Skip to main content
SpringerLink
Log in

Slow–fast analysis of a modified Leslie–Gower model with Holling type I functional response

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

A Correction to this article was published on 09 May 2022

This article has been updated

Abstract

In this paper, we consider a modified Leslie-type prey–generalist predator system with piecewise–smooth Holling type I functional response. Considering the reproduction rate of the prey population higher than that of the predator, the model becomes a slow–fast system that mathematically leads to a singular perturbation problem. To analyse the stability of the boundary equilibrium on the switching boundary, we use a generalized Jacobian that enables us to investigate how the eigenvalues jump at the boundary point. We investigate the slow–fast system by employing geometric singular perturbation theory and blow-up technique that reveal a wide range of interesting complicated dynamical phenomena. We have studied existence of saddle-node bifurcation, Bogdanov–Takens bifurcation, bistability, singular Hopf bifurcation, canard orbits, multiple relaxation oscillations, saddle-node bifurcation of limit cycle and boundary equilibrium bifurcations. Numerical simulations are carried out to substantiate 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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

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.

Change history

References

  1. Ai, S., Sadhu, S.: The entry-exit theorem and relaxation oscillations in slow-fast planar systems. J. Diff. Eq. 268(11), 7220–7249 (2020)

    Article  MathSciNet  Google Scholar 

  2. Arima, N., Okazaki, H., Nakano, H.: A generation mechanism of canards in a piecewise linear system. IEICE Trans. Fundament. Electron. Commun. Comput. Sci. 80(3), 447–453 (1997)

    Google Scholar 

  3. Atabaigi, A., Barati, A.: Relaxation oscillations and canard explosion in a predator-prey system of Holling and Leslie types. Nonlinear Anal. Real World Appl. 36, 139–153 (2017)

    Article  MathSciNet  Google Scholar 

  4. Baer, S.M., Erneux, T.: Singular Hopf bifurcation to relaxation oscillations. SIAM J. Appl. Math. 46(5), 721–739 (1986)

    Article  MathSciNet  Google Scholar 

  5. Buzzi, C.A., de Carvalho, T., Teixeira, M.A.: On 3-parameter families of piecewise smooth vector fields in the plane. SIAM J. Appl. Dyn. Syst. 11(4), 1402–1424 (2012)

    Article  MathSciNet  Google Scholar 

  6. Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, vol. 178. Springer (2008)

  7. De Maesschalck, P., Dumortier, F.: Bifurcations of multiple relaxation oscillations in polynomial Liénard equations. Proceed. Am. Math. Soc. 139(6), 2073–2085 (2011)

    Article  Google Scholar 

  8. Dumortier, F.: Techniques in the theory of local bifurcations: blow-up, normal forms, nilpotent bifurcations, singular perturbations. In: Bifurcations and Periodic Orbits of Vector Fields, 19–73. Springer (1993)

  9. Dumortier, F., Roussarie, R.: Geometric singular perturbation theory beyond normal hyperbolicity. In: Multiple-Time-Scale Dynamical Systems, 29–63. Springer (2001)

  10. Dumortier, F., Roussarie, R.: Multiple canard cycles in generalized liénard equations. J. Diff. Eq. 174(1), 1–29 (2001)

    Article  Google Scholar 

  11. Dumortier, F., Roussarie, R., Roussarie, R.H.: Canard Cycles and Center Manifolds, vol. 577. American Mathematical Soc (1996)

  12. Fenichel, N.: Asymptotic stability with rate conditions. Indiana Univ. Math. J. 23(12), 1109–1137 (1974)

    Article  MathSciNet  Google Scholar 

  13. Fenichel, N.: Asymptotic stability with rate conditions, II. Indiana Univ. Math. J. 26(1), 81–93 (1977)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Fenichel, N., Moser, J.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J. 21(3), 193–226 (1971)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  17. 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 

  18. Hsu, T.H.: Number and stability of relaxation oscillations for predator-prey systems with small death rates. SIAM J. Appl. Dyn. Syst. 18(1), 33–67 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Jones, C.K.: Geometric singular perturbation theory. In: Dynamical Systems, 44–118. Springer (1995)

  21. 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 

  22. Krupa, M., Szmolyan, P.: Extending slow manifolds near transcritical and pitchfork singularities. Nonlinearity 14(6), 1473 (2001)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  24. Kuehn, C.: Multiple Time Scale Dynamics, vol. 191. Springer (2015)

  25. Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1991)

  26. Li, S., Wang, X., Li, X., Wu, K.: Relaxation oscillations for Leslie-type predator–prey model with Holling type I response functional function. Appl. Math. Lett. 107328 (2021)

  27. Liu, W., Xiao, D., Yi, Y.: Relaxation oscillations in a class of predator-prey systems. J. Differ. Eq. 188(1), 306–331 (2003)

    Article  MathSciNet  Google Scholar 

  28. Ludwig, D., Jones, D.D., Holling, C.S.: Qualitative analysis of insect outbreak systems: the spruce budworm and forest. J. Anim. Ecol. 315–332 (1978)

  29. McDonald, R.A., Webbon, C., Harris, S.: The diet of stoats (Mustela erminea) and weasels (Mustela nivalis) in great britain. J. Zool. 252(3), 363–371 (2000)

