We model density-independent growth of an age-(or stage-) structured population, assuming that mortality and reproductive rates fluctuate as stationary time series. Analytical formulas are derived for the distribution of time to extinction and the cumulative probability of extinction before a certain time, which are determined by the initial age distribution, and by the infinitesimal mean and variance, # and a2, of a diffusion approximation for the logarithm of total population size. These parameters can be estimated from the average life history and the pattern of environmental fluctuations in the vital rates. We also show that the distribution of time to extinction (conditional on the event) depends on the magnitude but not the sign of t. When the environmental fluctuations in vital rates are small or moderate, the diffusion approximation gives accurate estimates of cumulative extinction probabilities obtained from computer simulations.The theory of extinction times for single populations has applications in diverse fields, including paleontology, island biogeography, community ecology, and conservation biology. Various analytical models have been constructed to estimate extinction times, probabilities of extinction, or both for a population subject to stochastic variation in demographic parameters (e.g., refs. 1-7). Two sources of random variation in population growth can be distinguished: first, demographic stochasticity caused by finite population size, in which each individual independently experiences age-specific probabilities of survival and reproduction; and second, environmental stochasticity, which affects the vital rates of all individuals similarly. Demographic stochasticity is important only in small populations, since chance variation in vital rates among individuals tends to average out in large populations (greater than about 100 individuals), whereas environmental factors can produce substantial fluctuations in demographic parameters in populations of any size. In natural populations subject to substantial abiotic and biotic perturbations, environmental stochasticity is usually far more important than demographic stochasticity in causing extinction (6, 7). Previous analytical models of stochastic population growth and extinction either lacked age structure or dealt only with demographic stochasticity (1-7).Here we analyze the extinction dynamics of a population subject to environmental fluctuations in age-specific birth and death rates. The theory of population growth developed by Cohen (8,9) and Tuljapurkar and Orzack (10, 11) is used to derive a diffusion approximation for the logarithm of total population size in a population subject to density-independent fluctuations in vital rates.
THE MODELAssumptions. A standard model of density-independent growth in an age-(or stage-) structured population specifies a recursion formula for the numbers of individuals in each age class or developmental stage at time t + 1 in terms of those at the previous time t, n(t + 1) = A(t)n(t), [1] where n...