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Nonlinear Dynamics

, Volume 80, Issue 1–2, pp 777–789 | Cite as

Fractional-order delayed predator–prey systems with Holling type-II functional response

  • F. A. Rihan
  • S. Lakshmanan
  • A. H. Hashish
  • R. Rakkiyappan
  • E. Ahmed
Original Paper

Abstract

In this paper, a fractional dynamical system of predator–prey with Holling type-II functional response and time delay is studied. Local and global stability of existence steady states and Hopf bifurcation with respect to the delay is investigated, with fractional-order \(0< \alpha \le 1\). It is found that Hopf bifurcation occurs when the delay passes through a sequence of critical values. Unconditionally, stable implicit scheme for the numerical simulations of the fractional-order delay differential model is introduced. The numerical simulations show the effectiveness of the numerical method and confirm the theoretical results. The presence of fractional order in the delayed differential model improves the stability of the solutions and enrich the dynamics of the model.

Keywords

Fractional calculus Predator–prey Hopf bifurcation Stability Time delay 

Notes

Acknowledgments

This work was supported by NRF Grant Project (UAE University). Dr. R. Rakkiyappan was supported by DST SERB Project # SB/FTP/MS-045/2013. The authors thank Prof. J. A. Tenreiro Machado and referees for their valuable comments.

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • F. A. Rihan
    • 1
  • S. Lakshmanan
    • 1
  • A. H. Hashish
    • 2
  • R. Rakkiyappan
    • 3
  • E. Ahmed
    • 4
  1. 1.Department of Mathematical Sciences, College of ScienceUAE UniversityAl-AinUAE
  2. 2.Department of Physics, College of ScienceUAE UniversityAl-AinUAE
  3. 3.Department of MathematicsBharathiar UniversityCoimbatoreIndia
  4. 4.Department of Mathematics, Faculty of ScienceMansoura UniversityMansouraEgypt

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