Spatio-temporal pattern formation in a nonlocal reaction-diffusion equation

S. A. Gourley, M. A. J. Chaplain, F. A. Davidson

    Research output: Contribution to journalArticle

    47 Citations (Scopus)

    Abstract

    We study a scalar reaction-diffusion equation which contains a nonlocal term in the form of an integral convolution in the spatial variable and demonstrate, using asymptotic, analytical and numerical techniques, that this scalar equation is capable of producing spatio-temporal patterns. Fisher's equation is a particular case of this equation. An asymptotic expansion is obtained for a travelling wavefront connecting the two uniform steady states and qualitative differences to the corresponding solution of Fisher's equation are noted. A stability analysis combined with numerical integration of the equation show that under certain circumstances nonuniform solutions are formed in the wake of this front. Using global bifurcation theory, we prove the existence of such non-uniform steady state solutions for a wide range of parameter values. Numerical bifurcation studies of the behaviour of steady state solutions as a certain parameter is varied, are also presented.
    Original languageEnglish
    Pages (from-to)173-192
    Number of pages20
    JournalDynamical Systems
    Volume16
    Issue number2
    DOIs
    Publication statusPublished - 2001

    Keywords

    • Analysis mathematics
    • Applied mechanics
    • Dynamical systems
    • Mathematical biology

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