Global Stability and Periodicity in a Glucose-Insulin Regulation Model with a Single Delay

TL;DR

This paper analyzes a delayed glucose-insulin model, establishing conditions for stability and demonstrating the existence of oscillations and multiple solutions through mathematical proofs and numerical simulations.

Contribution

It provides new criteria for global stability and shows the coexistence of stable equilibria and periodic solutions in a delayed biological system.

Findings

Conditions for global asymptotic stability of equilibrium.

Existence of slowly oscillating periodic solutions when unstable.

Numerical simulations support theoretical results.

Abstract

A two-dimensional system of differential equations with delay modelling the glucose-insulin interaction processes in the human body is considered. Sufficient conditions are derived for the unique positive equilibrium in the system to be globally asymptotically stable. They are given in terms of the global attractivity of the fixed point in a limiting interval map. The existence of slowly oscillating periodic solutions is shown in the case when the equilibrium is unstable. The mathematical results are supported by extensive numerical simulations. It is shown that typical behaviour in the system is the convergence to either a stable periodic solution or to the unique stable equilibrium. The coexistence of several periodic solutions together with the stable equilibrium is demonstrated as a possibility.