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

We study the Cauchy problem on the real line for the nonlocal Fisher-KPP equation in one spatial dimension,

u_t = Du_xx + u(1 − ϕ ∗ u),

where ϕ ∗ u is a spatial convolution with the top hat kernel, ϕ(y) ≡ H(1/4 - y²).

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 ∆₁ ≈ 0.00297 (the exact value is determined in section 3). We explicitly construct these spatially-periodic solutions as uniformly-valid asymptotic approximations for D ≪ 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 → 0⁺.

From numerical solutions, we find that for D ≥ ∆₁, permanent form travelling waves, with minimum wavespeed, 2√D, are generated, whilst for 0 < D < ∆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.