November  2019, 24(11): 5803-5830. doi: 10.3934/dcdsb.2019107

Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms

1. 

School of Mathematics and Statistics, Anhui Normal University, Wuhu, Anhui 241000, China

2. 

Departamento de Estatística, Análise Matemática e Optimización, Facultade de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

3. 

Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782, Santiago de Compostela, Spain

* Corresponding author: Juan J. Nieto

Received  July 2018 Revised  December 2018 Published  June 2019

Discontinuous system is playing an increasingly important role in terms of both theory and applications. This paper presents a hematopoiesis model with mixed discontinuous harvesting terms. By using differential inclusions theory, the non-smooth analysis theory with Lyapunov-like approach, some new sufficient criteria are given to ascertain the existence, uniqueness and globally exponential stability of the bounded positive almost periodic solutions for the addressed model. Some previously known results are extended and complemented. Moreover, simulation results of two topical numerical examples are also delineated to demonstrate the effectiveness of the established theoretical results.

Citation: Fanchao Kong, Juan J. Nieto. Almost periodic dynamical behaviors of the hematopoiesis model with mixed discontinuous harvesting terms. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5803-5830. doi: 10.3934/dcdsb.2019107
References:
[1]

M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, 2010. doi: 10.1007/978-1-4419-6581-3. Google Scholar

[2]

J. O. Alzabut, J. J. Nieto and G. T. Stamov, Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis, Boundary Value Problems, 2009 (2009), Art. ID 127510, 10 pp. doi: 10.1155/2009/127510. Google Scholar

[3]

J. O. Alzabut, Almost periodic solutions for an impulsive delay Nicholson's blowflies model, Journal of Computational and Applied Mathematics, 234 (2010), 233-239. doi: 10.1016/j.cam.2009.12.019. Google Scholar

[4]

J. Aubin and A. Cellina, Differential Inclusions, Berlin: Springer-Verlag, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar

[5]

Z. W. CaiL. H. HuangZ. Y. Guo and X. Y. Chen, On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33 (2012), 97-113. doi: 10.1016/j.neunet.2012.04.009. Google Scholar

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Google Scholar

[7]

C. Corduneanu, Almost Periodic Functions, 2nd edition, Chelsea, New york, 1989.Google Scholar

[8]

T. Diagana and H. Zhou, Existence of positive almost periodic solutions to the hematopoiesis model, Applied Mathematics and Computation, 274 (2016), 644-648. doi: 10.1016/j.amc.2015.10.029. Google Scholar

[9]

H. S. DingG. M. N'Guérékata and J. J. Nieto, Weighted pseudo almost periodic solutions for a class of discrete hematopoiesis model, Revista Matematica Complutense, 26 (2013), 427-443. doi: 10.1007/s13163-012-0114-y. Google Scholar

[10]

H. S. DingQ. L. Liu and J. J. Nieto, Existence of positive almost periodic solutions to a class of hematopoiesis model, Applied Mathematical Modelling, 40 (2016), 3289-3297. doi: 10.1016/j.apm.2015.10.020. Google Scholar

[11]

L. Duan, L. H. Huang and X. W. Fang, Finite-time synchronization for recurrent neural networks with discontinuous activations and time-varying delays, Chaos, 27 (2017), 013101, 10pp. doi: 10.1063/1.4966177. Google Scholar

[12]

A. F. Filippov, Mathematics and Its Applications (Soviet Series), Differential equations with discontinuous right-hand sides, Boston: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar

[13]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. Google Scholar

[14]

M. FortiP. Nistri and D. Papini, Global convergence of neural networks with discontinuous neuron activations, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50 (2003), 1421-1435. doi: 10.1109/TCSI.2003.818614. Google Scholar

[15]

Z. Y. Guo and L. H. Huang, Generalized Lyapunov method for discontinuous systems, Nonlinear Analysis, 71 (2009), 3083-3092. doi: 10.1016/j.na.2009.01.220. Google Scholar

[16]

I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon, 1991. Google Scholar

[17]

F. C. Kong, Positive piecewise pseudo almost periodic solutions of a generalized hematopoiesis model with harvesting terms and impulses, Fixed Point Theory, to appear.Google Scholar

