The asymptotic behavior of a tri-trophic food chain model in the chemostat is studied. The Monod-Herbert growth model is used for all trophic levels. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and of limit cycles with respect to two chemostat control parameters: the dilution rate of the chemostat and the concentration of input substrate. It is shown that the bifurcation structure of the food chain model has much in common with the bifurcation structure of a one-dimensional map with two turning points. This map is used to explain how attractors are created and destroyed under variation of the bifurcation parameters. It is shown that low as well as high concentration of input substrate can lead to extinction of the highest trophic level.