Nonlinear dynamics and chaos control of a discrete Rosenzweig-MacArthur prey-predator model
Article Sidebar
Main Article Content
Abstract
This paper investigates the nonlinear dynamics and chaos control of a discrete Rosenzweig-MacArthur predator-prey model. We first conduct a thorough dynamical analysis, identifying the system's equilibrium points and examining their stability conditions. The study reveals the occurrence of Flip and Neimark-Sacker bifurcations, which represent critical qualitative changes in population dynamics. Specifically, Flip bifurcations lead to period-doubling phenomena, while Neimark-Sacker bifurcations indicate the emergence of quasi-periodic oscillations, both of which are crucial for understanding the onset of complex behaviors such as cycles and oscillations in ecological systems. To address the chaotic dynamics induced by these bifurcations, two control strategies are applied: the Ott-Grebogi-Yorke (OGY) method and feedback control. The results demonstrate the effectiveness of both approaches in stabilizing the system's dynamics, with the OGY method proving to be more effective in achieving faster stabilization. These findings provide valuable insights into the management and preservation of ecological systems where predator-prey interactions exhibit instability and chaos.
Downloads
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
- Authors keep the rights and guarantee the Journal of Innovative Applied Mathematics and Computational Sciences the right to be the first publication of the document, licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License that allows others to share the work with an acknowledgement of authorship and publication in the journal.
- Authors are allowed and encouraged to spread their work through electronic means using personal or institutional websites (institutional open archives, personal websites or professional and academic networks profiles) once the text has been published.
References
[1] M.-S. Abdelouahab and R. Lozi, Hopf-like bifurcation and mixed mode oscillation in a fractional-order FitzHugh-Nagumo model, AIP Conference Proceedings 2183(1) (2019).
[2] M.-S. Abdelouahab and R. Lozi, Bifurcation analysis and chaos in simplest fractional-order electrical circuit, Proceedings of the 3rd International Conference on Control, Engineering & Information Technology (CEIT), IEEE, (2015).
[3] R. Amira, M.-S. Abdelouahab, N. Touafek, M. B. Mesmouli, H. N. Zaidi and T. S. Hassan, Nonlinear dynamic in a remanufacturing duopoly game: spectral entropy analysis and chaos control, AIMS Mathematics 9(3) (2024), 7711–7727.
[4] P. Auger, C. Lett and J. C. Poggiale, Modélisation mathématique en écologie-2e éd.: Cours et exercices corrigés, Dunod, Paris, (2015).
[5] M. Azioune and M.-S. Abdelouahab, Controlling chaos and mixed mode oscillations in a Bertrand duopoly game with homogeneous expectations and quadratic cost functions, Mathematics and Computers in Simulation 233 (2025), 553–566.
[6] M. Azioune, M.-S. Abdelouahab and R. Lozi, Bifurcation analysis of a Cournot triopoly game with bounded rationality and chaos control via the OGY method, International Journal of Bifurcation and Chaos 35(08) (2025), 2530019.
[7] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer, San Antonio, (2005).
[8] A. Q. Khan, A. Maqbool and T. D. Alharbi, Bifurcations and chaos control in a discrete Rosenzweig-Macarthur prey-predator model, Chaos: An Interdisciplinary Journal of Nonlinear Science 34(3) (2024).
[9] S. Bourafa, M.-S. Abdelouahab and R. Lozi, On periodic solutions of fractional-order differential systems with a fixed length of sliding memory, Journal of Innovative Applied Mathematics and Computational Sciences 1(1) (2021), 64–78.
[10] R. Bououden, M.-S. Abdelouahab, F. Jarad and Z. Hammouch, A novel fractional piecewise linear map: regular and chaotic dynamics, International Journal of General Systems 50(5) (2021), 501–526.
[11] N. Boudjerida, M.-S. Abdelouahab and R. Lozi, Modified projective synchronization of fractional-order hyperchaotic memristor-based Chua’s circuit, Journal of Innovative Applied Mathematics and Computational Sciences 2(3) (2022), 69–85.
[12] N. Boudjerida, M.-S. Abdelouahab and R. Lozi, Nonlinear dynamics and hyperchaos in a modified memristor-based Chua's circuit and its generalized discrete system, Journal of Difference Equations and Applications 29(9–12) (2023), 1369–1390.
[13] M.K. Hassani, Y. Yazlik, N. Touafek, M.-S. Abdelouahab, M.B. Mesmouli and F.E. Mansour, Dynamics of a higher-order three-dimensional nonlinear system of difference equations, Mathematics 12 (2024), 16.
[14] S. Kumar and H. Kharbanda, Chaotic behavior of predator-prey model with group defense and non-linear harvesting in prey, Chaos, Solitons and Fractals 119 (2019), 19–28.
[15] S. Kaouache and M.-S. Abdelouahab, Generalized synchronization between two chaotic fractional non-commensurate order systems with different dimensions, Nonlinear Dynamics and Systems Theory 18(3) (2018), 273–284.
[16] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd ed., Springer, New York, (1998).
[17] B. Labed, S. Kaouache and M.-S. Abdelouahab, Control of a novel class of uncertain fractional-order hyperchaotic systems with external disturbances via sliding mode controller, Nonlinear Dynamics and Systems Theory 20(2) (2020), 203–213.
[18] G. C. Layek, An Introduction to Dynamical Systems and Chaos, Springer, New Delhi, (2015).
[19] C. Lobry and T. Sari, Migrations in the Rosenzweig-MacArthur model and the "atto-fox" problem, Revue Africaine de Recherche en Informatique et Mathématiques Appliquées 20 (2015), 95–125.
[20] H. Meskine, M.-S. Abdelouahab and R. Lozi, Nonlinear dynamic and chaos in a remanufacturing duopoly game with heterogeneous players and nonlinear inverse demand functions, Journal of Difference Equations and Applications 29(9–12) (2023), 1503–1515.
[21] W. W. Murdoch, C. J. Briggs and R. M. Nisbet, Consumer-Resource Dynamics (MPB-36), Princeton University Press, (2003).
[22] K. Nadjah and M.-S. Abdelouahab, Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting, Electronic Research Archive 29(1) (2021), 1641–1660.
[23] G. A. Polis, C. A. Myers and R. D. Holt, The ecology and evolution of intraguild predation: potential competitors that eat each other, Annual Review of Ecology and Systematics 20(1) (1989), 297–330.
[24] H. L. Smith, The Rosenzweig-MacArthur predator-prey model, School of Mathematical and Statistical Sciences, (2008).
[25] Y. Wang, H. Wu and A. Wang, A predator-prey model characterizing negative effect of prey on its predator, Applied Mathematics and Computation 219(19) (2013), 9992–9999.