Stochastic differential equations harvesting policies: Allee effects, logistic-like growth and profit optimization

Abstract

In this article, stochastic differential equations are used to model the dynamics of a harvested population in the presence of weak Allee effects. Two optimal harvesting policies are presented, one with variable effort based on optimal control theory,which is for practical reasons inapplicable in a random environment, and the otherwith constant effort and easily applicable. For a logistic-like model with weak Allee effects, we show that the optimal policy based on constant effort implies, in a suitable range of effort values, the existence of a steady-state stochastic equilibrium with a stationary density, obtained explicitly here, for the population size.With this new result, we compare the performance of both policies in terms of the profit obtained over a finite time horizon. Using realistic data from a harvested population and a logistic-type growth model, we quantify the profit reduction when choosing the optimal policy based on constant effort instead of the optimal policy based on variable effort.We also study the influence of the Allee effects strength.