Skip to main content
SpringerLink
Log in

Singular Bautin bifurcation analysis of a slow–fast predator–prey system

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Over the past few decades, the study of complex oscillations in slow–fast systems has been a focal point of research. Within the realm of slow–fast systems theory, the determination of the singular Hopf bifurcation and maximal canard locations relies on computing the first Lyapunov coefficient, based on the genericity condition that it is nonzero. This manuscript seeks to broaden these results to scenarios where the first Lyapunov coefficient becomes zero. To achieve this, the analytical expression of the second Lyapunov coefficient is derived, and an exploration of the normal form for codimension-2 singular Bautin bifurcation in a singularly perturbed Holling type-III predator–prey system with a weak Allee effect is conducted, explicitly identifying locally invertible parameter-dependent transformations. Utilizing geometric singular perturbation theory, the normal form theory of slow–fast systems, and the blow-up technique, this research unveils a diverse array of rich and complex nonlinear dynamics. This includes, but is not limited to, the existence of relaxation oscillations, canard phenomena, singular Hopf bifurcation, singular Bautin bifurcation, and saddle-node bifurcation of the canard cycle. Furthermore, numerical simulations are carried out to substantiate the theoretical findings.

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

Data availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

References

  1. Ankur, Jiwari, R.: New multiple analytic solitonary solutions and simulation of (2+ 1)-dimensional generalized Benjamin-Bona-Mahony-Burgers model. Nonlinear Dyn. 1–29 (2023)

  2. Arnold, V.I.: Dynamical Systems V: Bifurcation Theory and Catastrophe Theory, vol. 5 (Encyclopaedia of Mathematical Sciences). Springer, Berlin (1994)

  3. Atabaigi, A.: Canard explosion, homoclinic and heteroclinic orbits in singularly perturbed generalist predator–prey systems. Int. J. Biomath. 14(01), 2150003 (2021)

    Article  MathSciNet  Google Scholar 

  4. Bai, D., Kang, Y., Ruan, S., Wang, L.: Dynamics of an intraguild predation food web model with strong Allee effect in the basal prey. Nonlinear Anal. Real World Appl. 58, 103206 (2021)

    Article  MathSciNet  Google Scholar 

  5. Courchamp, F., Berec, L., Gascoigne, J.: Allee Effects in Ecology and Conservation. Oxford University Press, Oxford (2008)

    Book  Google Scholar 

  6. Dumortier, F., Roussarie, R.: Canard Cycles and Center Manifolds, vol. 577. American Mathematical Society, Providence (1996)

    Google Scholar 

  7. Fenichel, N.: Geometric singular perturbation theory for ordinary differential equations. J. Differ. Equ. 31(1), 53–98 (1979)

    Article  MathSciNet  Google Scholar 

  8. FitzHugh, R.: Mathematical models of threshold phenomena in the nerve membrane. Bull. Math. Biophys. 17(4), 257–278 (1955)

    Article  Google Scholar 

  9. Garrett, B., Gian-Carlo, R.: Ordinary Differential Equations. Wiley, Boston (1982)

    Google Scholar 

  10. Getz, W.M.: A hypothesis regarding the abruptness of density dependence and the growth rate of populations. Ecology 77(7), 2014–2026 (1996)

    Article  Google Scholar 

  11. Han, X., Bi, Q.: Slow passage through canard explosion and mixed-mode oscillations in the forced Van der Pol’s equation. Nonlinear Dyn. 68(1–2), 275–283 (2012)

    Article  MathSciNet  Google Scholar 

  12. Hek, G.: Geometric singular perturbation theory in biological practice. J. Math. Biol. 60(3), 347–386 (2010)

    Article  MathSciNet  Google Scholar 

  13. Hsu, S.B., Hwang, T.W., Kuang, Y.: Global analysis of the Michaelis-Menten-type ratio-dependent predator-prey system. J. Math. Biol. 42, 489–506 (2001)

    Article  MathSciNet  Google Scholar 

  14. Hsu, S.B., Hwang, T.W., Kuang, Y.: Global dynamics of a predator–prey model with Hassell–Varley type functional response. Discrete Contin. Dyn. Syst. Ser. B 10(4), 857–871 (2008)

    MathSciNet  Google Scholar 

  15. Hsu, T.H., Ruan, S.: Relaxation oscillations and the entry-exit function in multidimensional slow-fast systems. SIAM J. Math. Anal. 53(4), 3717–3758 (2021)

    Article  MathSciNet  Google Scholar 

  16. Huang, J., Ruan, S., Song, J.: Bifurcations in a predator-prey system of Leslie type with generalized Holling type III functional response. J. Differ. Equ. 257(6), 1721–1752 (2014)

    Article  MathSciNet  Google Scholar 

  17. Kooi, B., Poggiale, J.: Modelling, singular perturbation and bifurcation analyses of bitrophic food chains. Math. Biosci. 301, 93–110 (2018)

