Optimal harvesting of a predator-prey system with marine reserve

Abstract

Fisheries in developing countries are often confronted with the problem of excessive harvesting of the marine resources as a result of lax management practices. This study formulates an optimal control problem involving a predator-prey model with partial harvesting of the prey species and a critical biomass level below which the fishery is not economically viable. The region of boundedness and the equilibrium points of the system are determined and discussed. Stability analysis of the model shows that the interior equilibrium point is both locally and globally asymptotically stable. The Bendixson–Dulac criterion establishes that the system does not possess any limit cycles or periodic orbits. Pontryagin’s maximum principle is employed to determine the necessary conditions for optimality of the model. Singularity analysis of the model, with the aid of the generalized Legendre–Clebsch condition, indicates that the existence of singular arcs cannot be ruled out. Therefore, the characterization of the optimal control gives rise to both bang-bang and singular controls. Furthermore, the relationship between the shadow price of fish stock, the shadow price of fishing effort and the price of landed fish as it relates to the optimal reserve area is explored. Numerical simulations are performed to validate the established theoretical results using empirical data on the Ghana sardinella fishery. The study reveals that the fishery can sustain an effort rate of more than twice the maximum sustainable (MSY) rate, provided that 40% of the current fish stock is in a marine reserve and the operational fish population is at least 30% of the carrying capacity of the ecosystem.