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.