This work focuses on a model of two bacteria growing and exchanging nutrients in a chemostat. Bacterial uptake rate is described via Michaelis-Menten equations; we assume a constant yield and metabolite production directly proportional to bacterial growth. We analyse the model via a reduced order system, then, conditions to determine existence and global stability of the equilibria are given in terms of the dilution rate. Finally, bacterial productivity is maximized; an interval detector is designed to estimate this productivity and an optimization strategy for periodic dilution rates is proposed.