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Bifurcation Analysis, Chaos Control and Complex Dynamics of a Discrete Prey-Predator System Incorporating an Ivlev Functional Response with an Allee Effect

  • Md. Mutakabbir Khan [1] ; Md. Jasim Uddin [1] ; Parvaiz Ahmad Naik [2] ; Zohreh Eskandari [3] ; Zhengxin Huang [2]
    1. [1] University of Dhaka

      University of Dhaka

      DCC (Kotwali), Bangladés

    2. [2] Youjiang Medical University for Nationalities
    3. [3] Fasa University
    Mostrar afiliaciones +
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 4, 2025
  • Idioma: inglés
  • Enlaces
  • Resumen

    • This study examines the discrete-time dynamics of a predator-prey model incorporating an Ivlev functional response with an Allee effect. Through rigorous algebraic analysis, we establish the occurrence of period-doubling (PD) and Neimark–Sacker (NS) bifurcations in the positive phase space. Using the center-manifold theorem and bifurcation theory, we provide a theoretical framework to understand these bifurcations. Numerical simulations confirm our findings, illustrating chaotic behavior through phase portraits, period-12 orbits, invariant closed curves, and chaotic attractors. Lyapunov exponents further validate the chaotic nature of the system, emphasizing the influence of parameter variations on dynamic transitions. To mitigate chaos, we implement an OGY control strategy, effectively stabilizing trajectories around an unstable equilibrium. Furthermore, we extend the analysis to a coupled predator-prey network, revealing that chaotic behavior emerges when the coupling strength exceeds a critical threshold. Stochastic simulations, employing the EulerMaruyama method, account for environmental uncertainties and explore system behavior under varying ecological conditions. This research advances the understanding of non-linear predator-prey interactions, bifurcations, and chaotic transitions in isolated and networked systems while demonstrating effective strategies for controlling instability in ecological dynamics.

