Bifurcations And Periodic Solutions For Several Kinds Of Predator-Prey Systems

This thesis is concerned with several predator-prey systems.By using the qualitative theory,bifurcation theory for differential equations and theory for functional differential equations,we study the effects of parameters on these systems(for example, time delay,diffusion,non-monotonic functional response and so on),which provide corresponding theory basis for explaining,predicting and controlling some phenomena arising in ecology.Concretely speaking,this paper have done the following works.Firstly,we study two dimension Lotka-Volterra predator-prey system with multiple discrete time delays and some conditions for the appearance of Hopf bifurcation are given.By applying the normal form theory and center manifold theorem,we discuss the property of bifurcation periodic solutions.Under suitable assumption,even if the time delays in predator-prey are different,the system has the same critical delay parameter at which the Hopf bifurcation appears.Furthermore,we give the global existence of bifurcation periodic solutions for the system.As to the general Holling type predator-prey system,we prove theoretically that Bogdanov-Takens singularity appears only in the predator-prey system with non-monotonic functional response.In view of this,we consider the Hopf bifurcation and Bogdanov-Takens bifurcation for the predator-prey system with delay and non-monotonic functional response,and investigate the direction of Hopf bifurcation and stability of bifurcation periodic solutions.What's more,we calculate the universal fold of Bogdanov-Takens bifurcation.Our results shows that there are different parameter values for which system has a limit cycle and homoclinic loop.Secondly,we consider Leslie type predator-prey system with delay and diffusion. By analyzing the linearized system at the positive constant steady state and the corresponding characteristic equation,we study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. By applying the normal form theory of partial functional differential equations and some known results,the property of spatially homogeneous Hopf bifurcation is discussed.Particularly,we investigate the effect of diffusion on Hopf bifurcation. Our results show that large diffusivity has no effect on the Hopf bifurcation of the corresponding functional differential equations,while small diffusivity can lead to the fact that the system bifurcates a spatially inhomogeneous periodic solutions at the positive equilibrium.Meantime,we obtain a formula which can determine the direction and stability of spatially inhomogeneous Hopf bifurcation.Finally,we consider the Leslie-Gower predator-prey system with non-monotonic functional response.Even if this system has no delays,it is difficult to discuss the dynamical behaviors of this system since the positive equilibrium can not be expressed explicitly.By applying the qualitative theory for differential equations,we discuss the Bogdanov-Takens bifurcation of this system.Numerical simulations show that non-monotonic functional response can lead to complex dynamical behavior for this system.For example,with the change of parameters,there are some new phenomena in this system,such as the coexistence of two limit cycles,or coexistence of limit cycle and homoclinic loop.