A proof of anomalous invasion speeds in a system of coupled Fisher-KPP equations

TL;DR

This paper rigorously proves that in a coupled Fisher-KPP system, certain initial conditions lead to faster-than-expected spreading speeds, confirming predictions from linear analysis.

Contribution

It provides a rigorous mathematical validation of anomalous spreading speeds in coupled Fisher-KPP equations using comparison principles and explicit sub and super solutions.

Findings

Initial data with compact support spreads at the anomalous speed

Validation of linear prediction for spreading speeds

Use of comparison principles and explicit solutions

Abstract

This article is concerned with the rigorous validation of anomalous spreading speeds in a system of coupled Fisher-KPP equations of cooperative type. Anomalous spreading refers to a scenario wherein the coupling of two equations leads to faster spreading speeds in one of the components. The existence of these spreading speeds can be predicted from the linearization about the unstable state. We prove that initial data consisting of compactly supported perturbations of Heaviside step functions spreads asymptotically with the anomalous speed. The proof makes use of a comparison principle and the explicit construction of sub and super solutions.