Two Types Of Discrete Smith-holling Type Predator Prey System Stability And Bifurcation Study

In this paper, we investigate the stability and bifurcation of a predator-prey system and discrete dynamical system with Holling. The whole paper consists of three chapters.In Chapter 1, the development of non-linear dynamical system and bifurcation are given.At the same time, fundamental theory of bifurcation are also introduced.In Chapter 2, the stability and bifurcation of a predator-prey system are investigated. First of all, the stability of the model is studied. It is found that there exists Neimark-Sacker bifurcations when the parameter passes a critical values. Then the explicit algorithm for determining the direction of the Neimark-Sacker bifurcation and the stability of the closed invariant curve bifurcating from the positive fixed point are derived. Finally, computer simulations are presented not only to illustrate our results with theoretical analysis, but also to exhibit the complex dynamical behaviors, such as period-5,8,10,11,15,16,26,30,37,48,63, cascade of period-doubling bifurcation in period-2,4,8,16 orbits quasi-periodic orbits and the chaotic sets.In Chapter 3, first, the stability of fixed point of a discrete dynamical systems is considered. The stability of Flip bifurcation and Neimark-Sacker bifurcation of the system is also discussed. Finally, computer simulations are presented not only to illustrate our results with theoretical analysis, but also to exhibit the complex dynamical behaviors, such as period-5,6,7,14,15,16,21,28,30,37,46,47,57,73 orbits, cascade of period-doubling bifurcation in period-2,4,8,16 orbits, quasi-periodic orbits and the chaotic sets.