The evolution problem for the 1D nonlocal Fisher-KPP equation with a top hat kernel. Part 3. The effect of perturbations in the kernel

In the third part of this series of papers, we address the same Cauchy problem that was considered in part 1 (see [5] ), namely for the nonlocal Fisher-KPP equation in one spatial dimension, u t = D u x x + u ( 1 − ϕ T ⁎ u ) , where ϕ T ⁎ u is a spatial convolution with the top hat kernel, ϕ T ( y ) ≡ H ( 1 4 − y 2 ) (with H ( ⋅ ) being the usual Heavisde function), except that now we include a specified perturbation to this kernel, which we denote as ϕ ‾ : R → R . Thus the top hat kernel ϕ T is now replaced by the perturbed kernel ϕ : R → R , where ϕ ( x ) = ϕ T ( x ) + ϕ ‾ ( x ) ∀ x ∈ R . When the magnitude of the kernel perturbation is small in a suitable norm, the situation is shown to be generally a regular perturbation problem when the diffusivity D is formally of O(1) or larger. However when D becomes small, and in particular, of the same order as the magnitude of the perturbation to the kernel, this becomes a strongly singular perturbation problem, with considerable changes in overall structure. This situation is investigated in detail. In terms of its generic interest, the 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. • We study the effect on the nonlocal Fisher-KPP equation with a top-hat kernel of a broad class of kernel perturbations. • Two specific and representative forms of kernel perturbation are considered in detail. • The regularity of kernel perturbations depends on the size of the diffusion coefficient ( D = O ( 1 ) or D = o ( 1 ) ). • A complicated network of secondary bifurcation structures is uncovered.