Electron. J. Diff. Equ., Vol. 2012 (2012), No. 160, pp. 1-30.

Pattern formation in a mixed local and nonlocal reaction-diffusion system

Evelyn Sander, Richard Tatum

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.

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Evelyn Sander
Department of Mathematical Sciences
George Mason University
4400 University Dr.
Fairfax, VA 22030, USA
email: esander@gmu.edu
Richard Tatum
Naval Surface Warfare Center Dahlgren Division
18444 Frontage Road Suite 327
Dahlgren, VA 22448-5161, USA
email: rchrd.ttm@gmail.com

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