Model-based on fishery management systems with selective harvest policies

Recently, marine reserves have become a widely advocated approach to marine conservation. However, these studies beg the question as to how the selection imposed by harvesting and the dispersal rates between two patches influence stock abundance and the maximum fishery yield from the perspective of mathematical model. In this paper, a prey–predator model with selective harvesting by incorporating two discrete state delays in age and size for both the species in unreserve area has been established. Considering the resources exploitation and ecological observation, a hybrid optimal problem with Lagrange function and state delays together with characteristic times is formulated. We then develop an efficient numerical method by the lights of optimal state delay control method and Pontryagin-type minimum (maximum) principle. Finally, we discuss the impacts of selection for harvesting ages and efforts as well as migration coefficients on the fishery abundance and maximal yield by a series of simulations. Our results manifest: (1) after establishing protected areas, the oscillating amplitudes of prey and predator in unprotected areas are lessened and the stocks of predator in both areas are greatly enhanced with increasing of dispersal rates. (2) selective harvest tactics is appropriate for the strong intensity harvest in order to increase the abundance of all populations; (3) at the specific ages, optimal harvest efforts can enhance greatly fishery yield for the harvested population. However, the too late harvest for populations can reduce fishery sustainable yield, and this undesirable effects should be considered alongside harvest efforts when developing hunting regulations or policies; (4) the appropriate age selections will bring about the higher yield when the over-exploitation is implemented; (5) creating reserved areas only for preys greatly enhances fishery stocks and the sustainable yield than the other two patterns. However, setting reserved areas merely for the predator population is undesirable for increasing the fishery stock and yield.