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arXiv · 2109.05420

Global Stability of the Periodic Solution of the Three Level Food Chain Model with Extinction of Top Predator

Kaijen Cheng·Hongming You·Ting-Hui Yang
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Abstract

In this work, we revisit the classical Holling type II three species food chain model from a different viewpoint. Two critical parameters {\lambda}1 and {\lambda}2 dependent on all parameters are defined. The existence and local stabilities of all equilibria can be reformulated by {\lambda}1 and {\lambda}2, and the complete classifications of parameters and its corresponding dynamics are given. Moreover, with the extinction of top-predator, there is an invariant two dimensional subsystem containing the prey and the intermediate predator. We prove the global stability of the boundary equilibrium in R3+ by differential inequality as well as Butler-McGehee lemma if it is stable. Alternatively, there is a unique limit cycle when the boundary equilibrium lost its stability, and we also show the global stability of the limit cycle in R3+ by differential inequality and computing the Floquet Multipliers. Finally, some interesting numerical simulations, the chaotic and the bi-stability phenomena, are presented numerically. A brief discussion and biological implications are also given.

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Kaijen Cheng, Hongming You, Ting-Hui Yang. 2021-09-12. Global Stability of the Periodic Solution of the Three Level Food Chain Model with Extinction of Top Predator. https://arxiv.org/abs/2109.05420

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