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Complex Dynamics of a Filippov Three-Species Food Chain Model
International Journal of Bifurcation and Chaos ( IF 2.2 ) Pub Date : 2021-04-29 , DOI: 10.1142/s0218127421500747
Soliman A. A. Hamdallah 1, 2 , Ayman A. Arafa 3, 4 , Sanyi Tang 1 , Yong Xu 3
Affiliation  

In order to avoid high extinction risks of prey and keep the stability of the three-species food chain model, we introduce a Filippov food chain model (FFCM) with Holling type II under threshold policy control. The threshold policy is designed to play a pivotal strategy for controlling the three species in the FFCM. With this strategy, no control is applied if the density of the prey population is less than the threshold, thus the exploitation is forbidden. However, the exploitation is permitted if the density of the prey population increases and exceeds the threshold. The dynamic behaviors and the bifurcation sets of this model including the existence and stability of different types of equilibria are discussed analytically and numerically. Moreover, the regions of sliding and crossing segments are analyzed. The dynamic behaviors of sliding mode including the bifurcation sets of pseudo-equilibria are investigated. Numerically, the bifurcation diagram and maximum Lyapunov exponents are computed and plotted to show the complex dynamics of FFCM, for instance, it has stable periodic, double periodic and chaotic solutions as well as double periodic sliding bifurcation. It is demonstrated that the threshold policy control can be easily implemented and used for stabilizing the chaotic behavior of FFCM.

中文翻译:

Filippov 三物种食物链模型的复杂动力学

为了避免猎物的高灭绝风险并保持三物种食物链模型的稳定性,我们引入了阈值政策控制下的Holling II型Filippov食物链模型(FFCM)。阈值政策旨在发挥控制 FFCM 中三个物种的关键策略。使用这种策略,如果猎物种群的密度小于阈值,则不进行控制,因此禁止利用。但是,如果猎物种群的密度增加并超过阈值,则允许利用。分析和数值讨论了该模型的动态行为和分岔集,包括不同类型平衡的存在和稳定性。此外,分析了滑动和交叉段的区域。研究了包括伪平衡分岔集在内的滑模动力学行为。在数值上,计算并绘制了分岔图和最大李雅普诺夫指数,以显示FFCM的复杂动力学,例如,它具有稳定的周期解、双周期解和混沌解以及双周期滑动分岔。证明了阈值策略控制可以很容易地实现并用于稳定FFCM的混沌行为。
更新日期:2021-04-29
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