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Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling

Published online by Cambridge University Press:  12 December 2012

M. Adimy  and
F. Crauste
M. Adimy
Affiliation:
INRIA Team Dracula, INRIA Grenoble Rhône-Alpes Center, France Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France
F. Crauste*
Affiliation:
INRIA Team Dracula, INRIA Grenoble Rhône-Alpes Center, France Université de Lyon, Université Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France
*
⋆ Corresponding author. E-mail: crauste@math.univ-lyon1.fr
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Abstract

We illustrate the appearance of oscillating solutions in delay differential equations modeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising as consequences of a destabilization of the system, for instance through a Hopf bifurcation. Models of hematopoietic stem cell dynamics are considered for their abilities to describe periodic hematological diseases, such as chronic myelogenous leukemia and cyclical neutropenia. After a review of delay models exhibiting oscillations, we focus on three examples, describing different delays: a discrete delay, a continuous distributed delay, and a state-dependent delay. In each case, we show how the system can have oscillating solutions, and we characterize these solutions in terms of periods and amplitudes.

Keywords

oscillationsdelay differential equationsHopf bifurcationhematological diseases
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 6: Biological oscillations , 2012 , pp. 1 - 22
Copyright
© EDP Sciences, 2012

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