Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models

Xiao Wang; Zhixiang Li

Open Mathematics (2007)

  • Volume: 5, Issue: 2, page 397-414
  • ISSN: 2391-5455

Abstract

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In this paper, we discuss the special diffusive hematopoiesis model ∂ P ( t , x ) ∂ t = Δ P ( t , x ) - γ P ( t , x ) + β P ( t - τ , x ) 1 + P n ( t - τ , x ) with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.

How to cite

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Xiao Wang, and Zhixiang Li. "Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models." Open Mathematics 5.2 (2007): 397-414. <http://eudml.org/doc/269168>.

@article{XiaoWang2007,
abstract = {In this paper, we discuss the special diffusive hematopoiesis model \[\frac\{\{\partial P(t,x)\}\}\{\{\partial t\}\} = \Delta P(t,x) - \gamma P(t,x) + \frac\{\{\beta P(t - \tau ,x)\}\}\{\{1 + P^n (t - \tau ,x)\}\}\] with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.},
author = {Xiao Wang, Zhixiang Li},
journal = {Open Mathematics},
keywords = {Global attractivity; Oscillation; Hopf bifurcation; delay differential equation; Neumann boundary condition; lower-upper solution pairs},
language = {eng},
number = {2},
pages = {397-414},
title = {Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models},
url = {http://eudml.org/doc/269168},
volume = {5},
year = {2007},
}

TY - JOUR
AU - Xiao Wang
AU - Zhixiang Li
TI - Global attractivity, oscillation and Hopf bifurcation for a class of diffusive hematopoiesis models
JO - Open Mathematics
PY - 2007
VL - 5
IS - 2
SP - 397
EP - 414
AB - In this paper, we discuss the special diffusive hematopoiesis model \[\frac{{\partial P(t,x)}}{{\partial t}} = \Delta P(t,x) - \gamma P(t,x) + \frac{{\beta P(t - \tau ,x)}}{{1 + P^n (t - \tau ,x)}}\] with Neumann boundary condition. Sufficient conditions are provided for the global attractivity and oscillation of the equilibrium for Eq. (*), by using a new theorem we stated and proved. When P(t, χ) does not depend on a spatial variable χ ∈ Ω, these results are also true and extend or complement existing results. Finally, existence and stability of the Hopf bifurcation for Eq. (*) are studied.
LA - eng
KW - Global attractivity; Oscillation; Hopf bifurcation; delay differential equation; Neumann boundary condition; lower-upper solution pairs
UR - http://eudml.org/doc/269168
ER -

