Stability And Bifurcation Analysis Of A Biological Dynamical System

The dynamic system is an important part of nonlinear disciplines.Nonlinear problems are presented in various disciplines and areas of life,such as mathematics,physics,biology,medicine,engineering,mechanics and economics which can be explained by the nonlinear power system.In particular,bio-mathematics,as an interdisciplinary subject of biology and mathematics,has developed rapidly in recent years.Such factors as time,space,time lag,randomicity,pulse and phase need to be considered to establish a realistic mathematical model.Therefore,a discrete epidemic model,a predator-prey model with time lag,and the dynamic behavior of a stochastic and discrete predator-prey model are studied.This thesis is divided into four chapters,as follows:1.In the first chaper,the development at home and abroad,purpose and significance of the biomechanical system are firstly described;then,basic definitions and theorems needed in the thesis are briefly explained;finally,the main work of the thesis is introduced.2.In the second chapter,the dynamic behavior of a discrete epidemic model SI system is analyzed.Firstly,conditions of fixed-point stability are obtained according to the roots about the characteristic equations;secondly,conditions of period-doubling bifurcation and Neimark-Sacker bifurcation of fixed points are obtained according to the center manifold theorem and the bifurcation theory;finally,the numerical simulation is applied to verify the correctness of the conclusion.3.In the third chapter,the dynamic behavior of a predator-prey model with double delays is analyzed.Firstly,the existence of Hopf bifurcation at the equilibrium point is determined according to the distribution of the characteristic roots;secondly,the direction and periodic solution of the Hopf bifurcation of the system are analyzed through use of the normative theory of the functional differential equation and the center manifold theorem;finally,the numerical simulation is applied to verify the correctness of the theory.4.In the fourth chapter,the asymptotic stability and Hopf bifurcation of the discrete predator-prey model system with randomicity and time delay are analyzed.Firstly,the stochastic discrete system is transformed into a deterministic one by use of the quadratic polynomial approximation;secondly,the critical value of Hopf bifurcation produced by the stochastic system is obtained according to Hopf bifurcation Theory,and is analyzed combined with the center manifold theorem;finally,the numerical simulation is applied to verify its correctness.