Replicator equation

Replikatorgleichungen are nichlineare differential equations describing the dynamics of a population in which successful individuals grow faster than less successful individuals. They are among the basic equations from evolutionary game theory and are used in theoretical biology and in evolutionary psychology in order, for example, patterns of behavior in animals or humans to be explained as a consequence of selection.

The concept of Replikatorgleichungen was introduced in 1978 to model the dynamic that leads to an evolutionarily stable state. This equilibrium states of Replikatorgleichungen are similar, but not identical, to the concept of evolutionarily stable strategy (ESS). The underlying model is a population of infinitely many individuals, which is spread over different types. How quickly reproduce the individuals of a type that depends on the fitness of the type. Most of these fitness is not constant, but results from the interaction with other individuals. In this case, it is assumed that each individual interacts with any other individual ( mean-field approximation). This approximation can be justified that interaction and reproduction take place usually on different time scales, that is, that each individual interacts with many others before it reproduces itself by it.

Equations

In a relatively general form are continuous Replikatorgleichungen of the form

With the proportion of a Replikatorspezies type in the overall population, distribution vector, fitness and average fitness of Replikatortyp.

A frequently made ​​to the model simplification assumption is that fitness depends linearly on the composition of the Replikatorpopulation:

The payoff matrix contains the fitness information on the population: the expected payoff can be written as and the average fitness of the population as a whole.

Generalizations

A generalization of the Replikatorgleichungen that takes into account mutations that provide replicator - mutator equations is:

Here the matrix are the transition probabilities of the mutation, according to Replikatortypen.