Differential and Integral Equations

Non-local dispersal

M. Grinfeld, G. Hines, V. Hutson, K. Mischaikow, and G. T. Vickers

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Abstract

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.

Article information

Source
Differential Integral Equations, Volume 18, Number 11 (2005), 1299-1320.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356059743

Mathematical Reviews number (MathSciNet)
MR2174822

Zentralblatt MATH identifier
1212.35484

Subjects
Primary: 35F30: Boundary value problems for nonlinear first-order equations
Secondary: 45K05: Integro-partial differential equations [See also 34K30, 35R09, 35R10, 47G20] 47G20: Integro-differential operators [See also 34K30, 35R09, 35R10, 45Jxx, 45Kxx]

Citation

Grinfeld, M.; Hines, G.; Hutson, V.; Mischaikow, K.; Vickers, G. T. Non-local dispersal. Differential Integral Equations 18 (2005), no. 11, 1299--1320. https://projecteuclid.org/euclid.die/1356059743


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