# American Institute of Mathematical Sciences

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  November 2019 Early access  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 and 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. [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. [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. [4] J. Aubin and A. Cellina, Differential Inclusions, Berlin: Springer-Verlag, 1984. doi: 10.1007/978-3-642-69512-4. [5] Z. W. Cai, L. H. Huang, Z. 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. [6] F. H. Clarke, Optimization and Nonsmooth Analysis, New York: Wiley, 1983. [7] C. Corduneanu, Almost Periodic Functions, 2nd edition, Chelsea, New york, 1989. [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. [9] H. S. Ding, G. 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. [10] H. S. Ding, Q. 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. [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. [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. [13] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. [14] M. Forti, P. 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. [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. [16] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon, 1991. [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. [18] F. C. Kong, X. 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. [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambrige University Press, 1982. [20] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control system, Sciences, 197 (1977), 287-289.  doi: 10.1126/science.267326. [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. [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. [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. [24] J. F. Wang, L. 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. [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. [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. [27] X. M. Wu, J. 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. [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. [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. [30] H. Zhang, M. 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. [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. [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. [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.

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##### References:
 [1] M. Akhmet, Principles of Discontinuous Dynamical Systems, Springer, 2010. doi: 10.1007/978-1-4419-6581-3. [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. [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. [4] J. Aubin and A. Cellina, Differential Inclusions, Berlin: Springer-Verlag, 1984. doi: 10.1007/978-3-642-69512-4. [5] Z. W. Cai, L. H. Huang, Z. 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. [6] F. H. Clarke, Optimization and Nonsmooth Analysis, New York: Wiley, 1983. [7] C. Corduneanu, Almost Periodic Functions, 2nd edition, Chelsea, New york, 1989. [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. [9] H. S. Ding, G. 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. [10] H. S. Ding, Q. 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. [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. [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. [13] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, vol. 377, Springer, Berlin, 1974. [14] M. Forti, P. 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. [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. [16] I. Györi and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon, 1991. [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. [18] F. C. Kong, X. 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. [19] B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambrige University Press, 1982. [20] M. C. Mackey and L. Glass, Oscillations and chaos in physiological control system, Sciences, 197 (1977), 287-289.  doi: 10.1126/science.267326. [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. [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. [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. [24] J. F. Wang, L. 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. [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. [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. [27] X. M. Wu, J. 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. [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. [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. [30] H. Zhang, M. 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. [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. [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. [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.
Discontinuous harvesting term for system (5.1)
Time-domain behavior of the state variables $x_1$ and $x_2$ for system (5.1) with random initial conditions
Phase plane behavior of the state variables $x_1$ and $x_2$ for system (5.1)
Three-dimensional trajectory of state variables $x_1$ and $x_2$ for system (5.1)
Discontinuous harvesting term for system (5.2)
Time-domain behavior of the state variables $x_1$ and $x_2$ for system (5.2) with random initial conditions
Phase plane behavior of the state variables $x_1$ and $x_2$ for system (5.2)
Three-dimensional trajectory of state variables $x_1$ and $x_2$ for system (5.2)
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