Geodesic
Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch
F. Crauste1
1 Université de Lyon, Université Lyon1, CNRS UMR 5208 Institut Camille Jordan - F - 69200 Villeurbanne Cedex, France
Mathematical modelling of natural phenomena, Tome 4 (2009) no. 2, pp. 28-47.

Voir la notice de l'article provenant de la source EDP Sciences

A nonlinear system of two delay differential equations is proposed to model hematopoietic stem cell dynamics. Each equation describes the evolution of a sub-population, either proliferating or nonproliferating. The nonlinearity accounting for introduction of nonproliferating cells in the proliferating phase is assumed to depend upon the total number of cells. Existence and stability of steady states are investigated. A Lyapunov functional is built to obtain the global asymptotic stability of the trivial steady state. The study of eigenvalues of a second degree exponential polynomial characteristic equation allows to conclude to the existence of stability switches for the unique positive steady state. A numerical analysis of the role of each parameter on the appearance of stability switches completes this analysis.
DOI : 10.1051/mmnp/20094202

F. Crauste 1

1 Université de Lyon, Université Lyon1, CNRS UMR 5208 Institut Camille Jordan - F - 69200 Villeurbanne Cedex, France 
@article{MMNP_2009_4_2_a1,
     author = {F. Crauste},
     title = {Delay {Model} of {Hematopoietic} {Stem} {Cell} {Dynamics:} {Asymptotic} {Stability} and {Stability} {Switch}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {28--47},
     publisher = {mathdoc},
     volume = {4},
     number = {2},
     year = {2009},
     doi = {10.1051/mmnp/20094202},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20094202/}
}
TY  - JOUR
AU  - F. Crauste
TI  - Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch
JO  - Mathematical modelling of natural phenomena
PY  - 2009
SP  - 28
EP  - 47
VL  - 4
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20094202/
DO  - 10.1051/mmnp/20094202
LA  - en
ID  - MMNP_2009_4_2_a1
ER  - 
%0 Journal Article
%A F. Crauste
%T Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch
%J Mathematical modelling of natural phenomena
%D 2009
%P 28-47
%V 4
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20094202/
%R 10.1051/mmnp/20094202
%G en
%F MMNP_2009_4_2_a1
F. Crauste. Delay Model of Hematopoietic Stem Cell Dynamics: Asymptotic Stability and Stability Switch. Mathematical modelling of natural phenomena, Tome 4 (2009) no. 2, pp. 28-47. doi : 10.1051/mmnp/20094202. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20094202/

[1] M. Adimy, F. Crauste Nonlinear Analysis 2003 1469 1491

[2] M. Adimy, F. Crauste Discret. Cont. Dyn. Sys. Ser. B 2007 19 38

[3] M. Adimy, F. Crauste, L. Pujo-Menjouet Discret. Cont. Dyn. Sys. Ser. A 2005 501 522

[4] M. Adimy, F. Crauste, S. Ruan Nonlinear Analysis: Real World Applications 2005 651 670

[5] M. Adimy, F. Crauste, S. Ruan J. Theo. Biol. 2006 288 299

[6] J. Bélair, M.C. Mackey, J.M. Mahaffy Math. Biosci. 1995 317 346

[7] E. Beretta, Y. Kuang SIAM J. Math. Anal. 2002 1144 1165

[8] S. Bernard, J. Bélair, M.C. Mackey J. Theor. Biol. 2003 283 298

[9] F.J. Burns, I.F. Tannock Cell Tissue Kinet. 1970 321 334

[10] C. Colijn, C. Foley, M.C. Mackey Exper. Hematol. 2007 898 907

[11] F. Crauste Math. Bio. Eng. 2006 325 346

[12] P. Fortin, M.C. Mackey Brit. J. Haematol. 1999 336 345

[13] J. Hale, S.M. Verduyn Lunel. Introduction to functional differential equations. Applied Mathematical Sciences 99. Springer-Verlag, New York, 1993.

[14] C. Haurie, D.C. Dale, M.C. Mackey Blood 1998 2629 2640

[15] L.G. Lajtha. On DNA labeling in the study of the dynamics of bone marrow cell populations, in: Stohlman, Jr., F. (Ed), The Kinetics of Cellular Proliferation, Grune and Stratton, New York (1959) 173–182.

[16] M.C. Mackey Blood 1978 941 956

[17] M.C. Mackey, R. Rudnicki J. Math. Biol. 1994 89 109

[18] M.C. Mackey, R. Rudnicki J. Math. Biol. 1999 195 219

[19] J.M. Mahaffy, J. Bélair, M.C. Mackey J. Theor. Biol. 1998 135 146

[20] L. Pujo-Menjouet, S. Bernard, M.C. Mackey SIAM J. Appl. Dyn. Systems 2005 312 332

[21] L. Pujo-Menjouet, M.C. Mackey C. R. Biologies 2004 235 244

[22] L.F. Shampine, S. Thompson Appl. Numer. Math. 2001 441 458

Cité par Sources :