Traveling fronts guided by the environment for reaction-diffusion equations

TL;DR

This paper investigates the existence and properties of traveling fronts in reaction-diffusion equations influenced by environmental heterogeneity, with applications to population dynamics and cortical spreading in the brain.

Contribution

It establishes conditions for the existence of traveling fronts guided by the medium in heterogeneous reaction-diffusion models, including thresholds for nonzero profiles and bistable cases.

Findings

Existence of a threshold for nonzero asymptotic profiles.

Traveling fronts exist when the reaction area is sufficiently large.

Non-existence of fronts when the reaction area is too small.

Abstract

This paper deals with the existence of traveling fronts guided by the medium for a KPP reaction-diffusion equation coming from a model in population dynamics in which there is spatial spreading as well as genetic mutation of a quantitative genetic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of a threshold value on the existence of a nonzero asymptotic profile (a stationary limiting solution). When a nonzero asymptotic profile exists, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case. We also study here the bistable case. The equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if the area where theere is reaction is large enough and the non-existence if it is too small.