Skip to main content
SpringerLink
Log in

Equilibrium Points and Their Stability Properties of a Multiple Delays Model

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

Abstract

In this paper, we present a mathematical model with delay which describes the dynamics of three fish populations. The equilibrium points and their stability properties of the non-delay model are studied using Eigenvalue analysis. For the system with single delay model and multiple delays model, the existence of the interior equilibrium point and its stability are studied using the characteristic equation for the linearized system.

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

Similar content being viewed by others

References

  1. Yi, F., Wei, J., Shi, J.: Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator–prey system. J. Differ. Equ. 246(5), 1944–1977 (2009)

    Article  MathSciNet  Google Scholar 

  2. Wang, J., Shi, J., Wei, J.: Dynamics and pattern formation in a diffusive predator–prey system with strong Allee effect in prey. J. Differ. Equ. 251(4), 1276–1304 (2011)

    Article  MathSciNet  Google Scholar 

  3. Yang, R., Wei, J.: Bifurcation analysis of a diffusive predator–prey system with nonconstant death rate and Holling III functional response. Chaos Solitons Fractals 70, 1–13 (2015)

    Article  MathSciNet  Google Scholar 

  4. Yan, X.P., Zhang, C.H.: Stability and turing instability in a diffusive predator–prey system with Beddington–DeAngelis functional response. Nonlinear Anal. RWA 20, 1–13 (2014)

    Article  MathSciNet  Google Scholar 

  5. Tang, X., Song, Y.: Stability, Hopf bifurcations and spatial patterns in a delayed diffusive predator–prey model with herd behavior. Appl. Math. Comput. 254, 375–391 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Yuan, R., Jiang, W., Wang, Y.: Saddle-node-Hopf bifurcation in a modified Leslie–Gower predator–prey model with time-delay and prey harvesting. J. Math. Anal. Appl. 422(2), 1072–1090 (2015)

    Article  MathSciNet  Google Scholar 

  7. Wang, X., Wei, J.: Diffusion-driven stability and bifurcation in a predator–prey system with Ivlev-type functional response. Appl. Anal. 92(4), 752–775 (2013)

    Article  MathSciNet  Google Scholar 

  8. 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 

  9. Li, X., Jiang, W., Shi, J.: Hopf bifurcation and Turing instability in the reaction–diffusion Holling–Tanner predator–prey model. IMA J. Appl. Math. 78(2), 287–306 (2013)

    Article  MathSciNet  Google Scholar 

  10. Moussaoui, A., Bassaid, S., Dads, E.L.H.A.: The impact of water level fluctuations on a delayed prey–predator model. Nonlinear Anal. RWA 21, 170–184 (2015)

    Article  MathSciNet  Google Scholar 

  11. Wijeratne, A.W., Yi, F., Wei, J.: Bifurcation analysis in the diffusive Lotka–Volterra system: an application to market economy. Chaos Solitons Fractals 40(2), 902–911 (2009)

    Article  MathSciNet  Google Scholar 

  12. Jiang, J., Song, Y.: Delay-induced Bogdanov–Takens bifurcation in a Leslie–Gower predator–prey model with nonmonotonic functional response. Commun. Nonlinear Sci. Numer. Simul. 19(7), 2454–2465 (2014)

    Article  MathSciNet  Google Scholar 

  13. Ma, Y.: Global Hopf bifurcation in the Leslie–Gower predator–prey model with two delays. Nonlinear Anal. RWA 13(1), 370–375 (2012)

    Article  MathSciNet  Google Scholar 

  14. Yafia, R., Adnani, F.E., Alaoui, H.T.: Limit cycle and numerical similations for small and large delays in a predator–prey model with modified Leslie–Gower and Holling-type II schemes. Nonlinear Anal. RWA 9(5), 2055–2067 (2008)

    Article  MathSciNet  Google Scholar 

  15. Nindjin, A.F., Aziz-Alaoui, M.A., Cadivel, M.: Analysis of a predator–prey model with modified Leslie–Gower and Hollingtype II schemes with time delay. Nonlinear Anal. RWA 7(5), 1104–1118 (2006)

    Article  Google Scholar 

  16. Hale, J., Lunel, S.V.: Introduction to Functional Differential Equations. Springer, New York (1993)

    Book  Google Scholar 

  17. Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, Boston (1993)

    MATH  Google Scholar 

  18. Wu, J.: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996)

    Book  Google Scholar 

  19. Taboas, P.: Periodic solutions of a planar delay equation. Proc. R. Soc. Edinb. Sect. A 116, 85–101 (1990)

    Article  MathSciNet  Google Scholar 

  20. Ruan, S., Wei, J.: Periodic solutions of planar systems with two delays. Proc. R. Soc. Edinb. Sect. A 129, 1017–1032 (1999)

    Article  MathSciNet  Google Scholar 

  21. Wei, J., Li, Y.: Global existence of periodic solutions in a Tri-Neuron Network model with delays. Phys. D 198, 106–119 (2004)

    Article  MathSciNet  Google Scholar 

  22. Leung, A.: Periodic solutions for a prey–predator differential delay equation. J. Differ. Equ. 26, 391–403 (1977)

    Article  MathSciNet  Google Scholar 

  23. Zhao, T., Kuang, Y., Smith, H.L.: Global existence of periodic solutions in a class of delayed Gause-type predator–prey systems. Nonlinear Anal. 28, 1373–1394 (1997)

    Article  MathSciNet  Google Scholar 

  24. Song, Y., Wei, J.: Local Hopf bifurcation and global periodic solutions in a delayed predator–prey system. J. Math. Anal. Appl. 301, 1–21 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Analysis, Modeling and Simulation Laboratory, Hassan II University, Casablanca, Morocco

    Youssef El Foutayeni

  2. Mathematical Populations Dynamics Laboratory, Cadi Ayyad University, Marrakesh, Morocco

    Mohamed Khaladi

  3. Unit for Mathematical and Computer Modeling of Complex Systems, IRD, Paris, France

    Youssef El Foutayeni & Mohamed Khaladi

  4. Faculté des Sciences Ben M’Sik, Av Driss El Harti B.P, 7955, Casablanca, Sidi Othmane, Morocco

    Youssef El Foutayeni

Authors
  1. Youssef El Foutayeni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mohamed Khaladi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Youssef El Foutayeni.

Additional information

The authors are sincerely grateful to the anonymous referees and the Corresponding Editor for their valuable comments, which lead to a substantial improvement in the contents of this paper.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

El Foutayeni, Y., Khaladi, M. Equilibrium Points and Their Stability Properties of a Multiple Delays Model. Differ Equ Dyn Syst 28, 255–272 (2020). https://doi.org/10.1007/s12591-016-0321-y

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-016-0321-y

Keywords

Search

Navigation