Understanding evolutionary and ecological dynamics using a continuum limit.

Peter Czuppon,Arne Traulsen

doi:10.1002/ece3.7205

Abstract

Continuum limits in the form of stochastic differential equations are typically used in theoretical population genetics to account for genetic drift or more generally, inherent randomness of the model. In evolutionary game theory and theoretical ecology, however, this method is used less frequently to study demographic stochasticity. Here, we review the use of continuum limits in ecology and evolution. Starting with an individual‐based model, we derive a large population size limit, a (stochastic) differential equation which is called continuum limit. By example of the Wright–Fisher diffusion, we outline how to compute the stationary distribution, the fixation probability of a certain type, and the mean extinction time using the continuum limit. In the context of the logistic growth equation, we approximate the quasi‐stationary distribution in a finite population.

Highlights

The huge computational power available today allows researchers to develop individual-based models of high complexity to explore dynamical processes in ecology and evolution
Even though similar in the questions they try to answer, evolutionary game theory and population genetics are developing in parallel, sometimes with little interaction between them. As this is partly arising from the different methods applied, here we aim to provide an introduction to the continuum limit for those less comfortable with these methods and hesitant to go into the extensive, more mathematical, literature
For the theoretical population geneticist with a probabilistic background, we provide a summary of some key results on stochastic differential equations; for the evolutionary game theorist, we give a new perspective on the derivations of results obtained when using discrete birth–death processes; and lastly, for the theoretical ecologist familiar with deterministic modeling, we outline how to derive and work with stochastic versions of classical ecological and evolutionary processes

Summary

| INTRODUCTION

The huge computational power available today allows researchers to develop individual-based models of high complexity to explore dynamical processes in ecology and evolution. The Moran process has become a popular model for stochastic dynamics in finite populations (Nowak et al, 2004). A continuum approximation for quantities that are known exactly may make limited sense at first sight—but it can provide a very useful new perspective Another important process in population genetics is the Wright–Fisher process—a model for allele frequency dynamics in a population of fixed size and nonoverlapping generations (Wright, 1931). While in the Wright–Fisher process generations are nonoverlapping and time is measured in discrete steps, generations in the Moran model are overlapping and measured in either discrete or continuous time Both processes describe the stochastic variation of allele frequencies due to finite population size effects referred to as genetic drift.

Conclusion

| DISCUSSION AND CONCLUSION