Modeling and asymptotic stability of a growth factor-dependent stem cell dynamics model with distributed delay

Discrete and Continuous Dynamical Systems - Series B

Volumn 8, Issue 1, 2007, Pages 19-38

  Adimy, Mostafa ^a   Crauste, Fabien ^b  

^a UNIVERSITÉ DE PAU ET DES PAYS DE L'ADOUR   (France)

^b UNIVERSITÉ LYON 1   (France)

Author keywords

Age structured model; Asymptotic stability; Blood cells model; Differential equations; Distributed delay; Growth factors; Hopf bifurcation; Lyapunov function; Stem cells

Indexed keywords

EID: 34548687470     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2007.8.19     Document Type: Conference Paper

References (38)

1. M. Adimy and F. Crauste, Global stability of a partial differential equation with distributed delay due to cellular replication, Nonlinear Analysis, 54 (2003), 1469-1491.
2. M. Adimy and F. Crauste, Existence, positivity and stability for a nonlinear model of cellular proliferation, Nonlinear Analysis: Real World Applications, 6 (2005), 337-366.
3. M. Adimy, F. Crauste, A. Halanay, M. Neamţu and D. Opriş, Stability of limit cycles in a pluripotent stem cell dynamics model, Chaos, Solitons and Fractals, 27 (2006), 1091-1107.
4. M. Adimy, F. Crauste and L. Pujo-Menjouet, On the stability of a maturity structured model of cellular proliferation, Discret. Cont. Dyn. Sys. Ser. A, 12 (2005), 501-522.
5. M. Adimy, F. Crauste and S. Ruan, Stability and Hopf bifurcation in a mathematical model of pluripotent stem cell dynamics, Nonlinear Analysis: Real World Applications, 6 (2005), 651-670.
6. M. Adimy, F. Crauste and S. Ruan, A mathematical study of the hematopoiesis process with applications to chronic myelogenous leukemia, SIAM J. Appl. Math., 65 (2005), 1328-1352.
7. M. Adimy, F. Crauste and S. Ruan, Modelling hematopoiesis mediated by growth factors with applications to periodic hematological diseases, Bull. Math. Biol., 68 (2006), 2321-2351.
8. M. Adimy and L. Pujo-Menjouet, Asymptotic behaviour of a singular transport equation modelling cell division, Discret. Cont. Dyn. Sys. Ser. B, 3 (2003), 439-456.
9. J. Bélair, M. C. Mackey and J. M. Mahaffy, Age-structured and two-delay models for erythropoiesis, Math. Biosci., 128 (1995), 317-346.
10. E. Beretta and Y. Kuang, Geometric stability switch criteria in delay differential systems with delay dependent parameters, SIAM J. Math. Anal., 33 (2002), 1144-1165.
11. S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, J. Theor. Biol., 223 (2003), 283-298.
12. S. Bernard, J. Bélair and M. C. Mackey, Bifurcations in a white-blood-cell production model, C. R. Biologies, 327 (2004), 201-210.
13. G. Bradford, B. Williams, R. Rossi and I. Bertoncello, Quiescence, cycling, and turnover in the primitive haematopoietic stem cell compartment, Exper. Hematol., 25 (1997), 445-453.
14. 0 phase in the cell cycle, Cell. Tissue Kinet., 19 (1970), 321-334.
15. C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis - I. Periodic chronic myelogenous leukemia, J. Theor. Biol., 237 (2005), 117-132.
16. C. Colijn and M.C. Mackey, A mathematical model of hematopoiesis - II. Cyclical neutropenia, J. Theor. Biol., 237 (2005), 133-146.
17. J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics, I: Basic theory., J. Math. Anal. Appl., 242 (2000), 93-104.
18. J. Dyson, R. Villella-Bressan and G. F. Webb, A nonlinear age and maturity structured model of population dynamics, II: Chaos., J. Math. Anal. Appl., 242 (2000), 255-270.
19. L. Granziero, P. Ghia, P. Circosta, D. Gottardi, G. Strola, M. Geuna, L. Montagna, P. Piccoli, M. Chilosi and F. Caligaris-Cappio, Survivin is expressed on CD40 stimulation and interfaces proliferation and apoptosis in B-cell chronic lymphocytic leukemia, Blood, 97 (2001), 2777-2783.
20. J. Hale and S.M. Verduyn Lunel, "Introduction to Functional Differential Equations," Applied Mathematical Sciences 99, Springer-Verlag, New York, 1993.
21. C. Haurie, D. Dale and M. C. Mackey, Cyclical neutropenia and other hematological disorders: A review of mechanisms and mathematical models, Blood, 92 (1998), 2629-2640.
22. M. Iiyama, K. Kakihana, T. Kurosu and O. Miura, Reactive oxygen species generated by hematopoietic cytokines play roles in activation of receptor-mediated signaling and in cell cycle progression, Cellular Signalling, 18 (2006), 174-182.
23. M. C. Mackey, Unified hypothesis of the origin of aplastic anaemia and periodic hematopoiesis, Blood, 51 (1978), 941-956.
24. M.C. Mackey, Dynamic hematological disorders of stem cell origin, in "Biophysical and Biochemical Information Transfer in Recognition" (ed. J.G. Vassileva-Popova and E.V. Jensen), Plenum Press, New-York, 941-956, 1979.
25. M.C. Mackey and A. Rey, Multistability and boundary layer development in a transport equation with retarded arguments, Can. Appl. Math. Quart., 1 (1993), 1-21.
26. M.C. Mackey and A. Rey, Propagation of population pulses and fronts in a cell replication problem: non-locality and dependence on the initial function, Physica D, 86 (1995), 373-395.
27. M.C. Mackey and A. Rey, Transitions and kinematics of reaction-convection fronts in a cell population model, Physica D, 80 (1995), 120-139.
28. M. C. Mackey and R. Rudnicki, Global stability in a delayed partial differential equation describing cellular replication, J. Math. Biol., 33 (1994), 89-109.
29. M. C. Mackey and R. Rudnicki, A new criterion for the global stability of simultaneous cell replication and maturation processes, J. Math. Biol., 38 (1999), 195-219.
30. J. M. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay, J. Theor. Biol., 190 (1998), 135-146.
31. L. Pujo-Menjouet, S. Bernard and M. C. Mackey, Long period oscillations in a Go model of hematopoietic stem cells, SIAM J. Appl. Dyn. Syst., 4 (2005), 312-332.
32. L. Pujo-Menjouet and M. C. Mackey, Contribution to the study of periodic chronic myelogenous leukemia, C. R. Biologies, 327 (2004), 235-244.
33. S. Ruan and J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 10 (2003), 863-874.
34. U. Testa, Apoptotic mechanisms in the control of erythropoiesis, Leukemia, 18 (2004), 1176-1199.
35. G. F. Webb, "Theory of Nonlinear Age-dependent Population Dynamics," Monographs and Textbooks in Pure and Applied Mathematics 89, Marcel Dekker Inc., New York, 1985.
36. J. Wei and S. Ruan, Stability and bifurcation in a neural network model with two delays, Physica D, 130 (1999), 255-272.
37. J. B. Weitzman, L. Fiette, K. Matsuo and M. Yaniv, JunD protects cells from p53-dependent senescence and apoptosis, Mol. Cell, 6 (2000), 1109-1119.
38. J. W. Williams, E. Beutler, A. J. Erslev and M. A. Lichtman, "Hematology," McGraw-Hill, 4th edition, 1990.