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The Cauchy problem for the infinitesimal model in the regime of small variance

Florian Patout

Vol. 16 (2023), No. 6, 1289–1350

We study the asymptotic behavior of solutions of the Cauchy problem associated to a quantitative genetics model with a sexual mode of reproduction. It combines trait-dependent mortality and a nonlinear integral reproduction operator, the infinitesimal model. A parameter describes the standard deviation between the offspring and the mean parental traits. We show that under mild assumptions upon the mortality rate m, when the deviations are small, the solutions stay close to a Gaussian profile with small variance, uniformly in time. Moreover, we characterize accurately the dynamics of the mean trait in the population. Our study extends previous results on the existence and uniqueness of stationary solutions for the model. It relies on perturbative analysis techniques with a sharp description of the correction from the Gaussian profile.

infinitesimal model, asymptotic analysis, quantitative genetics, sexual reproduction, perturbative analysis
Mathematical Subject Classification
Primary: 35B40, 35P20, 35P30, 35Q92, 47G20
Received: 2 June 2020
Revised: 20 September 2021
Accepted: 4 November 2021
Published: 23 August 2023
Florian Patout

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