Analysis of the steady solutions of the Fisher's infinitesimal model; a Hilbertian approach

TL;DR

This paper analyzes steady solutions of a nonlinear integro-differential equation modeling sexually reproducing populations under selection, using a Hilbertian approach and spectral analysis with Hermite polynomials.

Contribution

It introduces a novel spectral method based on Hermite polynomials to characterize and analyze steady states in the Fisher's infinitesimal model.

Findings

Characterization of steady states in the small segregational variance regime

Spectral analysis reveals stability properties of solutions

Framework extends understanding of nonlinear reproductive models

Abstract

We provide an asymptotic analysis of a nonlinear integro-differential equation which describes the evolutionary dynamics of a population which reproduces sexually and which is subject to selection and competition. The sexual reproduction is modeled via a nonlinear integral term, known as the Fisher's 'infinitesimal model'. We consider a small segregational variance regime, where a parameter in the infinitesimal model, which measures the deviation between the trait of the offspring and the mean parental trait, is small with respect to the selection variance. In this regime, we characterize the steady states of the problem and analyze their stability. Our method relies on a spectral analysis involving Hermite polynomials, highlighting the specific structure of the nonlinear reproduction term. We expect that the framework developed in this article will contribute to progress on several related problems that were out of reach with previous methods.