Qualitative analysis of impulsive state feedback control to an algae-fish system with bistable property

A kind of consumer-resource system is proposed to describe the bidirectional interactions of the algae and the fish in an eutrophic water body. The dynamical properties of the proposed continuous system are given. For the bistable case, an impulsive semidynamical system with state feedback control, which depends on the biomass of the algae, is formulated and investigated to consider the feasibility of state feedback control for the aim of maintaining two species coexisting. The impulsive semidynamical system has three cases corresponding to three kinds of control measures: releasing fish, spraying algaecide, integrated control combining releasing fish and spraying algaecide. The existences of order-1 periodic solutions of three models are discussed by using successor function, respectively. The conditions under which the order-1 periodic solution is stable are given by using the Poincaré map and the analogue of Poincaré criterion. Mathematical results show that, for every one of three control strategies, there exists a range of control parameter in which the corresponding control is feasible. Finally, those mathematical results are verified by numerical simulations and the practical meanings are given.