Non-local dispersal and bistability

V. Hutson, M. Grinfeld

Research output: Contribution to journalArticle

24 Citations (Scopus)

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$.
LanguageEnglish
Pages221-232
Number of pages11
JournalEuropean Journal of Applied Mathematics
Volume17
Early online date22 Feb 2006
DOIs
Publication statusPublished - 30 Apr 2006

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Bistability
Semiflow
Materials Science
Initial value problems
Diffusion Model
Materials science
Ecology
Asymptotic Behavior of Solutions
Bounded Linear Operator
Function Space
Initial Value Problem
Counterexample
Orbits
Orbit
Scalar
Nonlinearity
Converge
Model

Keywords

  • bistability
  • dispersal
  • mathematics
  • applied mathematics

Cite this

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Non-local dispersal and bistability. / Hutson, V.; Grinfeld, M.

In: European Journal of Applied Mathematics, Vol. 17, 30.04.2006, p. 221-232.

Research output: Contribution to journalArticle

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