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

Electronic Journal of Differential Equations

关键词: Reaction-diffusion system;    nonlocal equations;    Turing instability;    pattern formation;   

DOI  :  

来源: DOAJ

【 摘 要 】

Local and nonlocal reaction-diffusion models have been shown todemonstrate nontrivial steady state patterns known as Turingpatterns. That is, solutions which are initially nearly homogeneousform non-homogeneous patterns. This paper examines the patternselection mechanism in systems which contain nonlocal terms. Inparticular, we analyze a mixed reaction-diffusion system with Turinginstabilities on rectangular domains with periodic boundaryconditions. 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 thesize of the domain is given by a parameter $epsilon$. We associatethe nonlocal diffusion with a convolution kernel, such that thekernel 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 thelinearized equation. In contrast, if $heta = 1$ and $eta$ issmall, our numerics show that pattern selection is a fundamentallynonlinear process.

【 授权许可】

Unknown