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This study analyses the steady state behaviour of biological conversion systems with general kinetics, in which two consecutive reactions are carried out by two groups of micro-organisms. The model considered is a realistic description of wastewater treatment processes. A step-wise procedure is followed to reveal the mechanisms affecting the occurrence of steady states in terms of the process input variables. It is clearly demonstrated how taking into account inhibition effects by simply including additional inhibition terms to the kinetic expressions, a common practice, influences the model's long term behaviour. The overall steady state behaviour of the model has been summarized in easy-to-interpret operating diagrams, depicting the occurrence of steady states in terms of the reactor dilution rate and the influent substrate concentration, with well-defined boundaries between distinct * Corresponding author. Tel.: +32 9 264 61 29; fax: +32 9 264 62 35Email addresses: eveline.volcke@ugent.be (E.I.P. Volcke), mihaela@autoctrl.ugent.be (M. Sbarciog), erik.noldus@ugent.be (E.J.L. Noldus), bernard.debaets@ugent.be (B. De Baets), mia.loccufier@ugent.be (M. Loccufier) Preprint submitted to Mathematical BiosciencesSeptember 21, 2010 operating regions. This knowledge is crucial for modelers as steady state multiplicity -in the sense that more than one steady state can be reached depending on the initial conditions -may remain undetected during simulation. The obtained results may also serve for experimental design and for model validation based on experimental findings.

In this paper, the influence of microbial growth kinetics on the number and the stability of steady states for a nitrogen removal process is addressed. A two-step nitrification model is studied, in which the maximum growth rate of ammonium oxidizers is larger than the one of nitrite oxidizers. This model describes the behavior of a SHARON reactor for the treatment of wastewater streams with high ammonium concentrations. Steady states are identified through direct calculation using a canonical state space model representation, for several types of microbial kinetics. The stability of the steady states is assessed and the corresponding phase portraits are analyzed. Practical operation of a SHARON reactor aims at reaching ammonium conversion to nitrite while suppressing further conversion to nitrate. Regions in the input space are identified that result in this desired behavior, with only nitrite formation. It is demonstrated that not only the dilution rate plays a role, as is commonly known, but also the influent ammonium concentration. Besides, the type of microbial (inhibition) kinetics has a nonnegligible influence. While the results indicate that product inhibition does not affect the number of steady states of a (bio)reactor model, it is shown that substrate inhibition clearly yields additional steady states. Particular attention is devoted to the physical interpretation of these phenomena.