    Article  MathSciNet  Google Scholar 

  18. Kristiansen, K.U.: Geometric singular perturbation analysis of a dynamical target mediated drug disposition model. J. Math. Biol. 79(1), 187–222 (2019)

    Article  MathSciNet  Google Scholar 

  19. Krupa, M., Popović, N., Kopell, N.: Mixed-mode oscillations in three time-scale systems: a prototypical example. SIAM J. Appl. Dyn. Syst. 7(2), 361–420 (2008)

    Article  MathSciNet  Google Scholar 

  20. Krupa, M., Szmolyan, P.: Extending geometric singular perturbation theory to nonhyperbolic points–fold and canard points in two dimensions. SIAM J. Math. Anal. 33(2), 286–314 (2001)

    Article  MathSciNet  Google Scholar 

  21. Krupa, M., Szmolyan, P.: Relaxation oscillation and canard explosion. J. Differ. Equ. 174(2), 312–368 (2001)

    Article  MathSciNet  Google Scholar 

  22. Kuehn, C.: Multiple time scale dynamics, vol. 191 of Applied Mathematical Sciences. Springer, Cham (2015)

    Book  Google Scholar 

  23. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences. Springer, Berlin (1998)

    Google Scholar 

  24. Li, C., Zhu, H.: Canard cycles for predator-prey systems with Holling types of functional response. J. Differ. Equ. 254(2), 879–910 (2013)

    Article  MathSciNet  Google Scholar 

  25. Li, J., Quan, T., Zhang, W.: Bifurcation and number of subharmonic solutions of a 4D non-autonomous slow-fast system and its application. Nonlinear Dyn. 92(2), 721–739 (2018)

    Article  Google Scholar 

  26. Lu, M., Huang, J.: Global analysis in Bazykin’s model with Holling II functional response and predator competition. J. Differ. Equ. 280, 99–138 (2021)

    Article  MathSciNet  Google Scholar 

  27. Pal, P.J., Saha, T., Sen, M., Banerjee, M.: A delayed predator-prey model with strong Allee effect in prey population growth. Nonlinear Dyn. 68(1), 23–42 (2012)

    Article  MathSciNet  Google Scholar 

  28. Rinaldi, S., Muratori, S.: Slow-fast limit cycles in predator–prey models. Ecol. Model. 61(3–4), 287–308 (1992)

    Article  Google Scholar 

  29. Sadhu, S.: Analysis of the onset of a regime shift and detecting early warning signs of major population changes in a two-trophic three-species predator-prey model with long-term transients. J. Math. Biol. 85(4), 1–33 (2022)

    Article  MathSciNet  Google Scholar 

  30. Van Voorn, G.A., Hemerik, L., Boer, M.P., Kooi, B.W.: Heteroclinic orbits indicate overexploitation in predator-prey systems with a strong Allee effect. Math. Biosci. 209(2), 451–469 (2007)

    Article  MathSciNet  Google Scholar 

  31. Wang, J., Shi, J., Wei, J.: Predator-prey system with strong Allee effect in prey. J. Math. Biol. 62(3), 291–331 (2011)

    Article  MathSciNet  Google Scholar 

  32. Xia, Y., Zhang, Z., Bi, Q.: Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork-Hopf bifurcation. Nonlinear Dyn. (Online) (2020)

  33. Yaru, L., Shenquan, L.: Canard-induced mixed-mode oscillations and bifurcation analysis in a reduced 3D pyramidal cell model. Nonlinear Dyn. 1–37 (2020)

Download references

Acknowledgements

The authors extend their gratitude to the anonymous reviewers for their thorough examination, insightful comments, and valuable suggestions, which greatly contributed to improving the article’s presentation.

Funding

The first author acknowledges financial support from the Department of Science and Technology (DST), Govt. of India, under the “Fund for Improvement of S &T Infrastructure (FIST)” scheme (File No. SR/FST/MS-I/2019/41).

Author information

Authors and Affiliations

  1. Department of Mathematics, Presidency University, Kolkata, West Bengal, 700073, India

    Tapan Saha

  2. Department of Mathematics and Statistics, IIT Kanpur, Kanpur, Uttar Pradesh, India

    Pranali Roy Chowdhury & Malay Banerjee

  3. Department of Mathematics, Krishna Chandra College, Hetampur, Birbhum, West Bengal, 731124, India

    Pallav Jyoti Pal

Authors
  1. Tapan Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pranali Roy Chowdhury
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pallav Jyoti Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Malay Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pallav Jyoti Pal.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saha, T., Roy Chowdhury, P., Pal, P.J. et al. Singular Bautin bifurcation analysis of a slow–fast predator–prey system. Nonlinear Dyn 112, 7695–7713 (2024). https://doi.org/10.1007/s11071-024-09387-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-024-09387-0

Keywords

Search

Navigation