Math. Model. Nat. Phenom.
Volume 7, Number 6, 2012Biological oscillations
|Page(s)||1 - 22|
|Published online||12 December 2012|
Delay Differential Equations and Autonomous Oscillations in Hematopoietic Stem Cell Dynamics Modeling
1 INRIA Team Dracula,
INRIA Grenoble Rhône-Alpes Center, France
2 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@example.com
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.
Mathematics Subject Classification: 34A34 / 34C25 / 92C37
Key words: oscillations / delay differential equations / Hopf bifurcation / hematological diseases
© EDP Sciences, 2012
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