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In this thesis, we discuss the stability and bifurcation of two discrete-time variable-territory predator-prey systems with Holling response functions. It consists of three chapters.In Chapterl, we introduce the development of Ecological mathematics, the relative research result and the knowledge that was used in this paper, and give the relative knowledge about bifurcation theory.In Chapter2, we discuss the dynamics of a discrete-time Variable-territory predator-prey model with Holling III response function, including the existence and stability of the positive fixed point, the flip bifurcation and Neimark-Sacker bifurcation at the positive fixed point by using center manifold theorem and bifurcation theory. Numerical simulations not only illustrate our results we have gotten in this chapter, but also exhibit more complex dynamical behaviors than continual model, such as period-5,6,7,8,9,10,11,13,15,16,17,19,20,21,25,27,32,33 orbits, cascade period-doubling bifurcation in period-2,4,.8,16 orbits, quasi-periodic orbits and chaotic sets. Finally, the chaotic orbits are stabilized to the unstable fixed point via parameter delayed feedback control method.In Chapter3, we discuss the dynamic behavior of a discrete-time Variable-territory predator-prey model with Holling I response function, including the flip bifurcation and Neimark-Sacker bifurcation at the positive fixed point, and controlling chaos via parameter delayed feedback control method. Numerical simulations show more rich dynamic properties, such as period-4,5,6,9,11,13,14,21,22,30 orbits, cascade period-doubling bifurcation in period-2,4,8,16 orbits, quasi-periodic orbits and chaotic sets.
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