# American Institute of Mathematical Sciences

November  2019, 18(6): 3337-3349. doi: 10.3934/cpaa.2019150

## Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term

 1 School of Mathematics and Statistics, Changsha University of Science and Technology, Changsha 410114, Hunan, China 2 Hunan Provincial Key Laboratory of Mathematical Modeling, and Analysis in Engineering, Changsha 410114, Hunan, China

* Corresponding author

Received  October 2018 Revised  February 2019 Published  May 2019

Fund Project: The work is partially supported by the National Natural Science Foundation of China (Nos.71471020, 11771059, 51839002); Hunan Provincial Natural Science Foundation of China (No. 2016JJ1001); Scientific Research Fund of Hunan Provincial Education Department (Nos. 15A003, 16C0036).

This paper mainly investigates a class of almost periodic Nicholson's blowflies model involving a nonlinear density-dependent mortality term and time-varying delays. Combining Lyapunov function method and differential inequality approach, some novel assertions are established to guarantee the existence and exponential stability of positive almost periodic solutions for the addressed model, which generalize and refine the corresponding results in some recent published literatures. Particularly, an example and its numerical simulations are given to support the proposed approach.

Citation: Chuangxia Huang, Hua Zhang, Lihong Huang. Almost periodicity analysis for a delayed Nicholson's blowflies model with nonlinear density-dependent mortality term. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3337-3349. doi: 10.3934/cpaa.2019150
##### References:
 [1] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin, 1974. [2] C. Huang and H. Zhang, Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method, International Journal of Biomathematics, 12 (2019), 1950016. doi: 10.1142/S1793524519500165. [3] C. Zhang, Almost Periodic Type Functions and Ergodicity, Kluwer Academic/Science Press, Beijing, 2003.  doi: 10.1007/978-94-007-1073-3. [4] T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete & Continuous Dynamical Systems -A, 38 (2018), 4391-4419.  doi: 10.3934/dcds.2018191. [5] S. Chen and J. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete & Continuous Dynamical Systems -A, 38 (2018), 43-62.  doi: 10.3934/dcds.2018002. [6] T. Shibata, Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms, Communications on Pure & Applied Analysis, 17 (2018), 2139-2147.  doi: 10.3934/cpaa.2018102. [7] C. Huang, Y. Qiao, L. Huang and R. Agarwal, Dynamical behaviors of a food-chain model with stage structure and time delays, Advances in Difference Equations, 186 (2018), 1-26.  doi: 10.1186/s13662-018-1589-8. [8] Z. Yang, C. Huang and X. Zou, Effect of impulsive controls in a model system for age-structured population over a patchy environment, Journal of Mathematical Biology, 76 (2018), 1387-1419.  doi: 10.1007/s00285-017-1172-z. [9] C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, Journal of Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015. [10] B. Liu, Global exponential stability of positive periodic solutions for a delayed Nicholsons blowflies model, Journal of Mathematical Analysis and Applications, 412 (2014), 212-221.  doi: 10.1016/j.jmaa.2013.10.049. [11] B. Liu, New results on global exponential stability of almost periodic solutions for a delayed Nicholson blowflies model, Annales Polonici Mathematici, 113 (2015), 191-208.  doi: 10.4064/ap113-2-6. [12] W. Xiong, New results on positive pseudo-almost periodic solutions for a delayed Nicholson's blowflies model, Nonlinear Dynamics, 85 (2016), 563-571.  doi: 10.1007/s11071-016-2706-4. [13] Y. Xu, New stability theorem for periodic Nicholson's model with mortality term, Applied Mathematics Letters, (2019). doi: 10.1016/j.aml.2019.02.021. [14] B. Liu, Almost periodic solutions for a delayed Nicholson's blowflies model with a nonlinear density-dependent mortality term, Advances in Difference Equations, 72 (2014), 1-16.  doi: 10.1186/1687-1847-2014-72. [15] B. Liu, Positive periodic solutions for a nonlinear density-dependent mortality Nicholson's blowflies model, Kodai Mathematical Journal, 37 (2014), 157-173.  doi: 10.2996/kmj/1396008252. [16] L. Yao, Dynamics of Nicholson's blowflies models with a nonlinear density-dependent mortality, Applied Mathematical Modelling, 64 (2018), 185-195.  doi: 10.1016/j.apm.2018.07.007. [17] Y. Tang, Global attractivity of asymptotically almost periodic Nicholson's blowflies models with a nonlinear density-dependent mortality term, International Journal of Biomathematics, 11 (2018), 1850079. doi: 10.1142/S1793524518500791. [18] Y. Xu, Existence and global exponential stability of positive almost periodic solutions for a delayed Nicholson's blowflies model, Journal of the Korean Mathematical Society, 51 (2014), 473-493.  doi: 10.4134/JKMS.2014.51.3.473. [19] W. Chen and W. Wang, Almost periodic solutions for a delayed Nicholson's blowflies system with nonlinear density-dependent mortality terms and patch structure, Advances in Difference Equations, 205 (2014), 1-19.  doi: 10.1186/1687-1847-2014-205. [20] P. Liu, L. Zhang and et al, Global exponential stability of almost periodic solutions for Nicholson's Blowflies system with nonlinear density-dependent mortality terms and patch structure, Mathematical Modelling and Analysis, 22 (2017), 484-502.  doi: 10.3846/13926292.2017.1329171. [21] L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Applied Mathematical Modelling, 34 (2010), 1405-1417.  doi: 10.1016/j.apm.2009.08.027. [22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [23] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [24] L. Duan, X. Fang and C. Huang, Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting, Mathematical Methods in the Applied Sciences, 41 (2018), 1954-1965.  doi: 10.1002/mma.4722. [25] L. Duan and C. Huang, Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model, Mathematical Methods in the Applied Sciences, 40 (2017), 814-822.  doi: 10.1002/mma.4019.

