Evolutionary Games on the Torus

We study evolutionary games on the torus with N points in dimensions d¿3 with matrices of the form G¯=1+wG, where 1 is a matrix that consists of all 1's, and w is small. We show that there are three weak selection regimes (i) w¿N¿2/d, (ii) N¿2/d¿w¿N¿1, and (iii) there is a mutation rate ¿ so that ¿¿N¿1 and ¿¿w where in the last case we have introduced a mutation rate ¿ to make it nontrivial. In the first and second regimes the rescaled process converges to a PDE and an ODE respectively. In the third, which is the classical weak selection regime of population genetics, we give a new derivation of Tarnita's formula which describes how the equilibrium frequencies are shifted away from uniform due to the spatial structure.