Exotic plants threaten the biodiversity of natural habitats and the integrity of agricultural systems throughout the World. Therefore, understanding, predicting and controlling plant invasions became issues of great practical importance. In the present paper, a model for plant invasion based on allelopathic suppression is proposed and studied through analytical methods and numerical integration. Employing linear stability analysis the conditions for plant coexistence as well as one species extinction were determined for the spatially homogeneous system. These conditions demonstrate the advantage conferred to the alien plant by its phytotoxin. It was shown that the system exhibits bistability between two distinct fixed points, either associated to species coexistence or to the extinction of one species. Numerical simulation is also included to support such results. Further, the invasion spreading starting from a single, spatially localized initial focus was investigated by numerical integration of the model's equations. As obtained for the spatially homogeneous system, at strong interspecific competition the outcome is the extinction of one plant species. In contrast, at low interspecific competition, the rule is the coexistence between the invader and native plants. So, under weak competition alien species can invade, but genetic diversity can be sustained.