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Pattern formation in a mixed local and nonlocal reaction-diffusion systemKeywords: Reaction-diffusion system , nonlocal equations , Turing instability , pattern formation 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 $eta$ 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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