Optimal Harvesting of Fluctuating Populations with a Risk of Extinction

Optimal harvesting theories based on the concept of maximum sustained yield assume a stationary distribution of population size, ignoring that extinction is the eventual fate of all populations. We analyze the dynamics of populations at risk of extinction from demographic and environmental stochasticity as well as harvesting. Diffusion theory is used to derive approximate formulas for the mean time to extinction, T, the expected cumulative harvest before extinction, Y, and the mean and standard deviation of annual harvest, ȳ and σy. In numerical examples we compare the performance of different harvesting strategies in terms of these statistics. We derive optimal harvesting strategies that maximize Y or ȳ. The optimal strategies always involve a threshold function in which harvesting occurs at the maximum possible rate above a critical population size, c, with no harvest below c. For a broad class of stochastic population models we show that Y is maximized, with unlimited harvesting capability, when the critical size equals the carrying capacity of the population (c = K), and with limited harvesting capability when c < K. Optimizing ȳ produces a smaller value of c than optimizing Y. We also derive the optimal strategy that maximizes Y subject to a prescribed risk of extinction.