A comprehensive investigation of fractional glucose-insulin dynamics: existence, stability, and numerical comparisons using residual power series and generalized Runge-Kutta methods

The regulation of blood glucose levels involves complex interactions between glucose, insulin, and beta cells, where disruptions may lead to diabetes. Traditional integer-order models fail to capture the memory-dependent and hereditary properties of biological systems. This study develops a fractional-order glucose-insulin-beta cell model and investigates its dynamics using the Residual Power Series Method (RPSM) and the Generalized Runge-Kutta Method (GRKM). Theoretical analyses establish the model's existence, uniqueness, and boundedness of solutions, ensuring biological validity. Stability and bifurcation analyses reveal the critical role of fractional orders in shaping system dynamics. RPSM demonstrates computational efficiency with rapid convergence, while GRKM excels in stability and bifurcation studies. Comparative numerical simulations highlight the complementary strengths of these methods, providing a robust framework for predictive modeling in diabetes management. This work underscores the potential of fractional-order models to advance understanding and control of metabolic disorders.