Pattern Formation in a Mixed Local and Nonlocal Reaction-diffusion System

Local and nonlocal reaction-diffusion models have been shown to demonstrate nontrivial steady state patterns known as Turing patterns. That is, solutions which are initially nearly homogeneous form non-homogeneous patterns. This paper examines the pattern selection mechanism in systems which contain nonlocal terms. In particular, we analyze a mixed reactiondiffusion system with Turing instabilities on rectangular domains with periodic boundary conditions. This mixed system contains a homotopy parameter β to vary the effect of both local (β = 1) and nonlocal (β = 0) diffusion. The diffusion interaction length relative to the size of the domain is given by a parameter . We associate the nonlocal diffusion with a convolution kernel, such that the kernel is of order −θ in the limit as → 0. We prove that as long as 0 ≤ θ < 1, in the singular limit as → 0, the selection of patterns is determined by the linearized equation. In contrast, if θ = 1 and β is small, our numerics show that pattern selection is a fundamentally nonlinear process.