In this paper, we study an interspecific density-dependent model of two species competing on a single nutrient in a chemostat, taking into account the predator-prey relationship. Without mortality of species, we have proved previously that the system may have a multiplicity of positive steady states that can only appear or disappear through saddle-node or transcritical bifurcations. Including the mortality, we give a complete analysis for the existence and local stability of all steady states of the three-dimensional system which cannot be reduced to a two-dimensional one. We highlight the effect of mortality to destabilize the positive steady state with the emergence of stable limit cycles through supercritical Hopf bifurcations. To describe how the process behaves according to control parameters represented by the dilution rate and the input concentration of the substrate, we determine theoretically the operating diagram by plotting the various conditions of existence and stability and numerically by using MATCONT. The bifurcation diagram according to the input concentration shows the various types of bifurcations of steady states and the coexistence either around a positive steady state or sustained oscillations.