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

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.