Global dynamics and bifurcation in a stage structured prey–predator fishery model with harvesting

This paper describes a prey–predator model with stage structure for predator and selective harvesting effort on predator population. The Holling type II functional response function is taken into consideration. All the equilibria of the proposed system are determined and the behavior of the system is investigated near them. Local stability of the system is analyzed. Geometric approach is used to derive the sufficient conditions for global stability of the system. The occurrence of Hopf bifurcation of the model system in the neighborhood of the co-existing equilibrium point is shown through considering maximal relative increase of predation as bifurcation parameter. Fishing effort used to harvest predator population is considered as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent earned from the resource. Pontryagin’s maximum principle is used to characterize the optimal control. The optimal system is derived and then solved numerically using an iterative method with Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem.

► We describe a prey predator model with stage structure for predator and selective harvesting effort on predator population.
► Sufficient conditions for global stability of the system at the interior equilibrium are derived.
► Capture rate of mature predators may lead to the stable system to become unstable through Hopf bifurcation.
► Cannibalism of predator population ensures the existence of predator population without having prey population in the system.
► Solution of optimal control problem ensures sustainability of the system and maximizes the monetary social benefit.