Modulated traveling fronts for a nonlocal Fisher-KPP equation: a dynamical systems approach

TL;DR

This paper investigates a nonlocal Fisher-KPP equation, proving the existence of bifurcating stationary solutions and modulated traveling fronts near a Turing bifurcation using dynamical systems techniques.

Contribution

It introduces a rigorous analysis of bifurcations and traveling fronts in a nonlocal Fisher-KPP model, including spectral stability and asymptotic approximations.

Findings

Existence of bifurcating stationary periodic solutions

Spectral stability of these solutions

Existence of modulated traveling fronts near bifurcation

Abstract

We consider a nonlocal generalization of the Fisher-KPP equation in one spatial dimension. As a parameter is varied the system undergoes a Turing bifurcation. We study the dynamics near this Turing bifurcation. Our results are two-fold. First, we prove the existence of a two-parameter family of bifurcating stationary periodic solutions and derive a rigorous asymptotic approximation of these solutions. We also study the spectral stability of the bifurcating stationary periodic solutions with respect to almost co-periodic perturbations. Secondly, we restrict to a specific class of exponential kernels for which the nonlocal problem is transformed into a higher order partial differential equation. In this context, we prove the existence of modulated traveling fronts near the Turing bifurcation that describe the invasion of the Turing unstable homogeneous state by the periodic pattern established in the first part. Both results rely on a center manifold reduction to a finite dimensional ordinary differential equation.