 

• Evolutionary history of plants 

• Timeline of evolution 

• History of evolutionary thought 

Further reading 

• Cowen, R. (2004). History of Life (4th ed.). Blackwell Publishing Limited. ISBN 978-1405117562. 

• The Ancestor's Tale, A Pilgrimage to the Dawn of Life. Boston: Houghton Mifflin Company. 2004. 
ISBN 0-618-00583-8. 

• Richard Dawkins. (1990). The Selfish Gene. Oxford University Press. ISBN 0192860925. 

• Smith, John Maynard; Eors Szathmary (1997). The Major Transitions in Evolution. Oxfordshire: Oxford 
University Press. ISBN 0-198-50294-X. 

External links 

General information 

r2381 

• General information on evolution- Fossil Museum nav. 

[2391 

• Understanding Evolution from University of California, Berkeley 

• National Academies Evolution Resources 

• Evolution poster- PDF format "tree of life" 

[242] 

• Everything you wanted to know about evolution by New Scientist 

T2431 

• Howstuffworks.com — How Evolution Works 

• Synthetic Theory Of Evolution: An Introduction to Modern Evolutionary Concepts and Theories 
History of evolutionary thought 

• The Complete Work of Charles Darwin Online 

• Understanding Evolution: History, Theory, Evidence, and Implications 

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