This paper is concerned with the dynamics and the productivity of biomass in a continuous microalgae culture contaminated by predators. We propose a general model describing a cultivation system where photolimited microalgae are grazed by zooplankton. We state necessary and sufficient conditions for the existence of a coexistence equilibrium. If this equilibrium exists, then microalgae and predators converge either to it or to a globally asymptotically stable periodic solution. In absence of the coexistence equilibrium, any solution approaches either an equilibrium characterized by the presence of algae and the absence of predators, or the washout equilibrium. This description of the dynamics allows to define the microalgae productivity in the long-term (i.e. when an attracting set is reached). With the help of numerical simulations, we show that operating the system at optimal (constant) dilution rate triggers the extinction of predators.