Skip to main content
SpringerLink
Log in

Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

This paper discuss the cusp bifurcation of codimension 2 (i.e. Bogdanov-Takens bifurcation) in a Leslie-Gower predator-prey model with prey harvesting, which was not revealed by Zhu and Lan [Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates, Discrete and Continuous Dynamical Systems Series B. 14(1) (2010), 289–306]. It is shown that there are different parameter values for which the model has a limit cycle or a homoclinic loop.

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. Bogdanov, R. Bifurcations of a limit cycle for a family of vector fields on the plane. Selecta Math. Soviet., 1: 373–388 (1981)

    MATH  Google Scholar 

  2. Bogdanov, R. Versal deformations of a singular point on the plane in the case of zero eigen-values. Selecta Math. Soviet., 1: 389–421 (1988)

    Google Scholar 

  3. Broer H W, Naudot V, Roussarie R. Dynamics of a predator-prey model with non-monotonic response function. Discrete and Continuous Dynamical Systems, 18: 221–251 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  4. Etoua R M, Rousseau C. Bifurcation analysis of a Generalissed Gause model with prey harvesting and a generalized Holling response function of type III. J. Differential Equations, 249: 2316–2356 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  5. Huang J, Xiao D. Analyses of bifurcations and stability in a predator-prey system with holling type-IV functional response. Acta Mathematicae Applicatae Sinica (English Series), 20(1): 167–178 (2004)

    Article  MathSciNet  Google Scholar 

  6. Lamontagne, Y., Coutu, C., Rousseau, C. Bifurcation analysis of a predator-prey system with generalized holling type III functional response. J. Dynam. Differential Equations, 20(3): 535–571 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Li, B., Kuang, Y. Heteroclinic bifurcations in the Michaelis-Menten-type ratio-dependent predator-prey system. SIAM J. Appl. Math., 67(5): 1453–1464 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Perko, L. Differential Equations and Dynamical Systems. Springer-Verlag, New York, 1996

    Book  MATH  Google Scholar 

  9. Ruan, S., Xiao, D. Global analysis in a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math., 61(4): 1445–1472 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Takens, F. Forced Oscillations and Bifurcation. In: Applications of Global Analysis I, In: Comm. Math. Inst. Rijksuniversitat Utrecht, Vol.3, 1974, 1–59

    Google Scholar 

  11. Tang, Y., Zhang, W. Heteroclinic bifurcations in a ratio-dependent predator-prey system. J. Math. Biol., 50: 699–712 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Wolkowicz, G.S.K. Bifurcation analysis of a predator-prey system involving group defence. SIAM J. Appl. Math., 48: 592–606 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  13. Xiao, D., Ruan, S. Bogdanov-Takens bifurcations in harvested predator-prey systems. Fields Institute Communications, 21: 493–506 (1999)

    MathSciNet  Google Scholar 

  14. Xiao, D., Zhang, K.F. Multiple bifurcations of a predaotr-prey system. Discrete and Continuous Dynamical Systems Series B, 8: 417–433 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  15. Xiao, D., Zhu, H. Multiple focus and hopf bifurcations in a predaotr-prey system with nonmonotonic functional response. SIAM J. Appl. Math., 66: 802–819 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Zhu, H., Campbell, S.A., Wolkowicz, G.S.K. Bifurcation analysis of a predator-prey system with nonmonotonic functional response. SIAM J. Appl. Math., 63(2): 636–682 (2002)

    MathSciNet  Google Scholar 

  17. Zhu, C.R., Lan, K.Q. Phase portraits, Hopf bifurcation and limit cycles of Leslie-Gower predator-prey systems with harvesting rates. Discrete and Continuous Dynamical Systems Series B, 14(1): 289–306 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Yi-jun Gong & Ji-cai Huang

Authors
  1. Yi-jun Gong
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji-cai Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ji-cai Huang.

Additional information

Supported by the National Natural Science Foundation of China (No. 11101170), the Research Project of the Central China Normal University (No. CCNU12A01007), and the State Scholarship Fund of the China Scholarship Council (2011842509).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gong, Yj., Huang, Jc. Bogdanov-Takens bifurcation in a Leslie-Gower predator-prey model with prey harvesting. Acta Math. Appl. Sin. Engl. Ser. 30, 239–244 (2014). https://doi.org/10.1007/s10255-014-0279-x

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-014-0279-x

Keywords

2000 MR Subject Classification

Search

Navigation