# Rick Durrett : Evolutionary Games on the Torus

We study evolutionary games on the torus with N points in dimensions \(d\ge 3\) with matrices of the form \(\bar G = {\bf 1} + w G\), 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) \(w \gg N^{-2/d}\), (ii) \(N^{-2/d} \gg w \gg N^{-1}\), and (iii) there is a mutation rate \(\mu\) so that \(\mu \gg N^{-1}\) and \(\mu \gg w\) where in the last case
we have introduced a mutation rate \(\mu\) 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.

**Category**: Probability**Duration**: 01:44:35**Date**: October 8, 2015 at 4:25 PM**Views**: 117-
**Tags:**seminar, Probability Seminar

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