[18]

F. C. KongX. W. Fang and Z. T. Liang, Dynamic behavior of a class of neutral-type neural networks with discontinuous activations and time-varying delays, Applied Intelligence, 48 (2018), 4834-4854. doi: 10.1007/s10489-018-1240-0. Google Scholar

[19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambrige University Press, 1982. Google Scholar
[20]

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control system, Sciences, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar

[21]

J. X. Meng, Global exponential stability of positive pseudo-almost-periodic solutions for a model of hematopoiesis, Abstract and Applied Analysis, 2013 (2013), Art. ID 463076, 7 pp. doi: 10.1155/2013/463076. Google Scholar

[22]

S. H. Saker, Oscillation and global attractivity in Hematopoiesis model with periodic coefficients, Applied Mathematics and Computation, 142 (2003), 477-494. doi: 10.1016/S0096-3003(02)00315-6. Google Scholar

[23]

X. Wang and Z. X. Li, Dynamics for a class of general hematopoiesis model with periodic coefficients, Applied Mathematics and Computation, 186 (2007), 460–468. doi: 10.1016/j.amc.2006.07.109. Google Scholar

[24]

J. F. WangL. H. Huang and Z. Y. Guo, Dynamical behaviors of delayed Hopfield neural networks with discontinuous activations, Applied Mathematical Modelling, 33 (2009), 1793-1802. doi: 10.1016/j.apm.2008.03.023. Google Scholar

[25]

D. S. Wang and L. H. Huang, Almost periodic dynamical behaviors for generalized Cohen-Grossberg neural networks with discontinuous activations via differential inclusions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3857-3879. doi: 10.1016/j.cnsns.2014.02.016. Google Scholar

[26]

P. X. Weng, Global attractivity of periodic solution in a model of hematopoiesis, Computers and Mathematics with Applications, 44 (2002), 1019-1030. doi: 10.1016/S0898-1221(02)00211-0. Google Scholar

[27]

X. M. WuJ. W. Li and H. Q. Zhou, A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis, Computers and Mathematics with Applications, 54 (2007), 840-849. doi: 10.1016/j.camwa.2007.03.004. Google Scholar

[28]

Z. N. Xia and D. J. Wang, Pseudo-almost periodic solution for impulsive hematopoiesis model with infinite delays and linear harvesting term, International Jounal of Biomathematics, 9 (2016), 1650078, 17pp. doi: 10.1142/S1793524516500789. Google Scholar

[29]

Z. J. Yao, New results on existence and exponential stability of the unique positive almost periodic solution for hematopoiesis model, Applied Mathematical Modelling, 39 (2015), 7113-7123. doi: 10.1016/j.apm.2015.03.003. Google Scholar

[30]

H. ZhangM. Q. Yang and L. J. Wang, Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis, Applied Mathematics Letters, 26 (2013), 38-42. doi: 10.1016/j.aml.2012.02.034. Google Scholar

[31]

H. Zhang, New results on the positive pseudo almost periodic solutions for a generalized model of hematopoiesis, Electronic Journal of Qualitative Theory of Differential Equations, 24 (2014), 1-10. doi: 10.14232/ejqtde.2014.1.24. Google Scholar

[32]

H. Zhou, W. Wang and Z. F. Zhou, Positive almost periodic solution for a model of hematopoiesis with infinite time delays and a nonlinear harvesting term, Abstract and Applied Analysis, 2013 (2013), Art. ID 146729, 6 pp. doi: 10.1155/2013/146729. Google Scholar

[33]

H. Zhou and L. Yang, A new result on the existence of positive almost periodic solution for generalized hematopoiesis model, Journal of Mathematical Analysis and Applications, 462 (2018), 370-379. doi: 10.1016/j.jmaa.2018.01.075. Google Scholar

show all references

References:
[1]

M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, 2010. doi: 10.1007/978-1-4419-6581-3. Google Scholar

[2]

