Skip to main content
SpringerLink
Log in

Stability and Bifurcation in a Prey–Predator–Scavenger System with Michaelis–Menten Type of Harvesting Function

  • Original Research
  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

In this paper an ecological model consisting of prey–predator–scavenger involving Michaelis–Menten type of harvesting function is proposed and studied. The existence, uniqueness and uniformly bounded of the solution of the proposed model are discussed. The stability and persistence conditions of the model are established. Lyapunov functions are used to study the global stability of all equilibrium points. The possibility of occurrence of local bifurcation around the equilibrium points is investigated. Finally an extensive numerical simulation is carried out to validate the obtained analytical results and understand the effects of scavenger and harvesting on the model dynamics. It is observed that the proposed model is very sensitive for varying in their parameters values especially those related with scavenger and undergoes different types of local 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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Leslie, P.H.: Some further notes on the use of matrices in population mathematics. Biometrika 35, 213–245 (1948)

    Article  MathSciNet  Google Scholar 

  2. Solomon, M.: The natural control of animal populations. J. Anim. Ecol. 18(1), 1–35 (1949)

    Article  Google Scholar 

  3. Holling, C.S.: The components of predation as revealed by a study of small-mammal predation of the European pine sawfly. Can. Entomol. 91(5), 293–320 (1959)

    Article  Google Scholar 

  4. 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 

  5. Real, L.A.: The kinetics of functional response. Am. Nat. 111, 289–300 (1977)

    Article  Google Scholar 

  6. Berryman, A.: The origins and evolution of predator–prey theory. Ecology 73(5), 1530–1535 (1992)

    Article  Google Scholar 

  7. Hastings, A., Powell, T.: Chaos in a three-species food chain. Ecology 72(3), 896–903 (1991)

    Article  Google Scholar 

  8. Korobeinikov, A., Wake, G.: Global properties of the three-dimensional predator–prey Lotka–Volterra systems. J. Appl. Math. Decis. Sci. 3(2), 155–162 (1999)

    Article  MathSciNet  Google Scholar 

  9. Chauvet, E., Paullet, J.E., Previte, J.P., Walls, Z.: A Lotka–Volterra three-species food chain. Math. Mag. 75(4), 243–255 (2002)

    Article  MathSciNet  Google Scholar 

  10. Gakkhar, S., Naji, R.K.: Existence of chaos in two-prey one-predator system. Chaos Solitons Fractals 17, 639–649 (2003)

    Article  MathSciNet  Google Scholar 

  11. Gakkhar, S., Naji, R.K.: Order and chaos in a food web consisting of a predator and two independent preys. Commun. Nonlinear Sci. Numer. Simul. 10, 105–120 (2005)

    Article  MathSciNet  Google Scholar 

  12. Naji, R.K.: Global stability and persistence of three species food web involving omnivory. Iraqi J. Sci. 53(4), 866–876 (2012)

    Google Scholar 

  13. Upadhyay, R.K., Naji, R.K., Raw, S.N., Dubey, B.: The role of top predator interference on the dynamics of a food chain model. Commun. Nonlinear Sci. Numer. Simul. 18, 757–768 (2013)

    Article  MathSciNet  Google Scholar 

  14. Nolting, B., Paullet, J.E., Previte, J.P.: Introducing a scavenger onto a predator prey model. Appl. Math. E Notes 8, 214–222 (2008)

    MathSciNet  MATH  Google Scholar 

  15. Previte, J.P., Hoffman, K.A.: Period doubling cascades in a predator–prey model with a scavenger. Siam Rev. 55(3), 523–546 (2013)

    Article  MathSciNet  Google Scholar 

  16. Andayani, P., Kusumawinahyu, K.M.: Global stability analysis on a predator–prey model with omnivores. Appl. Math. Sci. 9(36), 1771–1782 (2015)

    Google Scholar 

  17. Jansen, J.E., Van Gorder, R.A.: Dynamics from a predator–prey–quarry–resource–scavenger model. Theor Ecol. 11, 19–38 (2018)

    Article  Google Scholar 

  18. Song, X.Y., Chen, L.S.: Optimal harvesting and stability for a two-species competitive system with stage structure. Math. Biosci. 170, 173–186 (2001)

    Article  MathSciNet  Google Scholar 

  19. Kar, T.K., Chaudhuri, K.S.: Harvesting in a two-prey one predator fishery: a bio economic model. ANZIAM J. 45, 443–456 (2004)

    Article  MathSciNet  Google Scholar 

  20. 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(8), 2101–2121 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Huang, J., Gong, Y., Chen, J.: Multiple bifurcations in a predator–prey system of Holling and Leslie type with constant-yield prey harvesting. Int. J. Bifurc. Chaos Appl. Sci. Eng. 23(10), 1350164 (2013)

    Article  MathSciNet  Google Scholar 

  22. Huang, J., Liu, S., Ruan, S., Zhang, X.: Bogdanov–Takens bifurcation of codimension 3 in a predator-prey model with constant-yield predator harvesting. Commun. Pure Appl. Anal. 15(3), 1041–1055 (2016)

    Article  MathSciNet  Google Scholar 

  23. Chen, J., Huang, J., Ruan, S., Wang, J.: Bifurcations of invariant tori in predator–prey models with seasonal prey harvesting. SIAM J. Appl. Math. 73(5), 1876–1905 (2013)

    Article  MathSciNet  Google Scholar 

  24. 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  Google Scholar 

  25. Feng, P.: Analysis of a delayed predator–prey model with ratio-dependent functional response and quadratic harvesting. J. Appl. Math. Comput. 44, 251–262 (2014)

    Article  MathSciNet  Google Scholar 

  26. Gupta, R.P., Chandra, P.: Dynamical properties of a prey–predator–scavenger model with quadratic harvesting. Commun. Nonlinear Sci. Numer. Simul. 49, 202–214 (2017)

    Article  MathSciNet  Google Scholar 

  27. Hu, D., Cao, H.: Stability and bifurcation analysis in a predator–prey system with Michaelis-Menten type predator harvesting. Nonlinear Anal. Real World Appl. 33, 58–82 (2017)

    Article  MathSciNet  Google Scholar 

  28. Freedman, H.I., Waltman, P.: Persistence in models of three interacting predator prey populations. Math. Biosci. 68, 213–231 (1984)

    Article  MathSciNet  Google Scholar 

  29. Perko, L.: Differential equation and dynamical systems, 3rd edn. Springer, New York (2001)

    Book  Google Scholar 

Download references

Acknowledgements

The authors are very much thankful to the reviewers for their valuable suggestions and constructive comments. Their useful comments have contributed to the improvement of the authors work.

Author information

Authors and Affiliations

  1. Department of Mathematics, College of Science, University of Baghdad, Baghdad, Iraq

    Huda Abdul Satar & Raid Kamel Naji

Authors
  1. Huda Abdul Satar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raid Kamel Naji
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raid Kamel Naji.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Satar, H.A., Naji, R.K. Stability and Bifurcation in a Prey–Predator–Scavenger System with Michaelis–Menten Type of Harvesting Function. Differ Equ Dyn Syst 30, 933–956 (2022). https://doi.org/10.1007/s12591-018-00449-5

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-018-00449-5

Keywords

Search

Navigation