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

Patout, Florian

doi:10.2140/apde.2023.16.1289

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

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

Florian Patout

Vol. 16 (2023), No. 6, 1289–1350

DOI: 10.2140/apde.2023.16.1289

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. 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.

Keywords

infinitesimal model, asymptotic analysis, quantitative genetics, sexual reproduction, perturbative analysis

Mathematical Subject Classification

Primary: 35B40, 35P20, 35P30, 35Q92, 47G20

Milestones

Received: 2 June 2020

Revised: 20 September 2021

Accepted: 4 November 2021

Published: 23 August 2023

Authors

Florian Patout
BioSP, INRAE Avignon France

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 16, No. 6, 2023