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 to vary the effect of both local and nonlocal 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 in the limit as . We prove that as long as , in the singular limit as , the selection of patterns is determined by the linearized equation. In contrast, if and 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
| Richard Tatum |
Naval Surface Warfare Center Dahlgren Division
18444 Frontage Road Suite 327
Dahlgren, VA 22448-5161, USA
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