Pure and Applied Mathematics Journal

Submit a Manuscript

Publishing with us to make your research visible to the widest possible audience.

Propose a Special Issue

Building a community of authors and readers to discuss the latest research and develop new ideas.

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

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.

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

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

Copyright © 2023 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. J. D. Murray, Mathematical biology II: Spatial models and biomedical applications, vol. 3. Springer New York, 2001.
2. S. N. Elaydi, Discrete chaos: with applications in science and engineering. Chapman and Hall/CRC, 2007.
3. E. S. Allman and J. A. Rhodes, Mathematical models in biology: an introduction. Cambridge University Press, 2004.
4. S. Gao and L. Chen, “The effect of seasonal harvesting on a single-species discrete population model with stage structure and birth pulses,” Chaos, Solitons & Fractals, vol. 24, no. 4, pp. 1013–1023, 2005.
5. M. Y. Hamada, T. El-Azab, and H. El-Metwally, “Allee effect in a ricker type predator-prey model,” Journal of Mathematics and Computer Science, vol. 29, no. 03, pp. 239–251, 2022.
6. C. Celik and O. Duman, “Allee effect in a discrete- time predator–prey system,” Chaos, Solitons & Fractals, vol. 40, no. 4, pp. 1956–1962, 2009.
7. S. Pal, S. K. Sasmal, and N. Pal, “Chaos control in a discrete-time predator–prey model with weak allee effect,” International Journal of Biomathematics, vol. 11, no. 07, p. 1850089, 2018.
8. L. Zhang and L. Zou, “Bifurcations and control in a discrete predator–prey model with strong allee effect,” International Journal of Bifurcation and Chaos, vol. 28, no. 05, p. 1850062, 2018.
9. A. Morozov, S. Petrovskii, and B.-L. Li, “Bifurcations and chaos in a predator-prey system with the allee effect,” Proceedings of the Royal Society of London. Series B: Biological Sciences, vol. 271, no. 1546, pp. 1407–1414, 2004.
10. J. Wang, J. Shi, and J. Wei, “Predator–prey system with strong allee effect in prey,” Journal of Mathematical Biology, vol. 62, no. 3, pp. 291–331, 2011.
11. A. J. Terry, “Predator–prey models with component allee effect for predator reproduction,” Journal of mathematical biology, vol. 71, no. 6, pp. 1325–1352, 2015.
12. C. Wang and X. Li, “Further investigations into the stability and bifurcation of a discrete predator– prey model,” Journal of Mathematical Analysis and Applications, vol. 422, no. 2, pp. 920–939, 2015.
13. M. Y. Hamada, T. El-Azab, and H. El-Metwally, “Bifurcations and dynamics of a discrete predator–prey model of ricker type,” Journal of Applied Mathematics and Computing, pp. 1–23, 2022.
14. S. M. Rana et al., “Chaotic dynamics and control of discrete ratio-dependent predator-prey system,” Discrete Dynamics in Nature and Society, vol. 2017, 2017.
15. Q. Cui, Q. Zhang, Z. Qiu, and Z. Hu, “Complex dynamics of a discrete-time predator-prey system with holling iv functional response,” Chaos, Solitons & Fractals, vol. 87, pp. 158–171, 2016.
16. M. Zhao, C. Li, and J. Wang, “Complex dynamic behaviors of a discrete-time predator-prey system,” Journal of Applied Analysis & Computation, vol. 7, no. 2, pp. 478–500, 2017.
17. M. Hamada, T. El-Azab, and H. El-Metwally, “Bifurcations and dynamics of a discrete predator–prey model of ricker type,” Journal of Applied Mathematics and Computing, vol. 69, no. 1, pp. 113–135, 2023.
18. A. Q. Khan and H. El-Metwally, “Global dynamics, boundedness, and semicycle analysis of a difference equation,” Discrete Dynamics in Nature and Society, vol. 2021, 2021.
19. M. Y. Hamada, T. El-Azab, and H. El-Metwally, “Bifurcation analysis of a two-dimensional discrete-time predator–prey model,” Mathematical Methods in the Applied Sciences, vol. 46, no. 4, pp. 4815–4833, 2023.
20. F. Courchamp, L. Berec, and J. Gascoigne, Allee effects in ecology and conservation. OUP Oxford, 2008.
21. Y. Ye, H. Liu, Y.-m. Wei, M. Ma, and K. Zhang, “Dynamic study of a predator-prey model with weak allee effect and delay,” Advances in Mathematical Physics, vol. 2019, 2019.
22. H. EL-METWALLY, A. KHAN, and M. HAMADA, “Allee effect in a ricker type discrete-time predator–prey model with holling type-II functional response,” Journal of Biological Systems, pp. 1–20, 2023.
23. Y. A. Kuznetsov, Elements of applied bifurcation theory, vol. 112. New York: Springer-Verlag, 2004.
24. S. Wiggins and M. Golubitsky, Introduction to applied nonlinear dynamical systems and chaos, vol. 2. Springer, 1990.
25. X. Liu and D. Xiao, “Complex dynamic behaviors of a discrete-time predator–prey system,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 80–94, 2007.
26. J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, vol. 42. Springer Science & Business Media, 2013.
27. Y. A. Kuznetsov and H. G. E. Meijer, Numerical Bifurcation Analysis of Maps: From Theory to Software. Cambridge University Press, 2019.
28. G. Iooss, Bifurcation of maps and applications. Elsevier, 1979.
29. J. D. Crawford, “Introduction to bifurcation theory,” Reviews of Modern Physics, vol. 63, no. 4, p. 991, 1991.
30. J. Carr, Applications of centre manifold theory, vol. 35. Springer Science & Business Media, 2012.
31. W.-B. Zhang, Discrete dynamical systems, bifurcations and chaos in economics. elsevier, 2006.