  Mathematical models of glucose, insulin, and pancreatic $\beta$-cell mass
dynamics are essential for understanding the physiological basis of type 2
diabetes. This paper investigates the Topp model's discrete-time dynamics to
represent these interactions. We perform a comprehensive analysis of the
system's trajectory, examining both local and global behavior. First, we
establish the invariance of the positive trajectory and analyze the existence
of fixed points. Then, we conduct a complete stability analysis, determining
the local and global asymptotic stability of these fixed points. Finally,
numerical examples validate the effectiveness and applicability of our
theoretical findings. Additionally, we provide biological interpretations of
our results.
