Hutson, V. and Grinfeld, M. (2006) Nonlocal dispersal and bistability. European Journal of Applied Mathematics, 17. pp. 221232. ISSN 09567925

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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 semiorbit 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 counterexample that this result does not hold in general for all $ ho$.
Item type:  Article 

ID code:  4546 
Keywords:  bistability, dispersal, mathematics, applied mathematics, Mathematics, Applied Mathematics 
Subjects:  Science > Mathematics 
Department:  Faculty of Science > Mathematics and Statistics 
Depositing user:  Strathprints Administrator 
Date Deposited:  01 Nov 2007 
Last modified:  20 Oct 2015 17:48 
URI:  http://strathprints.strath.ac.uk/id/eprint/4546 
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