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

M. Y. Hamada; Tamer El-Azab; H. El-Metwally

doi:doi:10.11648/j.pamj.20231204.11

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

M. Y. Hamada, Tamer El-Azab, H. El-Metwally

Published in Pure and Applied Mathematics Journal (Volume 12, Issue 4)

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.

Published in	Pure and Applied Mathematics Journal (Volume 12, Issue 4)
DOI	10.11648/j.pamj.20231204.11
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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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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TY  - JOUR
T1  - Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model
AU  - M. Y. Hamada
AU  - Tamer El-Azab
AU  - H. El-Metwally
Y1  - 2023/09/25
PY  - 2023
N1  - https://doi.org/10.11648/j.pamj.20231204.11
DO  - 10.11648/j.pamj.20231204.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 59
EP  - 71
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20231204.11
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
ER  -

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Author Information

M. Y. Hamada

Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
Tamer El-Azab

Mathematics Department, Faculty of Science, Mansoura University, Mansoura, Egypt
H. El-Metwally

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

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Plain Text BibTeX RIS

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}
}

Copy | Download

TY  - JOUR
T1  - Predator-Prey Interactions: Insights into Allee Effect Subject to Ricker Model
AU  - M. Y. Hamada
AU  - Tamer El-Azab
AU  - H. El-Metwally
Y1  - 2023/09/25
PY  - 2023
N1  - https://doi.org/10.11648/j.pamj.20231204.11
DO  - 10.11648/j.pamj.20231204.11
T2  - Pure and Applied Mathematics Journal
JF  - Pure and Applied Mathematics Journal
JO  - Pure and Applied Mathematics Journal
SP  - 59
EP  - 71
PB  - Science Publishing Group
SN  - 2326-9812
UR  - https://doi.org/10.11648/j.pamj.20231204.11
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
ER  -

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