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

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 transform φ̂(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: φ̂(ξ) > 0 or σ is sufficiently small. However, the wave is not monotonic for σ large.