Propagation and blocking in a two-patch reaction-diffusion model

TL;DR

This paper investigates how solutions propagate or are blocked in a two-patch reaction-diffusion model, revealing conditions for spreading, blocking, and stability across different patch configurations.

Contribution

It provides new insights into propagation phenomena in coupled two-patch reaction-diffusion systems, including conditions for spreading and blocking in KPP and bistable cases.

Findings

Propagation in KPP-KPP case resembles homogeneous KPP equation.

Various dynamics like blocking and virtual blocking are identified in KPP-bistable case.

Global stability is established when solutions propagate with positive speed.

Abstract

This paper is concerned with propagation phenomena for the solutions of the Cauchy problem associated with a two-patch one-dimensional reaction-diffusion model. It is assumed that each patch has a relatively well-defined structure which is considered as homogeneous. A coupling interface condition between the two patches is involved. We first study the spreading properties of solutions in the case when the per capita growth rate in each patch is maximal at low densities, a configuration which we call the KPP-KPP case, and which turns out to have some analogies with the homogeneous KPP equation in the whole line. Then, in the KPP-bistable case, we provide various conditions under which the solutions show different dynamics in the bistable patch, that is, blocking, virtual blocking (propagation with speed zero), or spreading with positive speed. Moreover, when propagation occurs with positive speed, a global stability result is proved. Finally, the analysis in the KPP-bistable frame is extended to the bistable-bistable case.