Limit cycles in a food-chain with inhibition responses

In this paper, by constructing a Lyapunov function, we show the global asymptotical stability of a three-dimensional food-chain model with inhibition response. We then using a corollary to center manifold theorem to show that the system undergoes a 3-D Hopf bifurcation, and obtain the existence of limit cycles for the three-dimensional model. The methods used here can be extended to many other 3-D differential systems.