Pure and Applied Mathematics Journal

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Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model

Received: 16 May 2023    Accepted: 19 July 2023    Published: 25 September 2023
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Abstract

This study investigates the dynamic properties of a discrete predator-prey model influenced by the Allee effect. Through rigorous analysis utilizing bifurcation theory and the center manifold theorem, we establish the stability of the system’s local equilibrium and reveal the intricate dynamical behaviors exhibited by the model, including period-doubling bifurcations at periods 2, 4, and 8, as well as the emergence of quasi-periodic orbits and chaotic sets. A notable finding is the significant role played by the parameter r in shaping the system’s behavior, as we identify a series of bifurcations, such as flip and Neimark-Sacker bifurcations, by systematically varying r while keeping other parameters fixed. These findings underscore the non-linear nature of the model and provide valuable insights into its complex dynamics. Our enhanced understanding of these bifurcations and resulting dynamical behaviors deepens our knowledge of the Allee effect’s implications for predator-prey models, contributing to our comprehension of population oscillations, stability transitions, and the emergence of chaotic dynamics in ecological systems under the Allee effect. Moreover, this study carries practical implications for population management and conservation strategies, as incorporating the Allee effect into predator-prey interactions allows for better insights into population dynamics and the development of more effective and sustainable management practices. Overall, this comprehensive analysis of the discrete predator-prey model under the Allee effect uncovers intricate dynamical behaviors and emphasizes the influential role of the parameter r in shaping system dynamics, with implications for both theoretical understanding and practical conservation management strategies.

DOI 10.11648/j.pamj.20231204.11
Published in Pure and Applied Mathematics Journal (Volume 12, Issue 4, August 2023)
Page(s) 59-71
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Discrete Predator-prey System, Allee Effect, Stability Analysis, Bifurcation Theory

References
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Cite This Article
  • APA Style

    M. Y. Hamada, Tamer El-Azab, H. El-Metwally. (2023). Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model. Pure and Applied Mathematics Journal, 12(4), 59-71. https://doi.org/10.11648/j.pamj.20231204.11

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    ACS Style

    M. Y. Hamada; Tamer El-Azab; H. El-Metwally. Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model. Pure Appl. Math. J. 2023, 12(4), 59-71. doi: 10.11648/j.pamj.20231204.11

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    AMA Style

    M. Y. Hamada, Tamer El-Azab, H. El-Metwally. Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model. Pure Appl Math J. 2023;12(4):59-71. doi: 10.11648/j.pamj.20231204.11

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  • @article{10.11648/j.pamj.20231204.11,
      author = {M. Y. Hamada and Tamer El-Azab and H. El-Metwally},
      title = {Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model},
      journal = {Pure and Applied Mathematics Journal},
      volume = {12},
      number = {4},
      pages = {59-71},
      doi = {10.11648/j.pamj.20231204.11},
      url = {https://doi.org/10.11648/j.pamj.20231204.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.pamj.20231204.11},
      abstract = {This study investigates the dynamic properties of a discrete predator-prey model influenced by the Allee effect. Through rigorous analysis utilizing bifurcation theory and the center manifold theorem, we establish the stability of the system’s local equilibrium and reveal the intricate dynamical behaviors exhibited by the model, including period-doubling bifurcations at periods 2, 4, and 8, as well as the emergence of quasi-periodic orbits and chaotic sets. A notable finding is the significant role played by the parameter r in shaping the system’s behavior, as we identify a series of bifurcations, such as flip and Neimark-Sacker bifurcations, by systematically varying r while keeping other parameters fixed. These findings underscore the non-linear nature of the model and provide valuable insights into its complex dynamics. Our enhanced understanding of these bifurcations and resulting dynamical behaviors deepens our knowledge of the Allee effect’s implications for predator-prey models, contributing to our comprehension of population oscillations, stability transitions, and the emergence of chaotic dynamics in ecological systems under the Allee effect. Moreover, this study carries practical implications for population management and conservation strategies, as incorporating the Allee effect into predator-prey interactions allows for better insights into population dynamics and the development of more effective and sustainable management practices. Overall, this comprehensive analysis of the discrete predator-prey model under the Allee effect uncovers intricate dynamical behaviors and emphasizes the influential role of the parameter r in shaping system dynamics, with implications for both theoretical understanding and practical conservation management strategies.},
     year = {2023}
    }
    

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    T1  - Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model
    AU  - M. Y. Hamada
    AU  - Tamer El-Azab
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    AB  - This study investigates the dynamic properties of a discrete predator-prey model influenced by the Allee effect. Through rigorous analysis utilizing bifurcation theory and the center manifold theorem, we establish the stability of the system’s local equilibrium and reveal the intricate dynamical behaviors exhibited by the model, including period-doubling bifurcations at periods 2, 4, and 8, as well as the emergence of quasi-periodic orbits and chaotic sets. A notable finding is the significant role played by the parameter r in shaping the system’s behavior, as we identify a series of bifurcations, such as flip and Neimark-Sacker bifurcations, by systematically varying r while keeping other parameters fixed. These findings underscore the non-linear nature of the model and provide valuable insights into its complex dynamics. Our enhanced understanding of these bifurcations and resulting dynamical behaviors deepens our knowledge of the Allee effect’s implications for predator-prey models, contributing to our comprehension of population oscillations, stability transitions, and the emergence of chaotic dynamics in ecological systems under the Allee effect. Moreover, this study carries practical implications for population management and conservation strategies, as incorporating the Allee effect into predator-prey interactions allows for better insights into population dynamics and the development of more effective and sustainable management practices. Overall, this comprehensive analysis of the discrete predator-prey model under the Allee effect uncovers intricate dynamical behaviors and emphasizes the influential role of the parameter r in shaping system dynamics, with implications for both theoretical understanding and practical conservation management strategies.
    VL  - 12
    IS  - 4
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Author Information
  • Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt; Mathematics Department, Faculty of Engineering, German International University, Cairo, Egypt

  • Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt

  • Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt

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