= Monod chemostat equilibrium and relaxation
{c}
For a dimensionless <chemostat> with <Monod equation> kinetics,
$$
n'=\alpha\frac{cn}{1+c}-n,\qquad c'=\beta-c-\frac{cn}{1+c},
$$
the total nutrient equivalent $z=c+n/\alpha$ satisfies $z'=\beta-z$. The positive <equilibrium> is $c_*=1/(\alpha-1)$, $n_*=\alpha(\beta-c_*)$, provided $\alpha>1$ and $\beta>c_*$. The <Jacobian matrix> has <eigenvalues> $-1$ and $-n_* /(1+c_*)^2$, so the positive <equilibrium> is <locally asymptotically stable>.
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