On the distribution of the time to extinction in the stochastic logistic population model

Abstract

The aim of this paper is to investigate the distribution of the extinction times, T, of the stochastic logistic process from both the numerical and the theoretical standpoint. The problem is approached first by deriving formulae for the moments of T; it is then shown that in most cases T is, very nearly, a gamma variate. Some simulated results are given and these agree well with the theory. Furthermore, a consideration of the process conditioned on non-extinction is shown to be an effective way of obtaining a large t (time) description of the unconditioned process. Finally, a more general form of the model in which the death-rate as well as the birth-rate is ‘density-dependent' is considered, and by comparison with the usual form of the model the effect on T of this additional factor is assessed.