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Wright-Fisher diffusions in stochastic spatial evolutionary games with death-birth updating. (English) Zbl 1404.60148

Summary: We investigate stochastic spatial evolutionary games with death-birth updating in large finite populations. Within growing spatial structures subject to appropriate conditions, the density processes of a fixed type are proven to converge to the one-dimensional Wright-Fisher diffusions. Convergence in the Wasserstein distance of the laws of the occupation measures also holds. The proofs study the convergences under certain voter models by an equivalence between their laws and the laws of the evolutionary games. In particular, the additional growing dimensions in minimal systems that close the dynamics of the game density processes are cut off in the limit. {
} As another application of this equivalence of laws, we consider a first-derivative test among the major methods for these evolutionary games in a large population of size \(N\). Requiring only the assumption that the stationary probabilities of the corresponding voting kernel are comparable to uniform probabilities, we prove that the test is applicable at least up to weak selection strengths in the usual biological sense [i.e., selection strengths of the order \(\mathcal{O}(1/N)\)].

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60F05 Central limit and other weak theorems
60J60 Diffusion processes
91A22 Evolutionary games
82C22 Interacting particle systems in time-dependent statistical mechanics
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