Stability And Bifurcation Analysis Of Tumor-immune Models With Diffusion Mechanisms

Based on the stochastic effect,diffusivity and time delay in the interaction between tumor cells and immune cells,the stochastic model of tumor immunity,tumor immune diffusion model and tumor immune diffusion model with time delay are established respectively.Using stochastic differential equation theory and functional differential equation theory,the dynamic properties of the three types of tumor immune models are analyzed to help understand the tumor growth mechanism from a mathematical perspective and thus provide new ideas for the effective containment of tumor growth.The specific contents are as follows:Firstly,the tumor immunity of cytokine assisted effector cells under the influence of random environmental noise during the process of anti-tumor immunity is studied.By using Ito formula and stochastic differential equation theory,the dynamic properties of tumor immune stochastic effect model are further analyzed,and sufficient conditions for tumor cell elimination and tumor cell persistence are obtained.According to the theoretical results obtained,and use Matlab numerical simulation to verify the state of tumor elimination and tumor persistence in the stochastic model of tumor immunity,and get some theoretical guidance and suggestions on tumor elimination and tumor growth inhibition.Secondly,a tumor immune reaction diffusion model with homogeneous Neumann boundary conditions is established to characterize the effect of the diffusion reaction on the interaction between tumor cells and immune cells during the anti-tumor process of dendritic cell helper T cells.Taking T cell inactivation rate and tumor cell elimination rate as parameters,the maximum parameter regions for the stability of the coexistence equilibrium solution are given.By analyzing the distribution of the characteristic roots of the linear model at the positive steady-state solution of the tumor and using the functional differential equation theory,the necessary and sufficient conditions for the positive steady-state solution of the tumor to undergo Turing instability,the conditions for the existence of the Wave bifurcation and the Hopf bifurcation are obtained.Using Matlab numerical simulation to verify the theorem conditions,the spatial and temporal diversification,such as periodic occurrence,occurred in the process of tumor immune diffusion was explained.Finally,according of the tumor immune diffusion model,and the characteristics of immune delay in the interaction between tumor and immune cells,a tumor immune diffusion model with time delay with homogeneous Neumann boundary condition is established to study the stability and bifurcation phenomenon of the model.The sufficient and necessary conditions for uniform instability and Hopf bifurcation conditions are obtained.The theoretical results reveal the common influence of diffusion and time delay on the model.Matlab simulation shows the model parameters in the event of a tumor and immune cells Turing instability and experience Hopf bifurcation near the critical value of time and space pattern of periodic solutions of a very few,and explaines the diffusion and time delay factors model dynamic behavior of the biomedical significance,provides the theoretical reference basis for the inhibition of tumor growth.