Extinction risk of a density-dependent population estimated from a time series of population size

Environmental threats, such as habitat size reduction or environmental pollution, may not cause immediate extinction of a population but shorten the expected time to extinction. We develop a method to estimate the mean time to extinction for a density-dependent population with environmental fluctuation. We first derive a formula for a stochastic differential equation model (canonical model) of a population with logistic growth with environmental and demographic stochasticities. We then study an approximate maximum likelihood (AML) estimate of three parameters (intrinsic growth rate r, carrying capacity K, and environmental stochasticity sigma(2)(e)) from a time series of population size. The AML estimate of r has a significant bias, but by adopting the Monte Carlo method, we can remove the bias very effectively (bias-corrected estimate). We can also determine the confidence interval of the parameter based on the Monte Carlo method. If the length of the time series is moderately long (with 40-50 data points), parameter estimation with the Monte Carlo sampling bias correction has a relatively small variance. However, if the time series is short (less than or equal to 10 data points), the estimate has a large variance and is not reliable. If we know the intrinsic growth rate r, however, the estimate of K and sigma(2)(e)and the mean extinction time T are reliable even if only a short time series is available. We illustrate the method using data for a freshwater fish, Japanese crucian carp (Carassius auratus subsp.) in Lake Biwa, in which the growth rate and environmental noise of crucian carp are estimated using fishery records.