Optimal harvesting for a predator-prey system with age structure and reserve area

Considering that the creation of marine reserves is one potential solution to the over exploitation of marine life and different population age implies different reproduction as well as survival capacities, we propose an age-structured predator-prey model for fishery resource with reserve area where fishing is strictly prohibited, and then consider the corresponding optimal harvesting problem where the objective functional represents the total economic benefits obtained from the harvesting process. First, the existence and uniqueness of the solution of the model are proved by means of the fixed point theorem. Then, the optimality conditions for achieving the optimal harvesting policy are given by solving the extremum point of the cost functional. Furthermore, the existence of a unique optimal harvesting strategy is demonstrated with the use of Ekeland's Principle. Finally, the numerical simulations indicate that the creation of marine reserves is indeed conducive to the recovery of fishery resources, increasing fish abundance and protecting biodiversity as well as ecosystem structure.