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Concentration in selection-mutation models: error estimates and asymptotic expansions

Caroline Guinet, Sepideh Mirrahimi, Jean-Michel Roquejoffre

TL;DR

This work analyzes a parabolic selection–mutation model with small mutational effects, recasting the density via the Hopf–Cole transform to obtain a Hamilton–Jacobi equation with a constraint. It proves a first-order asymptotic expansion for the solution and the environmental input, $u_\varepsilon=u+\varepsilon v+o(\varepsilon)$ and $I_\varepsilon=I+\varepsilon J+o(\varepsilon)$, and shows that the maximum trait and the phenotypic distribution admit precise corrections. By deriving detailed expansions for the moments of the phenotypic density, the authors connect the Hamilton–Jacobi framework to biologically meaningful quantities, including a generalized canonical equation with non-Gaussian variance dynamics. The results provide rigorous convergence and error estimates, offering a robust foundation for quantitative predictions of trait dynamics in evolving populations and enabling extensions to more complex, heterogeneous environments.

Abstract

In this paper, we study an integro-differential equation which describes the evolutionary dynamics of a population structured by a phenotypic trait. This population undergoes asexual reproduction, competition, selection, and mutation. We provide an asymptotic analysis of the model, assuming that the mutations have small effects. A standard approach for the analysis of the qualitative properties of the solutions of such an equation is to apply a logarithmic transformation, which yields a Hamilton-Jacobi equation with constraint. When the reproduction term is a concave function of the trait, it has been established that the solution is classical. We rigorously derive a first-order asymptotic expansion of the solution. This expansion allows us to approximate the moments of the phenotypic density. This result establishes a connection between the approximations of the phenotypic density obtained via the Hamilton-Jacobi approach and relevant biological quantities, which are more suitable from a modeling perspective.

Concentration in selection-mutation models: error estimates and asymptotic expansions

TL;DR

This work analyzes a parabolic selection–mutation model with small mutational effects, recasting the density via the Hopf–Cole transform to obtain a Hamilton–Jacobi equation with a constraint. It proves a first-order asymptotic expansion for the solution and the environmental input, and , and shows that the maximum trait and the phenotypic distribution admit precise corrections. By deriving detailed expansions for the moments of the phenotypic density, the authors connect the Hamilton–Jacobi framework to biologically meaningful quantities, including a generalized canonical equation with non-Gaussian variance dynamics. The results provide rigorous convergence and error estimates, offering a robust foundation for quantitative predictions of trait dynamics in evolving populations and enabling extensions to more complex, heterogeneous environments.

Abstract

In this paper, we study an integro-differential equation which describes the evolutionary dynamics of a population structured by a phenotypic trait. This population undergoes asexual reproduction, competition, selection, and mutation. We provide an asymptotic analysis of the model, assuming that the mutations have small effects. A standard approach for the analysis of the qualitative properties of the solutions of such an equation is to apply a logarithmic transformation, which yields a Hamilton-Jacobi equation with constraint. When the reproduction term is a concave function of the trait, it has been established that the solution is classical. We rigorously derive a first-order asymptotic expansion of the solution. This expansion allows us to approximate the moments of the phenotypic density. This result establishes a connection between the approximations of the phenotypic density obtained via the Hamilton-Jacobi approach and relevant biological quantities, which are more suitable from a modeling perspective.

Paper Structure

This paper contains 9 sections, 15 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Model and main question
  3. Motivation and the state of the art
  4. Assumptions
  5. Notations
  6. Previous results
  7. Main results
  8. Proof of Lemma \ref{['lemmeK']}
  9. Approximation of the moments -- Proof of Theorem \ref{['thm:moments']}

Key Result

Theorem 1.1

AL.SM.BP:10 Under assumptions aspsi--asuIni, as $\varepsilon \to 0$ and along subsequences, $(u_\varepsilon)_\varepsilon$ converges locally uniformly to $u\in L^\infty_{\mathrm loc} (R^+; W^{3,\infty}_{\mathrm loc}(\mathbb{R}^d) )\cap W^{1,\infty}_{\mathrm loc} (R^+; L^{\infty}_{\mathrm loc}(\mathbb and

Theorems & Definitions (17)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.3
  • Lemma 2.4
  • ...and 7 more