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 homogeneous models within one-dimensional or rectangular domains. However, it is recognised that in most applications nonhomogeneity, as well as other geom...

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Bibliographic Details
Published inMathematical and computer modelling Vol. 29; no. 4; pp. 53 - 66
Main Authors May, A., Firby, P.A., Bassom, A.P.
Format Journal Article
LanguageEnglish
Published Oxford Elsevier Ltd 01.02.1999
Elsevier Science
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
Two chemical reaction
Step-like diffusivity
Schnackenberg model
Bessel function
Circular domaine
Heterogeneity
Neumann problem
Chemical reaction
Coupled equations
Instability
Reaction diffusion equation
Partial differential equation
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Summary: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.
ISSN:0895-7177
1872-9479
DOI:10.1016/S0895-7177(99)00039-4