Combined harvesting of a stage structured prey–predator model incorporating cannibalism in competitive environment

In this paper, we propose a prey–predator system with stage structure for predator. The proposed system incorporates cannibalism for predator populations in a competitive environment. The combined fishing effort is considered as control used to harvest the populations. The steady states of the system are determined and the dynamical behavior of the system is discussed. Local stability of the system is analyzed and sufficient conditions are derived for the global stability of the system at the positive equilibrium point. The existence of the Hopf bifurcation phenomenon is examined at the positive equilibrium point of the proposed system. We consider harvesting effort as a control parameter and subsequently, characterize the optimal control parameter in order to formulate the optimal control problem under the dynamic framework towards optimal utilization of the resource. Moreover, the optimal system is solved numerically to investigate the sustainability of the ecosystem using an iterative method with a Runge–Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve sustainable ecosystem. Results are analyzed with the help of graphical illustrations.