Allelopathy refers to a type $0/-$ biological interaction that is neutral for a so-called inhibitory species but detrimental for a so-called susceptible species. To model this type of interaction in a spatially-structured environment, Durrett and Levin introduced a variant of the multitype contact process in which the death rate of the susceptible species is density-dependent, increasing with the local density of the inhibitory species. Their work combines mean-field analysis and simulations of the spatial model, and our main objective is to give rigorous proofs of some of their conjectures. In particular, we give a complete description of the behavior of the mean-field model, including the global stability of the fixed points. Our main results for the interacting particle system show the existence of two regimes depending on the relative fitness of the individuals. When the inhibitory species is the superior competitor, the inhibitory species always wins, whereas when the susceptible species is the superior competitor, the susceptible species wins if and only if the inhibitory effects do not exceed some critical threshold. We also prove that, at least in dimensions $d \geq 3$, the transition between these two regimes is continuous in the sense that, when both species are equally fit, the inhibitory species wins even in the presence of extremely weak inhibitory effects.