Non-local dispersal

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.