 Monod chemostat equilibrium and relaxation

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