References

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  1. [1] J. Chern and S. Huang: “Global stability in delay equation models of haematopoiesis”, Proc. Natl. Counc. Roc (A), 24(4), (2000), pp. 293–300. 
  2. [2] K. Cooke, P. van den Driessche and Xingfu Zou: “Interaction of maturation delay and nonlinear birth in population and epidemic models”, J. Math. Biol., Vol. 39, (1999), pp. 332–352. http://dx.doi.org/10.1007/s002850050194 Zbl0945.92016
  3. [3] T. Ding: “Asymptotic behavior of solution of some retarded differential equations”, Scientia Sinica (A), 25(4), (1982), pp. 363–370. Zbl0498.34056
  4. [4] K. Gopalsamy, N. Bantsur and S. Trofimchuk: “A note on global attractivity in models of hematopoiesis”, Ukrainian Math. J., Vol. 50, (1998), pp. 3–12. http://dx.doi.org/10.1007/BF02514684 Zbl0893.92012
  5. [5] K. Gopalsamy: Stability and oscillations in delay differential equations of population dynamics, Kluwer Academic Press, Boston, 1992. Zbl0752.34039
  6. [6] K. Gopalsamy, M.R.S. Kulenovic and G. Ladas: “Oscillation and global attractivity in models of hematopoiesis”, J. Dyn. Diff. Eqns, Vol. 2, (1990), pp. 117–132. http://dx.doi.org/10.1007/BF01057415 Zbl0694.34057
  7. [7] K. Gopalsamy and P. Weng: “Global attractivity and level crossing in model of Hematopoiesis”, Bulletin of the Institute of Mathematics, Academia Sinica, Vol. 22, (1994), pp. 341–360. Zbl0829.34067
  8. [8] J. Hale and N. Sternberg: “On set of chaos in differential delay equations”, J. Comput. Phys., Vol. 77, (1988), pp. 221–239. http://dx.doi.org/10.1016/0021-9991(88)90164-7 
  9. [9] J.K. Hale and S.M. Verduyn Lunel: Introduction to Function Differential Equations, Springer-verlag, 1993. 
  10. [10] B.D. Hassard, N.D. Kazarinoff and Y.H. Wan: Theory and Appliation of Hopf bifurcation, Cambridge University Press, London, 1981. Zbl0474.34002
  11. [11] D. Henry: Geometrictheoryofsemilinearparabolicequations, Lecture Notes in Math., Vol. 840, Springer-Verlag, Berlin-Heidelberg, 1981 
  12. [12] D.Q. Jiang and J.J. Wei: “Existence of positive periodic solutions for Volterra integro-differential equations”, Acta Mathematica Scientia, Vol. 21B(4), (2002), pp. 553–560. Zbl1035.45003
  13. [13] D.Q. Jiang and J.J. Wei: “Existence of positive periodic solutions of nonautonomous differential equations with delay(in Chinese)”, Chinese Annals of Mathematics, 20A(6), (1999), pp. 715–720. Zbl0948.34046
  14. [14] G. Karakostas, Ch.G. Philos and Y.G. Sficas: “Stable steady state of some population models”, J. Dyn. Diff. Eqns, Vol. 4(1), (1992), pp. 161–190. http://dx.doi.org/10.1007/BF01048159 Zbl0744.34071
  15. [15] Y. Kuang: Delay differential equations with applications in population dynamics, Academic Press, New York, 1993. 
  16. [16] M.R.S. Kulenovic and G. Ladas: “Linearized oscillations in population dynamics”, Bull. Math. Bio., Vol. 49, (1987), pp. 615–627. http://dx.doi.org/10.1016/S0092-8240(87)90005-X Zbl0634.92013
  17. [17] M.R.S. Kulenovic, G. Ladas and A. Meimaridou: “On oscillation of nonlinear delay differential equations”, Quart. Appl. Math., Vol. 45, (1987), pp. 155–164. Zbl0627.34076
  18. [18] G.S. Ladde, V. Lakshmikantham and B.G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987. Zbl0832.34071
  19. [19] G. Ladas and I.P. Stavroulakis: “Oscillations caused by several retarded and advanced argument”, J. Differential Equations, Vol. 44, (1982), pp. 134–152. http://dx.doi.org/10.1016/0022-0396(82)90029-8 Zbl0452.34058
  20. [20] E. Liz, M. Pinto, V. Tkachenko and S. Trofimchuk: “A global stability criterion for a family of delayed population models”, Quarterly of Applied Mathematics, Vol. 63, (2005), pp. 56–70. Zbl1093.34038
  21. [21] E. Liz, E. Trofimchuk and S. Trofimchuk: “Mackey-Glass type delay differential equations near the boundary of absolute stability”, Journal of Mathematical Analysis and Applications, Vol. 275, (2002), pp. 747–760. http://dx.doi.org/10.1016/S0022-247X(02)00416-X Zbl1022.34071
  22. [22] J. Luo and J. Yu: “Global asymptotic stability of nonautonomous mathematical ecological equations with distributed deviating arguments(in Chinese)”, Acta Mathematica Sinica, Vol. 41, (1998), pp. 1273–1282. Zbl1027.34088
