Predicting the time to quasi-extinction for populations far below their carrying capacity

Populations threatened by extinction are often far below their carrying cap acity. A population collapse or quasi-extinction is defined to occur when t he population size reaches some given lower density. If this density is cho sen to be large enough for the demographic stochasticity to be ignored comp ared to environmental stochasticity, then the logarithm of the population s ize may be modelled by a Brownian motion until quasi-extinction occurs. The normal-gamma mixture of inverse Gaussian distributions can then be applied to define prediction intervals for the time to quasi-extinction in such pr ocesses. A similar mixture is used to predict the population size at a fini te time for the same process provided that quasi-extinction has not occurre d before that time. Stochastic simulations indicate that the coverage of th e prediction interval is very close to the probability calculated theoretic ally. As an illustration, the method is applied to predict the time to exti nction of a declining population of white stork in southwestern Germany. (C ) 2000 Academic Press.