OPTIMAL MANAGEMENT OF A PREY-PREDATOR SYSTEM IN A POLLUTED ENVIRONMENT WITH EFFORT SHARED BETWEEN POLLUTION REDUCTION AND HARVESTING

OPTIMAL MANAGEMENT OF A PREY-PREDATOR SYSTEM IN A POLLUTED ENVIRONMENT WITH EFFORT SHARED BETWEEN POLLUTION REDUCTION AND HARVESTING

S. D. Zawka, Srinivasu P. D. N.

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Abstract

This paper is concerned with the optimal management of a prey-predator system in a polluted environment with e ort allocated between pollution reduction and prey harvesting. The growth rates of species and, consequently, the economic gain from the resources are impacted by the pollutants released from external sources (such as in- dustrial wastes). We assume that both the prey and predator populations have economic value. While revenue due to the prey comes from it's harvesting, it is from tourism that the revenue due to the predator is realised. The aim is to determine the optimal alloca- tion of the total e ort capacity between harvesting and pollution reduction to maximize the revenue. First, we study the qualitative behavior of the dynamical system and its de- pendence on the control parameter viz., e ort, followed by an application of Pontryagin's maximum principle to solve an optimal harvesting problem. The results indicate that the system has three possible equilibrium solutions whose existence and stability heavily depend on the level of e ort allocation made between harvesting and pollution reduc- tion. It underlines that the e ort allocation plays vital role in determining the eventual state of the system viz., permanence or extinction of the species. The result also shows that the optimal harvesting policy under pollution reduction e orts (as compared to one under no pollution reduction) calls for employing a lower level of optimal e ort, resulting in higher resource stock levels and increased income. On the other hand, when the stock bene t of the predator rises, the optimal harvest e ort falls, leading to increased stock levels in both the species.

Keywords

Prey-predator, pollution, optimal harvesting, stock bene t, pollution-reduction.