Bifurcation and chaos in a discrete Holling–Tanner model with Beddington–DeAngelis functional response

IF 2.3 Q1 MATHEMATICS
Run Yang, Jianglin Zhao
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引用次数: 0

Abstract

Abstract The dynamics of a discrete Holling–Tanner model with Beddington–DeAngelis functional response is studied. The permanence and local stability of fixed points for the model are derived. The center manifold theorem and bifurcation theory are used to show that the model can undergo flip and Hopf bifurcations. Codimension-two bifurcation associated with 1:2 resonance is analyzed by applying the bifurcation theory. Numerical simulations are performed not only to verify the correctness of theoretical analysis but to explore complex dynamical behaviors such as period-6, 7, 10, 12 orbits, a cascade of period-doubling, quasi-periodic orbits, and the chaotic sets. The maximum Lyapunov exponents validate the chaotic dynamical behaviors of the system. The feedback control method is considered to stabilize the chaotic orbits. These complex dynamical behaviors imply that the coexistence of predator and prey may produce very complex patterns.

Abstract Image

具有Beddington-DeAngelis函数响应的离散Holling-Tanner模型的分岔和混沌
研究了具有Beddington-DeAngelis泛函响应的离散Holling-Tanner模型的动力学问题。导出了模型的不动点的持久性和局部稳定性。利用中心流形定理和分岔理论证明了该模型可以发生翻转分岔和Hopf分岔。应用分岔理论分析了与1:2共振相关的共维二分岔。数值模拟不仅验证了理论分析的正确性,而且探索了复杂的动力学行为,如周期6,7,10,12轨道,周期倍级联,准周期轨道和混沌集。最大李雅普诺夫指数验证了系统的混沌动力学行为。采用反馈控制方法稳定混沌轨道。这些复杂的动态行为暗示着捕食者和猎物的共存可能会产生非常复杂的模式。
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