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arXiv · 1412.0567

Selection Mutation Differential Equations: Long-time Behavior of Measure-Valued Solutions

Azmy S. Akleh·John Cleveland·Horst R. Thieme
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Abstract

We study the long-time behavior of solutions to a measure-valued selection-mutation model that we formulated in \cite{CLEVACK}. We establish permanence results for the full model, and we study the limiting behavior even when there is more than one strategy of a given fitness; a case that arises in applications. We show that for the pure selection case the solution of the dynamical system converges to a Dirac measure centered at the fittest strategy class provided that the support of the initial measure contains a fittest strategy; thus we term this Dirac measure an Asymptotically Stable Strategy (ASS). We also show that when the strategy space is discrete, the selection-mutation model with small mutation has a locally asymptotically stable equilibrium that attracts all initial conditions that are positive at the fittest strategy.

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Azmy S. Akleh, John Cleveland, Horst R. Thieme. 2014-12-01. Selection Mutation Differential Equations: Long-time Behavior of Measure-Valued Solutions. https://arxiv.org/abs/1412.0567

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