On positive solutions and the Omega limit set for a class of delay differential equations

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November 2013, 18(9): 2487-2503. doi: 10.3934/dcdsb.2013.18.2487

On positive solutions and the Omega limit set for a class of delay differential equations

Changjing Zhuge ^1, , Xiaojuan Sun ^2, and Jinzhi Lei ^3,

1.	Zhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, China
2.	School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China
3.	Zhou Pei-Yuan Center for Applied Mathematics, MOE Key Laboratory of Bioinformatics, Tsinghua University, Beijing, 100084, China

Received March 2013 Revised May 2013 Published September 2013

Abstract
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This paper studies positive solutions of a class of delay differential equations of two delays that are originated from a mathematical model of hematopoietic dynamics. We give an optimal condition on initial conditions for $t\leq 0$ such that the solutions are positive for $t>0$. Long time behaviors of these positive solutions are also discussed through a dynamical system defined at a space of continuous functions. Characteristic description of the $\omega$ limit set of this dynamical system is obtained. This $\omega$ limit set provides informations for the long time behaviors of positive solutions of the delay differential equation.

Keywords: positive solution, $\omega$-limit set., Delay differential equation.

Mathematics Subject Classification: Primary: 34K90; Secondary: 92D2.

Citation: Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete and Continuous Dynamical Systems - B, 2013, 18 (9) : 2487-2503. doi: 10.3934/dcdsb.2013.18.2487

References:

[1]	J. Adamson, The relationship of erythropoietin and iron metabolism to red blood cell production in humans, Semin. Oncol., 21 (1974), 9-15.
[2]	J. Bélair and M. C. Mackey, Consumer memory and price fluctuations in commodity markets: An integrodifferential model, Journal of Dynamics and Differential Equations, 1 (1989), 299-325. doi: 10.1007/BF01053930.
[3]	D. L. Bellman and S. A. Gourley, Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insulin, Applied Mathematics and Computation, 151 (2004), 189-207. doi: 10.1016/S0096-3003(03)00332-1.
[4]	S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, Journal of Theoretical Biology, 223 (2003), 283-298. doi: 10.1016/S0022-5193(03)00090-0.
[5]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis-I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033.
[6]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: II. Cyclical neutropenia, Journal of Theoretical Biology, 237 (2005), 133-146. doi: 10.1016/j.jtbi.2005.03.034.
[7]	C. Colijn and M. C. Mackey, Bifurcation and bistability in a model of hematopoietic regulation, SIAM Journal on Applied Dynamical Systems, 6 (2007), 378-394. doi: 10.1137/050640072.
[8]	B. Dorizzi, B. Grammaticos, M. Le Berre, Y. Pomeau, E. Ressayre and A. Tallet, Statistics and dimension of chaos in differential delay systems, Physical Review A, 35 (1987), 328-339. doi: 10.1103/PhysRevA.35.328.
[9]	S. A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal on Applied Mathematics, 65 (2005), 550-566. doi: 10.1137/S0036139903436613.
[10]	J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM Journal on Applied Mathematics, 67 (2007), 387-407. doi: 10.1137/060650234.
[11]	J. Lei and M. C. Mackey, Deterministic Brownian motion generated from differential delay equations, Physical Review E, 84 (2011), 041105. doi: 10.1103/PhysRevE.84.041105.
[12]	J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, Journal of Theoretical Biology, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024.
[13]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[14]	J. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190 (1998), 135-146. doi: 10.1006/jtbi.1997.0537.
[15]	M. Silva, D. Grillot, A. Benito, C. Richard, G. Nunez and J. Fernandez-Luna, Erythropoietin can promote erythroid progenitor survival by repressing apoptosis through bcl-1 and bcl-2, Blood, 88 (1996), 1576-1582.
[16]	J. C. Sprott, A simple chaotic delay differential equation, Physics Letters A, 366 (2007), 397-402. doi: 10.1016/j.physleta.2007.01.083.

show all references

References:

[1]	J. Adamson, The relationship of erythropoietin and iron metabolism to red blood cell production in humans, Semin. Oncol., 21 (1974), 9-15.
[2]	J. Bélair and M. C. Mackey, Consumer memory and price fluctuations in commodity markets: An integrodifferential model, Journal of Dynamics and Differential Equations, 1 (1989), 299-325. doi: 10.1007/BF01053930.
[3]	D. L. Bellman and S. A. Gourley, Asymptotic properties of a delay differential equation model for the interaction of glucose with plasma and interstitial insulin, Applied Mathematics and Computation, 151 (2004), 189-207. doi: 10.1016/S0096-3003(03)00332-1.
[4]	S. Bernard, J. Bélair and M. C. Mackey, Oscillations in cyclical neutropenia: New evidence based on mathematical modeling, Journal of Theoretical Biology, 223 (2003), 283-298. doi: 10.1016/S0022-5193(03)00090-0.
[5]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis-I. Periodic chronic myelogenous leukemia, Journal of Theoretical Biology, 237 (2005), 117-132. doi: 10.1016/j.jtbi.2005.03.033.
[6]	C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: II. Cyclical neutropenia, Journal of Theoretical Biology, 237 (2005), 133-146. doi: 10.1016/j.jtbi.2005.03.034.
[7]	C. Colijn and M. C. Mackey, Bifurcation and bistability in a model of hematopoietic regulation, SIAM Journal on Applied Dynamical Systems, 6 (2007), 378-394. doi: 10.1137/050640072.
[8]	B. Dorizzi, B. Grammaticos, M. Le Berre, Y. Pomeau, E. Ressayre and A. Tallet, Statistics and dimension of chaos in differential delay systems, Physical Review A, 35 (1987), 328-339. doi: 10.1103/PhysRevA.35.328.
[9]	S. A. Gourley and Y. Kuang, A delay reaction-diffusion model of the spread of bacteriophage infection, SIAM Journal on Applied Mathematics, 65 (2005), 550-566. doi: 10.1137/S0036139903436613.
[10]	J. Lei and M. C. Mackey, Stochastic differential delay equation, moment stability, and application to hematopoietic stem cell regulation system, SIAM Journal on Applied Mathematics, 67 (2007), 387-407. doi: 10.1137/060650234.
[11]	J. Lei and M. C. Mackey, Deterministic Brownian motion generated from differential delay equations, Physical Review E, 84 (2011), 041105. doi: 10.1103/PhysRevE.84.041105.
[12]	J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia, Journal of Theoretical Biology, 270 (2011), 143-153. doi: 10.1016/j.jtbi.2010.11.024.
[13]	M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems, Science, 197 (1977), 287-289. doi: 10.1126/science.267326.
[14]	J. Mahaffy, J. Bélair and M. C. Mackey, Hematopoietic model with moving boundary condition and state dependent delay: Applications in erythropoiesis, Journal of Theoretical Biology, 190 (1998), 135-146. doi: 10.1006/jtbi.1997.0537.
[15]	M. Silva, D. Grillot, A. Benito, C. Richard, G. Nunez and J. Fernandez-Luna, Erythropoietin can promote erythroid progenitor survival by repressing apoptosis through bcl-1 and bcl-2, Blood, 88 (1996), 1576-1582.
[16]	J. C. Sprott, A simple chaotic delay differential equation, Physics Letters A, 366 (2007), 397-402. doi: 10.1016/j.physleta.2007.01.083.

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