  • Referencias bibliográficas
    • 1. Berryman, A.A.: The orgins and evolution of predator-prey theory. Ecology 73(5), 1530–1535 (1992)
    • 2. Lotka, A.J.: Elements of Physical Biology. Williams & Wilkins (1925)
    • 3. Volterra, V.: Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Società anonima tipografica" Leonardo...
    • 4. Singh, H., Dhar, J., Bhatti, H.S.: Discrete-time bifurcation behavior of a prey-predator system with generalized predator. Adv. Difference...
    • 5. Ghosh, U., Sarkar, S., Chakraborty, P.: Stability and bifurcation analysis of a discrete prey-predator model with mate-finding allee, holling...
    • 6. Sharma, V.S., Singh, A., Malik, P.: Bifurcation patterns in a discrete predator-prey model incorporating ratio-dependent functional response...
    • 7. Kuznetsov, Y.A., Meijer, H.G.: Numerical normal forms for codim-2 bifurcations of fixed points with at most two critical eigenvalues. SIAM...
    • 8. Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 112. Springer Science & Business Media, Germany (2013)
    • 9. Kuznetsov, Y.A., Meijer, H.G.: Numerical Bifurcation Analysis of Maps, 34. Cambridge University Press, United Kingdom (2019)
    • 10. Uddin, M.J., Rana, S.M.S.: Qualitative analysis of the discretization of a continuous fractional order prey-predator model with the effects...
    • 11. Mutakabbir, K.M., Jasim, U.M., Fahim, D., Islam, S., Sohel, R.S.M., Qadeer, K.A., Ali, S.N.: Complex dynamics of a discrete prey-predator...
    • 12. Naik, P.A., Javaid, Y., Ahmed, R., Eskandari, Z., Ganie, A.H.: Stability and bifurcation analysis of a population dynamic model with Allee...
    • 13. Holling, C.S.: The functional response of predators to prey density and its role in mimicry and population regulation. The Memoirs of...
    • 14. Andrews, J.F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng....
    • 15. Sokol, W., Howell, J.A.: Kinetics of phenol oxidation by washed cells. Biotechnol. Bioeng. 23(9), 2039–2049 (1981)
    • 16. Ivlev, V.S.: Experimental ecology of the feeding of fishes. (No Title) (1961)
    • 17. Cheng, K.S., Hsu, S.B., Lin, S.S.: Some results on global stability of a predator-prey system. J. Math. Biol. 12(1), 115–126 (1982)
    • 18. Guo, G., Li, B., Lin, X.: Qualitative analysis on a predator-prey model with ivlev functional response. Adv. Difference Equ. 2013, 1–14...
    • 19. Kooij, R.E., Zegeling, A.: A predator–prey model with ivlev’s functional response. J. Math. Anal. Appl. 198(2), 473–489 (1996)
    • 20. Naik, P.A., Ahmed, R., Faizan, A.: Theoretical and numerical bifurcation analysis of a discrete predatorprey system of ricker type with...
    • 21. Pearce, I.G., Chaplain, M.A., Schofield, P.G., Anderson, A.R., Hubbard, S.F.: Modelling the spatiotemporal dynamics of multi-species host–parasitoid...
    • 22. Preedy, K.F., Schofield, P.G., Chaplain, M.A., Hubbard, S.F.: Disease induced dynamics in host– parasitoid systems: chaos and coexistence....
    • 23. Jing, Z., Yang, J.: Bifurcation and chaos in discrete-time predator–prey system. Chaos, Solitons & Fractals 27(1), 259–277 (2006)
    • 24. Courchamp, F., Clutton-Brock, T., Grenfell, B.: Inverse density dependence and the allee effect. Trends Ecol. Evol. 14(10), 405–410 (1999)
    • 25. Allee, W.C.: Co-operation among animals. Am. J. Sociol. 37(3), 386–398 (1931)
    • 26. Stephens, P.A., Sutherland, W.J.: Consequences of the allee effect for behaviour, ecology and conservation. Trends Ecol. Evol. 14(10),...
    • 27. Ditta, A., Naik, P.A., Ahmed, R., Huang, Z.: Exploring periodic behavior and dynamical analysis in a harvested discrete-time commensalism...
    • 28. McCarthy, M.A.: The allee effect, finding mates and theoretical models. Ecol. Model. 103(1), 99–102 (1997)
    • 29. Poggiale, J.C.: From behavioural to population level: growth and competition. Math. Comput. Model. 27(4), 41–49 (1998)
    • 30. Dennis, B.: Allee effects: population growth, critical density, and the chance of extinction. Nat. Resour. Model. 3(4), 481–538 (1989)
    • 31. Kent, A., Doncaster, C.P., Sluckin, T.: Consequences for predators of rescue and allee effects on prey. Ecol. Model. 162(3), 233–245 (2003)
    • 32. Othmer, H.G., Scriven, L.E.: Instability and dynamic pattern in cellular networks. J. Theor. Biol. 32(3), 507–537 (1971)
    • 33. Plahte, E.: Pattern formation in discrete cell lattices. J. Math. Biol. 43(5), 411–445 (2001)
    • 34. Moore, P.K., Horsthemke, W.: Localized patterns in homogeneous networks of diffusively coupled reactors. Physica D 206(1–2), 121–144 (2005)
    • 35. Nakao, H., Mikhailov, A.S.: Turing patterns in network-organized activator–inhibitor systems. Nat. Phys. 6(7), 544–550 (2010)
    • 36. Fernandes, L.D., De Aguiar, M.A.M.: Turing patterns and apparent competition in predator-prey food webs on networks. Physical Review E-Statistical,...
    • 37. Asllani, M., Challenger, J.D., Pavone, F.S., Sacconi, L., Fanelli, D.: The theory of pattern formation on directed networks. Nat. Commun....
    • 38. Szolnoki, A., Mobilia, M., Jiang, L.L., Szczesny, B., Rucklidge, A.M., Perc, M.: Cyclic dominance in evolutionary games: a review. J....
    • 39. Kouvaris, N.E., Hata, S., Guilera, A.D.: Pattern formation in multiplex networks. Sci. Rep. 5(1), 10840 (2015)
    • 40. Aguirre, P., González-Olivares, E., Sáez, E.: Three limit cycles in a leslie-gower predator-prey model with additive allee effect. SIAM...
    • 41. Aguirre, P., González-Olivares, E., Sáez, E.: Two limit cycles in a leslie-gower predator–prey model with additive Allee effect. Nonlinear...
    • 42. Kloeden, P.E., Platen, E., Kloeden, P.E., Platen, E.: Stochastic Differential Equations, pp. 103–160. Springer, Berlin Heidelberg (1992)
    • 43. Chatibi, Y., El Kinani, E.H., Ouhadan, A.: Variational calculus involving nonlocal fractional derivative with mittag-leffler kernel. Chaos,...
    • 44. Fan, Y., Huang, X., Wang, Z., Li, Y.: Nonlinear dynamics and chaos in a simplified memristor-based fractional-order neural network with...
    • 45. El-Saka, H.A.A.: Backward bifurcations in fractional-order vaccination models. J. Egyptian Math. Soc. 23(1), 49–55 (2015)
    • 46. Javidi, M., Nyamoradi, N.: Dynamic analysis of a fractional order prey–predator interaction with harvesting. Appl. Math. Model. 37(20–21),...
    • 47. Cui, Z., Yang, Z.: Homotopy perturbation method applied to the solution of fractional lotka-volterra equations with variable coefficients....
    • 48. Mondal, S., Lahiri, A., Bairagi, N.: Analysis of a fractional order eco-epidemiological model with prey infection and type 2 functional...
    • 49. Uddin, M.J., Rana, S.S.: Chaotic dynamics of the fractional order schnakenberg model and its control. Math. Appl. Sci. Eng. 4(1), 40–60...
    • 50. Naik, P.A., Eskandari, Z., Yavuz, M., Huang, Z.: Bifurcation results and chaos in a two-dimensional predator-prey model incorporating...
    • 51. Yan, Y., Kou, C.: Stability analysis for a fractional differential model of HIV infection of CD4+ T-cells with time delay. Math. Comput....
    • 52. Leslie, P.H.: A stochastic model for studying the properties of certain biological systems by numerical methods. Biometrika 45(1–2), 16–31...
    • 53. Lynch, S., et al.: Dynamical Systems With Applications Using Mathematica. Springer, Germany (2007)
    • 54. Ott, E., Grebogi, C., Yorke, J.A.: Controlling chaos. Phys. Rev. Lett. 64(11), 1196 (1990)
    • 55. Uddin, M.J., Podder, C.N.: Fractional order prey-predator model incorporating immigration on prey: complexity analysis and its control....
    • 56. Khan, A.Q.,Maqbool, A., Uddin,M.J., Rana, S.M.S.: Dynamical analysis of a two-dimensional discrete predator–prey model. J. Comput. Appl....
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