The zebra mussel (Dreissena polymorpha) is a freshwater bivalve that has caused dramatic ecological and economic impacts worldwide in the last decades. Its success as ecosystem invader is mainly due to the species ability of spreading along rivers, together with the extremely high population densities that can be reached by local populations. Here we propose a spatially explicit model aimed at describing the population dynamics of mussel colonies in rivers represented as oriented graphs. We specifically apply the model to the Bilancino-Sieve-Arno water system (Tuscany, Italy), where D. polymorpha has been recently sighted. The model accounts for local scale demographic processes and along-stream transport mechanisms. Because of the different temporal scales involved, we couple continuous- and discrete-time dynamics. Our model produces spatiotemporal scenarios of invasion that present several distinctive features of the zebra mussel invasion patterns observed in nature, including the spatial patchiness and the temporal irregularity of local populations, and the so-called domino effect. The application of our model to the Arno case study shows that D. polymorpha could severely threaten the city of Florence within a few years. Control measures aimed at limiting the impact of the species can significantly alter the ecological and environmental parameters. We find that such variations may profoundly impact spatiotemporal invasion patterns, possibly with counter-intuitive effects. For instance, reduced larval input from Lake Bilancino can in some cases produce higher mussel densities in downstream colonies. Therefore, the control of D. polymorpha must be planned with care in order to prevent unwelcome outcomes.