Stability, Hopf Bifurcation, and Numerical Simulations of a Fractional-Order Glucose–Insulin System With Two Physiological Delays

This study develops and analyzes a fractional-order glucose–insulin interaction model incorporating two physiologically distinct time delays associated with the intravenous glucose tolerance test (IVGTT). Using Caputo fractional derivatives with orders α,β∈0,1, the proposed framework embeds metabolic memory and long-range temporal dependence into glucose–insulin regulation. We rigorously establish fundamental qualitative properties including existence, uniqueness, positivity, boundedness, and uniform persistence of solutions. The principal mathematical findings are as follows: (i) fractional-order dynamics increase critical delay thresholds by 12%–34% compared to integer-order formulations, with the effect scaling as 1/1−α for strong memory; (ii) insulin-related memory effects β produce significantly stronger stabilization than glucose-related memory effects α, with asymmetric configurations (0.95, 0.8) yielding 27.6% delay increases versus only 5.7% for (0.8, 0.95); (iii) equal-delay configurations generate synergistic memory effects, producing 16.8% increases compared to 12.0% for single-delay cases; (iv) supercritical Hopf bifurcations generate stable periodic oscillations with amplitude scaling τ−τ∗, quantitatively characterizing the transition from glucose homeostasis to abnormal metabolic oscillations; and (v) global asymptotic stability is established via fractional Krasovskii functionals with explicit Mittag-Leffler decay rates Vt≤V0Eα−λtα. Local stability analysis employs characteristic equations and linear matrix inequalities, yielding explicit conditions for asymptotic stability. The existence of Hopf bifurcations and periodic solutions is rigorously proven using a fixed-point framework based on Schauder’s theorem and the implicit function theorem, with geometric convergence of the iterative scheme. Numerical simulations using the fractional Adams–Bashforth–Moulton predictor–corrector scheme confirm all theoretical predictions with maximum relative error