Evolutionary Games on the Torus with Weak Selection

TL;DR

This paper investigates the limiting behavior of evolutionary games on high-dimensional tori under weak selection, deriving PDE and ODE limits and connecting to Tarnita's formula when mutations are introduced.

Contribution

It characterizes the asymptotic limits of evolutionary games on tori in different regimes of selection strength and introduces a connection to Tarnita's formula with mutations.

Findings

Limit is a PDE when selection strength dominates diffusion.

Limit is an ODE when selection is moderate.

Selection effects vanish when selection is very weak.

Abstract

We study evolutionary games on the torus with $N$ points in dimensions $d\ge 3$. The matrices have the form $\bar G = {\bf 1} + w G$, where ${\bf 1}$ 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 ${\bf R}^d$. 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.