The non-local Fisher–KPP equation: travelling waves and steady states

Abstract

We consider the Fisher-KPP equation with a nonlocal saturation effect defined through an interaction kernel φ(x) and investigate the possible differences with the standard Fisher-KPP equation. Our first concern is the existence of steady states. We prove that if the Fourier transformφ(ξ) is positive or if the length σ of the nonlocal interaction is short enough, then the only steady states are u ≡ 0 and u ≡ 1. Our second concern is the study of traveling waves. We prove that this equation admits traveling wave solutions that connect u = 0 to an unknown positive steady state u ∞ (x), for all speeds c ≥ c *. The traveling wave connects to the standard state u ∞ (x) ≡ 1 under the aforementioned conditions:φ(ξ) > 0 or σ is sufficiently small. However, the wave is not monotonic for σ large.

Nonlocal Diffusion Problems

“…Éste modelo ha sido previamente considerado en [30]. Los problemas con difusión local y reacción no local han sido considerados en [11], donde el término de reacción no local tiene en cuenta la saturación no local o los efectos de competición no local.…”

“…This model has been previously considered in [30]. The problem with local diffusion, (−∆), and nonlocal reaction has been considered in [11], where the nonlocal reaction term takes into account a nonlocal saturation, or nonlocal competition effects.…”

“…Éste modelo ha sido previamente considerado en [30]. Los problemas con difusión local y reacción no local han sido considerados en [11], donde el término de reacción no local tiene en cuenta la saturación no local o los efectos de competición no local.Otro tipo de modelos de difusión no local, es el que aparece en [12,19], dado pordonde Γ(u) = sign(u) |u| − 1 + . Este problema se llama problema de Stefan no local.…”

Preface to the Issue Nonlocal Reaction-Diffusion Equations

“…The proof of wave existence in the case of nonlocal equation becomes much more involved, and there are only partial results [2], [5], [6], [11], [15]. The notion of generalized travelling waves, which can be characterized as propagating solutions existing for all times from −∞ to ∞ [32], becomes appropriate here and allows the proof of wave existence without the assumption that the support of the kernel is sufficiently narrow [4], [11].…”

“…The notion of generalized travelling waves, which can be characterized as propagating solutions existing for all times from −∞ to ∞ [32], becomes appropriate here and allows the proof of wave existence without the assumption that the support of the kernel is sufficiently narrow [4], [11]. Numerical simulations show that nonmonotone travelling waves can be stable, and there exist periodic travelling waves [19], [22], [36].…”