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Figure (clik to enlarge) – Applying P(0)=0.6; r=4; N=100000; for(n=0;n<=N;n++) { P(n+1)=r*P(n)*(1-P(n)); } Robert May Population Dynamics equation [1974-76] (do check on Logistic maps) for several iterations (generations). After 780 iterations, P is attracted to 1 (max. population), and then suddenly, for the next generations the very same population is almost extinguish.

Not only in research, but also in the everyday world of politics and economics, we would all be better off if more people realised that simple non-linear systems do not necessarily possess simple dynamical properties.” ~ Robert M. May, “Simple Mathematical models with very complicated Dynamics”, Nature, Vol. 261, p.459, June 10, 1976.

(…) The fact that the simple and deterministic equation (1) can possess dynamical trajectories which look like some sort of random noise has disturbing practical implications. It means, for example, that apparently erratic fluctuations in the census data for an animal population need not necessarily betoken either the vagaries of an unpredictable environment or sampling errors: they may simply derive from a rigidly deterministic population growth relationship such as equation (1). This point is discussed more fully and carefully elsewhere [1]. Alternatively, it may be observed that in the chaotic regime arbitrarily close initial conditions can lead to trajectories which, after a sufficiently long time, diverge widely. This means that, even if we have a simple model in which all the parameters are determined exactly, long term prediction is nevertheless impossible. In a meteorological context, Lorenz [15] has called this general phenomenon the “butterfly effect“: even if the atmosphere could be described by a deterministic model in which all parameters were known, the fluttering of a butterfly’s wings could alter the initial conditions, and thus (in the chaotic regime) alter the long term prediction. Fluid turbulence provides a classic example where, as a parameter (the Reynolds number) is tuned in a set of deterministic equations (the Navier-Stokes equations), the motion can undergo an abrupt transition from some stable configuration (for example, laminar flow) into an apparently stochastic, chaotic regime. Various models, based on the Navier-Stokes differential equations, have been proposed as mathematical metaphors for this process [15,40,41] . In a recent review of the theory of turbulence, Martin [42] has observed that the one-dimensional difference equation (1) may be useful in this context. Compared with the earlier models [15,40,41] it has the disadvantage of being even more abstractly metaphorical, and the advantage of having a spectrum of dynamical behaviour which is more richly complicated yet more amenable to analytical investigation. A more down-to-earth application is possible in the use of equation (1) to fit data [1,2,3,38,39,43] on biological populations with discrete, non-overlapping generations, as is the case for many temperate zone arthropods. (…) in pp. 13-14, Robert M. May, “Simple Mathematical models with very complicated Dynamics“, Nature, Vol. 261, p.459, June 10, 1976 [PDF link].

Figure – Understanding the Brain as a Computational Network: significant neuronal motifs of size 3.  Most over-represented colored motifs of size 3 in the C. elegans complex neuronal network. Green: sensory neuron; blue: motor neuron; red: interneuron. Arrows represent direction that the signal travels between the two cells. (from Adami et al. 2011 [Ref. below])

Abstract: […] Complex networks can often be decomposed into less complex sub-networks whose structures can give hints about the functional organization of the network as a whole. However, these structural motifs can only tell one part of the functional story because in this analysis each node and edge is treated on an equal footing. In real networks, two motifs that are topologically identical but whose nodes perform very different functions will play very different roles in the network. Here, we combine structural information derived from the topology of the neuronal network of the nematode C. elegans with information about the biological function of these nodes, thus coloring nodes by function. We discover that particular colorations of motifs are significantly more abundant in the worm brain than expected by chance, and have particular computational functions that emphasize the feed-forward structure of information processing in the network, while evading feedback loops. Interneurons are strongly over-represented among the common motifs, supporting the notion that these motifs process and transduce the information from the sensor neurons towards the muscles. Some of the most common motifs identified in the search for significant colored motifs play a crucial role in the system of neurons controlling the worm’s locomotion. The analysis of complex networks in terms of colored motifs combines two independent data sets to generate insight about these networks that cannot be obtained with either data set alone. The method is general and should allow a decomposition of any complex networks into its functional (rather than topological) motifs as long as both wiring and functional information is available. […] from Qian J, Hintze A, Adami C (2011) Colored Motifs Reveal Computational Building Blocks in the C. elegans Brain, PLoS ONE 6(3): e17013. doi:10.1371/journal.pone.0017013

