The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 2. The initial-boundary value problem on a finite domain

Abstract

In the second part of this series of papers, we address the same evolution problem that was considered in part 1 (see [16]), namely the nonlocal Fisher-KPP equation in one spatial dimension, \begin{equation*} u_t = D u_{xx} + u(1-\phi *u), \end{equation*} 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 we modify this to an associated initial-boundary value problem on the finite spatial interval $[0,a]$ rather than the whole real line. Boundary conditions are required at the end points of the interval, and we address the situations when these are of either Dirichlet or Neumann type. This model is a natural extension of 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 their properties in detail, and to compare and contrast these with the well known properties of the classical Fisher-KPP model.

Keywords

• nonlocal reaction-diffusion equation
• singular perturbation
• bifurcations
• Fisher-KPP equation
• numerical solution
• 35R09
• 35B36
• 35B25
• 35K57