How successful are mutants in multiplayer games with fluctuating environments? Sojourn times, fixation and optimal switching.

Joseph W Baron,Tobias Galla

doi:10.1098/rsos.172176

Abstract

Using a stochastic model, we investigate the probability of fixation, and the average time taken to achieve fixation, of a mutant in a population of wild-types. We do this in a context where the environment in which the competition takes place is subject to stochastic change. Our model takes into account interactions which can involve multiple participants. That is, the participants take part in multiplayer games. We find that under certain circumstances, there are environmental switching dynamics which minimize the time that it takes for the mutants to fixate. To analyse the dynamics more closely, we develop a method by which to calculate the sojourn times for general birth–death processes in fluctuating environments.

Highlights

Models of evolutionary dynamics frequently involve randomness, and the timing of birth, death and mutation events is statistical
The modelling framework most commonly used in evolutionary dynamics, game theory, epidemiology and population dynamics is that of a Markovian birth–death process (e.g. [1,2])
We introduce an additional tool for the analysis of population dynamics in switching environments, and describe the calculation of mean sojourn times

Summary

Introduction

Models of evolutionary dynamics frequently involve randomness, and the timing of birth, death and mutation events is statistical. The modelling framework most commonly used in evolutionary dynamics, game theory, epidemiology and population dynamics is that of a Markovian birth–death process [1,2]) These processes capture the so-called intrinsic stochasticity (or demographic noise) in finite populations. Deterministic models are analysed relatively using tools from nonlinear dynamics; they do not capture fluctuation-driven phenomena such as fixation and extinction. These features can be characterized only within a stochastic model, and the analysis is frequently based on methods from non-equilibrium statistical mechanics [5,6]

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