Risks of Population Extinction from Demographic and Environmental Stochasticity and Random Catastrophes

TLDR

The study analyzes how demographic, environmental, and catastrophic stochasticity affect extinction risk in a single population. Relative risks are evaluated by comparing how the average extinction time scales with carrying capacity in a simple exponential‑growth model up to K, using the long‑run growth rate r̃ and considering how environmental variance and catastrophe magnitude/frequency influence the dynamics. When r̃ is positive, extinction time grows nearly exponentially with K under demographic stochasticity and as a power of K under environmental stochasticity or catastrophes, whereas for negative r̃ it grows only logarithmically; consequently, in large populations demographic stochasticity is less threatening than environmental or catastrophic factors, and modest populations can persist long if r̃ is substantially positive.

Abstract

Stochastic factors affecting the demography of a single population are analyzed to determine the relative risks of extinction from demographic stochasticity, environmental stochasticity, and random catastrophes. Relative risks are assessed by comparing asymptotic scaling relationships describing how the average time to extinction, T, increases with the carrying capacity of a population, K, under each stochastic factor alone. Stochastic factors are added to a simple model of exponential growth up to K. A critical parameter affecting the extinction dynamics is $$\tilde r,$$ the long-run growth rate of a population below K, including stochastic factors. If r̃ is positive, with demographic stochasticity T increases asymptotically as a nearly exponential function of K, and with either environmental stochasticity or random catastrophes T increases asymptotically as a power of K. If r̃ is negative, under any stochastic demographic factor, T increases asymptotically with the logarithm of K. Thus, for sufficiently large populations, the risk of extinction from demographic stochasticity is less important than that from either environmental stochasticity or random catastrophes. The relative risks of extinction from environmental stochasticity and random catastrophes depend on the mean and environmental variance of population growth rate, and the magnitude and frequency of catastrophes. Contrary to previous assertions in the literature, a population of modest size subject to environmental stochasticity or random catastrophes can persist for a long time, if r̃ is substantially positive.