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Evolutionary Games on the Torus with Weak Selection

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 $μ$ so that $μ/w \to \infty$ slowly enough then we arrive at Tarnita's formula that describes how the equilibrium frequencies are shifted due to selection.