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arXiv · 1911.11874

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

Alexander Roitershtein·Reza Rastegar·Robert S. Chapkin·Ivan Ivanov
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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.

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Alexander Roitershtein, Reza Rastegar, Robert S. Chapkin, Ivan Ivanov. 2019-11-26. Extinction scenarios in evolutionary processes: A Multinomial Wright-Fisher approach. https://arxiv.org/abs/1911.11874

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