Planar pulsating traveling wave solutions of non-cooperative Fisher--KPP systems in space-time periodic media

TL;DR

This paper establishes the existence and properties of pulsating traveling wave solutions in non-cooperative Fisher-KPP systems within space-time periodic media, revealing critical speeds for invasion fronts in various directions.

Contribution

It extends methods for scalar equations to systems with periodic coefficients, proving the existence of critical speeds and pulsating traveling waves in non-cooperative Fisher-KPP models.

Findings

Existence of a critical invasion speed in each spatial direction.

Rational directions admit pulsating traveling waves at speeds above the critical speed.

Homogeneous coefficients allow for time-periodic pulsating waves at all speeds above the critical.

Abstract

Non-cooperative Fisher-KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the invasion of open space by a persistent population at constant speed. These solutions are important in the understanding of long-time behaviors for the Cauchy problem. Adapting methods developed for scalar equations satisfying the comparison principle as well as methods developed for systems with homogeneous coefficients, we prove, in each spatial direction, the existence of a critical speed such that: there exists no almost planar generalized transition waves with a smaller speed; if the direction is rational, each rational speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time and transverse space periodicity; if the coefficients are homogeneous in space, each speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time periodicity.