Monodmodel

The Monod model, also called Monod kinetics, is a mathematical description of microbial growth as a function of substrate concentration. It posits that the specific growth rate mu depends on the concentration of a limiting substrate S according to mu = mu_max S / (K_s + S). In this formulation, mu is the specific growth rate (typically per hour), mu_max is the maximum specific growth rate, S is the limiting substrate concentration, and K_s is the half-saturation constant, the substrate level at which mu equals half of mu_max. The Monod equation is analogous to Michaelis–Menten kinetics for enzyme reactions and is widely used in microbiology and bioprocess engineering to model growth in batch, fed-batch, and continuous cultures.

In a simple batch culture, biomass concentration X grows according to dX/dt = mu X, with mu given by the Monod expression. The substrate is consumed according to dS/dt = -(1/Y_XS) mu X, where Y_XS is the biomass yield on substrate. These coupled differential equations are used to simulate growth curves, substrate depletion, and product formation in bioreactors, and they form a basis for design and control in wastewater treatment, fermentation, and other bioprocess applications.

Limitations of the basic Monod model include the assumption of a single limiting substrate, constant mu_max and K_s, no substrate inhibition or maintenance energy term, and no explicit lag phase. Extensions and alternative models incorporate multiple substrates, inhibition effects, substrate limitation, or lag behavior to address more complex microbial systems.