Evolution in a migrating population model

Włodzimierz Bąk ¹   ; Tadeusz Nadzieja ¹

¹ Instytut Matematyki i Informatyki Uniwersytet Opolski Oleska 48 45-052 Opole, Poland

Applicationes Mathematicae, Tome 39 (2012) no. 3, pp. 305-313

Résumé

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.

Détail

DOI : 10.4064/am39-3-5

Keywords: consider model migrating population occupying compact domain varomega plane assume malthusian growth population each point varomega mobility individuals depends varomega evolution probability density randomly chosen individual occupies varomega time described nonlocal linear equation int varomega varphi t dy varphi t where varphi given function characterizing mobility individuals living asymptotic behaviour infty depends properties varphi vicinity its zeros