Anomalous spreading in a system of coupled Fisher-KPP equations

TL;DR

This paper investigates anomalous spreading speeds in a coupled Fisher-KPP system, revealing how coupling influences wave propagation speeds through pole analysis of the Green's function.

Contribution

It identifies the mechanisms behind anomalous spreading in coupled Fisher-KPP equations and classifies the poles affecting wave speeds.

Findings

Anomalous spreading occurs when coupling causes one component to spread faster.

Poles of the Green's function determine the spreading behavior.

Certain poles are irrelevant for nonlinear wave speed selection.

Abstract

In this article, we report on the curious phenomena of anomalous spreading in a system of coupled Fisher-KPP equations. When a single parameter is set to zero, the system consists of two uncoupled Fisher-KPP equations which give rise to traveling fronts propagating with the unique, minimal KPP speed. When the coupling parameter is nonzero various behaviors can be observed. Anomalous spreading occurs when one component of the system spreads at a speed significantly faster in the coupled system than it does in isolation, while the speed of the second component remains unchanged. We study these anomalous spreading speeds and show that they arise due to poles of the pointwise Green's function corresponding to the linearizion about the unstable homogeneous state. These poles lead to anomalous spreading in the linearized system and come in two varieties -- one that persists and leads to anomalous spreading for the nonlinear system and one that does not. We describe the mechanisms leading to these two behaviors and prove that one class of poles are irrelevant as far as nonlinear wavespeed selection is concerned. Finally, we show that the same mechanism can give rise to anomalous spreading even when the slower component does not spread.