Nonlinear Modeling and Stability Analysis of Glucose–Insulin Homeostasis

Abstract

This study develops a nonlinear dynamic model for glucose-insulin regulation that includes glucose utilization, saturated insulin secretion, insulin clearance, and an auxiliary remote-effect variable capturing the delayed regulatory response. The proposed framework enables comprehensive investigation of glucose homeostasis from a dynamical systems perspective, emphasizing equilibrium structure, stability properties, and parameter-induced transitions. The positivity and bounds of the solution were established  ensuring the physical acceptability . Analytical characterization of the equilibrium points reveals that the steady-state   of glucose concentration satisfies a non-linear algebraic relationship, allowing parameter-dependent shifts in the metabolic regime. Local and global stability were investigated by Jacobian matrix analysis and the Lyapunov-cascade approach, showing that the system admits a unique globally asymptotically stable equilibrium under physically relevant conditions. Numerical simulations confirm the analytical results and show that variations in physiological parameters including external glucose input, insulin clearance rate, insulin secretion gain, and net insulin effectiveness consistently displace the equilibrium from normoglycemic to pathological hyperglycemic states without inducing oscillatory dynamics. It is important to mention that the transition to type 2 diabetes emerges as spontaneous equilibrium shifts and weakening of stability rather than bifurcation-driven oscillations as an mathematical simplification aiming to focus on the stability and equilibrium behavior of the system.These findings give a mathematically rigorous and physiologically interpretable explanation of metabolic flexibility and its gradual breakdown, providing a unifying framework for understanding glucose homeostasis and its impairment under adverse conditions.