Extinction dynamics of age-structured populations in a fluctuating environment.

Abstract

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...

Modelling population persistence on islands: mammal introductions in the New Zealand archipelago

“…Notably, the observed distribution of persistence times closely followed that predicted by stochastic population growth theory in which extinction results from fluctuations in population size caused by environmental and/or demographic stochasticity (Lande & Orzack 1988;Dennis et al 1991;Engen et al 2005; both the inverse Gaussian and the demographic distributions fitted the data well).…”

“…For small populations, such as the introductions considered here, fluctuations in population size due to demographic and environmental stochasticity are likely to be the major cause of extinction (Richter-Dyn & Goel 1972). For populations unaffected by density dependence and subject only to fluctuations caused by environmental stochasticity, stochastic population growth theory predicts that the distribution of persistence times will approach an inverse Gaussian distribution with parameters specifying the initial population size or viability, the long-term population growth rate and the variance due to environmental stochasticity (Lande & Orzack 1988;Dennis et al 1991). The inverse Gaussian distribution is highly skewed with a long right tail, so that the theory predicts that populations will either go extinct quickly or persist for a long time (Claessen et al 2005).…”

“…The probability that a population persists for a given time is a widely used measure of population viability and the subject of much theoretical work aimed at predicting the fate of endangered species (Lande & Orzack 1988;Dennis et al 1991;Mangel & Tier 1993;Foley 1994). For small populations, such as the introductions considered here, fluctuations in population size due to demographic and environmental stochasticity are likely to be the major cause of extinction (Richter-Dyn & Goel 1972).…”

Reproductive Value and Fluctuating Selection in an Age-Structured Population

“…The total reproductive value equals the total population size N L if the population is exactly at its stable age distribution, and Engen et al (2007) showed that ln N L undergoes stationary fluctuations around ln V L with a return time to equilibrium of about one generation. That the process ln V L has approximately white noise explains the success of the diffusion approximation for ln V L (and ln N L ), identified simply as the Wiener process with infinitesimal mean and variancer L and s 2 L (Lande and Orzack 1988).…”

“…Because a diffusion process is subject to white noise with no temporal autocorrelation, the approximation is most accurate if the noise in the underlying biological process is approximately uncorrelated among years. Despite temporal autocorrelation in total population size produced by age-structure fluctuations, the stochastic demography of age-structured populations over timescales of a generation or more can nevertheless be accurately approximated by a diffusion process (Tuljapurkar 1982;Lande and Orzack 1988;Engen et al 2005aEngen et al , 2007. The success of the diffusion approximation for total population size occurs because the noise in the total reproductive value is nearly white, with no temporal autocorrelation to first order, and the log of total population size fluctuates around the log of reproductive value with a return time to equilibrium on the order of a few generations (Engen et al 2007).…”