International Journal of Neutrosophic Science

Journal DOI

https://doi.org/10.54216/IJNS

Submit Your Paper

2690-6805ISSN (Online) 2692-6148ISSN (Print)

Volume 26 , Issue 2 , PP: 310-323, 2025 | Cite this article as | XML | Html | PDF | Full Length Article

Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation

Thwiba A. Khalid 1 *

  • 1 Department of Mathematics, Faculty of Science, Al-Baha University, Albaha 65525, Saudi Arabia; Academy of Engineering and Medical Sciences, Department of Mathematics, Khartoum, Sudan - (tabdulrhman@bu.edu.sa)
  • Doi: https://doi.org/10.54216/IJNS.260224

    Received: December 17, 2024 Revised: February 06, 2025 Accepted: March 03, 2025
    Abstract

    The study examines the dynamics of a predator-prey model that includes temporal delays, concentrating on the impact of these delays on system stability and behavior.It delineates criteria for the global stability of the positive equilibrium using a generalized Lyapunov function and the Razumkin-type theorem, emphasizing the significance of temporal delays in biological systems. The research highlights the Neymark-Saker (NS) bifurcation, examining the impact of fractional configurations on this bifurcation and the system’s overall dynamic stability. The research utilizes the Lyapunov-Razumihin approach to identify bifurcation points and forecast the system’s progression in intricate ecological settings. The research examines the presence of periodic solutions and local stability criteria related to the two delays in predator-prey interactions. Numerical simulations are used to substantiate the theoretical results, specifically for the periodic bifurcation solutions associated with the Neymark-Saker bifurcation.

    Keywords :

    Fixed Points , Bifurcation , Razumkin Theorem , Predator-Prey Dynamics , Time Delays

    References

    [1] Mishra, P., Raw, S. & Tiwari, B. Study of a Leslie–Gower predator-prey model with prey defense and mutual interference of predators. Chaos, Solitons Fractals. 120 pp. 1-16 (2019)

    [2] Toledo-Hernandez, R., Rico-Ramirez, V., Iglesias-Silva, G. & Diwekar, U. A fractional calculus approach to the dynamic optimization of biological reactive systems. Part I: Fractional models for biological reac- tions. Chemical Engineering Science. 117 pp. 217-228 (2014)

    [3] Zhang, Z. & Wang, Y. Qualitative analysis for a delayed epidemic model with latent and breaking-out over the Internet. Advances In Difference Equations. 2017 pp. 1-13 (2017)

    [4] Zhang, Z., Wei, R. & Xia, W. Dynamical analysis of a giving up smoking model with time delay. Ad- vances In Difference Equations. 2019 pp. 1-17 (2019)

    [5] Astakhov, O., Astakhov, S., Fadeeva, N. & Astakhov, V. Dynamics of the generator with three circuits in the feedback loop. Multistability formation and transition to chaos. Izvestiya Of Saratov University. Physics. 21, 21-28 (2021)

    [6] Aru˘gaslan C¸ inc¸in, D. Differential equations with discontinuities and population dynamics. (Middle East Technical University,2009)

    [7] Aru˘gaslan, D. & Zer, A. Stability analysis of a predator–prey model with piecewise constant argument of generalized type using Lyapunov functions. Journal Of Mathematical Sciences. 203 pp. 297-305 (2014)

    [8] Alsowait, N., Khalid, T., Alnoor, F., Mohammed, M. & Taha, N. Robust Control and Synchronization of Fractional-Order Unified Chaotic Systems. (2024)

    [9] Chekroun, A., Frioui, M., Kuniya, T. & Touaoula, T. Global stability of an age-structured epidemic model with general Lyapunov functional. Math. Biosci. Eng. 16, 1525-1553 (2019)

    [10] Song, Y., Li, Z. & Du, Y. Stability and Hopf bifurcation of a ratio-dependent predator-prey model with time delay and stage structure. Electronic Journal Of Qualitative Theory Of Differential Equations. 2016, 1-23 (2016)

    [11] Wang, C., Jia, L., Li, L. & Wei, W. Global stability in a delayed ratio-dependent predator-prey system with feedback controls. IAENG International Journal Of Applied Mathematics. 50, 1-9 (2020)

    [12] TANG, Y., Beretta, E., Solimano, F. & Others Stability analysis of a Volterra predator-prey system with two delays. Canad. Appl. Math. Quart. 9, 75-97 (2001)

    [13] Y. Lv, “The spatially homogeneous Hopf bifurcation induced jointly by memory and general delays in a diffusive system,” Journal of Applied Mathematics and Computing. 78, 213-232 (2020)

    [14] Y. Chen, X. Zeng, and B. Niu, “Equivariant Hopf bifurcation arising in circular-distributed predator-prey interaction with taxis,” Journal of Applied Mathematics and Computing. 82, 512-540 (2024)

