Predicting Extinction Times from Environmental Stochasticity and Carrying Capacity

Abstract

Managers of small populations often need to estimate the expected time to extinction T e of their charges. Useful models for extinction times must be ecologically realistic and depend on measurable parameters. Many populations become extinct due to environmental stochasticity, even when the carrying capacity K is stable and the expected growth rate is positive. A model is proposed that gives T e by diffusion analysis of the log population size n t (= log e N t ). The model population grows according to the equation N t+1 = R t N t , with K as a ceiling. Application of the model requires estimation of the parameters k = logK, r d = the expected change in n, v r = Variance(log R), and ϱ the autocorrelation of the r t . These are readily calculable from annual census data (r d is trickiest to estimate). General formulas for T e are derived. As a special case, when environmental fluctuations overwhelm expected growth (that is r d  0), T e = 2n o (k ‐ n o /2)/v r . If the r t are autocorrelated, then the effective variance is v re  v r (1 + ϱ)/(1 ‐ ϱ). The theory is applied to populations of checkerspot butterfly, grizzly bear, wolf, and mountain lion.