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Stability of Almost Periodic Nicholson’s Blowflies Model Involving Patch Structure and Mortality Terms

Part of: Qualitative theory Functional-differential and differential-difference equations

Published online by Cambridge University Press:  23 December 2019

Chuangxia Huang ,
Xin Long ,
Lihong Huang  and
Si Fu
Chuangxia Huang
Affiliation:
School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha410114, Hunan, P R China Email: cxiahuang@amss.ac.cnlhhuang@csust.edu.cn
Xin Long
Affiliation:
School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha410114, Hunan, P R China Email: cxiahuang@amss.ac.cnlhhuang@csust.edu.cn
Lihong Huang
Affiliation:
School of Mathematics and Statistics, Changsha University of Science and Technology, Hunan Provincial Key Laboratory of Mathematical Modeling and Analysis in Engineering, Changsha410114, Hunan, P R China Email: cxiahuang@amss.ac.cnlhhuang@csust.edu.cn
Si Fu
Affiliation:
College of Mathematics and Information Science, Jiangxi Normal University, Jiangxi330022, Nanchang, P R China
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Abstract

Taking into account the effects of patch structure and nonlinear density-dependent mortality terms, we explore a class of almost periodic Nicholson’s blowflies model in this paper. Employing the Lyapunov function method and differential inequality technique, some novel assertions are developed 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 recently published literatures. Particularly, an example and its numerical simulations are arranged to support the proposed approach.

Keywords

Nicholson’s blowflies modelpatch structuredensity-dependent mortality termalmost periodic solutionstability

MSC classification

Primary: 34C25: Periodic solutions
Secondary: 34K13: Periodic solutions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 405 - 422
Copyright
© Canadian Mathematical Society 2019

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Footnotes

This work was supported by the National Natural Science Foundation of China (Nos. 11971076, 11861037, 11771059, 51839002), the Scientific Research Fund of Hunan Provincial Education Department (No. 16C0036). Chuangxia Huang and Lihong Huang are the corresponding authors.

