Global extinction, dissipativity and persistence for a certain class of differential equations with state-dependent delay

TL;DR

This paper analyzes delay differential equations with state-dependent delays, focusing on their stability, persistence, and boundedness, motivated by a biological model of stem cell maturation.

Contribution

It introduces a novel analysis of a specific SD-DDE model for stem cell maturation, establishing conditions for stability, persistence, and dissipativity.

Findings

Global asymptotic stability of zero equilibrium

Persistence of solutions under certain conditions

Dissipativity and ultimate boundedness proved

Abstract

In this paper we study, at different levels of generality, certain systems of delay differential equations (DDE). One focus and motivation is a system with state-dependent delay (SD-DDE) that has been formulated to describe the maturation of stem cells. We refer to this system as the cell SD-DDE. In the cell SD-DDE, the delay is implicitly defined by a threshold condition. The latter is specified by the time at which the (also implicitly defined) solution of an external nonlinear ordinary differential equation (ODE), which is parametrised by a component of the SD-DDE, meets a given threshold value. We focus on the dynamical properties global asymptotic stability (GAS) of the zero equilibrium, persistence and dissipativity/ultimate boundedness.