Qualitative and Quantitative Features of Delay Differential Equations with Biological Systems

This paper proposes a delay differential model for glucose-insulin endocrine and metabolic regulation, incorporating beta-cell dynamics to maintain bloodstream insulin concentration. Two-time delays are involved in the model, which represent delayed insulin secretion and glucose reduction. Moderate hyperglycemia causes beta-cell growth (negative feedback), while severe hyperglycemia causes beta-cell reduction (positive feedback). A Hopf-bifurcation occurs when a time delay passes a bifurcation point. Based on biological findings, the model exhibits periodic oscillations. Numerical simulations have validated the theoretical results.