PREDICTING EXTINCTION TIMES FROM ENVIRONMENTAL STOCHASTICITY AND CARRYING-CAPACITY

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 rho 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) almost-equal-to 0), T(e) = 2n0(k - n0/2)/v(r). If the r(t) are autocorrelated, then the effective variance is v(re) almost-equal-to v(r) (1 + rho)/(1 - rho). The theory is applied to populations of checkerspot butterfly, grizzly bear, wolf and mountain lion.