Neimark–Sacker Bifurcation and Controlling Chaos in a Three-Species Food Chain Model through the OGY Method

Guo Feng,Li Jiacheng,Ding Yin,Chris Goodrich

doi:10.1155/2021/6316235

Guo Feng, Li Jiacheng + Show 2 more

Open Access

https://doi.org/10.1155/2021/6316235

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Abstract

The dynamics behavior of a discrete-time three-species food chain model is investigated. By using bifurcation theory, it is shown that the equilibrium point of the system loses its stability, and the system undergoes Neimark–Sacker bifurcation, which leads to chaos as the parameter changes. The chaotic motion is controlled on the stable periodic period-1 orbit using the implementation of the hybrid control strategy. The factor affecting the control time of chaos is also studied. Numerical simulations are consistent with the theoretical analysis. The results of this research prove that the chaos control method can be extended to the higher-dimensional biological model and can be realized.

Highlights

Predation and prey behaviors are common phenomena in the ecosystem
By studying chaos control of the model, we can better understand the dynamic behavior of the biological system
The complex chaotic behavior of this model shows the relationship of the different species, including the number population, reproduction rate, and survival rate, whether they can survive in a balanced state, or makes the population develop in disorder or chaos

Summary

Introduction

Predation and prey behaviors are common phenomena in the ecosystem. Since the last century when Volterra and Lotka constructed the predator-prey model, the predatorprey model has been concerned by many scholars. The chaotic dynamics of a three-species food chain model is investigated and the chaos is controlled. By studying chaos control of the model, we can better understand the dynamic behavior of the biological system. E discrete-time three-species food chain model produces a chaotic attractor when a 4.26, b 3.7, c 3, d 3.5, and r 3.8. The complex chaotic behavior of this model shows the relationship of the different species, including the number population, reproduction rate, and survival rate, whether they can survive in a balanced state, or makes the population develop in disorder or chaos

The Equilibrium Point and Local Stability Analysis of the Food Chain Model

Neimark–Sacker Bifurcation Analysis

The Controlling Chaos of the Three-Species Food Chain Model

Conclusion