IDEAS home Printed from https://ideas.repec.org/a/eee/chsofr/v180y2024ics0960077924000493.html
   My bibliography  Save this article

Analyzing multi-parameter bifurcation on a prey–predator model with the Allee effect and fear effect

Author

Listed:
  • Abbasi, Muhammad Aqib
  • Samreen, Maria

Abstract

In this research article, we present the dynamic analysis of the prey–predator model, adding the fear and the Allee effects. The main objective of this study is to analyze the periodicity and multi-parameter bifurcation of the model graphically. In particular, we studied the bifurcation between the two parameters. We critically analyzed the dynamics in the model with the help of detailed graphs of period ten and the Lyapunov exponent. Our study provides novel insights into the bifurcation behavior of the model, emphasizing the significance of Lyapunov exponents and the period-10 oscillation in understanding the dependencies among parameters. Through period ten oscillations, we confirm the complex dynamics, the coexistence of populations, and sensitivity to the initial conditions. Our analysis of the continuous-time model reveals that only Hopf bifurcation occurs at the positive fixed point, and we offer mathematical proof that no Hopf bifurcation occurs at other fixed points except the positive fixed point. From the numerical examples, we concluded that the crowding effect should be minimized for the stability of the model. Also, in the interior fixed point, when fear and Allee effects are taken as bifurcation parameters, backward bifurcations occur, which shows that in the presence of the crowding effect, the increase of the fear effect stabilizes the model.

