A Liouville-type result for non-cooperative Fisher--KPP systems and nonlocal equations in cylinders

TL;DR

This paper proves a Liouville-type uniqueness result for non-cooperative Fisher-KPP systems and nonlocal equations in cylinders, under natural assumptions, aiding the understanding of traveling wave behaviors.

Contribution

It establishes a novel Liouville-type theorem for non-cooperative reaction-diffusion systems and extends results to nonlocal equations, including the cane toads model.

Findings

Uniqueness of nonzero stationary states under natural conditions

Characterization of traveling wave wakes using the Liouville result

Extension of results to nonlocal reaction-diffusion equations

Abstract

We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.