## P13

dt y where Nt is the biomass at any given time. Measuring biomass at all given times is unrealistic, therefore Nt can be described in terms of initial biomass, N0, such that Nt = N0 + Nt - N0, and the above equation expands to dS = dt y

No Nt N0

yy and collecting terms dS dt

Expressing all terms as substrate, considering that the total amount of substrate consumed since t = 0 is S0 - St = (Nt - N0)/y, and letting X0 = N0/y where X0 becomes the amount of substrate needed to produce N0, then the differential form of the exponential equation for substrate depletion is dS dt

While the exponential equation links substrate depletion with microbial growth, it is well recognized that microbes do not grow exponentially at all times. Rather, microbial populations grow to a limit (K). As the population increases, the growth rate (p) decreases due to competition among individuals for increasingly scarce substrate resources. Microbial growth to a limit is expressed by the logistic equation

A similar exercise of algebra to express the logistic equation in terms of substrate, considering that K is the maximum biomass (original biomass plus that generated by converting all the original substrate into biomass), yields the differential equation for logistic growth in terms of substrate depletion: 