Stability Analysis Of A Kind Of Biological Immune System With Time Delay

The development trend or future of various systems does not only relate to present, but also relate to past more or less, which is named time delay phenomenon or genctic effect in natural science and social phenomenon research. Mathematical models of engineering, physics, mechanics, cybernetics, chemical reaction and biomedicine all have obvious delayed variables. In recent years, the time-delayed dynamical system has come to be the important objects of study in a variety of fields, such as medicine. Marchuk's model is a class of particular immune system with time delay. Marchuk's model of an immune system is one of the pioneering models in mathematical immunology. It was proposed by Marchuk in 1980. It is presented in the language of humoral immune reaction.Many researchers have made an exhaustive study of this model. (see e.g. [2, 3, 4, 5, 6, 7, 8, 9, 10,11, 12,47]). For example, in reference [12], the author studies the stability of system equilibrium with the method of latent root and the periodic solution of the Marchuk's model with the method which provided by reference [13]. In this paper, we still regard time lagr as a parameter and deeply study the properties of the solution in Marchuk's model of immune system based on different method. Firstly, we adopt Lyapunov function method, latent root method and analytical method to study the stability of equilibrium point in the system and the according area of existence. We find out there is a change in the stability of system equilibrium when time delay t experiences certain variation, i.e., when it changes from asymptotical stability (unstability) to unstability (stability), these r become the system's bifurcation value (the system has small-scale amplitude nontrivial periodic solution around these values), and then we obtain the bound of stability of equilibrium. Secondly, we make using of the method of HKW which based on the center manifold theorem and normal form theory, and list two calculation formulas which reflect the stability with Hopf bifurcating periodic solution, and bifurcating direction. Thus it provide a basis for numerical simulation calculation. Finally, the experiment result shows our conclusion with mathematical software matlab6.5.