Global dynamics and threshold control of a discontinuous fishery ecological system

In this paper, we investigate the global dynamics and sliding limit cycle analysis of a Filippov fishery ecological system with ratio variable predator threshold management policy. Different from the traditional Filippov threshold control model, a ratio variable predator threshold management policy is proposed to replace the single variable threshold control strategy here to make the model more realistic. Three layers goal are realized. First, we investigate the positiveness and boundedness of the solution and discuss the existence of sliding segments. Further, we analyze the sliding mode dynamics, and consider the existence and stability of three-type equilibria of controlled system (regular equilibrium point, virtual equilibrium and pseudo equilibrium). Moreover, we discuss nine kinds of possible dynamic phenomena of the system. We rule out that three situations are impossible to appear in the system. Then by constructing the Lyapunov function, using the successor function, and combining Poincaré-Bendixon Theorem, we obtain that the solution of the model will converge to either of two local equilibria (bistability), one local equilibrium or sliding limit cycle under different control values. Finally, we give some numerical simulations of seven examples to validate the correctness of the theoretical results.