Skip to main content
SpringerLink
Log in

Existence and Convergence Dynamics of Pseudo Almost Periodic Solutions for Nicholson’s Blowflies Model with Time-Varying Delays and a Harvesting Term

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

In this paper, an Nicholson’s blowflies model with time-varying delays and a harvesting term is investigated. By applying the fixed point theorem, the properties of pseudo almost periodic function, inequality analysis technique and constructing appropriate Lyapunov functionals, we establish some new criteria for the existence and convergence dynamics of pseudo almost periodic solutions for the model. An illustrative example with its numerical simulation is presented to demonstrate the effectiveness of the derived results. Our results complement some previous studies.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  1. Nicholson, A.L.: An outline of the dynamics of animal populations. Aust. J. Zool. 2(1), 9–65 (1954)

    Article  Google Scholar 

  2. Gurney, W.S., Blythe, S.P., Nisbet, R.M.: Nicholson’s blowflies revisited. Nature 287, 17–21 (1980)

    Article  Google Scholar 

  3. Ding, X.H., Li, W.X.: Stability and bifurcation of numerical discretization Nicholson’s blowflies equation with delay. Discrete Dyn. Nat. Soc. 2006, 19413 (2016), 12 pages

    MathSciNet  MATH  Google Scholar 

  4. Kulenovic, M.R.S., Ladas, G., Sficas, Y.G.: Global attractivity in population dynamics. Comput. Math. Appl. 18(10–11), 925–928 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. So, J.W.H., Yu, J.S.: On the stability and uniform persistence of a discrete model of Nicholson’s blowflies. J. Math. Anal. Appl. 193(1), 233–244 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. Saker, S.H., Zhang, B.G.: Oscillation in a discrete partial delay Nicholson’s blowflies model. Math. Comput. Model. 36(9–10), 1021–1026 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Li, J.W., Du, C.X.: Existence of positive periodic solutions for a generalized Nicholson’s blowflies model. J. Comput. Appl. Math. 221(1), 226–233 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhang, B.G., Xu, H.X.: A note on the global attractivity of a discrete model of Nicholson’s blowflies. Discrete Dyn. Nat. Soc. 3, 51–55 (1999)

    Article  MATH  Google Scholar 

  9. Wei, J.J., Li, M.Y.: Hopf bifurcation analysis in a delayed Nicholson’s blowflies equation. Nonlinear Anal. 60(7), 1351–1367 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  10. Li, W.T., Fan, Y.H.: Existence and global attractivity of positive periodic solutions for the impulsive delay Nicholson’s blowflies model. J. Comput. Appl. Math. 201(1), 55–68 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Saker, S.H., Agarwal, S.: Oscillation and global attractivity in a periodic Nicholson’s blowflies model. Math. Comput. Model. 35(7), 719–731 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Liu, B.W.: Global dynamic behaviors for a delayed Nicholson’s blowflies model with a linear harvesting term. Electron. J. Qual. Theory Differ. Equ. 45, 1–13 (2013)

    MATH  Google Scholar 

  13. Faria, T.: Global asymptotic behaviour for a Nicholson model with path structure and multiple delays. Nonlinear Anal. 74(18), 7033–7046 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hien, L.V.: Global asymptotic behaviour of positive solutions to a non-autonomous Nicholson’s blowflies model with delays. J. Biol. Dyn. 8(1), 135–144 (2014)

    Article  MathSciNet  Google Scholar 

  15. Alzabut, J.O.: Almost periodic solutions for an impulsive delay Nicholson’s blowflies model. J. Comput. Appl. Math. 234(1), 233–239 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  16. Amster, P., Deboli, A.: Existence of positive \(T\)-periodic solutions of a generalized Nicholson’s blowflies model with a nonlinear harvesting term. Appl. Math. Lett. 25(9), 1203–1207 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Zhou, Q.Y.: The positive periodic solution for Nicholson-type delay system with linear harvesting terms. Appl. Math. Model. 37(8), 5581–5590 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ding, H.S., Nieto, J.J.: A new approach for positive almost solutions to a class of Nicholson’s blowflies model. J. Comput. Appl. Math. 253, 249–254 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  19. Long, F.: Positive almost periodic solution for a class of Nicholson’s blowflies model with a linear harvesting term. Nonlinear Anal., Real World Appl. 13(2), 686–693 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tang, Y.: Global asymptotic stability for Nicholson’s blowflies model with a nonlinear density-dependent mortality term. Appl. Math. Comput. 250, 854–859 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Chen, W., Liu, B.: Positive almost periodic solution for a class of Nicholson’s blowflies model with multiple time-varying delays. J. Comput. Appl. Math. 235(8), 2090–2097 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  22. Alzabut, J.O.: Almost periodic solutions for an impulsive delay Nicholson’s blowflies model. J. Comput. Appl. Math. 234(1), 233–239 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Duan, L., Huang, L.H.: Pseudo almost periodic dynamics of delay Nicholson’s blowflies model with a linear harvesting term. Math. Methods Appl. Sci. 38(5), 1178–1189 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Liu, B.W., Tunç, C.: Pseudo almost periodic solutions for a class of nonlinear Duffing system with a deviating argument. J. Appl. Math. Comput. 49(1–2), 233–242 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liu, B.W., Tunç, C.: Pseudo almost periodic solutions for a class of first order differential iterative equations. Appl. Math. Lett. 40, 29–34 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  26. Khalil, E., Issa, Z.: Pseudo almost periodic solutions of infinite class for some functional differential equations. Appl. Anal. 92(8), 1627–1642 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhuang, R.K., Yuan, R.: The existence of pseudo-almost periodic solutions of third-order neutral differential equations with piecewise constant argument. Acta Math. Sin. 29(5), 943–958 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ezzinbi, K., Toure, H., Zabsonre, I.: Pseudo almost periodic solutions of class \(r\) for neutral partial functional differential equations. Afr. Math. 24(4), 691–704 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ding, H.S., Long, W., N’Guerekata, G.M.: Existence of pseudo almost periodic solutions for a class of partial functional differential equations. Electron. J. Differ. Equ. 2013(104), 1 (2013)

