Application Of Bifurcation Theory On Tumor Growth Model

In this paper,we apply center manifolds theory and local bifurcation theory to study a four dimensional Tumor Growth Model.First of all,we theoretically analyze the existence,type and stability of equilibrium points with the parameter changing.Secondly,the conditions of occurrence of static bifurcation or Hopf bifurcation of equilibrium points are discussed,as well as the type and stability of periodic orbits generated by Hopf bifurcation.At last,numerical simulations are done to analyze the dynamics of the model by observing the graph of equilibrium points,the phase diagrams and the bifurcation diagrams.The aim of our research is to find the factor which has great influence on tumor growth.The whole thesis is composed of four parts.The first chapter concentrates on introducing the application of bifurcation theory in various industries,the development of tumor growth model and the outcomes from former scholars.Related theories are also mentioned briefly.In the second chapter,we introduce a four dimensional tumor growth model and study the model with the parameter α₃₄ which means the growth rate of tumor cells stimulated by endothelial cells.Firstly,we compute the existence,type and stability of equilibrium points.Secondly,the system which is restricted on the local center manifold in the neighbourhood of non-hyperbolic equilibrium point is calculated in detail.Thirdly,we determine the type and stability of Hopf bifurcation occured at the equilibrium point.Lastly,the phase diagram of the periodic orbit produced by Hopf bifurcation,the time series graph and eigenvalue curves are drawn from simulations.It turns out that α₃₄ has little influence on dynamics of tumor growth model.In the third chapter,we continue our research on the same model studied in chapter 2,but we select the other parameter p3 which means the growth rate of tumor cells.Similar computations are done at first,but there is a difference that the system has some Transcritical bifurcations occured at equilibrium points.In addition,Equilibrium points graph and bifurcation diagrams are drawn to help the study.Finally,we find period-dobling bifurcation,the chaotic attractor and the coexistence of two steady states.It turns out that p3 has great influence on the tumor growth model and clinical treatment.The last chapter is a summary of the whole thesis.It not only points out my deficiencies,but also determines the direction and content of my future research.