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

Abstract

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" with a parameter describing 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 together with a sharp description of the correction measuring the departure from the Gaussian profile.

References in corpus (1)

• A class of Hamilton-Jacobi equations with constraint: uniqueness and constructive approach

Cited by in corpus (5)

• Ergodicity of the Fisher infinitesimal model with quadratic selection
• Dynamics of dirac concentrations in the evolution of quantitative alleles with sexual reproduction
• Uniform contractivity of the Fisher infinitesimal model with strongly convex selection
• On the Fisher infinitesimal model without variability
• The emergence of a birth-dependent mutation rate: causes and consequences