Chemostat washout ($D_c=\mu (s_{\mathrm{in}})$)

Chemostat washout means the culture's biomass concentration tends to zero because growth cannot sustain replacement of the cells leaving the vessel. In the ideal Monod equation chemostat with $D>0$, its equilibrium is $(s,x)=(s_{\mathrm{in}},0)$. Its Jacobian matrix has eigenvalues $-D$ and $\mu (s_{\mathrm{in}})-D$: a small inoculum grows when $D<\mu (s_{\mathrm{in}})$, whereas chemostat washout is locally asymptotically stable when $D>\mu (s_{\mathrm{in}})$. For nonnegative initial concentrations, chemostat washout occurs for $D\ge \mu (s_{\mathrm{in}})$; the boundary case also loses biomass asymptotically.

At $D=D_c$ the positive chemostat steady state meets the chemostat washout equilibrium, and one eigenvalue is zero, so linearization alone does not decide the boundary case. On the invariant nutrient-equivalent balance $s+x/Y=s_{\mathrm{in}}$, small positive $x$ obeys $\dot{x}=-[{\mu }^{\prime }(s_{\mathrm{in}})/Y]x^2+O(x^3)$: nutrient consumption lowers the specific growth rate below $D_c$, leading to slow decay rather than a positive biomass concentration. The continued mathematical branches exchange stability in a transcritical bifurcation; the positive-biomass branch becomes physically inadmissible beyond $D_c$.