    MathSciNet  MATH  Google Scholar 

  30. Adimy, M., Ezzinbi, K., Marquet, C.: Ergodic and weighted pseudo-almost periodic solutions for partial functional differential equations in fading memory spaces. J. Appl. Math. Comput. 44(1–2), 147–165 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Dads, E.A., Ezzinbi, K.: Pseudo almost periodic solutions of some delay differential equations. J. Math. Anal. Appl. 201(3), 840–850 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. Meng, J.X.: Global exponential stability of positive pseudo-almost periodic solutions for a model of Hematopoiesis. Abstr. Appl. Anal. 2013, 463076 (2013), 7 pages

    MathSciNet  MATH  Google Scholar 

  33. Chérif, F.: Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays. J. Appl. Math. Comput. 39(1–2), 235–251 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  34. Liu, B.W.: Pseudo almost periodic solution for CNNs with continuously distributed leakage delays. Neural Process. Lett. 42(1), 233–256 (2015)

    Article  Google Scholar 

  35. Cuevas, C., Sepulveda, A., Soto, H.: Almost periodic and pseudo-almost periodic solutions to fractional differential and integro-differential equation. Appl. Math. Comput. 218(5), 1735–1745 (2011)

    MathSciNet  MATH  Google Scholar 

  36. Dads, E.A., Lhachimi, L.: Pseudo almost periodic solutions for equation with piecewise constant argument. J. Math. Anal. Appl. 371(2), 842–854 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Diagana, T.: Existence and uniqueness of pseudo-almost periodic solutions to some classes of partial evolution equations. Nonlinear Anal. 66(2), 384–395 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  38. Pinto, M.: Pseudo-almost periodic solutions of neutral integral and differential equations with applications. Nonlinear Anal. 72(12), 4377–4383 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  39. Boukli-Hacene, N., Ezzinbi, K.: Weighted pseudo almost periodic solutions for some partial functional differential equations. Nonlinear Anal. 71(9), 3612–3621 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  40. Agarwal, R.P., Andrade, B., Cuevas, C.: Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations. Nonlinear Anal., Real World Appl. 11(5), 3532–3554 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  41. Hernández, E., Henriquez, H.: Pseudo-almost periodic solutions for non-autonomous neutral differential equations with unbounded delay. Nonlinear Anal., Real World Appl. 9(2), 430–437 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  42. Wang, W.T., Liu, B.W.: Global exponential stability of pseudo almost periodic solutions for SICNNs with time-varying leakage delays. Abstr. Appl. Anal. 2014, 967328 (2014), 17 pages

    MathSciNet  Google Scholar 

  43. Zhang, C.: Almost Periodic Type Functions and Ergodicity. Science Press, Beijing (2003)

    Book  MATH  Google Scholar 

  44. Fink, A.M.: Almost Periodic Differential Equations. Lecture Notes in Mathematics, vol. 377. Springer, Berlin (1974)

    MATH  Google Scholar 

  45. Zhang, H., Yang, M.: Global exponential stability of almost periodic solutions for SICNNs with continuously distributed leakage delays. Abstr. Appl. Anal. 2013, 307981 (2013), 14 pages

    MathSciNet  MATH  Google Scholar 

  46. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    Book  MATH  Google Scholar 

  47. Amir, B., Maniar, L.: Composition of pseudo almost periodic functions and Cauchy problems with operator nondense domain. Ann. Math. Blaise Pascal 6(1), 1–11 (1999)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, 550004, PR China

    Changjin Xu

  2. School of Mathematics and Physics, University of South China, Hengyang, 421001, PR China

    Maoxin Liao

  3. School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, 550004, PR China

    Yicheng Pang

Authors
  1. Changjin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maoxin Liao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yicheng Pang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Changjin Xu.

Additional information

This work is supported by National Natural Science Foundation of China (No. 11261010 and No. 11526063), Natural Science and Technology Foundation of Guizhou Province (J[2015]2025 and J[2015]2026), 125 Special Major Science and Technology of Department of Education of Guizhou Province ([2012]011) and Natural Science Foundation of the Education Department of Guizhou Province (KY[2015]482).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Xu, C., Liao, M. & Pang, Y. Existence and Convergence Dynamics of Pseudo Almost Periodic Solutions for Nicholson’s Blowflies Model with Time-Varying Delays and a Harvesting Term. Acta Appl Math 146, 95–112 (2016). https://doi.org/10.1007/s10440-016-0060-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-016-0060-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation