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 $eta$ to vary the effect of both local $(eta = 1)$ and nonlocal $(eta = 0)$ 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 $epsilon^{-heta}$ in the limit as $epsilon o 0$. We prove that as long as $0 le heta<1$, in the singular limit as $epsilon o 0$, the selection of patterns is determined by the linearized equation. In contrast, if $heta = 1$ and $eta$ is small, our numerics show that pattern selection is a fundamentally nonlinear process.

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pattern selection mechanism nontrivial steady state patterns diffusion interaction length process turing instabilities homotopy parameter linearized equation in contrast rectangular domains periodic boundary mixed reactiondiffusion system 0 convolution kernel 1

Evelyn Sander,Richard Tatum,.Pattern formation in a mixed local and nonlocal reaction-diffusion system. 2012 (160,),.

Evelyn Sander,Richard Tatum,"Pattern formation in a mixed local and nonlocal reaction-diffusion system" 2012

Evelyn Sander,Richard Tatum,"Pattern formation in a mixed local and nonlocal reaction-diffusion system"