A stochastic Lotka-Volterra Model with killing

Alexandru Hening, Martin Kolb

Published 2014-09-08Version 1

We study the long time behavior of a population which is modeled by a killed diffusion. The population evolves according to a stochastic version of a Lotka-Volterra equation with internal killing. The killing is chosen to be large when the population is small and small when the population is large. A possible biological interpretation would be that when the population is small there is a higher chance of having a catastrophic event resulting in extinction. For this system we study the existence and uniqueness of quasistationary distributions. Quasistationary distributions give us insight regarding the convergence of the population, conditioned on non-extinction, to an equilibrium. The killed diffusion does not fall into any previously studied frameworks because both of the boundary points are inaccessible (that is, the diffusion cannot reach them in finite time). By studying the spectral properties of the generator of the killed diffusion we are able to prove that there exists a quasistationary distribution which attracts all compactly supported probability measures on $(0,\infty)$.