A chemostat is a well-mixed tank of given volume $V_{0}$ that contains water in which lives a population $N(t)$ of bacteria that consume nutrient whose concentration is $C(t)$ per unit volume. An inflow pipe supplies a solution of nutrient at concentration $C_{0}$ and at a constant flow rate of $Q$ units of volume per unit time. The mixture flows out at the same rate through an outflow pipe. The bacteria consume the nutrient at a rate $N K(C)$, where

$$K(C)=\frac{K_{\max } C}{K_{0}+C}$$

and the bacterial population grows at a rate $\gamma N K(C)$, where $0<\gamma<1$.

Write down the differential equations for $N(t), C(t)$ and show that they can be rescaled into the following form:

$$\begin{aligned} &\frac{d n}{d \tau}=\alpha \frac{c n}{1+c}-n \\ &\frac{d c}{d \tau}=-\frac{c n}{1+c}-c+\beta \end{aligned}$$

where $\alpha, \beta$ are positive constants, to be found.

Show that this system of equations has a non-trivial steady state if $\alpha>1$ and $\beta>\frac{1}{\alpha-1}$, and that it is stable.