Dynamics Of The Delayed Glucose-insulin System And Diabetes Mechanism

Diabetes is a disease of high incidence, much complications and high cost of treat-ment, until now the pathogenesis of this disease is not clear.In order to provide effective solutions for prevention and treatment, we need to explore the pathogenesis of this dis-ease. However, methods such as clinical trials are high cost and poor result.Thus we want to study the pathogenesis by building glucose-insulin model. But the existing glucose-insulin model doesn’t consider the role of renal metabolism with hyperglycemia. In order to be closer to the reality, we consider the actual impact of hyperglycemia. In this dissertation, we discuss the effects of reaction delay and physiological delay on the dynamics of the glucose-insulin system and diabetes mechanism. Although the method of center manifold reduction and normal form has been widely applied to non-resonant double Hopf bifurcation of the time-delayed systems, this method is very abstract and difficult. So we try to seek a simple method. We first apply the method of multiple scales to the non-resonant double Hopf bifurcation for a generic system with two delays, we get the normal forms. Then we introduce the delayed glucose-insulin system, got the existence and bounded-ness of the solutions of the system, which shows the feasi-bility of the model:we also proved the existence and uniqueness of the equilibrium and the conditions for local stability and un-stability are obtained. Then we got its normal form by employing Center Manifold Reduction and normal form method and Method of Multiple Scales respectively, unfolded and classified the dynamical behavior near the non-resonant double Hopf bifurcation points. The results show the universality of the employed method of multiple scales to the study of the non-resonant double Hopf b-ifurcation in a dynamical system with two delays, and the method is simple, specific. Finally, we numerically simulated them by employing WinPP and verified the theoretical analysis. This dissertation can get two important conclusions:When the body takes in more glucose, it will result in a increase in the equilibrium coordinates of the system. The equilibrium corresponds to steady-state of the blood glucose. When steady-state is too high, it maybe leads to diabetes and some complications, such as blindness. So diabetic should pay more attention to the intake of glucose. Even if the physiological delay fluctuates in a small range, it will cause significant fluctuations in blood glucose, when the reaction delay is large enough. The large fluctuations in blood glucose is an important feature of diabetes, this may be the other cause of diabetes.