Survival, Extinction and Ergodicity in a Spatially Continuous Population Model

#### Julien Berestycki, A.M. Etheridge, M. Hutzenthaler

2009, v.15, №3, 265-288

ABSTRACT

We consider a model recently introduced by Barton & Etheridge for a population evolving in a spatial continuum, in which a succession of catastrophic events of varying intensity allows for the possibility of large-scale extinction and recolonisation. These reproduction events are based on a Poisson process of spatial events (rather than individuals) and the potential number of offspring produced during such an event is Poisson with a certain intensity. We show that if this intensity is sufficiently large the population, when started from a translation invariant initial condition in $R^d$, survives with probability one, whereas for low intensities the population dies out. Moreover we prove that ergodicity holds even in low dimensions. This contrasts sharply with the Dawson - Watanabe process and other traditional models in which reproduction of different individuals is uncorrelated.

Keywords: coalescence,ergodicity,survival,extinction,diffusion approximation