A Predator-prey System With Stage Structure And Time Delay For Prey And Its Impulsive

Predator-prey model is an important model in the dynamic of population and is paid close attentions. Here we study a predator-prey model and its application in pest management.Firstly, we introduce the research summary of prey-predator model and pest management, and some of the relevant definitions, theorems and lemmas, and then give the main works in this paper.Secondly, we propose and investigate a predator-prey system with time delay and stage structure for the prey. We discuss the positive and bounded of the solutions. By analyzing the corresponding characteristic equations, the existence and the local stability of a positive equilibrium and two boundary equilibria of the system are discussed, respectively. By using persistence theory on infinite dimensional systems and comparison argument, respectively, sufficient conditions are obtained for the global stability of the positive equilibrium and one of the boundary equilibria of the proposed system. Further, the existence of a Hopf bifurcation at the positive equilibrium is studied. Numerical simulations are carried out to illustrate the main results.Finally, we propose and study a pest management model with delayed and stage-structured predator-prey model and impulsive control. Using the theories and methods of impulsive delayed differential equations, we discuss the positive and bounded of the solutions, the existence and the global attractivity of the pest-extinction periodic solution and the uniform permanence of the system. Sufficient conditions for the global appeal of pest-extinction of periodic solutions and the permanence of the system are obtained. Numerical simulations are carried out to illustrate the main results.