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Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation

Anouar Jeddi

Abstract

In this paper we derive a constrained Hamilton-Jacobi equation with obstacle from a discrete non-linear integro-differential model of population dynamics, with exponentially decaying mutation kernel. The exponential decay of the kernel leads to a modification of the classical Hamilton-Jacobi equation obtained previously from continuous models in \cite{BMP}. We consider a population composed of individuals characterized by a quantitative trait, subject to selection, mutation and competition. In a regime of small mutations, small spatial discretization step and large time we prove that the WKB transformation of the density converges to a viscosity solution of a constrained Hamilton-Jacobi equation with obstacle.

Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation

Abstract

In this paper we derive a constrained Hamilton-Jacobi equation with obstacle from a discrete non-linear integro-differential model of population dynamics, with exponentially decaying mutation kernel. The exponential decay of the kernel leads to a modification of the classical Hamilton-Jacobi equation obtained previously from continuous models in \cite{BMP}. We consider a population composed of individuals characterized by a quantitative trait, subject to selection, mutation and competition. In a regime of small mutations, small spatial discretization step and large time we prove that the WKB transformation of the density converges to a viscosity solution of a constrained Hamilton-Jacobi equation with obstacle.

Paper Structure

This paper contains 11 sections, 12 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Model and main result
  3. The model
  4. Assumptions
  5. Main result
  6. Preliminary results and definitions
  7. Regularity estimates
  8. Convergence to a viscosity solution
  9. Viscosity supersolution
  10. Viscosity subsolution
  11. Proof of Theorem 2.1

Key Result

Theorem 2.1

Under Assumptions item:1-item:7, let $u^\varepsilon=(u^\varepsilon_i)_{i\in\mathbb{Z}}$ be the solution of eq:system of ODE and $\widetilde{u}^\varepsilon$ as in eq:interpolation. Then, as $\varepsilon\rightarrow 0,$ along a subsequence, $(I^\varepsilon)_\varepsilon$ converges almost everywhere to a Moreover, along a subsequence, the linear interpolation of $(n^\varepsilon_i)_{i\in \mathbb{Z}},$ d

Theorems & Definitions (20)

  • Theorem 2.1
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Proposition 3.3
  • Proposition 3.4
  • proof
  • Definition 3.5
  • Lemma 4.1
  • proof
  • ...and 10 more