= Chemostat washout
{title2=$D_c=\mu(s_{\mathrm{in}})$}
<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 $\dot x=-[\mu'(s_{\mathrm{in}})/Y]x^2+O(x^3)$: 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 $D_c$.
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