Skip to main content
SpringerLink
Log in

Dynamical Analysis of a Food Chain System with Two Delays

  • Published:
Qualitative Theory of Dynamical Systems Aims and scope Submit manuscript

Abstract

This paper is concerned with a three species food chain system with the Beddington–DeAngelis functional response and two delays. The local stability of the positive equilibrium and the existence of periodic solutions via Hopf bifurcation with respect to both delays at the positive equilibrium are established by analyzing the distribution of the roots of the associated characteristic equation. Further, the properties of the bifurcating periodic solutions such as the direction and the stability are determined by using the normal form method and center manifold argument. Numerical simulations are presented for supporting the analytical results.

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
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Liu, S.Q., Zhang, J.H.: Coexistence and stability of predator–prey model with Beddington–DeAngelis functional response and stage structure. J. Math. Anal. Appl 342(1), 446–460 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Liu, M., Wang, K.: Global stability of stage-structured predator–prey models with Beddington–DeAngelis functional response. Commun. Nolinear. Sci. Numer. Simulat. 16(9), 3791–3797 (2011)

    MathSciNet  MATH  Google Scholar 

  3. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44(1), 331–341 (1975)

    Article  Google Scholar 

  4. Zhang, J.F.: Bifurcation analysis of a modified Holling–Tanner predator–prey model with time delay. Appl. Math. Model. 36(3), 1219–1231 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Li, W.L., Wang, L.S.: Stability and bifurcation of a delayed three-level food chain model with Beddington–DeAngelis functional response. Nonlinear Anal. Real World Appl. 10(4), 2471–2477 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Jana, S., Kar, T.K.: Modeling and analysis of a prey–predator system with disease in the prey. Chaos Solitons Fractals 47, 42–53 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lian, D.: Periodic solutions for a neutral delay predator–prey model with nonmonotonic functional response. Electron. J. Qual. Theor. Diff. Equ. 2012(48), 1–15 (2012)

    MathSciNet  MATH  Google Scholar 

  8. Bianca, C., Guerrini, L.: On the Dalgaard–Strulik model with logistic population growth rate and delayed carrying capacity. Acta Appl. Math. 128(1), 39–48 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bianca, C., Ferrara, M., Guerrini, L.: Qualitative analysis of a retarded mathematical framework with applications to living systems. Abstract and Applied Analysis, vol. 2013, Article ID 736058, pp. 7 (2013)

  10. Bianca, C., Ferrara, M., Guerrini, L.: The Cai model with time delay: existence of periodic solutions and asymptotic analysis. Appl. Math. Inf. Sci. 7(1), 21–27 (2013)

    Article  MathSciNet  Google Scholar 

  11. Kar, T.K., Ghorai, A.: Dynamic behaviour of a delayed predator–prey model with harvesting. Appl. Math. Comput. 217(15), 9085–9104 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  12. Song, Y.L., Han, M.A., Peng, Y.H.: Stability and Hopf bifurcations in a competitive Lotka–Volterra system with two delays. Chaos Solitons Fractals 22(5), 1139–1148 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. Meng, X.Y., Huo, H.F., Xiang, H.: Hopf bifurcation in a three-species system with delays. J. Appl. Math. Comput. 35(1–2), 635–661 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhang, Z.Z., Yang, H.Z., Liu, J.: Stability and Hopf bifurcation in a modified Holling–Tanner predator–prey system with multiple delays. Abstract and Applied Analysis, vol. 2012, Aarticle ID 236484, pp. 19 (2012)

  15. Xu, C.J., Tang, X.H., Liao, M.X., He, X.F.: Bifurcation analysis in a delayed Lokta–Volterra predator–prey model with two delays. Nonlinear Dyn. 66(3–4), 169–183 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Cui, G.H., Yan, X.P.: Stability and bifurcation analysis on a three-species food chain system with two delays. Commun. Nonlinear Sci. Numer. Simulat. 16(9), 3704–3720 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gakkhar, S., Singh, A.: Complex dynamics in a prey predator system with multiple delays. Commun. Nonlinear Sci. Numer. Simulat. 17(2), 914–929 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Liao, M.X., Xu, C.J., Tang, X.H.: Dynamical behaviors for a competition and cooperation model of enterprises with two delays. Nonlinear Dyn. 75(1), 257–266 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Meng, X.Y., Huo, H.F., Zhang, X.B., Xiang, H.: Stability and Hopf bifurcation in a three-species system with feedback delays. Nolinear Dyn. 64(4), 349–364 (2011)

    Article  MathSciNet  Google Scholar 

  20. Hassard, B.D., Kazarinoff, N.D., Wan, Y.H.: Theory and Applications of Hopf Bifurcation. Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank the anonymous referees for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improve the presentation of this work.

Author information

Authors and Affiliations

  1. Department of Mathematics and Physics, Bengbu College, Bengbu, 233030, People’s Republic of China

    Juan Liu & Lijun Sun

Authors
  1. Juan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lijun Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Juan Liu.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, J., Sun, L. Dynamical Analysis of a Food Chain System with Two Delays. Qual. Theory Dyn. Syst. 15, 95–126 (2016). https://doi.org/10.1007/s12346-015-0152-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12346-015-0152-1

Keywords

Search

Navigation