Given:

The Monod model for the growth of a microbial population with density Xon a substrate in concentration S in a batch reactor is given by

$$\begin{eqnarray*}\frac{d} {dt} S & = & - j_S f X\\ \frac{d} {dt} X & = & \dot{r} X \end{eqnarray*}$$

where t stands for time, f for the scaled functional response $f = \frac{S} {S + K}$, j_S is the biomass-specific uptake rate, and $\dot{r}$ is the specific growth rate: $\dot{r} = y_{XS} \, j_S \, f$.