Stochastic differential equations for evolutionary dynamics with demographic noise and mutations

TL;DR

This paper introduces a stochastic differential equation framework to model evolutionary dynamics in finite populations, incorporating demographic noise and mutations, and demonstrates its effectiveness through a Rock-Scissors-Paper game example.

Contribution

The paper develops a general SDE-based approach for finite population evolution, including mutations, and validates it against simulations for various population sizes.

Findings

SDE framework accurately models demographic noise effects.

Mutations are effectively incorporated when mutation rates are not too small.

Excellent agreement with simulations for large populations, and small populations without mutations.

Abstract

We present a general framework to describe the evolutionary dynamics of an arbitrary number of types in finite populations based on stochastic differential equations (SDE). For large, but finite populations this allows to include demographic noise without requiring explicit simulations. Instead, the population size only rescales the amplitude of the noise. Moreover, this framework admits the inclusion of mutations between different types, provided that mutation rates, $\mu$, are not too small compared to the inverse population size 1/N. This ensures that all types are almost always represented in the population and that the occasional extinction of one type does not result in an extended absence of that type. For $\mu N\ll1$ this limits the use of SDE's, but in this case there are well established alternative approximations based on time scale separation. We illustrate our approach by a Rock-Scissors-Paper game with mutations, where we demonstrate excellent agreement with simulation based results for sufficiently large populations. In the absence of mutations the excellent agreement extends to small population sizes.