The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 2. The Cauchy problem on a finite interval

TL;DR

This paper investigates the behavior of a nonlocal Fisher-KPP equation with a top hat kernel on a finite interval, analyzing boundary conditions and comparing its properties to the classical local model.

Contribution

It extends the analysis of the nonlocal Fisher-KPP equation to finite domains with boundary conditions, highlighting differences from the classical model.

Findings

Boundary conditions significantly affect solution behavior.

Nonlocal effects alter wave propagation characteristics.

Comparison with classical Fisher-KPP reveals new dynamics.

Abstract

In the second part of this series of papers, we address the same Cauchy problem that was considered in part 1, namely 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, $\phi(y) \equiv H\left(\frac{1}{4}-y^2\right)$, except that now the spatial domain is the finite interval $[0,a]$ rather than the whole real line. Consequently boundary conditions are required at the interval end-points, and we address the situations when these boundary conditions are of either Dirichlet or Neumann type. This model forms a natural extension to the classical Fisher-KPP model, with the introduction of the simplest possible nonlocal effect into the saturation term. Nonlocal reaction-diffusion models arise naturally in a variety of (frequently biological or ecological) contexts, and as such it is of fundamental interest to examine its properties in detail, and to compare and contrast these with the well known properties of the classical Fisher-KPP model.