Simulation of Evolutionary Dynamics in Finite Populations

Abstract

In finite populations, evolutionary dynamics can no longer be described by deterministic differential equations, but require a stochastic formulation [1]. We show how Mathematica can be used to both simulate and visualize evolutionary processes in limited populations. The Moran process is introduced as the basic stochastic model of an evolutionary process in finite populations. This model is extended to mixed populations with relative fitness differences. We combine population ecology with game theoretic ideas, simulating evolutionary games in well-mixed and structured populations. ‡ The Moran Process The Moran process is a simple stochastic model to study selection in finite populations [2]. We consider a population of constant size with two types of individuals, type 1 and type 0. At each time step a single individual is allowed to reproduce a clone of the same type. Furthermore, to keep the population size constant, one individual must die. The Moran process is a birth-death update process. Individuals for reproduction and elimination are chosen randomly. If both random choices fall on the same individual, the individual will be replaced by its own identical offspring and the population remains unchanged. The variable i denotes the number of type 1 individuals in the population of size n. The number of type 0 individuals is therefore n-i. The Moran process is defined on the state space i = 0, …, n. The probability of choosing a type 1 individual is given by i ê n and the probability of choosing a type 0 individual is Hn-iL ê n.