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Bifurcation and Chaos in a Nonlinear Population Model with Delayed Nonlinear Interactions

Author

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  • Bashir Al-Hdaibat

    (Department of Mathematics, Faculty of Science, The Hashemite University, Zarqa 13133, Jordan)

  • A. Alameer

    (Department of Mathematics, University of Hafr Al Batin, Hafr Al Batin 31991, Saudi Arabia)

Abstract

This paper investigates the dynamic behavior of a second-order nonlinear rational difference equation modeling a population system with nonlinear interactions between current and previous population states. We derive analytical conditions for the stability of fixed points, explore codim-1 bifurcations, and compute the associated topological normal forms. The analysis also establishes the existence of period-2 solutions and reveals the potential for chaotic dynamics within specific parameter ranges. To validate the theoretical findings, we conduct numerical simulations and bifurcation analysis using the MATLAB package MatContM (version 5p4). Chaotic behavior is further confirmed through the computation of the largest Lyapunov exponent. The results offer new insights into the complex dynamics of delayed population models with nonlinear feedback, extending classical models and suggesting potential applications in stochastic systems and epidemiological modeling.

Suggested Citation

  • Bashir Al-Hdaibat & A. Alameer, 2025. "Bifurcation and Chaos in a Nonlinear Population Model with Delayed Nonlinear Interactions," Mathematics, MDPI, vol. 13(13), pages 1-23, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:13:p:2132-:d:1690639
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    References listed on IDEAS

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    1. Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Bifurcation Analysis for a Predator-Prey Model with Discrete and Distributed Delay," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-12, June.
    2. Al-Hdaibat, Bashir & Govaerts, Willy & Neirynck, Niels, 2015. "On periodic and chaotic behavior in a two-dimensional monopoly model," Chaos, Solitons & Fractals, Elsevier, vol. 70(C), pages 27-37.
    3. Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Hopf Bifurcation Analysis for a Stage‐Structured Predator‐Prey Model with Discrete and Distributed Delays," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    4. Mirela Garić-Demirović & Samra Moranjkić & Mehmed Nurkanović & Zehra Nurkanović, 2020. "Stability, Neimark–Sacker Bifurcation, and Approximation of the Invariant Curve of Certain Homogeneous Second-Order Fractional Difference Equation," Discrete Dynamics in Nature and Society, Hindawi, vol. 2020, pages 1-12, August.
    5. Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Hopf Bifurcation Analysis for a Stage-Structured Predator-Prey Model with Discrete and Distributed Delays," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-10, November.
    6. Ruiqing Shi & Junmei Qi & Sanyi Tang, 2013. "Stability and Bifurcation Analysis for a Predator‐Prey Model with Discrete and Distributed Delay," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
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