Stochastic growth and extinction in a spatial geometric Brownian population model with migration and correlated noise

A continuous spatial model for populations that are not density-regulated is analyzed. The model is a generalization of the geometric Brownian motion often used to describe populations at a single location. The locations are linked by migration and spatial correlation in the noise. At any point of time, the population size at a given location is lognormally distributed so the log population size constitutes a Gaussian field. The model is homogeneous in space but not in time. In particular, we analyze the case when the stochastic growth rate is negative and the total population approaches extinction. The properties of the extinction process is studied by considering local quasi-extinctions. A major conclusion is that migration tends to increase the time to extinction provided that there is no cost of migration. However, as the area occupied by the species starts to decrease, the decrease is faster for populations with larger migration.