A Liouville-Type Result for Non-cooperative Fisher–KPP Systems and Nonlocal Equations in Cylinders

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 long-time behavior of traveling waves. All results are extended to an analogous nonlocal reactiondiffusion equation that contains as a particular case the cane toads equation with bounded traits.

Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

“…In the scalar framework of KPP-type reaction-diffusion equations, λ 1 < 0 implies the locally uniform convergence of all solutions to the unique periodic and uniformly positive entire solution, whereas λ 1 ≥ 0 implies the uniform convergence of all solutions to 0, as proved by Nadin [56]. The study of entire solutions is much more delicate in the multidimensional setting, simply due to topological freedom [32,35,36,53], and their uniqueness and stability properties cannot in general be inferred from the linearization at 0. However, we will show in a sequel that in the multidimensional case, the results of Nadin [56] can be generalized in the following weak form: λ 1 < 0 implies the locally uniform persistence of all solutions and the existence of a periodic and uniformly positive entire solution, whereas λ 1 ≥ 0 implies the uniform convergence of all solutions to 0.…”

Addendum to ‘Non-cooperative Fisher–KPP systems: traveling waves and long-time behavior’

“…The Freidlin-Gärtner-type formula (Theorem 1.6). In this section, we assume that λ 1 < 0 and u ini is compactly supported, and we prove (11) and (12).…”

“…This is the main difficulty and novelty of this work. It is actually known that not all results of the scalar case can be generalized in this way; in particular, Liouville-type results on the uniformly positive entire solution are in general false even with constant coefficients [4,8,11,12,15]. In this regard, our intent is precisely to show what can be generalized to non-cooperative Fisher-KPP systems, and what cannot.…”