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**Global dynamics and bifurcation in a stage structured prey-predator fishery model with harvesting.**
*(English)*
Zbl 1246.92026

Summary: This paper describes a prey-predator model with stage structure for predator and selective harvesting effort on the 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. A geometric approach is used to derive sufficient conditions for global stability of the system. The occurrence of Hopf bifurcations of the model system in the neighborhood of the co-existing equilibrium point is shown through considering the maximal relative increase of predation as a bifurcation parameter. Fishing effort used to harvest the 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 a Runge-Kutta fourth-order scheme. Simulation results show that the optimal control scheme can achieve a sustainable ecosystem.

### MSC:

92D40 | Ecology |

49N90 | Applications of optimal control and differential games |

91B76 | Environmental economics (natural resource models, harvesting, pollution, etc.) |

37N25 | Dynamical systems in biology |

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\textit{K. Chakraborty} et al., Appl. Math. Comput. 218, No. 18, 9271--9290 (2012; Zbl 1246.92026)

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