zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Mathematical model of hematopoiesis dynamics with growth factor-dependent apoptosis and proliferation regulations. (English) Zbl 1171.34344
Summary: We consider a nonlinear mathematical model of hematopoietic stem cell dynamics, in which proliferation and apoptosis are controlled by growth factor concentrations. Cell proliferation is positively regulated, while apoptosis is negatively regulated. The resulting age-structured model is reduced to a system of three differential equations, with three independent delays, and existence of steady states is investigated. The stability of the trivial steady state, describing cells dying out with a saturation of growth factor concentrations is proven to be asymptotically stable when it is the only equilibrium. The stability analysis of the unique positive steady state allows the determination of a stability area, and shows that instability may occur through a Hopf bifurcation, mainly as a destabilization of the proliferative capacity control, when cell cycle durations are very short. Numerical simulations are carried out and result in a stability diagram that stresses the lead role of the introduction rate compared to the apoptosis rate in the system stability.
34K20Stability theory of functional-differential equations
34K60Qualitative investigation and simulation of models
92C37Cell biology
37N25Dynamical systems in biology
[1]Mackey, M. C.: Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis, Blood 51, 941-956 (1978)
[2]Pujo-Menjouet, L.; Mackey, M. C.: Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biologies 327, 235-244 (2004)
[3]Pujo-Menjouet, L.; Bernard, S.; Mackey, M. C.: Long period oscillations in a G0 model of hematopoietic stem cells, SIAM J. Appl. dyn. Syst. 4, No. 2, 312-332 (2005) · Zbl 1110.34050 · doi:10.1137/030600473
[4]Bernard, S.; Bélair, J.; Mackey, M. C.: Oscillations in cyclical neutropenia: new evidence based on mathematical modeling, J. theoret. Biol. 223, 283-298 (2003)
[5]Colijn, C.; Mackey, M. C.: A mathematical model of hematopoiesis – I. Periodic chronic myelogenous leukemia, J. theor. Biol. 237, 117-132 (2005)
[6]Colijn, C.; Mackey, M. C.: A mathematical model of hematopoiesis – II. Cyclical neutropenia, J. theor. Biol. 237, 133-146 (2005)
[7]Adimy, M.; Crauste, F.: Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear anal. 54, No. 8, 1469-1491 (2003) · Zbl 1028.35149 · doi:10.1016/S0362-546X(03)00197-4
[8]Adimy, M.; Crauste, F.: Existence, positivity and stability for a nonlinear model of cellular proliferation, Nonlinear anal. RWA 6, No. 2, 337-366 (2005) · Zbl 1073.35055 · doi:10.1016/j.nonrwa.2004.09.001
[9]Adimy, M.; Crauste, F.; Halanay, A.; Neamţu, M.; Opriş, D.: Stability of limit cycles in a pluripotent stem cell dynamics model, Chaos solitons fractals 27, No. 4, 1091-1107 (2006) · Zbl 1079.92022 · doi:10.1016/j.chaos.2005.04.083
[10]Adimy, M.; Crauste, F.; Pujo-Menjouet, L.: On the stability of a maturity structured model of cellular proliferation, Discrete contin. Dyn. syst. Ser. A 12, No. 3, 501-522 (2005) · Zbl 1064.35023 · doi:10.3934/dcds.2005.12.501
[11]Adimy, M.; Crauste, F.; Ruan, S.: Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear anal. RWA 6, No. 4, 651-670 (2005) · Zbl 1074.92010 · doi:10.1016/j.nonrwa.2004.12.010
[12]Adimy, M.; Crauste, F.; Ruan, S.: A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. math. 65, No. 4, 1328-1352 (2005) · Zbl 1090.34058 · doi:10.1137/040604698
[13]Adimy, M.; Pujo-Menjouet, L.: Asymptotic behaviour of a singular transport equation modelling cell division, Discrete contin. Dyn. syst. Ser. B 3, 439-456 (2003) · Zbl 1120.35305 · doi:10.3934/dcdsb.2003.3.439
[14]Bélair, J.; Mackey, M. C.; Mahaffy, J. M.: Age-structured and two-delay models for erythropoiesis, Math. biosci. 128, 317-346 (1995) · Zbl 0832.92005 · doi:10.1016/0025-5564(94)00078-E
[15]Mahaffy, J. M.; Bélair, J.; Mackey, M. C.: Hematopoietic model with moving boundary condition and state dependent delay, J. theor. Biol. 190, 135-146 (1998)
[16]Adimy, M.; Crauste, F.; Ruan, S.: Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases, Bull. math. Biol. 68, 2321-2351 (2006)
[17]Adimy, M.; Crauste, F.: Modelling and asymptotic stability of a growth factor-dependent stem cells dynamics model with distributed delay, Discrete contin. Dyn. syst. Ser. B 8, No. 1, 19-38 (2007) · Zbl 1139.34058 · doi:10.3934/dcdsb.2007.8.19
[18]Testa, U.: Apoptotic mechanisms in the control of erythropoiesis, Leukemia 18, No. 7, 1176-1199 (2004)
[19]Koury, M. J.; Bondurant, M. C.: Erythropoietin retards DNA breakdown and prevents programmed death in erythroid progenitor cells, Science 248, 378-381 (1990)
[20]Burns, F. J.; Tannock, I. F.: On the existence of a G0 phase in the cell cycle, Cell. tissue kinet. 19, 321-334 (1970)
[21]Webb, G. F.: Theory of nonlinear age-dependent population dynamics, Monographs and textbooks in pure and applied mathematics 89 (1985) · Zbl 0555.92014
[22]Hale, J.; Lunel, S. M. Verduyn: Introduction to functional differential equations, Applied mathematical sciences 99 (1993) · Zbl 0787.34002
[23]Hayes, N. D.: Roots of the transcendental equation associated with a certain differential difference equation, J. London math. Soc. 25, 226-232 (1950) · Zbl 0038.24102 · doi:10.1112/jlms/s1-25.3.226
[24]Wei, J.; Ruan, S.: Stability and bifurcation in a neural network model with two delays, Physica D 130, 255-272 (1999) · Zbl 1066.34511 · doi:10.1016/S0167-2789(99)00009-3
[25]Shampine, L. F.; Thompson, S.: Solving ddes in Matlab, Appl. numer. Math. 37, 441-458 (2001) · Zbl 0983.65079 · doi:10.1016/S0168-9274(00)00055-6