The Distribution of Population Survival Times

Abstract

A discrete unstructured population model with nonoverlapping generations is used to calculate the probability of extinction as a function of the number of generations elapsed and the initial population size. This formulation accounts for demographic effects, environmental variation, and catastrophic mortality. Accurate approximations are developed for large carrying capacity and large time. The main qualitative results are as follows: The probability of early extinction depends strongly on the initial population size. A substantial fraction of small starting populations become extinct within a short time. Those that survive the initial period typically survive for a long time. This skew in extinction times makes the expected time to extinction a misleading indicator. Probabilities of extinction are sensitive to environmental variation in the net reproductive rate and the shape of the distribution of disturbances, not just its mean. Probabilities of extinction are likewise sensitive to the rate of catastrophe occurrences and the shape of the catastrophe distribution, not just its mean. Diffusion approximations are likely to be inaccurate for small populations or for populations that show large changes in size in a single generation. These difficulties may be overcome by use of an extrapolation based on solutions for discrete population size and time.