Sustainability of Orange Roughy Population

Abstract

This article describes Holling type-III two-prey one-predator fishery model with harvesting and toxicity in unreserved area. Orange roughy is the predator species with prawn and small fish as prey species. Firstly, boundedness of the model has been examined. Existence of interior equilibrium point has been analyzed. Then sufficient conditions of permanence and extinction of populations have been obtained. It has been noted that very high rate of harvesting and toxicity are the main reasons of species like orange roughy being at the edge of extinction. But as per human need, some amount of harvesting is important. Keeping this in mind, we have reserved them with their main prey species, prawn and small fish, in a protected area where harvesting is not allowed and toxicity is ignorable. Using Pontryagin’s maximum principle, we have derived the optimal value of control parameter (partial closure m), for which total discounted net revenue gets maximized. Then we have investigated the sufficient condition for which the net revenue gets maximized and the orange roughy species can survive in future. Finally our analytical results have been verified by numerical simulation using MATLAB.