Suggested Citation

  • Abbasi, Muhammad Aqib & Samreen, Maria, 2024. "Analyzing multi-parameter bifurcation on a prey–predator model with the Allee effect and fear effect," Chaos, Solitons & Fractals, Elsevier, vol. 180(C).
    • Handle: RePEc:eee:chsofr:v:180:y:2024:i:c:s0960077924000493
    DOI: 10.1016/j.chaos.2024.114498
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0960077924000493
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.chaos.2024.114498?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Marotto, F.R., 2005. "On redefining a snap-back repeller," Chaos, Solitons & Fractals, Elsevier, vol. 25(1), pages 25-28.
    2. Garai, Shilpa & Pati, N.C. & Pal, Nikhil & Layek, G.C., 2022. "Organized periodic structures and coexistence of triple attractors in a predator–prey model with fear and refuge," Chaos, Solitons & Fractals, Elsevier, vol. 165(P2).
    3. Ruizhi Yang & Qiannan Song & Yong An, 2021. "Spatiotemporal Dynamics in a Predator–Prey Model with Functional Response Increasing in Both Predator and Prey Densities," Mathematics, MDPI, vol. 10(1), pages 1-15, December.
    4. Ye Xuan Li & Hua Liu & Yu Mei Wei & Ming Ma & Gang Ma & Jing Yan Ma & Ljubisa Kocinac, 2022. "Population Dynamic Study of Prey-Predator Interactions with Weak Allee Effect, Fear Effect, and Delay," Journal of Mathematics, Hindawi, vol. 2022, pages 1-15, February.
    5. Dhar, Joydip & Singh, Harkaran & Bhatti, Harbax Singh, 2015. "Discrete-time dynamics of a system with crowding effect and predator partially dependent on prey," Applied Mathematics and Computation, Elsevier, vol. 252(C), pages 324-335.
    6. Ruizhi Yang & Xiao Zhao & Yong An, 2022. "Dynamical Analysis of a Delayed Diffusive Predator–Prey Model with Additional Food Provided and Anti-Predator Behavior," Mathematics, MDPI, vol. 10(3), pages 1-18, January.
    7. Jie Yan & Chunli Li & Xueli Chen & Lishun Ren, 2016. "Dynamic Complexities in 2-Dimensional Discrete-Time Predator-Prey Systems with Allee Effect in the Prey," Discrete Dynamics in Nature and Society, Hindawi, vol. 2016, pages 1-14, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Dohtani, Akitaka, 2011. "Chaos resulting from nonlinear relations between different variables," Chaos, Solitons & Fractals, Elsevier, vol. 44(4), pages 290-297.
    2. Gardini, Laura & Sushko, Iryna & Avrutin, Viktor & Schanz, Michael, 2011. "Critical homoclinic orbits lead to snap-back repellers," Chaos, Solitons & Fractals, Elsevier, vol. 44(6), pages 433-449.
    3. Tamotsu Onozaki, 2018. "Nonlinearity, Bounded Rationality, and Heterogeneity," Springer Books, Springer, number 978-4-431-54971-0, June.
    4. Jiang, Guirong & Yang, Qigui, 2009. "Complex dynamics in a linear impulsive system," Chaos, Solitons & Fractals, Elsevier, vol. 41(5), pages 2341-2353.
    5. Chen, Shyan-Shiou & Shih, Chih-Wen, 2009. "Transiently chaotic neural networks with piecewise linear output functions," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 717-730.
    6. Elhafsi Boukhalfa & Elhadj Zeraoulia, 2014. "Existence of Super Chaotic Attractors in a General Piecewise Smooth Map of the Plane," Interdisciplinary Description of Complex Systems - scientific journal, Croatian Interdisciplinary Society Provider Homepage: http://indecs.eu, vol. 12(1), pages 92-98.
    7. Xiongxiong Du & Xiaoling Han & Ceyu Lei, 2022. "Behavior Analysis of a Class of Discrete-Time Dynamical System with Capture Rate," Mathematics, MDPI, vol. 10(14), pages 1-15, July.
    8. Gardini, Laura & Hommes, Cars & Tramontana, Fabio & de Vilder, Robin, 2009. "Forward and backward dynamics in implicitly defined overlapping generations models," Journal of Economic Behavior & Organization, Elsevier, vol. 71(2), pages 110-129, August.
    9. A. M. A. El-Sayed & S. M. Salman, 2019. "Dynamical analysis of a complex logistic-type map," Indian Journal of Pure and Applied Mathematics, Springer, vol. 50(2), pages 427-450, June.
    10. Zhang, Limin & Zhang, Chaofeng & He, Zhirong, 2019. "Codimension-one and codimension-two bifurcations of a discrete predator–prey system with strong Allee effect," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 162(C), pages 155-178.
    11. Liang, Wei & Lv, Xiaolin, 2022. "Li-Yorke chaos in a class of controlled delay difference equations," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    12. Salman, S.M. & Yousef, A.M. & Elsadany, A.A., 2016. "Stability, bifurcation analysis and chaos control of a discrete predator-prey system with square root functional response," Chaos, Solitons & Fractals, Elsevier, vol. 93(C), pages 20-31.
    13. He, Haoming & Xiao, Min & He, Jiajin & Zheng, Weixing, 2024. "Regulating spatiotemporal dynamics for a delay Gierer–Meinhardt model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 637(C).
    14. Li, Yan & Wang, Lidong, 2019. "Chaos in a duopoly model of technological innovation with bounded rationality based on constant conjectural variation," Chaos, Solitons & Fractals, Elsevier, vol. 120(C), pages 116-126.
    15. Kumbhakar, Ruma & Hossain, Mainul & Karmakar, Sarbari & Pal, Nikhil, 2024. "An investigation of the parameter space in a tri-trophic food chain model with refuge," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 217(C), pages 37-59.
    16. Ekaterina Ekaterinchuk & Jochen Jungeilges & Tatyana Ryazanova & Iryna Sushko, 2017. "Dynamics of a minimal consumer network with uni-directional influence," Journal of Evolutionary Economics, Springer, vol. 27(5), pages 831-857, November.
    17. Zhao, Yi & Xie, Lingli & Yiu, K.F. Cedric, 2009. "An improvement on Marotto’s theorem and its applications to chaotification of switching systems," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2225-2232.
    18. Haoming Shi & Fei Xu & Jinfu Cheng & Victor Shi, 2023. "Exploring the Evolution of the Food Chain under Environmental Pollution with Mathematical Modeling and Numerical Simulation," Sustainability, MDPI, vol. 15(13), pages 1-17, June.
    19. Rahman, Aminur & Blackmore, Denis, 2017. "Threshold voltage dynamics of chaotic RS flip-Flops," Chaos, Solitons & Fractals, Elsevier, vol. 103(C), pages 555-566.
    20. Sun, Huijing & Cao, Hongjun, 2007. "Bifurcations and chaos of a delayed ecological model," Chaos, Solitons & Fractals, Elsevier, vol. 33(4), pages 1383-1393.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:chsofr:v:180:y:2024:i:c:s0960077924000493. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Thayer, Thomas R. (email available below). General contact details of provider: https://www.journals.elsevier.com/chaos-solitons-and-fractals .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.