Dynamics, bifurcation, and chaos control in a fractional-order discrete multi-team predator–prey model

Mahmoud A. M. Abdelaziz; Elhadi E. Elamir; Ibrahim M. E. Abdelsatar; A. A. Al Qarni; Manal Alqhtani; Montaser O. S. Hilal; A. A. Elsadany; Mahmoud A. M. Abdelaziz; Elhadi E. Elamir; Ibrahim M. E. Abdelsatar; A. A. Al Qarni; Manal Alqhtani; Montaser O. S. Hilal; A. A. Elsadany

doi:10.3934/math.2026397

AIMS Mathematics

2026, Volume 11, Issue 4: 9587-9611. doi: 10.3934/math.2026397

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Research article

Dynamics, bifurcation, and chaos control in a fractional-order discrete multi-team predator–prey model

1.
Department of Mathematics, Faculty of Science and Arts, Najran University, Najran 66445, Saudi Arabia
2.
Science and Engineering Research Center, Najran University, Najran, Saudi Arabia
3.
Department of Management Information Systems, College of Business and Economics, Qassim University, Buraidah 51452, Saudi Arabia
4.
Department of Mathematics, College of Science, University of Bisha, P.O. Box 551, Bisha 61922, Saudi Arabia
5.
Department of Mathematics, College of Sciences and Humanities, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

Received: 03 December 2025 Revised: 26 February 2026 Accepted: 03 March 2026 Published: 10 April 2026
MSC : 39A30, 37G15

In this paper, a discrete-time fractional-order multi-team predator–prey model derived from a Caputo-type fractional framework was proposed and analyzed. The model describes the nonlinear interactions between two cooperating prey populations and a common predator species, where the fractional-order parameter modulates the influence of past states through the fractional discrete formulation. An algebraic stability criterion based on the characteristic polynomial coefficients of the Jacobian matrix was employed to investigate the local asymptotic stability of the coexistence equilibrium without relying on explicit eigenvalue computations. The necessary and sufficient analytical conditions for the occurrence of flip and Neimark–Sacker (NS) bifurcations were derived, and the associated codimension-two flip–NS bifurcation was rigorously characterized. Numerical simulations validated the theoretical results and revealed complex dynamical behaviors, including periodic oscillations, quasiperiodicity, and chaos. To suppress unstable oscillations, three chaos control strategies—Ott–Grebogi–Yorke (OGY) feedback control, hybrid control, and state feedback control—were implemented. A systematic quantitative comparison based on convergence speed, control effort, and stabilization robustness was conducted to evaluate their relative performance and to clarify their ecological management implications. The results showed that the fractional-order parameter significantly affects stability thresholds and bifurcation organization, thereby reshaping the qualitative structure of population dynamics. These findings highlight the regulatory role of fractional-order effects in discrete ecological systems and provide a rigorous framework for stability analysis and controlled ecosystem management.
- mathematical model,
- fractional difference equations,
- predator–prey model,
- algebraic stability criterion,
- flip bifurcation,
- Neimark–Sacker bifurcation,
- codimension-two bifurcation,
- discrete dynamical systems,
- chaos control
Citation: Mahmoud A. M. Abdelaziz, Elhadi E. Elamir, Ibrahim M. E. Abdelsatar, A. A. Al Qarni, Manal Alqhtani, Montaser O. S. Hilal, A. A. Elsadany. Dynamics, bifurcation, and chaos control in a fractional-order discrete multi-team predator–prey model[J]. AIMS Mathematics, 2026, 11(4): 9587-9611. doi: 10.3934/math.2026397

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Abstract

In this paper, a discrete-time fractional-order multi-team predator–prey model derived from a Caputo-type fractional framework was proposed and analyzed. The model describes the nonlinear interactions between two cooperating prey populations and a common predator species, where the fractional-order parameter modulates the influence of past states through the fractional discrete formulation. An algebraic stability criterion based on the characteristic polynomial coefficients of the Jacobian matrix was employed to investigate the local asymptotic stability of the coexistence equilibrium without relying on explicit eigenvalue computations. The necessary and sufficient analytical conditions for the occurrence of flip and Neimark–Sacker (NS) bifurcations were derived, and the associated codimension-two flip–NS bifurcation was rigorously characterized. Numerical simulations validated the theoretical results and revealed complex dynamical behaviors, including periodic oscillations, quasiperiodicity, and chaos. To suppress unstable oscillations, three chaos control strategies—Ott–Grebogi–Yorke (OGY) feedback control, hybrid control, and state feedback control—were implemented. A systematic quantitative comparison based on convergence speed, control effort, and stabilization robustness was conducted to evaluate their relative performance and to clarify their ecological management implications. The results showed that the fractional-order parameter significantly affects stability thresholds and bifurcation organization, thereby reshaping the qualitative structure of population dynamics. These findings highlight the regulatory role of fractional-order effects in discrete ecological systems and provide a rigorous framework for stability analysis and controlled ecosystem management.

References

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