Stability Analysis Of Circadian System Models With Delay

The theory of delay differential equations is applied in many fields, such as Mechanics, Physics, Life Science, Auto-control, Economics, Medical Science and so on. The last years, many new developments have occurred with the cross-subject appearance. The research of delay differential equation is very important in theory and in practice.The bifurcation phenomena can occur in the parameter dependent systems. When the parameters are varied, changes may occur in the qualitative structure of the solutions for certain parameter values. These changes are called bifurcation and the parameter values are called bifurcation values. The type of bifurcation that connects equilibrium state with periodic solution is called Hopf bifurcation. The study of bifurcation is a attractive branch in applied mathematics, especially in some realistic models such as ecological models, physiological models, and neural network models with delays.The bifurcation theories of delay differential equations are applied to analyze the stability of two kinds of circadian system models with delays, which are respectively described in the third and fourth chapter.In the third chapter, a kind of delay model for clock protein that inhibits its own expression through a feedback mechanism is discussed. The period oscillatory phenomenon existed in circadian system is explained from biological point of view. By studying the linear stability of the model, it is found that there exist Hopf bifurcations when the delay passes a sequence of critical values. Then the explicit algorithm is derived for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions by using the normal form method and center manifold theorem. And simulation examples are given to demonstrate the conclusion finally.The environmental light-dark cycle acts as one of the most potent influencing agents of circadian system. In the fourth chapter, the effects of light on the stability of the circadian system model with delay are studied. Firstly, the stability of the circadian system model without light is analyzed. Then the delay is used in circadian model. Using the delay as a argument, the distribution of the roots of the characteristic equation associated with the model under light is analyzed. Then the stable and Hopf bifurcation condition of the circadian model with delay is obtained. Thus it can be seen how the dynamical traits of the circadian system are influenced by light. Finally, some simulation examples of the model illustrate this conclusion with mathematical software Matlab.