  We study the combined impact of a colored environmental noise and demographic
noise on the extinction risk of a long-lived and well-mixed isolated stochastic
population which exhibits the Allee effect. The environmental noise modulates
the population birth and death rates. Assuming that the Allee effect is strong,
and the environmental noise is positively correlated and Gaussian, we derive a
Fokker-Planck equation for the joint probability distribution of the population
sizes and environmental fluctuations. In WKB approximation this equation
reduces to an effective two-dimensional Hamiltonian mechanics, where the most
likely path to extinction and the most likely environmental fluctuation are
encoded in an instanton-like trajectory in the phase space. The mean time to
extinction (MTE) is related to the mechanical action along this trajectory. We
obtain new analytic results for short-correlated, long-correlated and
relatively weak environmental noise. The population-size dependence of the MTE
changes from exponential for weak environmental noise to no dependence for
strong noise, implying a greatly increased extinction risk. The theory is
readily extendable to population switches between different metastable states,
and to stochastic population explosion, due to a combined action of demographic
and environmental noise.