[…] The role of evolution in producing these patterns is clear, said Adami. “Selection favors those motifs that impart high fitness to the organism, and suppresses those that work against the task at hand.” In this way, the efficient and highly functional motifs (such as the sensory neuron-interneuron-motor neuron motif) are very common in the nervous system, while those that would waste energy and give no benefit to, or even harm, the animal are not found in the network. “Adami and his team have used evolutionary computation to develop hypotheses about the evolution of neural circuits and find, for these nematode worms, that simplicity is the rule,” says George Gilchrist, program director in NSF’s Division of Environmental Biology (in the Directorate for Biological Sciences), which funds BEACON. “By including functional information about each node in the circuit, they have begun decoding the role of natural selection in shaping the architecture of neural circuits.” […] from Danielle J. Whittaker “Understanding the Brain as a Computational Network“, NSF, April 2011.

Book – Karl Sigmund, The Calculus of Selfishness, Princeton Series on Theoretical and Computational Biology, Princeton University Press,  ISBN: 978-1-4008-3225-5, 192 pp., 2009.

[…] Cooperation means that a donor pays a cost, c, for a recipient to get a benefit, b. In evolutionary biology, cost and benefit are measured in terms of fitness. While mutation and selection represent the main forces of evolutionary dynamics, cooperation is a fundamental principle that is required for every level of biological organization. Individual cells rely on cooperation among their components. Multicellular organisms exist because of cooperation among their cells. Social insects are masters of cooperation. Most aspects of human society are based on mechanisms that promote cooperation. Whenever evolution constructs something entirely new (such as multicellularity or human language), cooperation is needed. Evolutionary construction is based on cooperation. The five rules for cooperation which we examine in this chapter are: kin selection, direct reciprocity, indirect reciprocity, graph selection, and group selection. Each of these can promote cooperation if specific conditions are fulfilled. […], Martin A. Nowak, Karl Sigmund, How populations cohere: five rules for cooperation, in R. M. May and A. McLean (eds.) Theoretical Ecology: Principles and Applications, Oxford UP, Oxford (2007), 7-16. [PDF]

How does cooperation emerge among selfish individuals? When do people share resources, punish those they consider unfair, and engage in joint enterprises? These questions fascinate philosophers, biologists, and economists alike, for the “invisible hand” that should turn selfish efforts into public benefit is not always at work. The Calculus of Selfishness looks at social dilemmas where cooperative motivations are subverted and self-interest becomes self-defeating. Karl Sigmund, a pioneer in evolutionary game theory, uses simple and well-known game theory models to examine the foundations of collective action and the effects of reciprocity and reputation. Focusing on some of the best-known social and economic experiments, including games such as the Prisoner’s Dilemma, Trust, Ultimatum, Snowdrift, and Public Good, Sigmund explores the conditions leading to cooperative strategies. His approach is based on evolutionary game dynamics, applied to deterministic and probabilistic models of economic interactions. Exploring basic strategic interactions among individuals guided by self-interest and caught in social traps, The Calculus of Selfishness analyses to what extent one key facet of human nature–selfishness–can lead to cooperation. (from Princeton Press). [Karl Sigmund, The Calculus of Selfishness, Princeton Series on Theoretical and Computational Biology, Princeton University Press,  ISBN: 978-1-4008-3225-5, 192 pp., 2009.]

