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 | |  Population Dynamics | |
Population dynamics, sometimes referred to as “replicator dynamics”, offers a tractable way to model the evolution of populations over time under the kinds of selective pressures that can be characterized by static optimization models. This is often necessary, since the dynamics of such populations are often difficult to predict purely on the basis of static considerations of payoff differences. The so-called “replicator dynamics” were named by Taylor and Jonker (1978) and generalized by Schuster and Sigmund (1983) and Hofbauer and Sigmund (1988). They trace their source back to the seminal work of R.A. Fisher in the 1920's and 30's. The generalization covers evolutionary models used in population genetics, evolutionary game theory, ecology, and the study of prebiotic evolution. The models can be implemented either mathematically or computationally, and can model either stepwise (discrete) or continuous evolutionary change.Population dynamics models the evolution of populations. A population is a collection of individuals, which are categorized according to type. The types in genetics are genes, in evolutionary game theory, strategies. The types of interest in epistemological models would be types of cognitive apparatuses, or cognitive strategies — ways of responding to environmental cues, ways of manipulating representations, and so forth. Roughly, EEM models focus on the inherited and EET models focus on the learned. The evolution of the population consists in changes of the relative frequency of the different types within the population. Selection, typified by differential reproductive success, is represented as follows. Each type has a growth rate or “fitness”, designated by w, and a frequency designated by p. The frequency of type i at the next generation pi′ is simply the old frequency multiplied by the fitness and divided by the mean fitness of the population “[ltr]w[size=13]¯¯¯¯w¯[/ltr]”.[/size] - اقتباس :
- pi′ = pi · wi / [ltr]w[size=13]¯¯¯¯w¯[/ltr][/size]
Division by [ltr]w[size=13]¯¯¯¯w¯[/ltr] has the effect of “normalizing” the frequencies, so that they add up to one after each is multiplied by its fitness. It also makes evident that the frequency of a type will increase just in case its fitness is higher than the current population average.[/size] FitnessFitnesses, which should be understood simply as the aggregation of probable-growth factors that drive the dynamics of large populations, may depend on a variety of factors. Fitness components differ from variation components in that they affect population frequencies proportionally to those frequencies, that is to say, multiplicatively. Fitness component in biological evolution include mortality and reproductive rate. In cultural evolution, they include transmission probability and rejection probability. Within either sort of model, what matters is how fitnesses change as a result of other changing factors within the model. In the simplest cases, fitnesses are fixed and the type with the highest fitness inevitably dominates the population. In more complex cases, fitnesses may depend on variable factors like who one plays against, or the state of a variable environment. Most commonly, variable fitnesses are calculated using a payoff matrix like the one above. In general, to calculate the expected fitness of a type, one multiplies the fitness a type would have in each situation times the likelihood that individuals in the population will confront that situation and adds the resulting products. - اقتباس :
- wi = SA Pr(A) · WiA
where WiA is type i's fitness in situation A. This sort of calculation assumes that the effects of the various situations are additive. More complex situations can be modeled, of course, but additive matrices are the standard. It should be noted, however, that matrix-driven evolution can exhibit quite complex behavior. For instance, chaotic behavior is possible with as few as four strategies (Skyrms 1992).Some relationships may be represented without a matrix. Boyd and Richerson (1985), for instance, were interested in a special kind of frequency dependent transmission bias in culture, where being common conferred an advantage due to imitators “doing as the Romans do.” In such a case, the operative fitness of the type is just the fitness as calculated according to the usual factors, and then modified as a function of the frequency of the type.Continuity and ComputationThe conceptual bases of replicator dynamics are quite straightforward. Getting results typically requires one of two approaches. In order to prove more than rudimentary mathematical results, one typically needs to derive a continuous version of the dynamics. The basic form is - اقتباس :
- dpi /dt = p(wi − [ltr]w[size=13]¯¯¯¯w¯[/ltr])[/size]
with fitnesses calculated as usual. Mathematical approaches have been quite productive, though the bulk of theoretical results apply primarily to population genetics. See Hofbauer and Sigmund (1988) for a compendium of such results, as well as a reasonable graduate-level introduction to the mathematical study of evolutionary processes.The second approach is computational. With the increase in power of personal computers, computational implementation of evolutionary models become increasingly attractive. They require only rudimentary programming skills, and are in general much more flexible in the assumptions they require. The general strategy is to create an array to hold population frequencies and fitnesses, and then a series of procedures (or methods or functions) which[list="margin-top: 0.5em; color: rgb(26, 26, 26); font-family: serif; font-size: 16.5px; line-height: 21px; background-color: rgb(255, 255, 255);"] [*]calculate fitnesses, [*]update frequencies with the new fitnesses, and [*]manage interface details like outputting the new state of the population to a file or the screen. [/list] A loop then runs the routines in sequence, over and over again. Most modelers are happy to put their source code on the internet, which is probably the best place to find it.Modeling Cultural EvolutionPart of the difficulty in understanding cognitive behavior as the product of evolution is that there are at least three very different evolutionary processes involved. First, there is the biological evolution of cognitive and perceptual mechanisms via genetic inheritance. Second, there is the cultural evolution of languages and concepts. Third, there is the trial-and-error learning process that occurs during an individual's lifetime. Moreover, there is some reason to agree with Donald T. Campbell that understanding human knowledge fully will require understanding the interaction between these processes. This requires that we be able to model both processes of biological and cultural evolution. There are by now a number of well-established models of biological evolution. Cultural evolution presents more novelty. | |
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