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Delay Differential Equations and Autonomous Oscillations inHematopoietic 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 equationsmodeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising asconsequences of a destabilization of the system, for instance through a Hopf bifurcation.Models of hematopoietic stem cell dynamics are considered for their abilities to describeperiodic hematological diseases, such as chronic myelogenous leukemia and cyclicalneutropenia. After a review of delay models exhibiting oscillations, we focus on threeexamples, 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 oscillatingsolutions, and we characterize these solutions in terms of periods and amplitudes.

Keywords

oscillationsdelay differential equationsHopf bifurcationhematological diseases

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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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