Nongaussian fluctuations arising from finite populations: Exact results for the evolutionary Moran process

TL;DR

This paper derives exact stationary distributions for finite population Moran processes in evolutionary game theory, revealing non-Gaussian fluctuations and the impact of background fitness on dynamics.

Contribution

It provides an exact analytical solution for the stationary distribution of the Moran process in finite populations for 2x2 games, including effects of background fitness.

Findings

Finite size fluctuations can significantly deviate from Gaussian distributions.

Background fitness can be rescaled to a zero-background case without loss of generality.

Exact stationary distributions are derived for arbitrary 2x2 games.

Abstract

The appropriate description of fluctuations within the framework of evolutionary game theory is a fundamental unsolved problem in the case of finite populations. The Moran process recently introduced into this context [Nowak et al., Nature (London) 428, 646 (2004)] defines a promising standard model of evolutionary game theory in finite populations for which analytical results are accessible. In this paper, we derive the stationary distribution of the Moran process population dynamics for arbitrary $2\times{}2$ games for the finite size case. We show that a nonvanishing background fitness can be transformed to the vanishing case by rescaling the payoff matrix. In contrast to the common approach to mimic finite-size fluctuations by Gaussian distributed noise, the finite size fluctuations can deviate significantly from a Gaussian distribution.