Asymptotic spreading of KPP reactive fronts in heterogeneous shifting environments II: Flux-limited solutions

TL;DR

This paper investigates the asymptotic spreading behavior of Fisher-KPP reactive fronts in shifting heterogeneous environments, revealing how flux-limited solutions and dynamic junction conditions influence the rate function in complex environmental scenarios.

Contribution

It introduces a novel characterization of the rate function via Hamilton-Jacobi equations with dynamic junction conditions for non-monotone environments.

Findings

Rate function converges to Ishii solution under weak monotonicity.

Dynamic junction conditions describe the rate function in complex environments.

Results connect nonlocally pulled fronts with forced traveling waves.

Abstract

We consider the spreading dynamics of the Fisher-KPP equation in a shifting environment, by analyzing the limit of the rate function of the solutions. For environments with a weak monotone condition, it was demonstrated in a previous paper that the rate function converges to the unique Ishii solution of the underlying Hamilton-Jacobi equations. In case the environment does not satisfy the weak monotone condition, we show that the rate function is then characterized by the Hamilton-Jacobi equation with a dynamic junction condition, which depends additionally on the generalized eigenvalue derived from the environmental function. Our results applies to the case when the environment has multiple shifting speeds, and clarify the connection with previous results on nonlocally pulled fronts and forced traveling waves.