Study On Persistence And Stability Incorporating Hopf Bifurcation For Kinds Of Stage-structured Systems

This paper consists of four parts:Firstly, the persistence of nonautonomous Lotka-Volterra type stage structured predator-prey systems with infinite delay on Cg space was studied in this paper. By constructing Lyapunov function, the ultimately boundedness of positive solutions for system are investigated. Based on the comparison theorem and analyzing the right-hand functional of the systems, and under the same conditions of predator-prey systems with stage structure and infinite delay on Ch space, some sufficient conditions are derived for the persistence of nonautonomous ecosystems with infinite delay and stage structure. The obtained results show that death rate of mature predators and prey densities play a very important role on the persistence of predator-prey system.Secondly, the stability of the non-negative equilibrium of predator-prey models with stage structure incorporating prey refuge for both predators and preys is considered in this paper. By using of algebra method and characteristic roots theory, the local stability of non-negative equilibrium are derived, according to comparison theorem,iteration technique and geometric sequence properties, he global asymptotical stability of positive equilibrium is discussed, and some sufficient conditions for global asymptotical stability of three equilibrium are obtained. It is shown that refuge does not increase the stability of positive equilibrium, it also shows the effect on the equilibrium and the densities of the system equilibrium.Thirdly, stage-structured predator-prey systems with ratio-dependent type functional response and time delay is investigated in this paper. By analyzing the corresponding characteristic equations, the local stability of the equilibrium is investigated, moreover, it is found that time delay can cause a stable positive equilibrium to become unstable one, even occur Hopf bifurcation and produce a variety oscillation and periodic solutions, when time delay passes through some critical values. According to comparison theorem and iteration technique, the global asymptotical stability of positive equilibrium is discussed, and some sufficient conditions for global asymptotical stability of positive equilibrium are obtained. Numerical simulations are carried out to illustrate our main results.Finally, stage-structure predator-prey systems with Beddington-DeAngelis type functional response and time delay incorporating a prey refuge is investigated in this paper. By using of the characteristic roots theory, the local stability of the equilibrium is investigated, moreover, Hopf bifurcations occur at the positive equilibrium as the delay τ crosses some critical values. Further, the influence of prey refuge on densities of predator-prey systems is investigated. From the analysis of biological parameters, we notice that the density of prey species increases in the presence of prey refuge, and the density of predator species decreases. It shows that the existence of refuge has important effects on the coexistence of predator species and prey species.