Diffusion driven instability in an inhomogeneous circular domain

Classical reaction-diffusion systems have been extensively studied and are now well understood. Most of the work to date has been concerned with homog eneous models within one-dimensional or rectangular domains. However, it is recognised that in most applications nonhomogeneity, as well as other geom etries, are typically more important. In this paper, we present a two chemi cal reaction-diffusion process which is operative within a circular region and the model is made nonhomogeneous by supposing that one of the diffusion coefficients varies with the radial variable. Linear analysis leads to the derivation of a dispersion relation for the point of onset of instability and our approach enables both axisymmetric and nonaxisymmetric modes to be described. We apply our workings to the standard Schnackenberg activator-in hibitor model in the case when the variable diffusion coefficient takes on a step-function like profile. Some fully nonlinear simulations show that th e linear analysis captures the essential details of the behaviour of the mo del. (C) 1999 Elsevier Science Ltd. All rights reserved.