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

TL;DR

This paper derives a constrained Hamilton-Jacobi equation from a discrete population model with exponentially decaying mutation kernel, showing convergence of the transformed density to a viscosity solution in a specific regime.

Contribution

It introduces a new derivation of a constrained Hamilton-Jacobi equation accounting for exponential decay in mutation kernels, extending previous models.

Findings

Convergence of the WKB-transformed density to a viscosity solution.

Modification of classical Hamilton-Jacobi equation due to exponential decay.

Validation of the model in small mutation and discretization regimes.

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.