Geodesic
Existence and Stability of Limit Cycles in a Two-delays Model of Hematopoiesis Including Asymmetric Division
A. Halanay1 ; D. Cândea1 ; I. R. Rădulescu1
1 “POLITEHNICA” University of Bucharest, Department of Mathematics and Informatics, Splaiul Independentei 313 RO-060042 Bucharest, Romania
Mathematical modelling of natural phenomena, Tome 9 (2014) no. 1, pp. 58-78

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

A two dimensional two-delays differential system modeling the dynamics of stem-like cells and white-blood cells in Chronic Myelogenous Leukemia is considered. All three types of stem cell division (asymmetric division, symmetric renewal and symmetric differentiation) are present in the model. Stability of equilibria is investigated and emergence of periodic solutions of limit cycle type, as a result of a Hopf bifurcation, is eventually shown. The stability of these limit cycles is studied using the first Lyapunov coefficient.
DOI : 10.1051/mmnp/20149105

A. Halanay 1 ; D. Cândea 1 ; I. R. Rădulescu 1

1 “POLITEHNICA” University of Bucharest, Department of Mathematics and Informatics, Splaiul Independentei 313 RO-060042 Bucharest, Romania 
@article{10_1051_mmnp_20149105,
     author = {A. Halanay and D. C\^andea and I. R. R\u{a}dulescu},
     title = {Existence and {Stability} of {Limit} {Cycles} in a {Two-delays} {Model} of {Hematopoiesis} {Including} {Asymmetric} {Division}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {58--78},
     publisher = {mathdoc},
     volume = {9},
     number = {1},
     year = {2014},
     doi = {10.1051/mmnp/20149105},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149105/}
}
TY  - JOUR
AU  - A. Halanay
AU  - D. Cândea
AU  - I. R. Rădulescu
TI  - Existence and Stability of Limit Cycles in a Two-delays Model of Hematopoiesis Including Asymmetric Division
JO  - Mathematical modelling of natural phenomena
PY  - 2014
SP  - 58
EP  - 78
VL  - 9
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149105/
DO  - 10.1051/mmnp/20149105
LA  - en
ID  - 10_1051_mmnp_20149105
ER  - 
%0 Journal Article
%A A. Halanay
%A D. Cândea
%A I. R. Rădulescu
%T Existence and Stability of Limit Cycles in a Two-delays Model of Hematopoiesis Including Asymmetric Division
%J Mathematical modelling of natural phenomena
%D 2014
%P 58-78
%V 9
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20149105/
%R 10.1051/mmnp/20149105
%G en
%F 10_1051_mmnp_20149105
A. Halanay; D. Cândea; I. R. Rădulescu. Existence and Stability of Limit Cycles in a Two-delays Model of Hematopoiesis Including Asymmetric Division. Mathematical modelling of natural phenomena, Tome 9 (2014) no. 1, pp. 58-78. doi: 10.1051/mmnp/20149105

[1] M. Adimy J. of Math. Anal. and Appl. 1993 125 134

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

[3] M. Adimy, F. Crauste, S. Ruan SIAM Journal on Applied Mathematics 2005 1328 1352

[4] M. Adimy, F. Crauste, S. Ruan J. Theor. Biol. 2006 288 299

[5] M. Adimy, F. Crauste, A. Halanay, M. Neamtu, D. Opris Chaos, Solitons and Fractals 2006 1091 1107

[6] M. Adimy, F. Crauste Math. Model. Nat. Phenom. 2012 1 22

[7] J. Belair, M.C. Mackey Annals of the New York Academy of Sciences 1987 280 282

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

[9] S. Bernard, J. Belair, M.C. Mackey J. Theor. Biology 2003 283 298

[10] N. Chafee J. Math. Anal. and Appl. 1991 312 348

[11] C. Colijn, A.C. Fowler, M.C. Mackey Journal of mathematical biology 2006 499 519

[12] C. Colijn, M.C. Mackey J. Theor. Biology 2005 117 132

[13] K. Cooke, Z. Grossman J. Math. Anal. Appl. 1982 592 627

[14] K. Cooke, P. Van Den Driessche Funkcialaj Ekvacioj 1986 77 90

[15] M.W.N. Deininger, J.M. Goldman, J.V. Melo Blood 2000 3343 3356

[16] I. Drobnjak, A.C. Fowler Bulletin of mathematical biology 2011 2983 3007

[17] L.E. El’sgol’ts, S.B. Norkin. Introduction to the theory of differential equations with deviating arguments. (in Russian). Nauka, Moscow, 1971.

[18] K. Engelborghs, T. Luzyanina, D. Roose ACM Transactions on Mathematical Software (TOMS) 2002 1 21

[19] A. Friedman Math. Model. Nat. Phenom. 2012 3 28

[20] J.M. Goldman, J.V. Melo New England Journal of Medicine 2003 1451 1464

[21] J. Hale. Introduction to Functional Differential Equations. Springer, New York, 1977.

[22] J. Hale, S.M. Verduyn Lunel. Theory of Functional Differential Equations, Springer, New York, 1993.

[23] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan. Theory and Applications of Hopf Bifurcation. London Mathem. Soc. Lecture Note Series 41, Cambridge University Press, 1981.

[24] C. Haurie, D.C. Dale, R. Rudnicki, M.C. Mackey Journal of theoretical biology 2000 505 519

[25] N.L. Komarova, D. Wodarz Proceedings of the National Academy of Sciences of the United States of America 2005 9714 9719

[26] M.C. Mackey. Mathematical models of hematopoietic cell replication and control. Case Studies in Mathematical Modeling–Ecology, Physiology and Cell Biology. New Jersey, Prentice-Hall, 151-182, 1997.

[27] J.M. Mahaffy, J. Belair, M.C. Mackey Journal of theoretical biology 1998 135 146

[28] A. Marciniak-Czochra, T. Stiehl, W. Wagner Aging 2009 723 732

[29] F. Michor, T.P. Hughes, Y. Iwasa, S. Branford, N.P. Shah, C.L. Sawyers, M.A. Nowak Nature 2005 1267 1270

[30] F. Michor, Y. Iwasa, M.A. Nowak Nature Reviews Cancer 2004 197 205

[31] H. Ozbay, C. Bonnet, H. Benjelloun, J. Clairambault Math. Model. Nat. Phenom. 2012 203 234

[32] L. Pujo-Menjouet, S. Bernard, M. C. Mackey SIAM J. Appl. Dynam. Sys. 2005 312 332

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

[34] I. Roeder, M. Horn, I. Glauche, A. Hochhaus, M.C. Mueller, M. Loeffler Nature medicine 2006 1181 1184

[35] C.L. Sawyers New England Journal of Medicine 1999 1330 1340

[36] L. Shampine, S. Thompson Applied Numerical Mathematics 2001 441 458

[37] T. Stiehl, A. Marciniak-Czochra Math. Model. Nat. Phenom. 2012 166 202

[38] C. Tomasetti, D. Levy PNAS 2010 16766 16771

Cité par Sources :