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

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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.

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Notes

  1. We may notice that the characteristic equation of a system with delay has infinite roots.

  2. One definition of the stiffness is that the global accuracy of the numerical solution is determined by stability rather than local error, and implicit methods are more appropriate for it.

  3. Method of steps is not universal, as it cannot be applied with time-varying delays that vanish in some points.

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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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Authors and Affiliations

  1. Department of Mathematical Sciences, College of Science, UAE University, Al-Ain, 15551, UAE

    F. A. Rihan & S. Lakshmanan

  2. Department of Physics, College of Science, UAE University, Al-Ain, 15551, UAE

    A. H. Hashish

  3. Department of Mathematics, Bharathiar University, Coimbatore, 641 046, India

    R. Rakkiyappan

  4. Department of Mathematics, Faculty of Science, Mansoura University, Mansoura, Egypt

    E. Ahmed

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  1. F. A. Rihan
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  2. S. Lakshmanan
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  3. A. H. Hashish
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  4. R. Rakkiyappan
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  5. E. Ahmed
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Correspondence to F. A. Rihan.

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Rihan, F.A., Lakshmanan, S., Hashish, A.H. et al. Fractional-order delayed predator–prey systems with Holling type-II functional response. Nonlinear Dyn 80, 777–789 (2015). https://doi.org/10.1007/s11071-015-1905-8

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