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

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.

Show me the PDF file (1581 KB), TEX file, and other files for this article.

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

Return to the EJDE web page