    Article  Google Scholar 

  30. Pokrovskii, A., Rachinskii, D., Sobolev, V., Zhezherun, A.: Topological degree in analysis of canard-type trajectories in 3-D systems. Appl. Anal. 90(7), 1123–1139 (2011)

    Article  MathSciNet  Google Scholar 

  31. Van der Pol, B.: On relaxation-oscillations. Lond. Edinburgh Dublin Philosoph. Magaz. J. Sci. 2(11), 978–992 (1926)

    Article  Google Scholar 

  32. Prohens, R., Teruel, A., Vich, C.: Slow-fast n-dimensional piecewise linear differential systems. J. Differ. Eq. 260(2), 1865–1892 (2016)

    Article  MathSciNet  Google Scholar 

  33. Roberts, A.: Canard explosion and relaxation oscillation in planar, piecewise-smooth, continuous systems. SIAM J. Appl. Dyn. Syst. 15(1), 609–624 (2016)

    Article  MathSciNet  Google Scholar 

  34. Roberts, A., Gendinning, P.: Canard-like phenomena in piecewise-smooth Van der Pol systems, Chaos: An Interdisciplinary. J. Nonlinear Sci. 24(2), 023138 (2014)

    MATH  Google Scholar 

  35. Rotstein, H.G., Coombes, S., Gheorghe, A.M.: Canard-like explosion of limit cycles in two-dimensional piecewise-linear models of Fitzhugh-Nagumo type. SIAM J. Appl. Dyn. Syst. 11(1), 135–180 (2012)

  36. Saha, T., Pal, P.J., Banerjee, M.: Relaxation oscillation and canard explosion in a slow-fast predator-prey model with Beddington-DeAngelis functional response. Nonlinear Dyn. 103(1), 1195–1217 (2021)

    Article  Google Scholar 

  37. Scheffer, M.: Ecology of Shallow Lakes, vol. 22. Springer (2004)

  38. Seo, G., DeAngelis, D.L.: A predator-prey model with a Holling type I functional response including a predator mutual interference. J. Nonlinear Sci. 21(6), 811–833 (2011)

    Article  MathSciNet  Google Scholar 

  39. Seo, G., Kot, M.: A comparison of two predator-prey models with Holling’s type I functional response. Math. Biosci. 212(2), 161–179 (2008)

    Article  MathSciNet  Google Scholar 

  40. Simpson, D.J.W.: Bifurcations in Piecewise-Smooth Continuous Systems, vol. 70. World Scientific (2010)

  41. Stenseth, N.C., Falck, W., Bjørnstad, O.N., Krebs, C.J.: Population regulation in snowshoe hare and Canadian lynx: asymmetric food web configurations between hare and lynx. Proceed. National Acad. Sci. 94(10), 5147–5152 (1997)

    Article  Google Scholar 

  42. Szmolyan, P., Wechselberger, M.: Relaxation oscillations in \(\mathbb{R}^3\). J. Differ. Eq. 200(1), 69–104 (2004)

    Article  Google Scholar 

  43. Turchin, P.: Complex Population Dynamics: a Theoretical/Empirical Synthesis, vol. 35. Princeton University Press (2003)

  44. Turchin, P., Hanski, I.: An empirically based model for latitudinal gradient in vole population dynamics. Am. Natur. 149(5), 842–874 (1997)

    Article  Google Scholar 

  45. Vakakis, A.F.: Relaxation oscillations, subharmonic orbits and chaos in the dynamics of a linear lattice with a local essentially nonlinear attachment. Nonlinear Dyn. 61(3), 443–463 (2010)

    Article  MathSciNet  Google Scholar 

  46. Verhulst, F.: Singular perturbation methods for slow-fast dynamics. Nonlinear Dyn. 50(4), 747–753 (2007)

    Article  MathSciNet  Google Scholar 

  47. Wang, Z., Zhang, Z., Bi, Q.: Relaxation oscillations in a nonsmooth oscillator with slow-varying external excitation. Int. J. Bifurc. Chaos 29(07), 1930019 (2019)

    Article  MathSciNet  Google Scholar 

  48. Xia, Y., Zhang, Z., Bi, Q.: Relaxation oscillations and the mechanism in a periodically excited vector field with pitchfork-Hopf bifurcation. Nonlinear Dyn. 101(1), 37–51 (2020)

  49. Yaru, L., Shenquan, L.: Canard-induced mixed-mode oscillations and bifurcation analysis in a reduced 3D pyramidal cell model. Nonlinear Dyn. 101(1), 531–567 (2020)

Download references

Acknowledgements

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

Author information

Authors and Affiliations

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

    Tapan Saha

  2. Department of Mathematics, Krishna Chandra College, Hetampur, 731124, Birbhum, India

    Pallav Jyoti Pal

  3. Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur, 208016, Uttar Pradesh, India

    Malay Banerjee

Authors
  1. Tapan Saha
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pallav Jyoti Pal
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Malay Banerjee
    View author publications

    You can also search for this author in PubMed Google Scholar

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.

Appendix

Appendix

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Saha, T., Pal, P.J. & Banerjee, M. Slow–fast analysis of a modified Leslie–Gower model with Holling type I functional response. Nonlinear Dyn 108, 4531–4555 (2022). https://doi.org/10.1007/s11071-022-07370-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07370-1

Keywords

Search

Navigation