Analysis of a Slow-Fast System for Glucose-Insulin Regulatory with β Cell Function

A kind of mathematical model of the glucose-insulin regulatory system with the mass of β cells as variables is proposed. Firstly, the stability of the positive equilibrium for the fast subsystem is discussed, and then the full system behaviors are studied, including the existence of the system equilibrium, the stability of the boundary equilibrium, the local stability of the positive equilibrium and the Hopf bifurcation phenomenon. The effects of β cell mass on the regulatory system are further discussed physiologically.