KPP transition fronts in a one-dimensional two-patch habitat

TL;DR

This paper establishes the existence of a transition front in a one-dimensional two-patch habitat model with KPP reaction terms, connecting different fronts across patches and characterized by asymptotic speeds.

Contribution

It provides the first example of a transition front for a KPP two-patch model with interface conditions, using super- and subsolutions and limiting arguments.

Findings

Existence of a transition front connecting two patches.

Construction of super- and subsolutions based on leading edges.

Identification of asymptotic past and future speeds.

Abstract

This paper is concerned with the existence of transition fronts for a one-dimensional twopatch model with KPP reaction terms. Density and flux conditions are imposed at the interface between the two patches. We first construct a pair of suitable super-and subsolutions by making full use of information of the leading edges of two KPP fronts and gluing them through the interface conditions. Then, an entire solution obtained thanks to a limiting argument is shown to be a transition front moving from one patch to the other one. This propagating solution admits asymptotic past and future speeds, and it connects two different fronts, each associated with one of the two patches. The paper thus provides the first example of a transition front for a KPP-type two-patch model with interface conditions.