Stability And Bifurcation Analysis Of The Two Types Of Discrete Biological Mathematical Model

In this paper, the stability and bifurcation of two discrete biomathe-matical models, named discrete Baleen-Wale model and Holling-Tanner predator-prey system, are discussed.In Chapter1, the preliminaries of dynamic system are presented, including the development of biological mathematics and basic bifurcati-on and chaos theory. Specifically, the center manifold theorem is introduced, which plays an important role in reducing the highdimensi-onal system to lower one.In Chapter2, we mainly discuss the Neimark-Sacker (N-S) bifurc-ation of the discrete Baleen-Wale biological model. The stability conditions of equilibria are obtained via analysing the property of eigenvalues. Moreover, the stability and Neimark-Sacker (N-S) bifurcati-on orientation are investigated, and the bifurcation coefficients are calculated via Maple. Finally, the numerical simulations are carried out which illustrate the theoretical analysis in previous sections.In Chapter3, we study the discrete Holling-Tanner predator-prey system. First, stability of the equilibria are discussed via discussing the eigenvalues. Then, the Neimark-Sacker bifurcation and Flip bifurcation are drawn, and their stability conditions and orientation are deduced. Finally, numerical simulations show that the theoretical results are valid, and it displays that the Holling-Tanner system can exhibit many complex dynamic behaviours, such as8,11,19,22-periodic trajectories, inverse periodic-doubling, invariant cycle, cascade of period-doubling, and the system tends to be chaotic eventually.