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Non-local dispersal and bistability

Hutson, V. and Grinfeld, M. (2006) Non-local dispersal and bistability. European Journal of Applied Mathematics, 17. pp. 221-232. ISSN 0956-7925

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    Abstract

    The scalar initial value problem [ u_t = ho Du + f(u), ] is a model for dispersal. Here $u$ represents the density at point $x$ of a compact spatial region $Omega in mathbb{R}^n$ and time $t$, and $u(cdot)$ is a function of $t$ with values in some function space $B$. $D$ is a bounded linear operator and $f(u)$ is a bistable nonlinearity for the associated ODE $u_t = f(u)$. Problems of this type arise in mathematical ecology and materials science where the simple diffusion model with $D=Delta$ is not sufficiently general. The study of the dynamics of the equation presents a difficult problem which crucially differs from the diffusion case in that the semiflow generated is not compactifying. We study the asymptotic behaviour of solutions and ask under what conditions each positive semi-orbit converges to an equilibrium (as in the case $D=Delta$). We develop a technique for proving that indeed convergence does hold for small $ ho$ and show by constructing a counter-example that this result does not hold in general for all $ ho$.

    Item type: Article
    ID code: 4546
    Keywords: bistability, dispersal, mathematics, applied mathematics, Mathematics
    Subjects: Science > Mathematics
    Department: Faculty of Science > Mathematics and Statistics
    Related URLs:
    Depositing user: Strathprints Administrator
    Date Deposited: 01 Nov 2007
    Last modified: 16 Mar 2012 00:47
    URI: http://strathprints.strath.ac.uk/id/eprint/4546

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