Extinction scenarios in evolutionary processes: A Multinomial Wright-Fisher approach

Abstract: We study a generalized discrete-time multi-type Wright-Fisher population process. The mean-field dynamics of the stochastic process is induced by a general replicator difference equation. We prove several results regarding the asymptotic behavior of the model, focusing on the impact of the mean-field dynamics on it. One of the results is a limit theorem that describes sufficient conditions for an almost certain path to extinction, first eliminating the type which is the least fit at the mean-field equilibrium. The effect is explained by the metastability of the stochastic system, which under the conditions of the theorem spends almost all time before the extinction event in a neighborhood of the equilibrium. In addition, to limit theorems, we propose a variation of Fisher's maximization principle, fundamental theorem of natural selection, for a completely general deterministic replicator dynamics and study implications of the deterministic maximization principle for the stochastic model.