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

TL;DR

This paper analyzes extinction scenarios in a generalized Wright-Fisher model with multiple types, establishing conditions for almost certain extinction and exploring the influence of mean-field dynamics and a generalized Fisher's maximization principle.

Contribution

It introduces a limit theorem for extinction in a multi-type Wright-Fisher process and extends Fisher's maximization principle to general deterministic replicator dynamics.

Findings

Conditions for almost certain extinction of the least fit type.

Metastability explains the system's behavior before extinction.

Generalized Fisher's maximization principle for deterministic dynamics.

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.