Skip to main content
SpringerLink
Log in

Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

In this paper, a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten type prey harvesting is investigated. It is shown that the model exhibits several bifurcations of codimension 1 viz. Neimark–Sacker bifurcation, transcritical bifurcation and flip bifurcation on varying one parameter. Bifurcation theory and center manifold theory are used to establish the conditions for the existence of these bifurcations. Moreover, existence of Bogdanov–Takens bifurcation of codimension 2 (i.e. two parameters must be varied for the occurrence of bifurcation) is exhibited. The non-degeneracy conditions are determined for occurrence of Bogdanov–Takens bifurcation. The extensive numerical simulation is performed to demonstrate the analytical findings. The system exhibits periodic solutions including flip bifurcation and Neimark–Sacker bifurcation followed by the wide range of dense chaos.

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

References

  1. Agiza, H.N., ELabbasy, E.M., EL-Metwally, H., Elsadany, A.A.: Chaotic dynamics of a discrete prey-predator model with Holling type II. Nonlinear Anal. Real World Appl. 10, 116–129 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ajaz, M.B., Saeed, U., Din, Q., Ali, I., Siddiqui, M.I.: Bifurcation analysis and chaos control in discrete-time modified Leslie–Gower prey harvesting model. Adv. Differ. Equ. 24, 45 (2020)

    Article  MathSciNet  Google Scholar 

  3. Arnold, V.I.: Geometrical Methods in the Theory of Ordinary Differential Equations. Grundlehren Math. Wiss. Springer, Berlin (1983)

    Book  Google Scholar 

  4. Aziz-Alaoui, M.A., Daher Okiye, M.: Boundedness and global stability for a predator-prey model with modified Leslie–Gower and Holling-Type II schemes. Appl. Math. Lett. 16, 1069–1075 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bogdanov, R.I.: Versal deformation of a singular point of a vector field on the plane in the case of zero eigenvalues. Funkc. Anal. i Priložen. 9, 63 (1975)

    MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  7. Broer, H.W., Roussarie, R., Simó, C.: On the Bogdanov–Takens bifurcation for planar diffeomorphisms. In: International Conference on Differential Equations. 1, 2 (Barcelona, 1991), pp. 81–92. World Sci. Publ, River Edge, NJ (1993)

  8. Broer, H., Roussarie, R., Simó, C.: Invariant circles in the Bogdanov–Takens bifurcation for diffeomorphisms. Ergod. Theory Dyn. Syst. 16, 1147–1172 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chen, Q., Teng, Z.: Codimension-two bifurcation analysis of a discrete predator–prey model with nonmonotonic functional response. J. Differ. Equ. Appl. 23, 2093–2115 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chow, S.N., Hale, J.K.: Methods of Bifurcation Theory. Springer, New York (1982)

    Book  MATH  Google Scholar 

  11. Chow, S., Li, C., Wang, D.: Normal Forms and Bifurcation of Planar Vector Fields. Cambridge University Press, Cambridge (1994)

    Book  MATH  Google Scholar 

  12. Clark, C.W.: Mathematical models in the economics of renewable resources. SIAM Rev. 21, 81–99 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  13. Collings, J.B.: The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model. J. Math. Biol. 36, 149–168 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Din, Q.: Complexity and chaos control in a discrete-time prey–predator model. Commun. Nonlinear Sci. Numer. Simul. 49, 113–134 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  15. Du, Y., Peng, R., Wang, M.: Effect of a protection zone in the diffusive Leslie predator–prey model. J. Differ. Equ. 246, 3932–3956 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dumortier, F., Roussarie, R., Sotomayor, J., Zoladek, H.: Bifurcations of Planar Vector Fields. Lecture Notes in Math. Springer, New York (1991)

    Book  MATH  Google Scholar 

  17. Elabbasy, E.M., Elsadany, A.A., Zhang, Y.: Bifurcation analysis and chaos in a discrete reduced Lorenz system. Appl. Math. Comput. 228, 184–194 (2014)

    MathSciNet  MATH  Google Scholar 

  18. Elaydi, S.N.: Discrete Chaos: With Applications in Science and Engineering. Chapman and Hall/CRC, Boca Raton (2007)

    Book  Google Scholar 

  19. Gakkhar, S., Singh, A.: Control of chaos due to additional predator in the Hastings–Powell food chain model. J. Math. Anal. Appl. 385, 423–438 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Gakkhar, S., Singh, A.: Complex dynamics in a prey predator system with multiple delays. Commun. Nonlinear Sci. Numer. Simul. 17, 914–929 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (1983)

    Book  MATH  Google Scholar 

  22. Gupta, R.P., Chandra, P.: Bifurcation analysis of modified Leslie–Gower predator–prey model with Michaelis–Menten type prey harvesting. J. Math. Anal. Appl. 398, 278–295 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hadeler, K.P., Gerstmann, I.: The discrete Rosenzweig model. Math. Biosci. 98, 49–72 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  24. Huang, J., Xiao, D.: Analyses of bifurcations and stability in a predator–prey system with Holling type-IV functional response. Acta Math. Appl. Sin. Eng. Ser. 20, 167–178 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Huang, J.: Bifurcations and chaos in a discrete predator-prey system with Holling type-IV functional response. Acta Math. Appl. Sin. Engl. Ser. 21, 157–176 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  26. Huang, J., Gong, Y., Ruan, S.: Bifurcation analysis in a predator–prey model with constant-yield predator harvesting. Discrete Contin. Dyn. Syst. Ser. B 18, 2101–2121 (2013)

