Two Types Of Stability And Bifurcation Analysis Of Delay Difference Equations

In this thesis, we investigate the stability and bifurcation of a discrete θ-logistic model with delay, and a delay discrete-time predator-prey system of Leslie-Gower type. It consists of three chapters as follows:Chapter1introduces the background knowledge and the recent work of the nonlinear dynamic systems simply.Chapter2studies a kind of discrete6-logistic model with delay. First of all, the stability of positive fixed point of the system is investigated. Neimark-Sacker bifurcation occurs when the delay passes a sequence of critical values, and then the direction and stability of the Neimark-Sacker bifurcation is derived by using the normal form method and center manifold theorem. Finally, numerical simulations are performed to illustrate the results obtained.Chapter3discusses the dynamics of a discrete-time predator-prey model of Leslie-Gower type with delay. First of all, the stability of positive fixed point of the system is studied. It is found that the system can undergo flip bifurcation when the parameter passes a critical value, and then the direction and stability of the flip bifurcation is derived by using the bifurcation theory. Numerical simulations are performed not only to illustrate our results, but also to show the complex dynamical behaviors, such as a cascade of period-doubling bifurcation in period-2,4,8orbits, the chaotic sets.