Stability Analysis Of A Tumor-immune System And A Population Dynamic Model With Time Delay

Delay effect and spatial migration phenomenon widespreadly exist in nature,such as resource regeneration and feedback delay in biological population models and immune response delay in tumor immune models.The delay differential equation is a kind of important mathematical model,which reflects the effect of time lag on the state of the system.In many practical control systems,the existence of time delay may reduce the performance of the system and even lead to the essential change of dynamic properties of the system.The time delay can not be easily ignored.In addition,the time delay and spatial diffusion usually exist simultaneously in nature.It is of great importance to consider these factors in the mathematical model.In order to study the role of time delay in tumor immune dynamics and population dynamics,two kinds of mathematical models are established to discuss the dynamical behavior of the two situation.In the second chapter,a three dimensional dynamic model with time delay is established to describe the interaction among virus,tumor and immune cells.The stimulation of tumor cells with virus infection requires a certain response time before the immune response.This process can be more accurately described by a time delay term in the corresponding differential models.In the first section,we gave a tumor immune cell dynamics model with time delay and virus infection first.Secondly,we analyzed the stability of every equilibrium point of the system without delay,and verified it by numerical simulation.The stability condition of the positive equilibrium point is gained by employing Routh Hurwitz stability criterion and the existence of Hopf bifurcation are assessed by applying the transversal condition.Finally,the conclusion is verified by numerical simulation.In the biological population model,the digestion process of predators will be delayed for a certain time.In the third chapter,a predator-prey reaction diffusion model with digestive delay is established.In this paper,we are focused on the stability of co-existence equilibria of the system,and the conditions for the Turing instability of the model with time delay are obtained.A large number of numerical simulation results are used to verify the effect of time delay on the pattern dynamics of the reaction diffusion predator-prey model.The results show that the common effect of the time delay and diffusion may enrich the results of the patterns generated by the reaction diffusion predator-prey model.Finally,we give some conclusions and discussions.