OPTIMAL HARVEST CONTROL OF PREDATOR–PREY SYSTEMS IN FISHERIES WITH STAGE STRUCTURE AND ADDITIONAL FOOD SUPPLY

In this paper, we consider a fishery predator–prey system with stage structure and additional food incorporating fluctuations in resource prices influenced by supply and demand. The fishery system can be regarded as a slow–fast system under some suitable assumptions. Through variable aggregation methods, we derive a simplified four-dimensional model governing the density of prey fish, the juvenile and adult predator densities, and fishing effort in the fishery. We perform a stability analysis of the simplified system and give the optimal harvesting policy by using Pontryagin’s maximum principle. In our analysis, we find that the catastrophe equilibrium point corresponding to overfishing leading to fish extinction is unstable, which indicates that the predator stage structure has an impact on the stability of the system. Moreover, we find that due to the additional food item the system appears to have prey-free equilibrium points, which indicates that the system does not collapse due to the extinction of one prey item. We also give thresholds for the amount of additional food needed to maintain the balance of the system under different circumstances.