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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: [email protected]
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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
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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