Stability And Impulsive Control Of A Ratio-Dependent Predator-Prey Model With Stage Structure And Time Delay

In this paper,a ratio-dependent predator-prey model described by Holling type ?functional response with time delay and stage structure for the prey is investigated.By using the corresponding mathematical theory,the model is analyzed and numerical simulations are carried out to illustrate the main theoretical results.The main work of this paper includes:Firstly,a ratio-dependent predator-prey model described by Holling type ? functional response with time delay and stage structure for the prey is investigated.By analyzing the corresponding characteristic equations,the local stability of the coexistence equilibrium of the model is discussed and the existence of Hopf bifurcations at the coexistence equilibrium is established.By using the persistence theory on infinite dimensional systems,it is proven that the system is permanent if the coexistence equilibrium exists.By introducing some new lemmas and the comparison theorem,sufficient conditions are obtained for the global stability of the coexistence equilibrium.Numerical simulations are carried out to illustrate the main results.Secondly,based on the biological control strategy,a ratio-dependent predator-prey model with time delay and stage structure for the prey(pest)is investigated,by releasing regularly the predator(natural)to control the pests.Using the theories and methods of impulsive and delayed differential equations,sufficient conditions are obtained,which guarantee the global attractivity of pest-eradication periodic solution and permanence of the system.Numerical simulations are carried out to illustrate the main results.