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

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...

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Bibliographic Details
Published in:Mathematical modelling of natural phenomena 2012-01, Vol.7 (6), p.1-22
Main Authors: Adimy, M., Crauste, F.
Format: Article
Language:English
Subjects:
34A34
34C25
92C37
Cancer
Cellular Biology
delay differential equations
Differential equations
Dynamical Systems
hematological diseases
Hopf bifurcation
Life Sciences
Mathematical models
Mathematics
oscillations
Stem cells
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Summary: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.
ISSN:0973-5348
1760-6101
DOI:10.1051/mmnp/20127601