The Effect of an Upper Limit to Population Size on Persistence Time

For a population with density-independent vital rates in a randomly varying environment, previous authors have calculated the probability that population size will first drop to some specified (arbitrary) low level at a given time (the first passage time distribution (FPTD), which may be interpreted as a distribution of extinction times). In this paper, we study the FPTD For a stochastic model of density-independent population growth which includes a hard upper limit to population size. We discuss the conditions under which this distribution may be approximated by the FPTD of a Wiener process with a reflecting boundary condition, for which an exact calculation is presented in an appendix. We compare the FPTD of the new model with its counterpart in the model without an upper limit. The most important effects of introducing the upper limit are: (a) ultimate extinction becomes certain; (b) if the long run growth rate in the absence of the upper boundary was small but positive, extinction within ecologically significant times is likely; (c) for larger values of the long run growth rate, persistence over ecologically significant times is almost certain. We discuss the implications of result (b) for conservation. Result (c) establishes that "density-vague" regulation can produce persistent, but bounded, populations.