    MathSciNet  MATH  Google Scholar 

  27. Huang, J., Gong, Y., Chen, J.: Multiple bifurcations in a predator–prey system of Holling and Leslie type with constant-yield prey harvesting. J. Bifurc. Chaos Appl. Sci. Eng. 23, 1350164 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Huang, J., Liu, S., Ruan, S., Xiao, D.: Bifurcations in a discrete predator–prey model with nonmonotonic functional response. J. Math. Anal. Appl. 464, 201–230 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ji, C., Jiang, D., Shi, N.: Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 359, 482–498 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ji, C., Jiang, D., Shi, N.: A note on a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation. J. Math. Anal. Appl. 377, 435–440 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  31. Kong, L., Zhu, C.: Bogdanov–Takens bifurcations of codimension 2 and 3 in a Leslie–Gower predator–prey model with Michaelis–Menten-type prey harvesting. Math. Methods Appl. Sci. 40, 1–17 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  32. Krishna, S.V., Srinivasu, P.D.N., Kaymakcalan, B.: Conservation of an ecosystem through optimal taxation. Bull. Math. Biol. 60, 569–584 (1998)

    Article  MATH  Google Scholar 

  33. Kuznetsov, Y.: Elements of Applied Bifurcation Theory. Springer, New York (1998)

    MATH  Google Scholar 

  34. Levine, S.H.: Discrete time modeling of ecosystems with applications in environmental enrichment. Math. Biosci. 24, 307–317 (1975)

    Article  MATH  Google Scholar 

  35. Li, S., Zhang, W.: Bifurcations of a discrete prey–predator model with Holling type II functional response. Discrete Contin. Dyn. Syst. Ser B 14, 159–176 (2010)

    MathSciNet  MATH  Google Scholar 

  36. Li, L., Wang, Z.J.: Global stability of periodic solutions for a discrete predator–prey system with functional response. Nonlinear Dyn. 72, 507–516 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, X., Xiao, D.: Bifurcations in a discrete time Lotka–Volterra predator–prey system. Discrete Contin. Dyn. Syst. Ser. B. 69, 559–572 (2006)

    MathSciNet  MATH  Google Scholar 

  38. Liu, Z., Magal, P., Xiao, D.: Bogdanov–Takens bifurcation in a predator–prey model. Z. Angew. Math. Phys. 67, 1–29 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  39. Liu, Y., Liu, Z., Wang, R.: Bogdanov–Takens bifurcation with codimension three of a predator–prey system suffering the additive Allee effect. Int. J. Biomath. 10, 1750044 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  40. Liu, W., Cai, D.: Bifurcation, chaos analysis and control in a discrete-time predator–prey system. Adv. Differ. Equ. 2019, 11 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  41. May, R.M.: Biological populations with nonoverlapping generations: stable points, stable cycles, and chaos. Science 186, 645–647 (1974)

    Article  Google Scholar 

  42. Singh, A., Gakkhar, S.: Stabilization of modified Leslie–Gower prey–predator model. Differ. Equ. Dyn. Syst. 22, 239–249 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  43. Singh, A., Elsadany, A.A., Elsonbaty, A.: Complex dynamics of a discrete fractional-order Leslie–Gower predator–prey model. Math. Methods Appl. Sci. 42, 3992–4007 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  44. Singh, A., Deolia, P.: Dynamical analysis and chaos control in discrete-time prey–predator model. Commun. Nonlinear Sci. Numer. Simul. 90, 105313 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  45. Smith, J.M.: Mathematical Ideas in Biology. Cambridge University Press, Cambridge (1968)

    Book  Google Scholar 

  46. Takens, F.: Forced oscillations and bifurcations. Comm. Math. Inst. Rijksuniv. Utrecht 2, 1–111 (1974)

    MathSciNet  MATH  Google Scholar 

  47. Takens, F.: Singularities of vector fields. Publ. Math. Inst. Hautes Etudes Sci. 43, 47–100 (1974)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Xiao, D., Ruan, S.: Bogdanov–Takens bifurcations in predator–prey systems with constant rate harvesting. Fields Inst. Commun. 21, 493–506 (1999)

    MathSciNet  MATH  Google Scholar 

  50. Xiang, C., Huang, J., Ruan, S., Xiao, D.: Bifurcation anlysis in a host-generalist parasitoid model with Holling II functional response. J. Differ. Equ. 268, 4618–4662 (2020)

    Article  MATH  Google Scholar 

  51. Yagasaki, K.: Melnikov’s method and codimension-two bifurcations in forced oscillations. J. Differ. Equ. 185, 1–24 (2002)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ABV-Indian Institute of Information Technology and Management Gwalior, Gwalior, M.P., India

    Anuraj Singh

  2. Shree Guru Gobind Singh Tricentenary University, Gurugram, Haryana, India

    Pradeep Malik

Authors
  1. Anuraj Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pradeep Malik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anuraj Singh.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Singh, A., Malik, P. Bifurcations in a modified Leslie–Gower predator–prey discrete model with Michaelis–Menten prey harvesting. J. Appl. Math. Comput. 67, 143–174 (2021). https://doi.org/10.1007/s12190-020-01491-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-020-01491-9

Keywords

Search

Navigation