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

Abstract

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension, \[ u_t = D u_{xx} + u(1-φ*u), \] where $φ*u$ is a spatial convolution with the top hat kernel, $φ(y) \equiv H\left(\frac{1}{4}-y^2\right)$. 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 $Δ_1 \approx 0.00297$. 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 $D \geq Δ_1$, permanent form travelling waves, with minimum wavespeed, $2 \sqrt{D}$, are generated, whilst for $0 < D < Δ_1$, 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.

Figures (15)

• Figure 1: The numerical solution of (IBVP) for various values of $D$ with $A =0.01$ (solid black line), along with the minimum speed travelling wave (broken blue line).
• Figure 2: The numerically-calculated position of the wavefront for various values of $D$. The broken line has slope $2 \sqrt{D}$, the minimum wavespeed.
• Figure 3: The numerical solution of (IBVP) for Gaussian initial data with $A=0.01$ and width $w = 0.04$, and $D = 10^{-8}$, when $t = 6000$. The upper panel shows $\log_{10} u$. New spikes are initiated ahead of the wave at the point where $u$ is close to $10^{-700}$, which can only be captured accurately by solving for $\log u$ instead of $u$.
• Figure 4: The numerically-calculated position of the wavefront for $D = 10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-7}$, $10^{-8}$ and $10^{-9}$, with $w = 0.1$ and $A=0.01$. The broken line is the function $x_f(t)$, defined in (\ref{['eqn_xf']}).
• Figure 5: The wavelength of the spatially-periodic steady state left behind the wavefront, calculated numerically as a function of $D$. The broken line shows the most unstable wavelength given by the linearized theory.

Theorems & Definitions (6)

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