Skip to main content
SpringerLink
Log in

Analysis on Bifurcation for a Predator-Prey Model with Beddington-DeAngelis Functional Response and Non-Selective Harvesting

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

The biomathematical model is an important tool in exploring the dynamic behavior of different species. In this paper, we deal with a predator-prey model with Beddington-DeAngelis functional response and non-selective harvesting. We discuss the existence and stability of the local bifurcation solution, which emanates from the semi-trivial solution. Moreover, by using the boundedness of positive solutions and the global bifurcation theory, we find that the local bifurcation branch can be extended to the global bifurcation.

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. Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)

    Book  MATH  Google Scholar 

  2. Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91, 293–320 (1959)

    Article  Google Scholar 

  3. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. Mem. Entomol. Soc. Can. 97, 5–60 (1965)

    Article  Google Scholar 

  4. Holling, C.S.: The functional response of invertebrate predators to prey density. Mem. Entomol. Soc. Can. 98, 5–86 (1966)

    Article  Google Scholar 

  5. Li, C.: Global existence of solutions to a cross-diffusion predator-prey systems with Holling type-II functional response. Comput. Math. Appl. 65, 1152–1162 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Ko, W., Ryu, K.: Qualitative analysis of a predator-prey model with Holling type II functional response incorporating a prey refuge. J. Differ. Equ. 231, 534–550 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Jia, Y., Wu, J., Nie, H.: The coexistence states of a predator-prey model with nonmonotonic functional response and diffusion. Acta Appl. Math. 108, 413–428 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Zhou, J., Shi, J.: The existence, bifurcation and stability of positive stationary solutions of a diffusive Leslie-Gower predator-prey model with Holling-type II functional responses. J. Math. Anal. Appl. 405, 618–630 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ma, Z.-P., Li, W.-T.: Bifurcation analysis on a diffusion Holling-Tanner predator-prey model. Appl. Math. Model. 37, 4371–4384 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Xu, C., Liao, M.: Bifurcation behaviors in a delayed three-species food-chain model with Holling type-II functional response. Appl. Anal. 92, 2468–2486 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Prasad, B.S.R.V., Banergee, M., Srinivasu, P.D.N.: Dynamics of additional food provided predator-prey system with mutually interfering predator. Math. Biosci. 246, 176–190 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Shi, H.-B., Li, W.-T., Lin, G.: Positive steady states of a diffusive predator-prey system with modified Holling-Tanner functional response. Nonlinear Anal., Real World Appl. 11, 3711–3721 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fan, M., Kuang, Y.: Dynamics of a nonautonomous predator-prey system with the Beddington-DeAngelis functional response. J. Math. Anal. Appl. 295, 15–39 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  14. Zhou, J., Mu, C.: Positive solutions for a three-trophic food chain model with diffusion and Beddington-DeAngelis functional response. Nonlinear Anal., Real World Appl. 12, 902–917 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, W., Wang, M.: Qualitative analysis of predator-prey models with Beddington-DeAngelis functional response and diffusion. Math. Comput. Model. 42, 31–44 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Guo, G., Wu, J.: Multiplicity and uniqueness of positive solutions for a predator-prey model with B-D functional response. Nonlinear Anal. 72, 1632–1646 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  18. Chakraborty, K., Das, S., Kar, T.K.: On non-selective harvesting of a multispecies fishery incorporating partial closure for the populations. Appl. Math. Comput. 221, 581–597 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Chaudhuri, K.S., Saha, R.S.: On the combined harvesting of a prey-predator system. J. Biol. Syst. 4, 373–389 (1996)

    Article  Google Scholar 

  20. Fister, K.R.: Optimal control of harvesting coupled with boundary control in a predator-prey system. Appl. Anal. 77, 11–28 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kumar, S., Srivastava, S.K., Chingakham, P.: Hopf bifurcation and stability analysis in a harvested one-predator-two-prey model. Appl. Math. Comput. 129, 107–118 (2002)

    MathSciNet  MATH  Google Scholar 

  22. Sadhukhan, D., Sahoo, L.N., Mondal, B., Maiti, M.: Food chain model with optimal harvesting in fuzzy environment. J. Appl. Math. Comput. 34, 1–18 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Xiao, D., Ruan, S.: Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting. In: Differential Equations with Applications to Biology, Halifax, NS, 1997. Fields Inst. Commun., vol. 21, pp. 493–506. Am. Math. Soc., Providence (1999)

    Google Scholar 

  24. Jia, Y., Wu, J., Xu, H.-K.: On qualitative analysis for a two competing fish species model with a combined non-selective harvesting effort in the presence of toxicity. Commun. Pure Appl. Anal. 12, 1927–1941 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  25. Brown, K.J.: Nontrivial solutions of predator-prey systems with diffusion. Nonlinear Anal. 11, 685–689 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  26. Li, L.: Coexistence theorems of steady states for predator-prey interacting systems. Trans. Am. Math. Soc. 305, 143–166 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  27. Pao, C.V.: Nonlinear Parabolic and Elliptic Equations. Plenum Press, New York (1992)

    MATH  Google Scholar 

  28. Figueiredo, D.G., Gossez, G.P.: Strict monotonicity of eigenvalues and unique continuation. Commun. Partial Differ. Equ. 17, 339–346 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  29. Blat, J., Brown, K.J.: Global bifurcation of positive solutions in some systems of elliptic equations. SIAM J. Math. Anal. 17, 1339–1353 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  30. Smoller, J.: Shock Waves and Reaction-Diffusion Equation. Springer, New York (1983)

    Book  MATH  Google Scholar 

  31. Crandall, M.G., Rabinowitz, P.H.: Bifurcation from simple eigenvalues. J. Funct. Anal. 8, 321–340 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  32. López-Gómeza, J., Molina-Meyerb, M.: Bounded components of positive solutions of abstract fixed point equations: mushrooms, loops and isolas. J. Differ. Equ. 209, 416–441 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to the referees for their suggestions and comments which improved the presentation of this manuscript.

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xi’an, Shaanxi, 710062, China

    Rong Wang & Yunfeng Jia

Authors
  1. Rong Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yunfeng Jia
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yunfeng Jia.

Additional information

The work was supported in part by the National Science Foundation of China (11271236, 11401356), by the Program of New Century Excellent Talents in University of Ministry of Education of China (NCET-12-0894), by the Shaanxi New-star Plan of Science and Technology (2015KJXX-21), and also by the Natural Science Basic Research Plan in Shaanxi Province of China (2015JQ1023).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, R., Jia, Y. Analysis on Bifurcation for a Predator-Prey Model with Beddington-DeAngelis Functional Response and Non-Selective Harvesting. Acta Appl Math 143, 15–27 (2016). https://doi.org/10.1007/s10440-015-0025-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-015-0025-2

Keywords

Mathematics Subject Classification

Search

Navigation