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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

Chuangxia Huang 1,2,, , Hua Zhang 1,2, and  Lihong Huang 1,2,

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.

Keywords: Nicholson's blowflies model, density-dependent mortality term, almost periodic solution, stability.
Mathematics Subject Classification: 34C25; 34K13.
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 & 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.  Google Scholar

[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.  Google Scholar

[3] C. Zhang, Almost Periodic Type Functions and Ergodicity, Kluwer Academic/Science Press, Beijing, 2003.  doi: 10.1007/978-94-007-1073-3.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[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.  Google Scholar

[3] C. Zhang, Almost Periodic Type Functions and Ergodicity, Kluwer Academic/Science Press, Beijing, 2003.  doi: 10.1007/978-94-007-1073-3.  Google Scholar
[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Numerical solutions $ x(t) $ to example (4.1) with initial values: $ 2, 4, 6 $
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