References

Berezansky, L., Braverman, E., and Idels, L., Nicholson’s blowflies differential equations revisited: Main results and open problems. Appl. Math. Model. 34(2010), 1405–1417. https://doi.org/10.1016/j.apm.2009.08.027CrossRefGoogle Scholar
Chen, W., Permanence for Nicholson-type delay systems with patch structure and nonlinear density-dependent mortality terms. Electron. J. Qual. Theory Differ. Equ. 73(2012), 1–14. https://doi.org/10.14232/ejqtde.2012.1.73Google Scholar
Chen, W. and Wang, W., Almost periodic solutions for a delayed Nicholsons blowflies system with nonlinear density-dependent mortality terms and patch structure. Adv. Difference Equ. 2014 205, 1–19. https://doi.org/10.1186/1687-1847-2014-205Google Scholar
Diagana, T., Pseudo almost periodic functions in Banach spaces. Nova Science Publishers, New York, 2007.Google Scholar
Duan, L., Fang, X., and Huang, C., Global exponential convergence in a delayed almost periodic nicholsons blowflies model with discontinuous harvesting. Math. Methods Appl. Sci. 41(2017), 1954–1965. https://doi.org/10.1002/mma.4722CrossRefGoogle Scholar
Duan, L., Huang, L., Guo, Z., and Fang, X., Periodic attractor for reaction diffusion high-order hopfield neural networks with time-varying delays. Comput. Math. Appl. 73(2017), 233–245. https://doi.org/10.1016/j.camwa.2016.11.010CrossRefGoogle Scholar
Fink, A. M., Almost periodic differential equations, Lecture Notes in Mathematics, Vol. 377, Springer-Verlag, Berlin-New York, 1974.CrossRefGoogle Scholar
Hale, J. K. and Verduyn Lunel, S. M., Introduction to functional differential equations. In: Applied Mathematical Sciences. Vol. 99, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4612-4342-7Google Scholar
Hu, H., Yi, T., and Zou, X., On spatial-temporal dynamics of Fisher-KPP equation with a shifting environment. Proc. Amer. Math. Soc. 2019. https://doi.org/10.1090/proc/14659Google Scholar
Huang, C., Liu, B., Tian, X., Yang, L., and Zhang, X., Global convergence on asymptotically almost periodic SICNNs with nonlinear decay functions. Neural Processing Letters 49(2019), 625–641. https://doi.org/10.1007/s11063-018-9835-3CrossRefGoogle Scholar
Huang, C., Qiao, Y., Huang, L., and Agarwal, R., Dynamical behaviors of a food-chain model with stage structure and time delays. Adv. Difference Equ. 186(2018), 1–26. https://doi.org/10.1186/s13662-018-1589-8Google Scholar
Huang, C., Yang, Z., Yi, T., and Zou, X., On the basins of attraction for a class of delay differential equations with non-monotone bistable nonlinearities. J. Differential Equations 256(2014), 2101–2114. https://doi.org/10.1016/j.jde.2013.12.015CrossRefGoogle Scholar
Huang, C., Zhang, H., Cao, J., and Hu, H., Stability and Hopf bifurcation of a delayed prey-predator model with disease in the predator. Internat. J. Bifur. Chaos Appl. Sci. Engrg. 29(2019), 1950091, 23. https://doi.org/10.1142/S0218127419500913Google Scholar
Huang, C., Zhang, H., and Huang, L., Almost periodicity analysis for a delayed Nicholson’s blowflies model with nonlinear density-dependent mortality term. Commun. Pure Appl. Anal. 18(2019), 3337–3349.CrossRefGoogle Scholar
Li, Y., Zhang, T., and Ye, Y., On the existence and stability of a unique almost periodic sequence solution in discrete predator-prey models with time delays. Appl. Math. Model. 35(2011), 5448–5459. https://doi.org/10.1016/j.apm.2011.04.034CrossRefGoogle Scholar
Liu, B., Almost periodic solutions for a delayed Nicholsons blowflies model with a nonlinear density-dependent mortality term. Adv. Difference Equ. 2014 72, 1–16. https://doi.org/10.1186/1687-1847-2014-72Google Scholar
Liu, B., Global exponential stability of positive periodic solutions for a delayed Nicholsons blowflies model. J. Math. Anal. Appl. 412(2014), 212–221. https://doi.org/10.1016/j.jmaa.2013.10.049CrossRefGoogle Scholar
Long, X. and Gong, S., New results on stability of Nicholson’s blowflies equation with multiple pairs of time-varying delays. Appl. Math. Lett. 100(2020), 106027. https://doi.org/10.1016/j.aml.2019.106027CrossRefGoogle Scholar
Smith, H. L., An introduction to delay differential equations with applications to the life sciences. Springer, New York, 2011. https://doi.org/10.1007/978-1-4419-7646-8CrossRefGoogle Scholar
Son, D. T., Hien, L. V., and Anh, T. T., Global attractivity of positive periodic solution of a delayed Nicholson model with nonlinear density-dependent mortality term. J. Qual. Theory Differ. Equ. 2019 8, 1–21.Google Scholar
Tan, Y., Huang, C., Sun, B., and Wang, T., Dynamics of a class of delayed reaction-diffusion systems with Neumann boundary condition. J. Math. Anal. Appl. 458(2018), 1115–1130.CrossRefGoogle Scholar
Tang, Y. and Xie, S., Global attractivity of asymptotically almost periodic Nicholson’s blowflies models with a nonlinear density-dependent mortality term. Int. J. Biomath. 11(2018), 1850079, 15. https://doi.org/10.1142/S1793524518500791CrossRefGoogle Scholar
Xu, Y., Existence and global exponential stability of positive almost periodic solutions for a delayed Nicholson’s blowflies model. J. Korean Math. Soc. 51(2014), 473–493. https://doi.org/10.4134/JKMS.2014.51.3.473CrossRefGoogle Scholar
Xu, Y., New stability theorem for periodic Nicholson’s model with mortality term. Appl. Math. Lett. 94(2019), 59–65. https://doi.org/10.1016/j.aml.2019.02.021CrossRefGoogle Scholar
Yao, L., Dynamics of Nicholson’s blowflies models with a nonlinear density-dependent mortality. Appl. Math. Model. 64(2018), 185–195. https://doi.org/10.1016/j.apm.2018.07.007CrossRefGoogle Scholar
Yang, Z., Huang, C., and Zou, X., Effect of impulsive controls in a model system for age-structured population over a patchy environment. J. Math. Biol. 76(2018), 1387–1419. https://doi.org/10.1007/s00285-017-1172-zCrossRefGoogle Scholar
Zhang, C., Almost periodic type functions and ergodicity. Science Press Beijing, Beijing; Kluwer Academic Publishers, Dordrecht, 2003. https://doi.org/10.1007/978-94-007-1073-3CrossRefGoogle Scholar