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Replicator-mutator equations with quadratic fitness

Authors: Matthieu Alfaro and Rémi Carles
Journal: Proc. Amer. Math. Soc. 145 (2017), 5315-5327
MSC (2010): Primary 92D15, 35K15, 45K05, 35C05
Published electronically: August 29, 2017
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Abstract: This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is nonpositive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is nonnegative (inverted harmonic potential), solutions always become extinct in finite time.

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Additional Information

Matthieu Alfaro
Affiliation: CNRS and Université de Montpellier, IMAG, CC 051, 34095 Montpellier, France

Rémi Carles
Affiliation: CNRS and Université de Montpellier, IMAG, CC 051, 34095 Montpellier, France

Keywords: Evolutionary genetics, nonlocal reaction diffusion equation, explicit solution, long time behaviour, extinction in finite time
Received by editor(s): November 18, 2016
Received by editor(s) in revised form: January 15, 2017
Published electronically: August 29, 2017
Communicated by: Catherine Sulem
Article copyright: © Copyright 2017 American Mathematical Society

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