Nonvanishing biomass

The added factor 1/(1 + s/X, s) in Equ. 5.88 represents the toxicity of the substrate at higher concentrations. Let us recall that the condition for calculation of the stationary state with nonvanishing biomass concentration is the relation fx(s) = D. This equation has only one solution if fi(s) is a monotonic function. But with characteristics as in Equ. 5.88, there are two solutions. Together with the washout state ( x, s) we have three stationary states. Two of them are stable ( x, and x, s), one of them is unstable ( x, s). Thus, we have a bistable system. The stationary values of the stable and the unstable stationary state are shown as a function of D in Fig. 6.11. Hysteresis may occur in shift experiments. Figure 6.12 shows how the final biomass concentration depends on the initial concentration. Figure 6.13 demonstrates that the phase plane is divided into two attraction domains. Both domains are touched by a separatrix in which the unstable stationary state lies. Note that, after an external disturbance, the system can cross over the separatrix and shift from one steady state to the other. This bistable behavior is a serious problem in, for example, waste treatment It takes place if substrates such as alcohols, phenols, or hydrocarbons occur in such high concentrations that the utilization of these substrates is inhibited. 

The second stationary solution is characterized by a nonvanishing biomass concentration ... 

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