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Bazykin’s Predator–Prey Model Includes a Dynamical Analysis of a Caputo Fractional Order Delay Fear and the Effect of the Population-Based Mortality Rate on the Growth of Predators

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Abstract

In this paper, we investigate a system of two differential equations of fractional order for the fear effect in prey-predator interactions, in which the density of predators controls the mortality pace of the prey population. The non-integer order differential equation is interpreted in terms of the Caputo derivative, and the development of the non-integer order scheme is described in terms of the influence of memory on population increase. The primary goal of existing research is to explore how the changing aspects of the current scheme are impacted by various types of parameters, including time delay, fear effect, and fractional order. The solutions’ positivity, existence-uniqueness, and boundedness are established with precise mathematical conclusions. The requirements necessary for the local asymptotic stability of different equilibrium points and the global stability of coexistence equilibrium are established. Hopf bifurcation occurs in the system at various delay times. The model’s fractional-order derivatives enhance the model behaviours and provide stability findings for the solutions. We have observed that fractional order plays an important role in population dynamics. Also, Hopf bifurcation for the proposed system have been observed for certain values of order of derivatives. Thus, the stability conditions of the equilibrium points may be changed by changing the order of the derivatives without changing other parametric values. Finally, a numerical simulation is run to verify our conclusions.

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Acknowledgements

The authors are grateful to all the reviewers for their careful reading, valuable comments and helpful suggestions, which have helped us to improve the presentation of this work significantly. Also, the authors Aziz Khan and Thabet Abdeljawad would like to thank Prince Sultan University, Saudi Arabia for paying the APC and the support through TAS research lab.

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Authors and Affiliations

  1. Department of Mathematics, Anurag University, Venkatapur, Hyderabad, Telangana, 500088, India

    G. Ranjith Kumar & K. Ramesh

  2. Department of Mathematics and Sciences, Prince Sultan University, P.O. Box 66833, 11586, Riyadh, Saudi Arabia

    Aziz Khan & Thabet Abdeljawad

  3. Department of Mathematics, Vidya Jyothi Institute of Technology, Hyderabad, Telangana, 500075, India

    K. Lakshminarayan

  4. Department of Medical Research, China Medical University, Taichung, 40402, Taiwan

    Thabet Abdeljawad

  5. Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, 02447, Korea

    Thabet Abdeljawad

  6. Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, 0204, Medusa, South Africa

    Thabet Abdeljawad

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  1. G. Ranjith Kumar
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All authors contributed equally and significantly in writing this paper and typed, read, and approved the final manuscript.

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Correspondence to K. Ramesh or Thabet Abdeljawad.

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Kumar, G.R., Ramesh, K., Khan, A. et al. Bazykin’s Predator–Prey Model Includes a Dynamical Analysis of a Caputo Fractional Order Delay Fear and the Effect of the Population-Based Mortality Rate on the Growth of Predators. Qual. Theory Dyn. Syst. 23, 130 (2024). https://doi.org/10.1007/s12346-024-00981-6

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