A Stochastic Logistic Growth Model with Predation: An Overview of the Dynamics and Optimal Harvesting

We consider a logistic growth model with predation and a stochastic perturbation given by a diffusive term with power-type coefficient. The resulting stochastic differential equation (SDE) has the particularity that the standard conditions for the existence and uniqueness of solutions of SDEs do not hold for a large subset of parameter space. Thus, we start by discussing the well posedness of the problem at hand, leading to a detailed characterization for the existence and uniqueness of solutions. We then provide criteria ensuring extinction and persistence of such population. Additionally, we list subsets of parameter space where (absolutely continuous) stationary measures for the SDE under consideration are guaranteed to exist, providing a description for the corresponding densities. We conclude with an application to the optimal management of resources. We consider a real asset such as, for instance, a farm or an aquaculture facility, devoted to the exploration of a unique culture or population whose growth follows a SDE such as described above, and look for the optimal harvesting strategy associated with such culture or population.