    [15] Wang, L., Xu, R. & Feng, G. Global dynamics of a delayed predator–prey model with stage structure and holling type II functional response. Journal Of Applied Mathematics And Computing. 47, 73-89 (2015)

    [16] Jankovic, M. Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems. IEEE Transactions On Automatic Control. 46, 1048-1060 (2001)

    [17] Nekhoroshikh, A., Efimov, D., Polyakov, A., Perruquetti, W. & Furtat, I. Hyperexponential and fixed- time stability of time-delay systems: Lyapunov–Razumikhin method. IEEE Transactions On Automatic Control. 68, 1862-1869 (2022)

    [18] Gokul, P. & Rakkiyappan, R. New finite-time stability for fractional-order time-varying time-delay linear systems: A Lyapunov approach. Journal Of The Franklin Institute. 359, 7620-7631 (2022)

    [19] Zhang, S., Tang, M., Li, X. & Liu, X. Stability and stabilization of fractional-order non-autonomous systems with unbounded delay. Communications In Nonlinear Science And Numerical Simulation. 117 pp. 106922 (2023)

    [20] Palanisamy, G., Kashkynbayev, A. & Rajan, R. Finite-Time Stability of Fractional-Order Discontinuous Nonlinear Systems With State-Dependent Delayed Impulses. IEEE Transactions On Systems, Man, And Cybernetics: Systems. (2023)

    [21] Chen, Z. Bifurcations and mixed mode oscillations in a bi-stable plasma model with slow parametric excitation. Chaos: An Interdisciplinary Journal Of Nonlinear Science. 34 (2024)

    [22] Pandey, V. & Singh, S. Bifurcations emerging from a double Hopf bifurcation for a BWR. Progress In Nuclear Energy. 117 pp. 103049 (2019)

    [23] Govaerts, W. & Ghaziani, R. Numerical bifurcation analysis of a nonlinear stage structured cannibalism population model. Journal Of Difference Equations And Applications. 12, 1069-1085 (2006)

    [24] Robbio, F., Alonso, D. & Moiola, J. Detection of limit cycle bifurcations using harmonic balance meth- ods. International Journal Of Bifurcation And Chaos. 14, 3647-3654 (2004)

    [25] Wang, X. & Wang, Y. Novel dynamics of a predator–prey system with harvesting of the predator guided by its population. Applied Mathematical Modelling. 42 pp. 636-654 (2017)

    [26] Khalid, T., Alnoor, F., Babeker, E., Ahmed, E. & Mustafa, A. Legendre Polynomials and Techniques for Collocation in the Computation of Variable-Order Fractional Advection-Dispersion Equations. Interna- tional Journal Of Analysis And Applications. 22 pp. 185-185 (2024)

    [27] Abdulrhman, T. Stability Analysis of Fractional Chaotic and Fractional-Order Hyperchain Systems Using Lyapunov Functions. European Journal Of Pure And Applied Mathematics. 18, 5576-5576 (2025)

    [28] Adouane, S., Laadjal, B. & Menacer, T. Hopf bifurcation analysis in cai system with fractional order. Studies In Engineering And Exact Sciences. 5, e11803-e11803 (2024)

    [29] Boulkroune, A. & Ladaci, S. Advanced synchronization control and bifurcation of chaotic fractional- order systems. (IGI Global,2018)

    [30] Sun, X., Dimirovski, G., Zhao, J. & Wang, W. Exponential stability for switched delay systems based on average dwell time technique and Lyapunov function method. 2006 American Control Conference. pp. 5-pp (2006)

    Cite This Article As :
    A., Thwiba. Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation. International Journal of Neutrosophic Science, vol. , no. , 2025, pp. 310-323. DOI: https://doi.org/10.54216/IJNS.260224
    A., T. (2025). Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation. International Journal of Neutrosophic Science, (), 310-323. DOI: https://doi.org/10.54216/IJNS.260224
    A., Thwiba. Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation. International Journal of Neutrosophic Science , no. (2025): 310-323. DOI: https://doi.org/10.54216/IJNS.260224
    A., T. (2025) . Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation. International Journal of Neutrosophic Science , () , 310-323 . DOI: https://doi.org/10.54216/IJNS.260224
    A. T. [2025]. Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation. International Journal of Neutrosophic Science. (): 310-323. DOI: https://doi.org/10.54216/IJNS.260224
    A., T. "Dynamics of Predator-Prey Interactions, Analyzing the Effects of Time Delays and Neymark-Saker Bifurcation," International Journal of Neutrosophic Science, vol. , no. , pp. 310-323, 2025. DOI: https://doi.org/10.54216/IJNS.260224