Evolutionary Games on the Torus with Weak Selection

Abstract

We study evolutionary games on the torus with $N$ points in dimensions $d\ge 3$. The matrices have the form $\bar{G}=1+wG$, where $1$ is a matrix that consists of all 1's, and $w$ is small. As in Cox Durrett and Perkins \cite{CDP} we rescale time and space and take a limit as $N\to \infty$ and $w\to 0$. If (i) $w\gg N^{-2/d}$ then the limit is a PDE on ${\mathbf{R}}^d$. If (ii) $N^{-2/d}\gg w\gg N^{-1}$, then the limit is an ODE. If (iii) $w\ll N^{-1}$ then the effect of selection vanishes in the limit. In regime (ii) if we introduce a mutation $\mu$ so that $\mu /w\to \infty$ slowly enough then we arrive at Tarnita's formula that describes how the equilibrium frequencies are shifted due to selection.

References in corpus (2)

• Evolutionary games on graphs
• Voter Model Perturbations and Reaction Diffusion Equations

Cited by in corpus (1)

• Latent Voter Model on Locally Tree-Like Random Graphs