Initially, S = S0 and the reaction rate is governed primarily by X0. As the microbial population grows, S decreases such that the first term in the equation above decreases while the second term increases. Therefore, two competing trends govern the substrate depletion rate. The rate of substrate depletion is maximized at -dS/dtmax = p(S0 + X0)/4. The differential form of the logistic equation can be integrated and solved for St to give

Monod kinetics are the most general kinetic expressions because they relate substrate depletion both to changes in population density and to changes in substrate concentration. The basic relationship is

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