The Monod equation cannot describe either the so called lag phase or the process of cell division. However, it is not a problem to design many models taking these phenomena into account.

[Excel sheet]

Let as assume that the cells (microbes) are spheres of diameter D, which consume either a nutrient transferred from the surroundings (substrate S) or their own resources in hungry days. Further on we will assume that the amount of nutrients needed for a cell is directly proportional to the volume of the cell (volume of sphere =(D³/6) and that the mass flowrate from the surroundings is proportional to the concentration of substrate, [S], and to the surface of the cell (surface of sphere =(D²). This sentence can be translated into an equation describing the time change of the cell size (diameter D)
(11.10)
where the coefficients k_S and k_D are related to substrate transfer and consumption, respectively. The substrate transfer term, multiplied by the number of cells [X], represents the consumption of substrate,
(11.11)
We need not write an equation for [X], assuming that the division of cells occurs in the time instant when the diameters of the cells reach a prescribed value D_max (synchronised division), and this is the last (third) model parameter. A solution of Eq. (11.10-11) can be obtained in the same way as previously (e.g., Example 7.3, 11.1.1) and an example of results is shown in Fig.11.14 (time courses of substrate and biomass concentrations).