Interacting Predator-Prey Chains: A Complex Systems Approach to Stability and Chaos

Abstract

This paper presents a detailed analysis of the dynamics of a three-species food chain model, emphasizing the influence of population mobility on system behavior. The equilibrium points and bifurcation structures are analyzed to characterize transitions between regular and chaotic regimes. Numerical investigations reveal the occurrence of both Neimark–Sacker and flip bifurcations, leading to stable periodic oscillations and chaotic trajectories under parameter variations. The sensitivity of the system to initial conditions is quantified through the maximum Lyapunov exponent, confirming the presence of chaos. To suppress chaotic behavior, the Ott–Grebogi–Yorke (OGY) control method is implemented, where appropriate adjustments of regulator poles successfully stabilize period-1 orbits. Numerical simulations validate the theoretical predictions and demonstrate the effectiveness and robustness of the proposed control strategy.

Keywords

• Chaos
• Discrete FCM
• Flip bifurcation
• Neimark-sacker bifurcation
• OGY
• Maximum lyapunov exponent

References

1. Abarbanel, H. D. I., 1996 Analysis of Observed Chaotic Data. Springer Science and Business Media.
2. Abbas, A. and A. Khaliq, 2023 Analyzing predator–prey interaction in chaotic and bifurcating environments. Chaos Theory and Applications 5: 207–218.
3. Abbas, A. and A. Khaliq, 2024 Chaotic dynamics of a three–species food chain model. Physica Scripta 99.
4. Ahmad, A., U. Atta, A. Akgül, et al., 2025 A study on mathematical modeling and control of leptospirosis transmission dynamics during a hurricane with asymptomatic measures. Modeling Earth Systems and Environment 11: 361.
5. Alalhareth, F. K., U. Atta, A. H. Ali, A. Ahmad, and M. H. Alharbi, 2023 Alexandria Engineering Journal 80: 372–382.
6. Alasty, A. and H. H. Salarie, 2007 Nonlinear feedback control of chaotic pendulum in presence of saturation effect. Chaos, Solitons and Fractals 31: 292–304.
7. Alligood, K. T., T. D. Sauer, and J. A. Yorke, 1997 Chaos: An Introduction to Dynamical Systems. Springer.
8. Arneodo, A., P. Coullet, and C. Tresser, 1980 Occurrence of strange attractors in three-dimensional volterra equations. Physics Letters A 79: 259–263.

9. Berkovich, Y. et al., 2014 Controlling chaotic dynamics in ecological systems: A practical guide. Ecology Letters 17: 672–681.
10. Chakraborty, K. and T. K. Kar, 2012 Optimal harvesting in a prey– predator fishery with stage structure and nonlinear controls. Biosystems 109: 123–136.
11. Chen, L. and G. Chen, 2007 Controlling chaos in an economics model. Physica A: Statistical Mechanics and Its Applications 374: 349–358.
12. Duarte, C. M. et al., 2014 The role of nonlinearity in predicting ecological responses to global change. Science 345: 1517–1520.
13. Feng, G., 2020a Chaotic dynamics and chaos control of hassell–type recruitment population model. Discrete Dynamics in Nature and Society 2020: Article ID 8148634.
14. Feng, G., 2020b Controlling chaos of the ricker population model. American Journal of Bioscience and Bioengineering 16: 424–431.
15. Gomes, A. A., E. Manica, and M. C. Varriale, 2006 Applications of chaos control techniques to a three–species food chain. Chaos, Solitons and Fractals 36: 1097–1107.
16. Guckenheimer, J. and P. Holmes, 1983 Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer-Verlag.
17. Guo, F., J. Xie, and L. Yuan, 2014 Chaos control of lauwerier mapping. Journal of Southwest Jiaotong University 49: 525–529.
18. Hashemi, S., M. Ali Pourmina, S. Mobayen, and M. R. Alagheband, 2020 Design of a secure communication system between base transmitter station and mobile equipment based on finite–time chaos synchronisation. International Journal of Systems Science, Taylor and Francis Journals 51: 1969–1986.
19. Hastings, A. and T. Powell, 1991 Chaos in a three-species food chain. Ecology 72: 896–903.
20. Holyst, J. A. and K. Urbanowicz, 2000 Chaos control in economical model by time-delayed feedback control. Physica A: Statistical Mechanics and its Applications 287: 587–598.
21. Hu, Z., Z. Teng, and L. Zhang, 2011 Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response. Nonlinear Analysis: Real World Applications 12: 2356–2377.
22. Kacarev, L. and U. Parlitz, 1996 General approach for chaotic synchronization, with application to communication. Physical Review Letters 74: 5028–5031.
23. Li, X., S. Dong, and H. Fan, 2025 Bifurcation analysis of a class of food chain model with two time delays. Mathematics 13: 1307.
24. May, R. M., 1976 Simple mathematical models with very complicated dynamics. Nature 261: 459–467.
25. Mchich, R., M. Chakir, and S. Petrovskii, 2011 Complex behavior in a four species food-web model. Journal of Biological Dynamics 5: 524–549.
26. Mobayen, S., K. V. Christos, and S. Kaçar, 2018 A chaotic system with infinite number of equilibria located on an exponential curve and its chaos–based engineering application. International Journal of Bifurcation and Chaos 28: Article ID 1850112.
27. Ott, E., C. Grebogi, and J. A. Yorke, 1990 Controlling chaos. Physical Review Letters 64: 1196–1199.
28. Paine, R. T., 1980 Food webs: Linkage, interaction strength, and community infrastructure. The Journal of Animal Ecology 49: 667–685.
29. Pimm, S. L. and R. L. Kitching, 1987 The structure of food webs. In Theoretical Ecology, pp. 335–363, Sinauer Associates.
30. Pyragas, K., 1992 Continuous control of chaos by self–controlling feedback. Physics Letters A 170: 421–428.
31. Rinaldi, S. and M. Scheffer, 2000 Geometric analysis of ecological models. Springer Science and Business Media.
32. Romeiras, F. J., C. Grebogi, E. Ott, and W. P. Dayawansa, 1992 Controlling chaotic dynamical systems. Physica D Nonlinear Phenomena 58: 165–192.
33. Scheffer, M., S. Carpenter, J. A. Foley, C. Folke, and B. Walker, 2009 Catastrophic shifts in ecosystems. Nature 413: 591–596.
34. Vaseghi, B., S. Mobayen, S. S. Hashemi, and A. Fekih, 2020 Fast reaching finite time synchronization approach for chaotic systems with application in medical image encryption. IEEE Access 9: 25911–25925.
35. Vaseghi, B., M. A. Pourmina, and S. Mobayen, 2017 Finite-time chaos synchronization and its application in wireless sensor networks. Transactions of the Institute of Measurement and Control 40: 3788–3799.
36. Zhao, Y., Y. Chen, J. Wu, and T. Zhang, 2021 Appearance of temporal and spatial chaos in an ecological system. Frontiers in Ecology and Evolution 9: 630554.
37. Znegui, W., H. Gritli, and S. Belghith, 2020a Stabilization of the passive biped dynamic locomotion using the controlled poincaré map. IEEE 101: 1061–1091.
38. Znegui, W., H. Gritli, and S. Belghith, 2020b Stabilization of the passive biped dynamic locomotion using the controlled poincaré map. IEEE 101: 1061–1091.
39. Znegui,W., H. Gritli, and S. Belghith, 2021 A new poincaré map for investigating the complex walking behavior of the compass–gait biped robot. Applied Mathematical Modelling 94: 534–557.