Pattern formation in the doubly-nonlocal Fisher-KPP equation

TL;DR

This paper investigates the existence, bifurcations, and stability of stationary solutions in the doubly-nonlocal Fisher-KPP equation, revealing complex parameter-dependent behaviors and connecting to classical local models.

Contribution

It provides a detailed classification of stationary solutions and bifurcation phenomena in the doubly-nonlocal Fisher-KPP equation, extending previous local and nonlocal models.

Findings

Bifurcation from homogeneous states can occur under certain parameters.

Generated patterns are locally asymptotically stable.

Parameter regimes exist where no bifurcations occur.

Abstract

We study the existence, bifurcations, and stability of stationary solutions for the doubly-nonlocal Fisher-KPP equation. We prove using Lyapunov-Schmidt reduction that under suitable conditions on the parameters, a bifurcation from the non-trivial homogeneous state can occur. The kernel of the linearized operator at the bifurcation is two-dimensional and periodic stationary patterns are generated. Then we prove that these patterns are, again under suitable conditions, locally asymptotically stable. We also compare our results to previous work on the nonlocal Fisher-KPP equation containing a local diffusion term and a nonlocal reaction term. If the diffusion is approximated by a nonlocal kernel, we show that our results are consistent and reduce to the local ones in the local singular diffusion limit. Furthermore, we prove that there are parameter regimes, where no bifurcations can occur for the doubly-nonlocal Fisher-KPP equation. The results demonstrate that intricate different parameter regimes are possible. In summary, our results provide a very detailed classification of the multi-parameter dependence of the stationary solutions for the doubly-nonlocal Fisher-KPP equation.