Tumor Growth Estimation And Hopf Bifurcation Of Tumor-Lymphatic Model

There exits the time delay in course of killing tumor cells by immune cells.There exits the time delay in course of the tumor growth.The effects of tumor growth delay and the transition delay of mature lymphocyte on the dynamic behavior of tumor and lymphocyte growth are investigated.Firstly,the mechanisms of tumor growth are collected.According to the tumor growth characteristics,tumor name are given.The current research hotspots of tumor growth are obtained by co-word analysis method,which shows the internal relationship of various disciplines for tumor growth.The growth data of breast tumors is extracted from the published literature.Based on the biological mechanism of the exponential growth and the logistic growth,the tumor growth coefficient and the maximum volume of tumor growth are estimated by the least square method.The obtained estimated parameters are substituted into the tumor growth model to perform data curve fitting.Secondly,take the time delay in the transition of immature lymphocytes to mature lymphocytes as a parameter to analyze the local stability about the equilibrium of the model by using characteristic root distribution of the linear equation at the equilibrium.We explore the Hopf bifurcation conditions of the tumor-lymphatic model at the equilibrium.The properties of Hopf bifurcation are obtained by the differential equation bifurcation theorem.Take the tumor growth delay as a parameter,the local stability about the equilibrium point of the tumor-lymphatic model and the conditions of Hopf bifurcation are analyzed.The Matlab numerical simulations shows the regular oscillation.Finally,we investigate the local stability of equilibriums,as well as,Hopf bifurcation of the tumor--lymphatic model with the tumor growth delay and the transition delay of mature lymphocytes.In addition,the periodic solutions properties of Hopf bifurcation are derived by central manifold and norm form theory.Matlab numerical simulations show the regular and irregular oscillations.The results provide a theoretical guidance for exploring the inhibit tumor growth methods by using the delay factors of tumor cell growth and lymphocyte transformation.