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.
Categories: math.PR