Non-cooperative Fisher--KPP systems: traveling waves and long-time behavior

TL;DR

This paper investigates non-cooperative Fisher--KPP systems, establishing extinction and persistence conditions, existence of traveling waves with minimal speeds, and their relation to spreading speeds, thus extending scalar KPP results to systems.

Contribution

It introduces new results on traveling waves and long-term behavior for non-cooperative Fisher--KPP systems, including existence of minimal speeds and their relation to spreading speeds.

Findings

Extinction and persistence dichotomy established.

Existence of traveling waves with a positive minimal speed.

Minimal wave speed equals spreading speed for the Cauchy problem.

Abstract

This paper is concerned with non-cooperative parabolic reaction--diffusion systems which share structural similarities with the scalar Fisher--KPP equation. These similarities make it possible to prove, among other results, an extinction and persistence dichotomy and, when persistence occurs, the existence of a positive steady state, the existence of traveling waves with a half-line of possible speeds and a positive minimal speed and the equality between this minimal speed and the spreading speed for the Cauchy problem. Non-cooperative KPP systems can model various phenomena where the following three mechanisms occur: local diffusion in space, linear cooperation and su-perlinear competition.