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, \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, $\phi (y) \equiv H\left (\frac{1}{4}-y^2\right )$ . After observing 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 $\Delta _1 \approx 0.00297$ (the exact value is determined in Section 3). 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 $0 \lt D \lt \Delta _1$ , the wavefronts generated separate the regions where $u=0$ from a region where a steady periodic solution is created via a distinct periodic shedding mechanism acting immediately to the rear of the advancing front, with this mechanism becoming more pronounced with decreasing $D$ . The structure of these transitional travelling wave forms is examined in some detail.