Predicting the Time to Quasi-extinction for Populations far below their Carrying Capacity

Populations threatened by extinction are often far below their carrying capacity. A population collapse or quasi-extinction is defined to occur when the population size reaches some given lower density. If this density is chosen to be large enough for the demographic stochasticity to be ignored compared to environmental stochasticity, then the logarithm of the population size 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 processes. A similar mixture is used to predict the population size at a finite time for the same process provided that quasi-extinction has not occurred before that time. Stochastic simulations indicate that the coverage of the prediction interval is very close to the probability calculated theoretically. As an illustration, the method is applied to predict the time to extinction of a declining population of white stork in southwestern Germany.