Comptes Rendus
no. 2
Mathematical analysis/Partial differential equations
A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations
[Méthode de Hamilton–Jacobi pour décrire des équilibres évolutifs dans les environnements hétérogènes avec des mutations non évanescentes]

Présenté par : Haïm Brézis

Sylvain Gandon1 ; Sepideh Mirrahimi2
1 Centre d'écologie fonctionnelle et évolutive (CEFE), UMR CNRS 5175, 34293 Montpellier cedex 5, France
2 CNRS, Institut de mathématiques (UMR CNRS 5219), Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France
Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 155-160.

In this note, we characterize the solution to a system of elliptic integro-differential equations describing a phenotypically structured population subject to mutation, selection, and migration. Generalizing an approach based on the Hamilton–Jacobi equations, we identify the dominant terms of the solution when the mutation term is small (but nonzero). This method was initially used, for different problems arisen from evolutionary biology, to identify the asymptotic solutions, while the mutations vanish, as a sum of Dirac masses. A key point is a uniqueness property related to the weak KAM theory. This method allows us to go further than the Gaussian approximation commonly used by biologists, and is an attempt to fill the gap between the theories of adaptive dynamics and quantitative genetics.

Dans cette note, nous étudions un système d'équations intégro-différentielles elliptiques, décrivant une population structurée par trait phénotypique soumise à des mutations, à la sélection et à des migrations. Nous généralisons une approche basée sur des équations de Hamilton–Jacobi pour détérminer les termes dominants de la solution lorsque les effets des mutations sont petits (mais non nuls). Cette méthode était initialement utilisée, pour différents problèmes venant de la biologie évolutive, pour identifier les solutions asymptotiques, lorsque les effets des mutations tendent vers 0, sous forme de sommes de masses de Dirac. Un point-clé est une propriété d'unicité en rapport avec la théorie de KAM faible. Cette méthode nous permet d'aller au-delà des approximations gaussiennes habituellement utilisées par les biologistes, et contribue ainsi à relier les théories de la dynamique adaptative et de la génétique quantitative.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.12.001

Sylvain Gandon 1 ; Sepideh Mirrahimi 2

1 Centre d'écologie fonctionnelle et évolutive (CEFE), UMR CNRS 5175, 34293 Montpellier cedex 5, France
2 CNRS, Institut de mathématiques (UMR CNRS 5219), Université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse cedex, France 
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Sylvain Gandon; Sepideh Mirrahimi. A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations. Comptes Rendus. Mathématique, Volume 355 (2017) no. 2, pp. 155-160. doi : 10.1016/j.crma.2016.12.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.12.001/

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