AM  Vol.5 No.17 , October 2014
Traveling Wavefronts of a Diffusive Hematopoiesis Model with Time Delay
Abstract: In this paper, a reaction-diffusion equation with discrete time delay that describes the dynamics of the blood cell production is analyzed. The existence of the traveling wave front solutions is demonstrated using the technique of upper and lower solutions and the associated monotone iteration.
Cite this paper: Ling, Z. and Zhu, L. (2014) Traveling Wavefronts of a Diffusive Hematopoiesis Model with Time Delay. Applied Mathematics, 5, 2712-2718. doi: 10.4236/am.2014.517258.

[1]   Kolmogorov, A., Petrovskii, I. and Piskunov, N. (1937) Study of a Diffusion Equation That Is Related to the Growth of a Quality of Matter and Its Application to a Biological Problem. Moscow University Mathematics Bulletin, 1, 1-26.

[2]   Fisher, R.A. (1937) The Wave of Advance of Advantageous Gene. Annals of Eugenics, 7, 355-369.

[3]   Murray, J.D. (1989) Mathematical Biology. Springer, New York.

[4]   Volpert, A.I. and Volpert, V.A. (1994) Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs. American Mathematical Society, RI, 140.

[5]   Gomez, A. and Trofimchuk, S. (2011) Monotone Traveling Wavefronts of the KPP-Fisher Delayed Equation. Journal of Differential Equations, 250, 1767-1787.

[6]   Schaaf, K. (1987) Asymptotic Behavior and Traveling Wave Solutions for Parabolic Functional Differential Equations. Transactions of the American Mathematical Society, 302, 587-615.

[7]   Huang, J. and Zou, X. (2003) Existence of Travelling Wave Fronts of Delayed Reaction Diusion Systems without Monotonicity. Discrete and Continuous Dynamical Systems, 9, 925-936.

[8]   Ma, S. (2001) Travelling Wavefronts for Delayed Reaction-Diusion Systems via a Fixed Point Theorem. Journal of Differential Equations, 171, 294-314.

[9]   Mackey, M.C. and Glass, L. (1977) Oscillation and Chaos in Physiological Control System. Science, 197, 287-289.

[10]   Mackey, M.C. (1978) Unified Hypothesis for the Origin of Aplastic Anemia and Periodic Hematopoiesis. Blood, 51, 941-956.

[11]   Mackey, M.C. (1981) Some Models in Hemapoiesis: Predictions and Problems. In: Biomathematics and Cell Kinetzcs, Elservier, North Holland, 23-38.

[12]   Liz, E., Tkachenko, V. and Trofimchuk, S. (2003) A Global Stability Criterion for Scalar Functional Differential Equations. SIAM Journal on Mathematical Analysis, 35, 596-622.

[13]   Weng, P.X. and Dai, Z.P. (2001) Global Atractivity for a Model of Hemapoiesis. Journal of South China Normal University, 2, 12-19.

[14]   Wu, X.M., Li, J.W. and Zhou, H.Q. (2007) A Necessary and Sufficient Condition for the Existence of Positive Periodic Solutions of a Model of Hematopoiesis. Computers & Mathematics with Applications, 54, 840-849.

[15]   Wang, X. and Li, Z.X. (2007) Dynamics for a Type of General Reaction-Diffusion Model. Nonlinear Analysis: Theory, Methods & Applications, 67, 2699-2711.

[16]   Gopalsamy, K., Kulenvic, M.R. and Ladas, G. (1990) Oscillations and Global Attractivity in Models of Hematopoiesis. Journal of Dynamics and Differential Equations, 2, 117-132.

[17]   Cheng, S.S. and Zhang, G. (2001) Existence of Positive Periodic Solutions for Non-Autonomaous Functional Differential Equations. Electronic Journal of Differential Equations, 59, 1-8.

[18]   Jiang, D.Q., Wei, J.J. and Zhang, B. (2002) Positive Periodic Solutions of Functional Differential Equations and Population Model. Electronic Journal of Differential Equations, 71, 1-13.

[19]   Wu, J.H. and Zou, X.F. (2001) Traveling Wave Fronts of Reaction-Diffusion Systems. Journal of Dynamics and Differential Equations, 13, 651-687.