On the existence of oscillatory behavior in unstructured models of bioreactors

Abstract

A complete analysis of the static and dynamic behavior of the basic unstructured model of bioreactors with wall attachment, substrate inhibition and cell decay was carried out using elementary principles of the singularity theory and continuation techniques. The analysis of dynamic bifurcation showed that, for the case of constant yield coefficient, the model cannot predict periodic behavior for any growth rate. This results shows the fundamental weakness of the basic unstructured model for both homogeneous and inhomogeneous environment in predicting the transient behavior of continuous cultures. Elementary principles of the singularity theory and continuation techniques allowed the derivation of analytical conditions for the existence of Hopf points when the yield coefficient is allowed to vary with the substrate. The classification of the static and dynamic behavior for the case of linear dependence of the yield coefficient revealed that the pitchfork is the only singularity the model can predict. A total of four qualitatively different behavior were delineated. For some combination of kinetic parameters and dilution rate, the model predicted autonomous sustained oscillation while for some range of dilution rate, these oscillations coexist with wash-out conditions. The model also predicted a region where a static branch coexists with wash-out line but with no periodic behavior. In the last region the model predicted a stable Monod-like behavior.