  23. [23] A. Makroglou and Yang Kuang: “Some analytical and numerical results for a nonlinear Volterra integro-differential equation with periodic solution modeling hematopoiesis”, Proceedings of Hercma, 2005. Zbl1244.65242
  24. [24] M.C. Mackey and L. Glass: “Oscillation and chaos in physiological control system”, Science, Vol. 197, (1977), pp. 287–289. http://dx.doi.org/10.1126/science.267326 
  25. [25] R. Redlinger: “On Volterra's population equation with diffusion”, SIAM J. Math. Anal, Vol. 16, (1985), pp. 135–142. http://dx.doi.org/10.1137/0516008 Zbl0593.92014
  26. [26] R. Redlinger: “Exitence-theorems for Semiliner parabolic Systems with Functional”, Nonlinear Anal. TMA., Vol.8, (1984), pp. 667–682. http://dx.doi.org/10.1016/0362-546X(84)90011-7 
  27. [27] S.H. Saker: “Oscillation and global attractivity in hematopoiesis model with periodic coefficients”, Applied Mathematics and Computation, Vol. 142, (2003), pp. 477–494. http://dx.doi.org/10.1016/S0096-3003(02)00315-6 Zbl1048.34114
  28. [28] S.H. Saker: “Oscillation and global attractivity in hematopoiesis model with delay time”, Applied Mathematics and Computation, Vol. 136, (2003), pp. 241–250. http://dx.doi.org/10.1016/S0096-3003(02)00035-8 Zbl1026.34082
  29. [29] K. Schmitt: “Oscillation in nonlinear differential-delay equations, Resear Notes in Mathematics”, In: F. Kappel and W. Schappacher (Eds.): Abstract Cauchy Problems and Functional Differential Equations,Pitman Advanced Publishing Program, Boston, Lonton, Melbourne, 1981, pp. 191–211. 
  30. [30] M.M.A. Sheikh, Zaghrout and A. Ammar: “Oscillation and global attractivity in delay equation of population dynamics”, Appl. Math. Comput., Vol. 77(2–3), (1996), pp. 195–204. Zbl0848.92018
  31. [31] H. Smith: Monotone Dynamical Systems, AMS Surveys andMonographs Providence, Hode Island, 1995. 
  32. [32] C.C. Travis and G.F. Webb: “Existence and stability for partial functional differential equations”, Trans. Amer. Math. Soc., Vol. 200, (1974), pp. 395–418. http://dx.doi.org/10.2307/1997265 Zbl0299.35085
  33. [33] C.C. Travis and G.F. Webb: “Partial differential equations with deviating arguments in the time variable”, J. Math. Anal. Appl., Vol. 56, (1976), pp. 397–409. http://dx.doi.org/10.1016/0022-247X(76)90052-4 Zbl0349.35071
  34. [34] A. Wan, D.Q. Jiang and X.J. Xu: “A new existence theory for positive periodic solutions to functional differential equations”, Computers and Mathematics with Applications, Vol. 47, (2004), pp. 1257–1262. http://dx.doi.org/10.1016/S0898-1221(04)90120-4 Zbl1073.34082
  35. [35] P. Weng and M. Liang: “The existence and behavior of periodic solution of Hematopoiesis model”, Mathematica Applicate, Vol. 8(4), (1995), pp. 434–439. Zbl0949.34517
  36. [36] P. Weng: “Asymptotic stability for a class of integro-differential equations with infinite delay”, Matheamtica Applicta, Vol. 14(1), (2001), pp. 22–27. Zbl1007.45007
  37. [37] P. Weng: “Global attractivity of periodic solution in a model of Hematopoiesis”, Computers and Mathematics with Applications, Vol. 44, (2002), pp. 1019–1030. http://dx.doi.org/10.1016/S0898-1221(02)00211-0 Zbl1035.45004
  38. [38] P. Weng and Z. Dai: “Global attractivity for a model of hematopoiesis(in Chinese)”, Journal Of Southchina Normal University, Vol. 2, (2001), pp. 12–19. Zbl0973.92010
  39. [39] Jianhong Wu: Theory and Appliation of Partial Functional Differential Equation, Applied Mathematical Sciences, Vol. 119, Springer, 1996. 
  40. [40] Yuanjie Yang and Joseph. W.H. So: “Dynamics for the Diffusive Nicholson's Blowflies Equation. Proceeding of the International Conference on Dynamical Systems and Differential Equation” (Springfiled, Missouri, U.S.A. M29-June, 1996), Volume II: Wenxiong Chen and Shou Chucan Hu (Eds.): An added Volumeto discrete and continuous Dynamical Systems, 1998, pp. 333–352. Reaction-Diffusion Equation with time delay:Theory, Appliation and Numericalsimulation. 
  41. [41] Qixiao Ye and Zhengyuan Li: Introduction to the reaction-diffusion equation(in Chinese), Scienc Press, Benjing, 1990. Zbl0784.35050
  42. [42] Xiaoqiang Zhao: Dynamical Systems in Population Biology, Springer, New York, 2003. 

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