Asymptotic analysis of a quantitative genetics model with nonlinear integral operator
Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 537-579.

We study the asymptotic behavior of stationary solutions to a quantitative genetics model with trait-dependent mortality and a nonlinear integral reproduction operator with a parameter describing the deviation between the offspring and the mean parental trait. Our asymptotic analysis encompasses the case when the parameter is typically small. Under suitable regularity and growth conditions on the mortality rate, we prove existence and local uniqueness of a stationary profile that gets concentrated around a local optimum of mortality, with a Gaussian shape having small variance. Our approach is based on perturbative analysis techniques that require to describe accurately the correction to the Gaussian leading order profile. Our result extends previous results obtained with linear reproduction operator, but using an alternative methodology.

Nous étudions le comportement asymptotique des solutions stationnaires d’un modèle de génétique quantitative. La sélection agit sur le trait, l’opérateur de reproduction est intégral et non linéaire, avec un paramètre décrivant la déviation du descendant par rapport à la moyenne du trait des parents. Nous étudions le régime où ce paramètre est petit. Nous prouvons alors l’existence et l’unicité locale d’un profil stationnaire ressemblant à une distribution gaussienne avec une petite variance. Notre approche est basée sur des techniques d’analyse perturbative pour mesurer précisément la déviation par rapport à l’ordre principal qu’est le profil gaussien.

Accepted:
Published online:
DOI: 10.5802/jep.100
Classification: 35P20,  35P30,  35Q92,  35B40,  47G20
Keywords: Non linear spectral theory, asymptotic analysis, integro-differential equations, quantitative genetics
Calvez, Vincent 1; Garnier, Jimmy 2; Patout, Florian 3

1 ICJ, UMR 5208 CNRS & Université Claude Bernard Lyon 1, Lyon, France
2 LAMA, UMR 5127 CNRS & Univ. Savoie Mont-Blanc, Chambéry, France
3 UMPA, UMR 5669 CNRS & Ecole Normale Supérieure de Lyon, Lyon, France
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Calvez, Vincent; Garnier, Jimmy; Patout, Florian. Asymptotic analysis of a quantitative genetics model with nonlinear integral operator. Journal de l’École polytechnique - Mathématiques, Volume 6 (2019), pp. 537-579. doi : 10.5802/jep.100. http://www.numdam.org/articles/10.5802/jep.100/

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