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

M Hillairet, S Mirrahimi

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.

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

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.
Paper Structure (32 sections, 22 theorems, 350 equations)

This paper contains 32 sections, 22 theorems, 350 equations.

Table of Contents

  1. Introduction
  2. Model and question
  3. Biological motivation
  4. Expected shape of solutions
  5. A dimensionless parametrization of the model
  6. Assumptions and notations
  7. Main results
  8. State of the art
  9. Plan of the article
  10. The Hermite-polynomials framework
  11. Hermite-polynomial expansions vs moments analysis.
  12. Rephrasing our main results with Hermite-polynomial expansions.
  13. Hermite-polynomial expansion of source term $m$
  14. The concentration property of the steady states: the proof of Theorem \ref{['thm:concentration']}-(i)
  15. Preliminaries.
  16. ...and 17 more sections

Key Result

Theorem 1.1

Assume As:m andAs:m2 and fix $\delta \in (0,1)$. Let $\varepsilon >0$ and $\overline q_\varepsilon \in L^1(\mathbb R,(1+x^2)dx)$ be a steady solution to eq_qepsform satisfying: (i) We have $\overline{q}_{\varepsilon} \in L^1((1+x^{2})^{\ell}dx)$ for all $\ell \in \mathbb N$ and there exists positive constants $C_k$, for $k=1,2,...$, which may depend on $\delta$ but not on $\varepsilon$ such that

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • ...and 26 more