The Study Of Asymptotic Behavior Of Stochastic Delay Biological Model

The evolution of complex biological systems,from microscopic to macroscopic perspectives,can be abstracted by differential equation models.There are some familiar models such as predator-prey model,infectious disease model,and cellular immunity model.Considering that the biological environment in nature is inevitably affected by stochastic factors and that the evolution of some biological systems depends on past states,stochastic functional differential equations(SFDEs)provide a more accurate representation of real systems.In this thesis,we focus on the stochastic Kuznetsov-Taylor model with constant time delay.This is due to the fact that inevitable random disturbances,such as oxygen concentration,temperature,radiation,etc.,affect the mortality of utility cells(ECs)and the multiplication rate of tumor cells(TCs)in tumor tissues,and the time delay is caused by the time it takes for the immune system to recognize non-self cells and generate a response.Therefore,this model is a more complex and accurate representation of the real tumor immune model,and the study of the evolutionary behavior of this model can give some reference to tumor immunotherapy.In this thesis,the existence and uniqueness of the global positive solution of the system are given.Then,by constructing the functional Lyapunov function,the stochastic moment bounded conditions of TCs and ECs at infinite time are given,which are affected by the noise intensity and time delay.Since SFDEs are not Markov,the Khasminskii theorem cannot prove the existence of the stationary distribution of the system.Before proving the existence of stationary distribution,the initial conditions are extended so that there is a unique global non-negative solution for the initial conditions of the system in the space of complete nonnegative continuous functions.Subsequently,we prove the tightness of the Krylov-Bogoliubovc measure sequence of the transfer function of SFDEs,where any weak limit point of this measure sequence is a stationary distribution of the system,thus the condition for the existence of a stationary distribution of the system is given.In this process,it is necessary to ensure that the space is complete,which is also the purpose of the extension of the initial conditions.This thesis also gives the conditions for the persistence and extinction of the tumor immune system under different intensities of random disturbances and time delay.The thesis ended with numerical simulations,giving examples to illustrate the theoretical results.The results show that if the system are subjected to a weak random perturbations,the TCs and ECs can persist.If TCs are subjected to strong random perturbations,the concentration of TCs decreases exponentially and the number of ECs converges to a stationary distribution.The persistence and extinction of the stochastic constant time delay Kuznetsov-Taylor model studied in this thesis are of significance for tumor immunotherapy.This thesis uses functional Lyapunov functions to prove the boundedness of moments and the existence of stationary distribution of the system,which is an innovation in the bio-mathematical theory of stochastic time delay models,and other stochastic time delay biological models can be studied by similar methods.