J. O. Alzabut, J. J. Nieto and G. T. Stamov, Existence and exponential stability of positive almost periodic solutions for a model of hematopoiesis, Boundary Value Problems, 2009 (2009), Art. ID 127510, 10 pp. doi: 10.1155/2009/127510. Google Scholar

[3]

J. O. Alzabut, Almost periodic solutions for an impulsive delay Nicholson's blowflies model, Journal of Computational and Applied Mathematics, 234 (2010), 233-239. doi: 10.1016/j.cam.2009.12.019. Google Scholar

[4]

J. Aubin and A. Cellina, Differential Inclusions, Berlin: Springer-Verlag, 1984. doi: 10.1007/978-3-642-69512-4. Google Scholar

[5]

Z. W. CaiL. H. HuangZ. Y. Guo and X. Y. Chen, On the periodic dynamics of a class of time-varying delayed neural networks via differential inclusions, Neural Networks, 33 (2012), 97-113. doi: 10.1016/j.neunet.2012.04.009. Google Scholar

[6]

F. H. Clarke, Optimization and Nonsmooth Analysis, New York: Wiley, 1983. Google Scholar

[7]

C. Corduneanu, Almost Periodic Functions, 2nd edition, Chelsea, New york, 1989.Google Scholar

[8]

T. Diagana and H. Zhou, Existence of positive almost periodic solutions to the hematopoiesis model, Applied Mathematics and Computation, 274 (2016), 644-648. doi: 10.1016/j.amc.2015.10.029. Google Scholar

[9]

H. S. DingG. M. N'Guérékata and J. J. Nieto, Weighted pseudo almost periodic solutions for a class of discrete hematopoiesis model, Revista Matematica Complutense, 26 (2013), 427-443. doi: 10.1007/s13163-012-0114-y. Google Scholar

[10]

H. S. DingQ. L. Liu and J. J. Nieto, Existence of positive almost periodic solutions to a class of hematopoiesis model, Applied Mathematical Modelling, 40 (2016), 3289-3297. doi: 10.1016/j.apm.2015.10.020. Google Scholar

[11]

L. Duan, L. H. Huang and X. W. Fang, Finite-time synchronization for recurrent neural networks with discontinuous activations and time-varying delays, Chaos, 27 (2017), 013101, 10pp. doi: 10.1063/1.4966177. Google Scholar

[12]

A. F. Filippov, Mathematics and Its Applications (Soviet Series), Differential equations with discontinuous right-hand sides, Boston: Kluwer Academic Publishers, 1988. doi: 10.1007/978-94-015-7793-9. Google Scholar

[13]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. Google Scholar

[14]

M. FortiP. Nistri and D. Papini, Global convergence of neural networks with discontinuous neuron activations, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 50 (2003), 1421-1435. doi: 10.1109/TCSI.2003.818614. Google Scholar

[15]

Z. Y. Guo and L. H. Huang, Generalized Lyapunov method for discontinuous systems, Nonlinear Analysis, 71 (2009), 3083-3092. doi: 10.1016/j.na.2009.01.220. Google Scholar

[16]

I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon, 1991. Google Scholar

[17]

F. C. Kong, Positive piecewise pseudo almost periodic solutions of a generalized hematopoiesis model with harvesting terms and impulses, Fixed Point Theory, to appear.Google Scholar

[18]

F. C. KongX. W. Fang and Z. T. Liang, Dynamic behavior of a class of neutral-type neural networks with discontinuous activations and time-varying delays, Applied Intelligence, 48 (2018), 4834-4854. doi: 10.1007/s10489-018-1240-0. Google Scholar

[19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambrige University Press, 1982. Google Scholar
[20]

M. C. Mackey and L. Glass, Oscillations and chaos in physiological control system, Sciences, 197 (1977), 287-289. doi: 10.1126/science.267326. Google Scholar

[21]

J. X. Meng, Global exponential stability of positive pseudo-almost-periodic solutions for a model of hematopoiesis, Abstract and Applied Analysis, 2013 (2013), Art. ID 463076, 7 pp. doi: 10.1155/2013/463076. Google Scholar

[22]

