Dynamics of a Mathematical Hematopoietic Stem-Cell Population Model

TL;DR

This paper investigates a delay differential equation model of human hematopoietic stem cell populations, revealing complex dynamics including stable periodic orbits, quasi-periodic behavior, and chaos, which may explain instability in blood diseases.

Contribution

It provides a detailed bifurcation analysis of a delay differential equation model, uncovering mechanisms like canard explosions and exotic dynamics relevant to blood cell regulation.

Findings

Stable periodic dynamics with periods from one week to ten years.

Identification of canard explosions near slow manifolds.

Presence of quasi-periodic and chaotic solutions in the model.

Abstract

We explore the bifurcations and dynamics of a scalar differential equation with a single constant delay which models the population of human hematopoietic stem cells in the bone marrow. One parameter continuation reveals that with a delay of just a few days, stable periodic dynamics can be generated of all periods from about one week up to one decade! The long period orbits seem to be generated by several mechanisms, one of which is a canard explosion, for which we approximate the dynamics near the slow manifold. Two-parameter continuation reveals parameter regions with even more exotic dynamics including quasi-periodic and phase-locked tori, and chaotic solutions. The panoply of dynamics that we find in the model demonstrates that instability in the stem cell dynamics could be sufficient to generate the rich behaviour seen in dynamic hematological diseases.