Evolution in a migrating population model
We consider a model of migrating population occupying a compact domain $\varOmega $ 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.