[Méthode de Hamilton–Jacobi pour décrire des équilibres évolutifs dans les environnements hétérogènes avec des mutations non évanescentes]
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.
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.
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@article{CRMATH_2017__355_2_155_0,
author = {Gandon, Sylvain and Mirrahimi, Sepideh},
title = {A {Hamilton{\textendash}Jacobi} method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations},
journal = {Comptes Rendus. Math\'ematique},
pages = {155--160},
publisher = {Elsevier},
volume = {355},
number = {2},
year = {2017},
doi = {10.1016/j.crma.2016.12.001},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2016.12.001/}
}
TY - JOUR AU - Gandon, Sylvain AU - Mirrahimi, Sepideh TI - A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations JO - Comptes Rendus. Mathématique PY - 2017 SP - 155 EP - 160 VL - 355 IS - 2 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2016.12.001/ DO - 10.1016/j.crma.2016.12.001 LA - en ID - CRMATH_2017__355_2_155_0 ER -
%0 Journal Article %A Gandon, Sylvain %A Mirrahimi, Sepideh %T A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations %J Comptes Rendus. Mathématique %D 2017 %P 155-160 %V 355 %N 2 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2016.12.001/ %R 10.1016/j.crma.2016.12.001 %G en %F CRMATH_2017__355_2_155_0
Gandon, Sylvain; Mirrahimi, Sepideh. A Hamilton–Jacobi method to describe the evolutionary equilibria in heterogeneous environments and with non-vanishing effects of mutations. Comptes Rendus. Mathématique, Tome 355 (2017) no. 2, pp. 155-160. doi : 10.1016/j.crma.2016.12.001. http://www.numdam.org/articles/10.1016/j.crma.2016.12.001/
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