Bifurcations in a delayed fractional model of glucose–insulin interaction with incommensurate orders

Abstract

Abstract This paper proposes a delayed fractional-order model of glucose–insulin interaction in the sense of the Caputo fractional derivative with incommensurate orders. This fractional-order model is developed from the first-order model of glucose–insulin interaction. Firstly, we investigate the non-negativity and the boundedness of solutions of the fractional-order model. Secondly, the stability and the bifurcation of the model are studied by separating the associated characteristic equation of the model into its real and imaginary parts and taking a time delay as the bifurcation parameter. The asymptotic stability and the Hopf bifurcation are discussed via the condition of creation of the bifurcation. Furthermore, it is shown that the onset of the bifurcation is related to the fractional orders of the model. Finally, some numerical simulations of the model using the Adam–Bashforth–Moulton predictor corrector scheme are demonstrated to support our obtained theoretical results.

Keywords

• Hopf bifurcation
• Glucose-Insulin
• Fractional model
• Time delay
• Stability analysis