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Asymptotic and transient behaviour for a nonlocal problem arising in population genetics

Part of: Miscellaneous topics - Partial differential equations Partial differential equations Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  18 September 2018

J.-B. BURIE ,
R. DJIDJOU-DEMASSE  and
A. DUCROT
J.-B. BURIE*
Affiliation:
University of Bordeaux, IMB, UMR 5251, F-33400 Talence, France emails: jean-baptiste.burie@u-bordeaux.fr; arnaud.ducrot@u-bordeaux.fr CNRS, IMB, UMR 5251, F-33400 Talence, France
R. DJIDJOU-DEMASSE
Affiliation:
UMR 1065, INRA, Villenave d’Ornon F-33882, France email: ddramses@gmail.com
A. DUCROT
Affiliation:
University of Bordeaux, IMB, UMR 5251, F-33400 Talence, France emails: jean-baptiste.burie@u-bordeaux.fr; arnaud.ducrot@u-bordeaux.fr CNRS, IMB, UMR 5251, F-33400 Talence, France
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Abstract

This work is devoted to the study of an integro-differential system of equations modelling the genetic adaptation of a pathogen by taking into account both mutation and selection processes. First, we study the asymptotic behaviour of the system and prove that it eventually converges to a stationary state. Next, we more closely investigate the behaviour of the system in the presence of multiple EAs. Under suitable assumptions and based on a small mutation variance asymptotic, we describe the existence of a long transient regime during which the pathogen population remains far from its asymptotic behaviour and highly concentrated around some phenotypic value that is different from the one described by its asymptotic behaviour. In that setting, the time needed for the system to reach its large time configuration is very long and multiple evolutionary attractors may act as a barrier of evolution that can be very long to bypass.

Keywords

Nonlocal equationasymptotic behaviourtransient behaviourpopulation genetics

MSC classification

Primary: 35B40: Asymptotic behavior of solutions
Secondary: 35R09: Integro-partial differential equations 35Q92: PDEs in connection with biology and other natural sciences 92D10: Genetics
Type
Papers
Information
European Journal of Applied Mathematics , Volume 31 , Issue 1 , February 2020 , pp. 84 - 110
Copyright
© Cambridge University Press 2018 

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