Model of Phytoplanktonic Cell Growth

The models used to describe the growth of phytoplanktonic cells (biomass $x_2$) on a substrate (of concentration $x_1$) assume usually that the growth is a function of a variable ($x_3$) called internal quota, representing the nutrient stored in the cells:

$$\begin{align*} \dot{x}_1 &= u(t)(1-x_1)-\rho(x_1)x_2\\ \dot{x}_2 &= (\mu(x_3)-u(t))x_2\\ \dot{x}_3 &= \rho(x_1)-\mu(x_3)x_3. \end{align*}$$

The input $u(t)$ is the dilution rate of the continuously stirred bioreactor (we suppose $u(t) \geq u \geq 0$). The functions $\rho$ and $\mu$ represent the absorption rate and the growth rate:

$\rho(x_1)=a_1\dfrac{x_1}{a_2+x_1};$ $\mu(x_3)=a_3\left(1-\dfrac{a_4}{x_3}\right)$.