Dissertation
Dissertation > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Ordinary Differential Equations > Stability theory

Two discrete Ivlev density dependent predator -prey system stability and bifurcation analysis

Author GuiJingFeng
Tutor HeZhiMin
School Central South University
Course Applied Mathematics
Keywords predator-prey system discrete dynamical system chaos periodic orbit flip bifurcation Neimark-Sacker bifurcation delayed feedback control
CLC O175.13
Type Master's thesis
Year 2011
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In this thesis, we investigate the stability and bifurcation of two discrete-time Lotka-Volterra predator-prey systems with Ivlev-type functional response. It consists of three chapters.The first Chapter called introduction describes the production and development of Lotka-Volterra model. At the same time, some preliminary knowledge related to this article is listed.In Chapter 2, the stability and bifurcation of a discrete-time Smith-Ivlev predator-prey system are investigated. First we discuss the direction and stability of the system’s flip bifurcation and Neimark-Sacker bifurcation by using center manifold theorem and bifurcation theory. The system exist flip bifurcation or Neimark-Sacker bifurcation when the parameter passes some critical values. And afterwards numerical simulations are presented not only to illustrate our results with the theoretical analysis, but also to exhibit the complex dynamical behaviors, such as orbits of period-7,8,13,14,20,24,26, cascade of period-doubling bifurcation in orbits of period-2,4,8,16, quasi-periodic orbits and chaotic sets. Finally, we have stabilized the chaotic orbits to an unstable fixed point using the delayed feedback control method.In Chapter 3, the stability and bifurcation of a discrete-time Rosenzweig-Ivlev predator-prey system are considered. The direction and stability of the system’s Neimark-Sacker bifurcation is discussed, and the chaotic orbits are stabilized to an unstable fixed point by using the delayed feedback control method. Numerical simulations exhibit the system exists period-9,10,12,20,36 orbits, quasi-periodic orbits and the chaotic sets.

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