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Evolutionary games on the torus with weak selection. (English) Zbl 1348.91029

Summary: We study evolutionary games on the torus with \(N\) points in dimensions \(d \geq 3\). The matrices have the form \(\overline{G} = \mathbf{1} + wG\), where \(\mathbf{1}\) is a matrix that consists of all 1’s, and \(w\) is small. As in [the authors et al., Voter model perturbations and reaction diffusion equations. Paris: Société Mathématique de France (2013; Zbl 1277.60004)] 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 \(\mathbb{R}^d\). 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 mutations at rate \(\mu\) 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.

MSC:

91A22 Evolutionary games
60K35 Interacting random processes; statistical mechanics type models; percolation theory
91B12 Voting theory

Citations:

Zbl 1277.60004
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References:

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