Evolutionary Games on the Torus with Weak Selection
Ted Cox and Rick Durrett
Abstract.
We study evolutionary games on the torus with N points in dimensions d ≥ 3.
The matrices have the form 1 + w G, where 1 is a matrix that consists of all 1's,
and w is small. As in Cox Durrett and Perkins 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) N-1 >> w then the effect of selection vanishes in the limit.
In regime (ii) if we introduce a mutation 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.
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