Extinction rate of a population under both demographic and environmental stochasticity

We examined the asymptotic rate of population extinction beta when the population experiences density-dependent population regulation, demographic stochasticity, and environmental stochasticity. We assume discrete-generation population dynamics, in which some parameters fluctuate between years. The fluctuation of parameters can be of any magnitude, including both fluctuation traditionally treated as diffusion processes and fluctuation from catastrophes within a single scheme. We develop a new approximate method of calculating the asymptotic rate of population extinction per year, beta=integralinfinity0 exp(-x) u(x) dx, where u(x) is the stationary distribution of adult population size from the continuous-population model including environmental stochasticity and population-regulation but neglecting demographic stochasticity. The method can be regarded as a perturbation expansion of the transition operator for population size. For several sets of population growth functions and probability distributions of environmental fluctuation, the stationary distributions can be calculated explicitly. Using these, we compare the predictions of this approximate method with that using a full transition operator and with the results of a direct Monte Carlo simulation. The approximate formula is accurate when the intrinsic rate of population increase is relatively large, though the magnitude of environmental fluctuation is also large. This approximation is complementary to the diffusion approximation.