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We study the combined impact of a colored environmental noise and demographicnoise on the extinction risk of a long-lived and well-mixed isolated stochasticpopulation which exhibits the Allee effect. The environmental noise modulatesthe population birth and death rates. Assuming that the Allee effect is strong,and the environmental noise is positively correlated and Gaussian, we derive aFokker-Planck equation for the joint probability distribution of the populationsizes and environmental fluctuations. In WKB approximation this equationreduces to an effective two-dimensional Hamiltonian mechanics, where the mostlikely path to extinction and the most likely environmental fluctuation areencoded in an instanton-like trajectory in the phase space. The mean time toextinction (MTE) is related to the mechanical action along this trajectory. Weobtain new analytic results for short-correlated, long-correlated andrelatively weak environmental noise. The population-size dependence of the MTEchanges from exponential for weak environmental noise to no dependence forstrong noise, implying a greatly increased extinction risk. The theory isreadily extendable to population switches between different metastable states,and to stochastic population explosion, due to a combined action of demographicand environmental noise.

Impact of Colored Environmental Noise on the Extinction of a Long-Lived Stochastic Population: Role of the Allee Effect