Evolution in a migrating population model
Volume 39 / 2012
Abstract
We consider a model of migrating population occupying a compact domain in the plane. We assume the Malthusian growth of the population at each point x\in \varOmega and that the mobility of individuals depends on x\in \varOmega . The evolution of the probability density u(x,t) that a randomly chosen individual occupies x\in \varOmega at time t is described by the nonlocal linear equation u_t=\int _\varOmega \varphi (y)u(y,t) \, dy-\varphi (x)u(x,t), where \varphi (x) is a given function characterizing the mobility of individuals living at x. We show that the asymptotic behaviour of u(x,t) as t\to \infty depends on the properties of \varphi in the vicinity of its zeros.