Diffusion driven instability in an inhomogeneous circular domain

• Published in: Mathematical and computer modelling 1999-02, Vol.29 (4), p.53-66
• Main Authors: May, A., Firby, P.A., Bassom, A.P.
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Mathematical analysis; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Partial differential equations; Partial differential equations, initial value problems and time-dependant initial-boundary value problems; Schnackenberg model; Sciences and techniques of general use; Step-like diffusivity; Two chemical reaction
• ISSN: 0895-7177; 1872-9479
• DOI: 10.1016/S0895-7177(99)00039-4

Classical reaction-diffusion systems have been extensively studied and are now well understood. Most of the work to date has been concerned with homogeneous models within one-dimensional or rectangular domains. However, it is recognised that in most applications nonhomogeneity, as well as other geometries, are typically more important. In this paper, we present a two chemical 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-inhibitor model in the case when the variable diffusion coefficient takes on a step-function like profile. Some fully nonlinear simulations show that the linear analysis captures the essential details of the behaviour of the model.