One of the main reasons for failing of the sedimentation process in activated sludge waste water treatment systems is the phenomenon of filamentous bulking. This occurs when filamentous bacteria overgrow Roc-forming bacteria, in this paper mathematical models are discussed which describe the competition of Aocs and filaments based on the kinetic selection theory. It is proven with a stability analysis for a continuous reactor, that, in order to control filamentous bulking, the dilution rate plays a crucial role. Moreover, it is shown that coexistence of both organisms is generically not possible. Afterwards the continuous reactor model is extended in order to describe a standard waste water treatment system. Due to the complexity of the extended model it is difficult to perform the stability analysis analytically. It is proven that the model can be reduced without loss of stability properties. For the reduced model it is proven with a stability analysis that the transport terms play a crucial role in the survival of one or the other organism. As for the continuous reactor model it is shown that coexistence of both organisms is generically not possible. (C) 2000 Elsevier Science Ltd. All rights reserved.