Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 160, pp. 1-30.
Title: Pattern formation in a mixed local and nonlocal reaction-diffusion system
Authors: Evelyn Sander (George Mason Univ., Fairfax, VA, USA
Richard Tatum (Naval Surface Warfare Center, Dahlgren VA, USA)
Abstract:
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 reaction-diffusion system with Turing
instabilities on rectangular domains with periodic boundary
conditions. This mixed system contains a homotopy parameter $\beta$
to vary the effect of both local $(\beta = 1)$ and nonlocal $(\beta
= 0)$ diffusion. The diffusion interaction length relative to the
size of the domain is given by a parameter $\epsilon$. We associate
the nonlocal diffusion with a convolution kernel, such that the
kernel is of order $\epsilon^{-\theta}$ in the limit as $\epsilon \to 0$.
We prove that as long as $0 \le \theta<1$, in the singular limit as
$\epsilon \to 0$, the selection of patterns is determined by the
linearized equation. In contrast, if $\theta = 1$ and $\beta$ is
small, our numerics show that pattern selection is a fundamentally
nonlinear process.
Submitted March 19, 2012. Published September 20, 2012.
Math Subject Classifications: 35B36, 35K57
Key Words: Reaction-diffusion system; nonlocal equations;
Turing instability; pattern formation.