What follows comes partly from chapter 1, available here:

THE SOCIAL ANIMAL: Aristotle classified humans as social animals, along with other species, such as ants and bees. Since then, countless authors have compared cities or states with bee hives and ant hills: for instance, Bernard de Mandeville, who published his The Fable of the Bees more than three hundred years ago. Today, we know that the parallels between human communities and insect states do not reach very far. The amazing degree of cooperation found among social insects is essentially due to the strong family ties within ant hills or bee hives. Humans, by contrast, often collaborate with non-related partners. Cooperation among close relatives is explained by kin selection. Genes for helping offspring are obviously favouring their own transmission. Genes for helping brothers and sisters can also favour their own transmission, not through direct descendants, but indirectly, through the siblings’ descendants: indeed, close relatives are highly likely to also carry these genes. In a bee hive, all workers are sisters and the queen is their mother. It may happen that the queen had several mates, and then the average relatedness is reduced; the theory of kin selection has its share of complex and controversial issues. But family ties go a long way to explain collaboration. The bee-hive can be viewed as a watered-down version of a multicellular organism. All the body cells of such an organism carry the same genes, but the body cells do not reproduce directly, any more than the sterile worker-bees do. The body cells collaborate to transmit copies of their genes through the germ cells – the eggs and sperm of their organism. Viewing human societies as multi-cellular organisms working to one purpose is misleading. Most humans tend to reproduce themselves. Plenty of collaboration takes place between non-relatives. And while we certainly have been selected for living in groups (our ancestors may have done so for thirty million years), our actions are not as coordinated as those of liver cells, nor as hard-wired as those of social insects. Human cooperation is frequently based on individual decisions guided by personal interests. Our communities are no super-organisms. Former Prime Minister Margaret Thatcher pithily claimed that “there is no such thing as society“. This can serve as the rallying cry of methodological individualism – a research program aiming to explain collective phenomena bottom-up, by the interactions of the individuals involved. The mathematical tool for this program is game theory. All “players” have their own aims. The resulting outcome can be vastly different from any of these aims, of course.

THE INVISIBLE HAND: If the end result depends on the decisions of several, possibly many individuals having distinct, possibly opposite interests, then all seems set to produce a cacophony of conflicts. In his Leviathan from 1651, Hobbes claimed that selfish urgings lead to “such a war as is every man against every man“. In the absence of a central authority suppressing these conflicts, human life is “solitary, poor, nasty, brutish, and short“. His French contemporary Pascal held an equally pessimistic view: : “We are born unfair; for everyone inclines towards himself…. The tendency towards oneself is the origin of every disorder in war, polity, economy etc“. Selfishness was depicted as the root of all evil. But one century later, Adam Smith offered another view.An invisible hand harmonizes the selfish efforts of individuals: by striving to maximize their own revenue, they maximize the total good. The selfish person works inadvertently for the public benefit. “By pursuing his own interest he frequently promotes that of the society more effectually than when he really intends to promote it“. Greed promotes behaviour beneficial to others. “It is not from the benevolence of the butcher, the brewer, or the baker, that we expect our dinner, but from their regard to their own self-interest. We address ourselves, not to their humanity but to their self-love, and never talk to them of our own necessities but of their advantages“. A similar view had been expressed, well before Adam Smith, by Voltaire in his Lettres philosophiques: “Assuredly, God could have created beings uniquely interested in the welfare of others. In that case, traders would have been to India by charity, and the mason would saw stones to please his neighbour. But God designed things otherwise….It is through our mutual needs that we are useful to the human species; this is the grounding of every trade; it is the eternal link between men“. Adam Smith (who knew Voltaire well) was not blind to the fact that the invisible hand is not always at work. He merely claimed that it frequently promotes the interest of the society, not that it always does. Today, we know that there are many situations – so-called social dilemmas – where the invisible hand fails to turn self-interest to everyone’s advantage.

[...] People should learn how to play Lego with their minds. Concepts are building bricks [...] V. Ramos, 2002.

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