2005 Non-local dispersal
M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, G. T. Vickers
Differential Integral Equations 18(11): 1299-1320 (2005). DOI: 10.57262/die/1356059743


Equations with non-local dispersal have been used extensively as models in material science, ecology and neurology. We consider the scalar model $$ \frac{\partial u}{\partial t}(x,t)=\rho \Big \{\int_\Omega \beta (x,y)u(y,t)dy- u(x,t) \Big \}+f(u(x,t)),$$ where the integral term represents a general form of spatial dispersal and $u(x,t)$ is the density at $x\in\Omega$, the spatial region, and time $t$ of the quantity undergoing dispersal. We discuss the asymptotic dynamics in the bistable case and contrast these with those for the corresponding reaction-diffusion model. First, we note that it is easy to show for large $\rho$ that the behavior is similar to that of the reaction-diffusion system; in the case of the analogue of zero Neumann conditions, the dynamics are governed by the ODE $\dot u=f(u)$. However, for small $\rho$, it is known that this is not the case, the set of equilibria being uncountably infinite and not compact in $L^p~(1\le p\le\infty)$. Our principal aim in this paper is to enquire whether every orbit converges to an equilibrium, regardless of the size of $\rho$. The lack of compactness is a major technical obstacle, but in a special case we develop a method to show that this is indeed true.


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M. Grinfeld. G. Hines. V. Hutson. K. Mischaikow. G. T. Vickers. "Non-local dispersal." Differential Integral Equations 18 (11) 1299 - 1320, 2005. https://doi.org/10.57262/die/1356059743


Published: 2005
First available in Project Euclid: 21 December 2012

zbMATH: 1212.35484
MathSciNet: MR2174822
Digital Object Identifier: 10.57262/die/1356059743

Primary: 35F30
Secondary: 45K05 , 47G20

Rights: Copyright © 2005 Khayyam Publishing, Inc.


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Vol.18 • No. 11 • 2005
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