Evolutionary games on the torus with weak selection

We study evolutionary games on the torus with N points in dimensions d≥3. The matrices have the form Ḡ=1+wG, where 1 is a matrix that consists of all 1's, and w is small. As in Cox Durrett and Perkins (2011) we rescale time and space and take a limit as N→∞ and w→0. If (i) w≫N−2/d then the limit is a PDE on Rd. If (ii) N−2/d≫w≫N−1, then the limit is an ODE. If (iii) w≪N−1 then the effect of selection vanishes in the limit. In regime (ii) if we introduce mutations at rate μ so that μ/w→∞ slowly enough then we arrive at Tarnita's formula that describes how the equilibrium frequencies are shifted due to selection.