Diabetes mellitus is one of the most extensive diseases in the world. The mathematical models are prevalent to study the dynamics of glucose, insulin, and β-cells non-invasively. Therefore, to study the impact of β-cells on the glucose-insulin regulatory system, a non-linear three-dimensional mathematical model is proposed. The dynamics of the glucose-insulin regulatory system comprising of boundedness of solutions, existence, and stability condition of equilibria are explored theoretically. Additionally, the conditions for saddle-node, transcritical, and Hopf-bifurcation are also examined. The results illustrate that the glucose-insulin regulatory system offers various dynamics in distinct circumstances. The proposed model is in good agreement with the real-life physical significance of glucose-insulin dynamics. Different types of diabetes conditions such as type 2 diabetes and hyperinsulinemia are also observed through the bifurcation analysis.

糖尿病是世界上范围最广的疾病之一。数学模型普遍用于非侵入性地研究葡萄糖、胰岛素和β细胞的动力学。因此，研究β的影响-细胞对葡萄糖-胰岛素调节系统的影响，提出了非线性三维数学模型。从理论上探讨了由溶液的有界性、平衡的存在性和稳定性条件组成的葡萄糖-胰岛素调节系统的动力学。此外，还检查了鞍形节点、跨临界和 Hopf 分叉的条件。结果表明，葡萄糖-胰岛素调节系统在不同情况下提供各种动态。所提出的模型与葡萄糖 - 胰岛素动力学的现实物理意义非常吻合。通过分叉分析还观察到不同类型的糖尿病状况，例如 2 型糖尿病和高胰岛素血症。