  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.
