Skip to main content
SpringerLink
Log in

The Study of Global Stability of a Periodic Beddington–DeAngelis and Tanner Predator-Prey Model

  • Published:
Results in Mathematics Aims and scope Submit manuscript

Abstract

Of concern is the global dynamics of a Beddington–DeAngelis and Tanner type predator-prey model with periodic coefficients. By establishing sufficient conditions, we prove that, despite the high degree of coupling in general, the existence and global asymptotic stability of a positive solution. As pointwise estimate, the stability conditions, in average form, are first provided.

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

Similar content being viewed by others

References

  1. Ahmad, S., Stamova, I.M.: Lotka–Volterra and Related Systems: Recent Development in Population Dynamics. De Gruyter, Berlin (2013)

    Book  Google Scholar 

  2. Arditi, R., Ginzburg, L.R.: Coupling in predator-prey dynamics: ratio-dependence. J. Theor. Biol. 139, 311–326 (1989)

    Article  Google Scholar 

  3. Beddington, J.R.: Mutual interference between parasites or predators and its effect on searching efficiency. J. Anim. Ecol. 44, 331–340 (1975)

    Article  Google Scholar 

  4. Cantrell, R.S., Cosner, C.: On the dynamics of predator-prey models with the Beddington–DeAngelis functional response. J. Math. Anal. Appl. 257, 206–222 (2001)

    Article  MathSciNet  Google Scholar 

  5. Chen, S., Shi, J.: Global stability in a diffusive Holling–Tanner predator-prey model. Appl. Math. Lett. 25, 614–618 (2012)

    Article  MathSciNet  Google Scholar 

  6. DeAngelis, D.L., Goldstein, R.A., O’Neill, R.V.: A model for tropic interaction. Ecology 56, 881–892 (1975)

    Article  Google Scholar 

  7. Fan, Y.H., Li, W.T.: Global asymptotic stability of a ratio-dependent predator-prey system with diffusion. J. Comput. Appl. Math. 188, 205–227 (2006)

    Article  MathSciNet  Google Scholar 

  8. Fan, Y.H., Li, W.T.: Permanence in delayed ratio-dependent predator-prey models with monotonic functional responses. Nonlinear Anal. Real World Appl. 8, 424–434 (2007)

    Article  MathSciNet  Google Scholar 

  9. Fan, Y.H., Wang, L.L.: On a generalized discrete ratio-dependent predator-prey system. Discrete Dyn. Nat. Soc. 2009(3), 332–337 (2009)

    MathSciNet  Google Scholar 

  10. Fan, Y.H., Wang, L.L.: Multiplicity of periodic solutions for a delayed ratio-dependent predator-prey model with Holling type III functional response and harvesting terms. J. Math. Anal. Appl. 365, 525–540 (2010)

    Article  MathSciNet  Google Scholar 

  11. Fan, Y.H., Wang, L.L.: Average conditions for the permanence of a bounded discrete predator-prey system. Discrete Dyn. Nat. Soc. 2013(3), 1375–1383 (2013)

    MathSciNet  Google Scholar 

  12. Freedman, H.I., Mathsen, R.M.: Persistence in predator-prey systems with ratio-dependent predator influence. Bull. Math. Biol. 55(4), 817–827 (1993)

    Article  Google Scholar 

  13. Haque, M.: A detailed study of the Beddington–DeAngelis predator-prey model. Math. Biosci. 234, 1–16 (2011)

    Article  MathSciNet  Google Scholar 

  14. Holling, C.S.: The functional response of predator to prey density and its role in mimicry and population regulations. Mem. Entomol. Soc. Can. 45, 1–60 (1965)

    Google Scholar 

  15. Hwang, T.W.: Global analysis of the predator-prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 281, 395–401 (2003)

    Article  MathSciNet  Google Scholar 

  16. Hwang, T.W.: Uniqueness of limit cycles of the predator-prey system with Beddington–DeAngelis functional response. J. Math. Anal. Appl. 290, 113–122 (2004)

    Article  MathSciNet  Google Scholar 

  17. Lee, J., Baek, H.: Dynamics of a Beddington–DeAngelis-type predator-prey system with constant rate harvesting. Electron. J. Qual. Theory Differ. Equ. 2017(1), 1–20 (2017)

    Article  MathSciNet  Google Scholar 

  18. Ling, L., Wang, W.: Dynamics of a Ivlev-type predator-prey system with constant rate harvesting. Chaos Solitons Fractals 41(4), 2139–2153 (2009)

    Article  MathSciNet  Google Scholar 

  19. Lisena, B.: Global stability of a periodic Holling–Tanner predator-prey model. Math. Methods Appl. Sci. 44(9), 3270–3281 (2018)

    Article  MathSciNet  Google Scholar 

  20. Liu, N., Li, N.: Global stability of a predator-prey model with Beddington–DeAngelis and Tanner functional response. Electron. J. Qual. Theory Differ. Equ. 2017(35), 1–8 (2017)

    Article  MathSciNet  Google Scholar 

  21. Liu, B., Zhang, Y., Chen, L.: Dynamic complexities in a Lotka–Volterra predator-prey model concerning impulsive control strategy. Int. J. Bifurcat. Chaos Appl. Sci. Eng. 15(2), 517–531 (2005)

    Article  MathSciNet  Google Scholar 

  22. Liu, B., Zhang, Y., Chen, L.: The dynamical behaviors of a Lotka–Volterra predator-prey model concerning integrate pest management. Nonlinear Anal. Real World Appl. 6, 227–243 (2005)

    Article  MathSciNet  Google Scholar 

  23. Shi, H.B., Li, Y.: Global asymptotic stability of a diffusive predator-prey model with ratio-dependent functional response. Appl. Math. Comput. 25, 71–77 (2015)

    MathSciNet  MATH  Google Scholar 

  24. Skalski, G.T., Gilliam, J.F.: Functional responses with predator interference: viable alternatives to the Holling type II mode. Ecology 82, 3083–3092 (2001)

    Article  Google Scholar 

  25. Song, X., Neumann, A.: Global stability and periodic solution of the viral dynamics. J. Math. Anal. Appl. 329, 281–297 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

I would like to take the opportunity to express my sincere thanks to my friend from School of Economics and Commerce, Guangdong University of Technology, Tian Fang, for her smile and constant encouragement. The author also thank the referees for their careful reading of the paper and their useful reports which helped improve the paper.

Author information

Authors and Affiliations

  1. School of Mathematics, Sun Yat-sen University, Guangzhou, 510275, People’s Republic of China

    Demou Luo

Authors
  1. Demou Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Demou Luo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported in part by the National Natural Science Foundation of China under Grant No. 61673121, in part by the Natural Science Foundation of Guangdong Province under Grant No. 2014A030313507, and in part by the Projects of Science and Technology of Guangzhou under Grant No. 201508010008.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Luo, D. The Study of Global Stability of a Periodic Beddington–DeAngelis and Tanner Predator-Prey Model. Results Math 74, 101 (2019). https://doi.org/10.1007/s00025-019-1016-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00025-019-1016-9

Keywords

Mathematics Subject Classification

Search

Navigation