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Population invasion with bistable dynamics and adaptive evolution: The evolutionary rescue

Authors: Matthieu Alfaro and Arnaud Ducrot
Journal: Proc. Amer. Math. Soc. 146 (2018), 4787-4799
MSC (2010): Primary 35K45, 92B05, 92D15
Published electronically: July 23, 2018
MathSciNet review: 3856146
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Abstract: We consider the system of reaction-diffusion equations proposed by Kanarek and Webb (2010) as a population dynamics model. The first equation stands for the population density and models the ecological effects, namely dispersion and growth with an Allee effect (bistable nonlinearity). The second one stands for the Allee threshold, seen as a trait mean, and accounts for evolutionary effects. Precisely, the Allee threshold is submitted to three main effects: dispersion (mirroring ecology), asymmetrical gene flow, and selection. The strength of the latter depends on the population density and is thus coupling ecology and evolution. Our main result is to mathematically prove evolutionary rescue: any small initial population that would become extinct in the sole ecological context will persist and spread thanks to evolutionary factors.

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Additional Information

Matthieu Alfaro
Affiliation: IMAG, Université de Montpellier, CC051, 34 095 Montpellier, France

Arnaud Ducrot
Affiliation: IMB, Université de Bordeaux, UMR 5251, 33400 Talence, France

Keywords: Reaction-diffusion system, Allee effect, long-time behaviour, energy method, evolutionary rescue
Received by editor(s): January 22, 2018
Received by editor(s) in revised form: February 24, 2018
Published electronically: July 23, 2018
Communicated by: Wenxian Shen
Article copyright: © Copyright 2018 American Mathematical Society