The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 1. The Cauchy problem on the real line

TL;DR

This paper analyzes the one-dimensional nonlocal Fisher-KPP equation with a top hat kernel, demonstrating bifurcation of periodic solutions as diffusivity decreases, and characterizing wavefronts and their structures through asymptotic methods and numerical analysis.

Contribution

It provides the first explicit construction of spatially-periodic solutions and detailed analysis of wavefront structures for the nonlocal Fisher-KPP equation with a top hat kernel.

Findings

Periodic solutions bifurcate from the uniform steady state at a critical diffusivity.

Explicit asymptotic approximations for solutions when diffusivity is small.

Transition from traveling waves to steady periodic solutions as diffusivity varies.

Abstract

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-\phi*u), \] where $\phi*u$ is a spatial convolution with the top hat kernel, $\phi(y) \equiv H\left(\frac{1}{4}-y^2\right)$. After showing that the problem is globally well-posed, we demonstrate that positive, spatially-periodic solutions bifurcate from the spatially-uniform steady state solution $u=1$ as the diffusivity, $D$, decreases through $\Delta_1 \approx 0.00297$. We explicitly construct these spatially-periodic solutions as uniformly-valid asymptotic approximations for $D \ll 1$, over one wavelength, via the method of matched asymptotic expansions. These consist, at leading order, of regularly-spaced, compactly-supported regions with width of $O(1)$ where $u=O(1)$, separated by regions where $u$ is exponentially small at leading order as $D \to 0^+$. From numerical solutions, we find that for $D \geq \Delta_1$, permanent form travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst for $0 < D < \Delta_1$, the wavefronts generated separate the regions where $u=0$ from a region where a steady periodic solution is created. The structure of these transitional travelling waves is examined in some detail.