The replicator equation in stochastic spatial evolutionary games

TL;DR

This paper demonstrates that in large stochastic spatial evolutionary games, the strategy densities follow the replicator equation and fluctuations resemble a Gaussian process, confirming a biological conjecture about graph structures.

Contribution

It establishes the connection between stochastic spatial models and the replicator equation, extending understanding to non-regular graphs and large populations.

Findings

Strategy densities follow the replicator equation in large populations.

Normalized fluctuations converge to a Gaussian process with Wright-Fisher covariance.

Confirms a biological conjecture about the approximation of the replicator equation on various graphs.

Abstract

We study the multi-strategy stochastic evolutionary game with death-birth updating in expanding spatial populations of size $N\to \infty$. The model is a voter model perturbation. For typical populations, we require perturbation strengths satisfying $1/N\ll w\ll 1$. Under appropriate conditions on the space, the limiting density processes of strategy are proven to obey the replicator equation, and the normalized fluctuations converge to a Gaussian process with the Wright-Fisher covariance function in the limiting densities. As an application, we resolve in the positive a conjecture from the biological literature that the expected density processes approximate the replicator equation on many non-regular graphs.