In this paper, we consider a classical food chain model describing predator-prey interaction in a chemostat. The Michaelis-Menten kinetics is used as the uptake for both predator and prey. We observe the dynamical behavior of the model around each of the equilibria and points out the exchange of stability. We use Lyapunov function in the study of the global stability of predator-free equilibrium. Using removal rate of prey as the bifurcation parameter, we prove that the model undergoes a Hopf bifurcation around interior equilibrium. It has been found that the dynamical behavior of the model is very sensitive to the parameter values. With the aid of numerical simulations we analyze the model equations and illustrate the key points of analytical findings, and determine the effects of operating parameters of the chemostat on the dynamics of the system.