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_tN_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.