Global Bifurcation Analysis Of Several Cancer Models

As we all know,cancer has always been one of the major diseases threatening life and health,and its intrinsic pathogenesis is an important research topic that many scientists focus on.In the field of biomedicine,more experiments are used to discover potential mechanism and evolutionary processes that may exist.Studying generation,development,and elimination of canmcer from the perspective of mathematical modeling is a very important research field.By analyzing the different dynamical behaviors in the established models,we can discover the effect of different parameter on the dynamical behavior of the full system,and do benefit to strengthen active prevention,accurate diagnosis,and treatment of cancer.In recent years,scientists have attempted to accurately describe the interrelationships between tumor cells,host cells,and immune cells through modeling.However,the Allee effect of biological models also plays an important role in the growth process of tumors.It is worth noting that in the process of cancer treatment,it is necessary to introduce external therapies to control cancer conditions.However,the role of external therapy has not yet been fully revealed.In this thesis,we will make full use of bifurcation theory of dynamical systems and numerical simulations to study various dynamics related to cancer,striving to perfectly reveal existing medical phenomena,and attempting to predict the future development trend of the disease.The full thesis is divided into four chapters: the first chapter is an introduction,which first describes the research background and current situation at home and abroad,briefly introduces the main research content,and gives the basic definitions and theorems involved in this thesis.In Chapter 2,we use the dynamical system approach to study a cancer growth model with Allee effect involving tumor cells,healthy host cells,and effector immune cell populations,and obtain the existence and type of equilibrium points,Hopf bifurcation,chaotic dynamics,and Lyapunov exponent and Lyapunov dimension.It is found that there are saddle node bifurcation of limit cycle,snake bifurcation of limit cycle,homoclinic cycle,and three-point heteroclinic cycle in this model,it is shown that the cancer model has complex dynamical behavior.It is noteworthy that we first discovered a snake bifurcation curve of limit cycle,which indicates that the codimension of the Hopf bifurcation is ∞.When the Allee effect parameter is taken as the main bifurcation parameter,it reveals the role of the Allee effect.All these theoretical and numerical results will help to understand the rich dynamical behavior in cancer models.However,due to the complexity of the dynamical behavior of cancer model,we only provide one-parameter bifurcation diagram,and cannot provide multiple parameters bifurcation diagram.More phenomena deserve further research.In Chapter 3,we mainly study the dynamics of a two-dimensional cancer model involving tumor cells and immune effector cells.We obtain the number and type of equilibrium points,the Hopf bifurcation of codimension 2 and the Bogdanov-Takens bifurcation of codimension 2.Finally,the conclusions obtained are verified by numerical simulations.In Chapter 4,we comprehensively summarize the research results and look forward to the future research direction.