S. H. Saker, Oscillation and global attractivity in Hematopoiesis model with periodic coefficients, Applied Mathematics and Computation, 142 (2003), 477-494. doi: 10.1016/S0096-3003(02)00315-6. Google Scholar

[23]

X. Wang and Z. X. Li, Dynamics for a class of general hematopoiesis model with periodic coefficients, Applied Mathematics and Computation, 186 (2007), 460–468. doi: 10.1016/j.amc.2006.07.109. Google Scholar

[24]

J. F. WangL. H. Huang and Z. Y. Guo, Dynamical behaviors of delayed Hopfield neural networks with discontinuous activations, Applied Mathematical Modelling, 33 (2009), 1793-1802. doi: 10.1016/j.apm.2008.03.023. Google Scholar

[25]

D. S. Wang and L. H. Huang, Almost periodic dynamical behaviors for generalized Cohen-Grossberg neural networks with discontinuous activations via differential inclusions, Communications in Nonlinear Science and Numerical Simulation, 19 (2014), 3857-3879. doi: 10.1016/j.cnsns.2014.02.016. Google Scholar

[26]

P. X. Weng, Global attractivity of periodic solution in a model of hematopoiesis, Computers and Mathematics with Applications, 44 (2002), 1019-1030. doi: 10.1016/S0898-1221(02)00211-0. Google Scholar

[27]

X. M. WuJ. W. Li and H. Q. Zhou, A necessary and sufficient condition for the existence of positive periodic solutions of a model of hematopoiesis, Computers and Mathematics with Applications, 54 (2007), 840-849. doi: 10.1016/j.camwa.2007.03.004. Google Scholar

[28]

Z. N. Xia and D. J. Wang, Pseudo-almost periodic solution for impulsive hematopoiesis model with infinite delays and linear harvesting term, International Jounal of Biomathematics, 9 (2016), 1650078, 17pp. doi: 10.1142/S1793524516500789. Google Scholar

[29]

Z. J. Yao, New results on existence and exponential stability of the unique positive almost periodic solution for hematopoiesis model, Applied Mathematical Modelling, 39 (2015), 7113-7123. doi: 10.1016/j.apm.2015.03.003. Google Scholar

[30]

H. ZhangM. Q. Yang and L. J. Wang, Existence and exponential convergence of the positive almost periodic solution for a model of hematopoiesis, Applied Mathematics Letters, 26 (2013), 38-42. doi: 10.1016/j.aml.2012.02.034. Google Scholar

[31]

H. Zhang, New results on the positive pseudo almost periodic solutions for a generalized model of hematopoiesis, Electronic Journal of Qualitative Theory of Differential Equations, 24 (2014), 1-10. doi: 10.14232/ejqtde.2014.1.24. Google Scholar

[32]

H. Zhou, W. Wang and Z. F. Zhou, Positive almost periodic solution for a model of hematopoiesis with infinite time delays and a nonlinear harvesting term, Abstract and Applied Analysis, 2013 (2013), Art. ID 146729, 6 pp. doi: 10.1155/2013/146729. Google Scholar

[33]

H. Zhou and L. Yang, A new result on the existence of positive almost periodic solution for generalized hematopoiesis model, Journal of Mathematical Analysis and Applications, 462 (2018), 370-379. doi: 10.1016/j.jmaa.2018.01.075. Google Scholar

Figure 1.  Discontinuous harvesting term for system (5.1)
Figure 2.  Time-domain behavior of the state variables $ x_1 $ and $ x_2 $ for system (5.1) with random initial conditions
Figure 3.  Phase plane behavior of the state variables $ x_1 $ and $ x_2 $ for system (5.1)
Figure 4.  Three-dimensional trajectory of state variables $ x_1 $ and $ x_2 $ for system (5.1)
Figure 5.  Discontinuous harvesting term for system (5.2)
Figure 6.  Time-domain behavior of the state variables $ x_1 $ and $ x_2 $ for system (5.2) with random initial conditions
Figure 7.  Phase plane behavior of the state variables $ x_1 $ and $ x_2 $ for system (5.2)
Figure 8.  Three-dimensional trajectory of state variables $ x_1 $ and $ x_2 $ for system (5.2)
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