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

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

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.
Citation: Changjing Zhuge, Xiaojuan Sun, Jinzhi Lei. On positive solutions and the Omega limit set for a class of delay differential equations. Discrete & 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. Google Scholar

[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. doi: 10.1007/BF01053930. Google Scholar

[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. doi: 10.1016/S0096-3003(03)00332-1. Google Scholar

[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. doi: 10.1016/S0022-5193(03)00090-0. Google Scholar

[5]

C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis-I. Periodic chronic myelogenous leukemia,, Journal of Theoretical Biology, 237 (2005), 117. doi: 10.1016/j.jtbi.2005.03.033. Google Scholar

[6]

C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: II. Cyclical neutropenia,, Journal of Theoretical Biology, 237 (2005), 133. doi: 10.1016/j.jtbi.2005.03.034. Google Scholar

[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. doi: 10.1137/050640072. Google Scholar

[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. doi: 10.1103/PhysRevA.35.328. Google Scholar

[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. doi: 10.1137/S0036139903436613. Google Scholar

[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. doi: 10.1137/060650234. Google Scholar

[11]

J. Lei and M. C. Mackey, Deterministic Brownian motion generated from differential delay equations,, Physical Review E, 84 (2011). doi: 10.1103/PhysRevE.84.041105. Google Scholar

[12]

J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia,, Journal of Theoretical Biology, 270 (2011), 143. doi: 10.1016/j.jtbi.2010.11.024. Google Scholar

[13]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326. Google Scholar

[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. doi: 10.1006/jtbi.1997.0537. Google Scholar

[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. Google Scholar

[16]

J. C. Sprott, A simple chaotic delay differential equation,, Physics Letters A, 366 (2007), 397. doi: 10.1016/j.physleta.2007.01.083. Google Scholar

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. Google Scholar

[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. doi: 10.1007/BF01053930. Google Scholar

[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. doi: 10.1016/S0096-3003(03)00332-1. Google Scholar

[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. doi: 10.1016/S0022-5193(03)00090-0. Google Scholar

[5]

C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis-I. Periodic chronic myelogenous leukemia,, Journal of Theoretical Biology, 237 (2005), 117. doi: 10.1016/j.jtbi.2005.03.033. Google Scholar

[6]

C. Colijn and M. C. Mackey, A mathematical model of hematopoiesis: II. Cyclical neutropenia,, Journal of Theoretical Biology, 237 (2005), 133. doi: 10.1016/j.jtbi.2005.03.034. Google Scholar

[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. doi: 10.1137/050640072. Google Scholar

[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. doi: 10.1103/PhysRevA.35.328. Google Scholar

[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. doi: 10.1137/S0036139903436613. Google Scholar

[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. doi: 10.1137/060650234. Google Scholar

[11]

J. Lei and M. C. Mackey, Deterministic Brownian motion generated from differential delay equations,, Physical Review E, 84 (2011). doi: 10.1103/PhysRevE.84.041105. Google Scholar

[12]

J. Lei and M. C. Mackey, Multistability in an age-structured model of hematopoiesis: Cyclical neutropenia,, Journal of Theoretical Biology, 270 (2011), 143. doi: 10.1016/j.jtbi.2010.11.024. Google Scholar

[13]

M. C. Mackey and L. Glass, Oscillation and chaos in physiological control systems,, Science, 197 (1977), 287. doi: 10.1126/science.267326. Google Scholar

[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. doi: 10.1006/jtbi.1997.0537. Google Scholar

[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. Google Scholar

[16]

J. C. Sprott, A simple chaotic delay differential equation,, Physics Letters A, 366 (2007), 397. doi: 10.1016/j.physleta.2007.01.083. Google Scholar

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