A Hamilton-Jacobi Approach to Evolution of Dispersal

TL;DR

This paper analyzes the evolution of dispersal traits in populations using a Hamilton-Jacobi framework, characterizing the asymptotic behavior of solutions in the rare mutation limit.

Contribution

It introduces a Hamilton-Jacobi approach to study the evolution of dispersal, extending previous models to characterize time-dependent solutions.

Findings

Population concentrates on minimal dispersal trait in the rare mutation limit

Asymptotic behavior of solutions is characterized under convexity assumptions

Provides a mathematical framework for evolution of dispersal traits

Abstract

The evolution of dispersal is a classical question in evolutionary biology, and it has been studied in a wide range of mathematical models. A selection-mutation model, in which the population is structured by space and a phenotypic trait, with the trait acting directly on the dispersal (diffusion) rate, was formulated by Perthame and Souganidis [Math. Model. Nat. Phenom. 11 (2016), 154-166] to study the evolution of random dispersal towards the evolutionarily stable strategy. For the rare mutation limit, it was shown that the equilibrium population concentrates on a single trait associated to the smallest dispersal rate. In this paper, we consider the corresponding evolution equation and characterize the asymptotic behaviors of the time-dependent solutions in the rare mutation limit, under mild convexity assumptions on the underlying Hamiltonian function.