Optimal harvesting for a logistic growth model with predation and a constant elasticity of variance

Abstract

In this paper, we address the problem of optimal management of renewable resources such as agricultural commodities and fishery production. For that purpose, we consider the population associated with such commodities and assume that its size evolves according to a logistic growth model with a predation term given by a Holling type-n functional response. Additionally, we assume that such population is subject to random fluctuations, modeled by a diffusive term driven by a one-dimensional Brownian motion and having a power-type coefficient, thus endowing the model under consideration with the property of having constant elasticity of variance. Since the stochastic differential equation associated with this model does not fit the standard assumptions in the stochastic optimal control literature, namely sublinear growth, we develop an appropriate version of the dynamic programming principle for the problem under consideration herein, proceeding also to provide a characterization of the optimal harvesting strategies and discuss some qualitative properties of the corresponding value function.