Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments

Abstract

We study the asymptotic spreading of Kolmogorov-Petrovsky-Piskunov (KPP) fronts in heterogeneous shifting habitats, with any number of shifting speeds, by further developing the method based on the theory of viscosity solutions of Hamilton-Jacobi equations. Our framework addresses both reaction-diffusion equations and integro-differential equations with a distributed time-delay. The latter leads to a class of limiting equations of Hamilton-Jacobi-type depending on the variable x/t and in which the time and space derivatives are coupled together. We first establish uniqueness results for these Hamilton-Jacobi equations using elementary arguments, and then characterize the spreading speed in terms of a reduced equation on a one-dimensional domain in the variable s=x/t. In terms of the standard Fisher-KPP equation of reaction-diffusion type, we give explicit formulas of the spreading speed when the environment has one or two shifting speeds. As a byproduct, we also introduce a novel class of “asymptotically homogeneous” environments which share the same spreading speed with the corresponding homogeneous environments.