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##### References:
 [1] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin, 1974. [2] C. Huang and H. Zhang, Periodicity of non-autonomous inertial neural networks involving proportional delays and non-reduced order method, International Journal of Biomathematics, 12 (2019), 1950016. doi: 10.1142/S1793524519500165. [3] C. Zhang, Almost Periodic Type Functions and Ergodicity, Kluwer Academic/Science Press, Beijing, 2003.  doi: 10.1007/978-94-007-1073-3. [4] T. M. Touaoula, Global stability for a class of functional differential equations (Application to Nicholson's blowflies and Mackey-Glass models), Discrete & Continuous Dynamical Systems -A, 38 (2018), 4391-4419.  doi: 10.3934/dcds.2018191. [5] S. Chen and J. Yu, Stability and bifurcation on predator-prey systems with nonlocal prey competition, Discrete & Continuous Dynamical Systems -A, 38 (2018), 43-62.  doi: 10.3934/dcds.2018002. [6] T. Shibata, Global behavior of bifurcation curves for the nonlinear eigenvalue problems with periodic nonlinear terms, Communications on Pure & Applied Analysis, 17 (2018), 2139-2147.  doi: 10.3934/cpaa.2018102. [7] C. Huang, Y. Qiao, L. Huang and R. Agarwal, Dynamical behaviors of a food-chain model with stage structure and time delays, Advances in Difference Equations, 186 (2018), 1-26.  doi: 10.1186/s13662-018-1589-8. [8] Z. Yang, C. Huang and X. Zou, Effect of impulsive controls in a model system for age-structured population over a patchy environment, Journal of Mathematical Biology, 76 (2018), 1387-1419.  doi: 10.1007/s00285-017-1172-z. [9] C. Huang, Z. Yang, T. Yi and X. Zou, On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities, Journal of Differential Equations, 256 (2014), 2101-2114.  doi: 10.1016/j.jde.2013.12.015. [10] B. Liu, Global exponential stability of positive periodic solutions for a delayed Nicholsons blowflies model, Journal of Mathematical Analysis and Applications, 412 (2014), 212-221.  doi: 10.1016/j.jmaa.2013.10.049. [11] B. Liu, New results on global exponential stability of almost periodic solutions for a delayed Nicholson blowflies model, Annales Polonici Mathematici, 113 (2015), 191-208.  doi: 10.4064/ap113-2-6. [12] W. Xiong, New results on positive pseudo-almost periodic solutions for a delayed Nicholson's blowflies model, Nonlinear Dynamics, 85 (2016), 563-571.  doi: 10.1007/s11071-016-2706-4. [13] Y. Xu, New stability theorem for periodic Nicholson's model with mortality term, Applied Mathematics Letters, (2019). doi: 10.1016/j.aml.2019.02.021. [14] B. Liu, Almost periodic solutions for a delayed Nicholson's blowflies model with a nonlinear density-dependent mortality term, Advances in Difference Equations, 72 (2014), 1-16.  doi: 10.1186/1687-1847-2014-72. [15] B. Liu, Positive periodic solutions for a nonlinear density-dependent mortality Nicholson's blowflies model, Kodai Mathematical Journal, 37 (2014), 157-173.  doi: 10.2996/kmj/1396008252. [16] L. Yao, Dynamics of Nicholson's blowflies models with a nonlinear density-dependent mortality, Applied Mathematical Modelling, 64 (2018), 185-195.  doi: 10.1016/j.apm.2018.07.007. [17] Y. Tang, Global attractivity of asymptotically almost periodic Nicholson's blowflies models with a nonlinear density-dependent mortality term, International Journal of Biomathematics, 11 (2018), 1850079. doi: 10.1142/S1793524518500791. [18] Y. Xu, Existence and global exponential stability of positive almost periodic solutions for a delayed Nicholson's blowflies model, Journal of the Korean Mathematical Society, 51 (2014), 473-493.  doi: 10.4134/JKMS.2014.51.3.473. [19] W. Chen and W. Wang, Almost periodic solutions for a delayed Nicholson's blowflies system with nonlinear density-dependent mortality terms and patch structure, Advances in Difference Equations, 205 (2014), 1-19.  doi: 10.1186/1687-1847-2014-205. [20] P. Liu, L. Zhang and et al, Global exponential stability of almost periodic solutions for Nicholson's Blowflies system with nonlinear density-dependent mortality terms and patch structure, Mathematical Modelling and Analysis, 22 (2017), 484-502.  doi: 10.3846/13926292.2017.1329171. [21] L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Applied Mathematical Modelling, 34 (2010), 1405-1417.  doi: 10.1016/j.apm.2009.08.027. [22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [23] H. Smith, An Introduction to Delay Differential Equations with Applications to the Life Sciences, Springer-Verlag, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [24] L. Duan, X. Fang and C. Huang, Global exponential convergence in a delayed almost periodic Nicholson's blowflies model with discontinuous harvesting, Mathematical Methods in the Applied Sciences, 41 (2018), 1954-1965.  doi: 10.1002/mma.4722. [25] L. Duan and C. Huang, Existence and global attractivity of almost periodic solutions for a delayed differential neoclassical growth model, Mathematical Methods in the Applied Sciences, 40 (2017), 814-822.  doi: 10.1002/mma.4019.
Numerical solutions $x(t)$ to example (4.1) with initial values: $2, 4, 6$
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