Non-local dispersal. (English) Zbl 1212.35484

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

Differ. Integral Equ. 18, No. 11, 1299-1320 (2005).

Summary: 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 \(L^p\) \((1\leq p\leq \infty)\). Our principal aim in this paper is to inquire 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.

MSC:

35Q92	PDEs in connection with biology, chemistry and other natural sciences
45K05	Integro-partial differential equations
47G20	Integro-differential operators

Keywords:

reaction-diffusion model