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Stability of fractional-order prey–predator system with time-delay and Monod–Haldane functional response

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Abstract

In this paper, we study the dynamics of a fractional-order delayed prey–predator system with Monod–Haldane response function. The model describes the interaction between prey and two populations of predator species: immature or juvenile and mature or adult predator. A single time delay is incorporated in the model to justify the gestation period of adult predator. The existence of solutions, steady states, and sufficient conditions to ensure the asymptotic stability of the steady states are investigated. Global stability around the interior equilibrium point by constructing the suitable Lyapunov functional is also investigated. The system displays Hopf bifurcation depending on the conversion coefficient (prey to immature predator) and time delay. The fractional-order derivatives in the model develop the stability results of solutions and improve the model behaviors. Finally, some numerical simulations are given to verify the success of derived theoretical results.

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Notes

  1. A functional response is the rate at which predator species capture prey individuals.

  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.

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Acknowledgements

The support of UAE University to execute this work is highly acknowledged and appreciated.

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  1. Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al-Ain, 15551, United Arab Emirates

    Rajivganthi Chinnathambi & Fathalla A. Rihan

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  1. Rajivganthi Chinnathambi
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  2. Fathalla A. Rihan
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Correspondence to Rajivganthi Chinnathambi or Fathalla A. Rihan.

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Chinnathambi, R., Rihan, F.A. Stability of fractional-order prey–predator system with time-delay and Monod–Haldane functional response. Nonlinear Dyn 92, 1637–1648 (2018). https://doi.org/10.1007/s11071-018-4151-z

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  • DOI: https://doi.org/10.1007/s11071-018-4151-z

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