Analysis and constrained optimal impulsive control of a Holling-II type trophic system with two sources

In this paper, we study a multi-trophic prey-predator system composed by top- and intermediate-level consumers and two food sources. Biomass fluxes across trophic levels are modeled by means of Holling-II type functional responses and one of the sources is subject to periodic impulsive events consisting in biomass injections. Referring to this system, we analyze the local and global stability properties of periodic eradicated solutions and establish permanence properties for the populations in terms of relations between the model parameters and the intensity of the impulses. Secondly, we formulate an optimal control problem for this system as well as an Iterative Dynamic Programming (IDP) scheme for its solution. By means of the proposed algorithm we specify the intensity of the impulsive controls to be applied periodically in order to regulate the biomass levels along time. In our formulation, the intensity of the periodic impulsive events depends on the biomass of the upper- level consumer population, as we can often see in an agricultural framework where in the quantity of human workforce can significantly affect the size of crop periodically collected.