Stability And Bifurcation Of Two Discrete Dynamical Systems

Discrete phenomena exist widely in nature,and the data in economy,ecology and production are counted by day,month and year,therefore it is a very natural choice to use discrete system to analyze and describe.In addition,the discretization of differential equations has very rich and complex dynamic behavior,so it is necessary to study the discrete system.This paper discusses the bifurcation of two discrete biological models.The full text consists of four chapters,which main contents are as follows:First chapter mainly expound the background and significance of differential equations as well as the main work of this research.The second chapter briefly summarizes the basic concepts of dynamical systems and the center manifold theorem.In Chapter 3,a discrete two-species competitive model with Michaelies-Menten type harvesting in the first species is studied.Firstly,the existence of the fixed points of this model under different parameters is considered.Then the local stability of the fixed points is discussed,and the parameter conditions for the occurrences of its pitchfork bifurcation,transcritical bifurcation and period-doubling bifurcation are obtained.Finally,use Matlab software to carry out numerical simulations as well as give the corresponding bifurcation diagrams and Lyapunov diagrams.For Chapter 4,the existence and local stability of the fixed points of a discrete two-trophic plant-herbivore model are analyzed,and compute the parameter conditions for the occurrences of period-doubling bifurcation and transcritical bifurcation.Finally,use Matlab software to carry out numerical simulations as well as give the corresponding bifurcation diagrams and Lyapunov diagrams.