# Droop model

## Model description:

The behavior of phytoplankton cells in a continuous reactor is usually described by the Droop model. Cell growth is limited by a nutrient with concentration $S$. The biomass has a concentration $N$ and $Q$ represents the cell quota of assimilated nutrient, expressed as the amount of intracellular nutrient per biomass unit. The dilution rate $D$ corresponds to the flow rate of renewal medium over the volume of the reactor, and $D$ is the input of the system.

We denote $D = D_0 + u$, and the system fits

$$\sum_D \begin{cases} \dot{x}_i = f(x) + ug(x)\\ y=h(x_1) \end{cases}$$

with

$f(x)=\begin{pmatrix} a_2\left(1-\dfrac{1}{x_2}\right)x_1 - D_0x_1\\ a_3\dfrac{x_3}{a_1+x_3} - a_2(x_2 - 1)\\ D_0(1-x_3)-\dfrac{x_1x_3}{a_1+x_3} \end{pmatrix}$

$g(x)=\begin{pmatrix} -x_1\\ 0\\ 1-x_3 \end{pmatrix}$, and $h(x_1)=x_1$, where

$x_1 = (\rho_m N/S_i);\\ x_2 = (Q/K_Q);\\ x_3 = (S/S_i);\\ a_1 = (K_{\rho}/S_i);\\ a_2 = \mu_m;\\ a_3 = (\rho_m/K_Q).$

3

## Publication details:

 Title Nonlinear observers for a class of biological systems: application to validation of a phytoplanktonic growth model Publication Type Journal Article Authors Bernard, O., Sallet G., and Sciandra A.