Computation and Simulation of Evolutionary Game Dynamics in Finite Populations

Laura Hindersin,Arne Traulsen,Bin Wu,Julian García

doi:10.1038/s41598-019-43102-z

Abstract

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. The computational methods for simulation and numerical approximation of the relevant quantities are diverging without being compared for accuracy and performance. We thoroughly investigate these algorithms in order to propose a reliable standard. For expositional clarity we focus on symmetric 2 × 2 games leading to one-dimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases. We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process. We provide efficient implementations that substantially improve wall times over naive or immediate implementations. Implications are also discussed for the Wright-Fisher process, as well as structured populations and multiple types.

Highlights

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored
For expositional clarity we focus on symmetric 2 × 2 games leading to onedimensional processes, noting that extensions can be straightforward and lessons will often carry over to more complex cases
We provide time-complexity analysis and systematically compare three families of methods to compute fixation probabilities, fixation times and long-term stationary distributions for the popular Moran process

Summary

Introduction

The study of evolutionary dynamics increasingly relies on computational methods, as more and more cases outside the range of analytical tractability are explored. Theoretical models of evolutionary games in finite populations typically require numerical procedures or simulations[1–5]. This is even the case when analytical results exist, as these are often difficult to interpret or confined to specific limits[6–13]. The Moran process[14] and the Wright-Fisher process[15] have become popular models to describe how phenotypes change over time by evolution Both processes have their roots in population genetics. In each time step of the Moran process, an individual is selected proportional to its fitness and produces an identical offspring. Another randomly chosen individual is removed from the population. If two B’s interact, they both get payoff d

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