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Mathematical Description of the Chemostat: The volume of the solution V in the culture vessel is constant. Inflow and outflow rates, F are equal and F is a constant in volume$/$time$.$ The bacterial concentration at time t is given with N$($t$)$ in mass$/$volume$.$ The growth medium concentration is supplied with a constant $($C$\_(0))$ and the concentration in the vessel at time t is given with C$($t$)$ in mass$/$volume$.$

The concentrations in the culture tank are well$-$mixed, meaning that all the ingredients in the tank are homogeneously distributed. We will use N and C for N$($t$)$ and C$($t$),$ respectively, for the sake of simplicity.

The chemostat is modeled with the following coupled differential equations as follows:

$($dN$)/($dt$)=$r$($C$)$N$-($F$)/($V$)$N $($eq$\mathrm{.}1)$

$($dC$)/($dt$)=-\backslash$mu r$($C$)$N$-($F$)/($V$)$C$+($F$)/($V$)$C$\_(0)($eq$\mathrm{.}2)$

where $\backslash$mu is the nutrition consumption and r$($C$)$ is the growth rate of the bacteria population depending on the availability of the nutrient C$.$ Michaelis$-$Menten Kinetics is one of the best$-$known models of

enzyme kinetics that describes the rate of enzymatic reactions by

r$($C$)=($k$\_($max$)$C$)/($k$\_($n$)+$C$)($eq$\mathrm{.}3)$

where k$\_($max$)$ and k$\_($n$)$ are constants that can be learned from experimental data. If we assume that Michaelis$-$Menten Kinetics is an appropriate choice for our r$($C$),$ then Eqs. $(1)$ and $(2)$ can be written as follows:

$($dN$)/($dt$)=\backslash$alpha $($C$)/(1+$C$)$N$-$N $($eq$\mathrm{.}4)$

$($dC$)/($dt$)=-($C$)/(1+$C$)$N$-$C$+\backslash$beta $($eq$\mathrm{.}5)$

where $\backslash$alpha $=($Vk$\_($max$))/($F$)$ and $\backslash$beta $=($C$\_(0))/($k$\_($n$)).$

$!$ Question $!$:

A scientist sets up a chemostat experiment which is governed by Eq$.(4)$ and $(5).$ The scientist

experimentally learned the constants and initial conditions as given: alpha;$1\mathrm{.}43,$ beta;$19\mathrm{.}80,$ N$0$;$11\mathrm{.}64,$ C$0$;$5\mathrm{.}99.$

$($a$)$ Simulate this system and plot the variables N$($t$)$ and C$($t$)$ of the experiment versus time t$.$

$($b$)$ Plot the orbit $($N$($t$),$ C$($t$))$ in the state space for the given initial condition.

$($please do not sent just an meanless gpt answer..$)$