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The biomass concentration is the mass of biological material per unit culture volume; here it is measured as dry cell mass in $\mathrm{g}{\mathrm{L}}^{-1}$. A biomass concentration differs from a cell count per unit volume because cell sizes can change. In the ideal chemostat, biomass leaves the vessel at rate $Qx$, giving a removal term $-Dx$ in the ordinary differential equation for $x$.

A chemostat maintains a well-mixed culture at constant volume $V$ by feeding fresh nutrient solution and removing culture at the same volumetric flow $Q$. Its dilution rate is $D=Q/V$. Mixing makes the outflow's nutrient concentration and biomass concentration equal to those inside the vessel; continuous replacement supplies nutrient and also removes growing cells.

The simplest one-nutrient chemostat uses nutrient concentration $s(t)$, biomass concentration $x(t)$, constant feed nutrient concentration $s_{\mathrm{in}}>0$, and constant biomass yield $Y>0$. Assume a cell-free feed, one growth-limiting nutrient, constant volume, and no cell retention, death or additional nutrient consumption for maintenance. With specific growth rate $\mu (s)$, its ordinary differential equations areThe nutrient balance contains inflow $D{s}_{\mathrm{in}}$, outflow $Ds$, and consumption $\mu (s)x/Y$; the biomass balance contains production $\mu (s)x$ and removal $Dx$. If $s$ is in grams of nutrient per litre and $x$ in grams of dry biomass per litre, $Y$ is grams of dry biomass produced per gram of nutrient consumed. Thus both balances have units of their respective concentration per time. The Monod equation supplies an empirical choice of $\mu (s)$; the balances themselves express the stated production, consumption and transport assumptions.

The appeal of a chemostat is that, at a positive chemostat steady state, adjusting the pump sets the specific growth rate to $D$. The residual nutrient concentration then adjusts to that rate, and the available nutrient together with biomass yield sets the biomass concentration. This gives a controlled way to compare cultures at selected growth rates and a simple quantitative warning of chemostat washout.

Two 1950 publications on continuous culture are Novick and Szilard's Description of the Chemostat, Science 112(2920), 715-716, dated 15 December, and Monod's La technique de culture continue. Théorie et applications, Annales de l'Institut Pasteur 79, 390-410. The latter's original bibliographic details are reproduced in the contents of Monod's collected papers. These references identify contemporary publications without establishing priority between them. ^[ref]. ^[ref].

A chemostat steady state has constant nutrient concentration and biomass concentration even while nutrient and cells flow through the vessel. It is an equilibrium of the chemostat ordinary differential equations, rather than a culture in which all growth has stopped. Assume $D>0$ and the Monod equation. Setting both derivatives to zero gives the always-existing chemostat washout equilibrium $(s_{\ast },x_{\ast })=(s_{\mathrm{in}},0)$. A positive biomass concentration instead requires growth to balance removal, $\mu (s_{\ast })=D$, whenceThe positive chemostat steady state exists exactly whenThe weaker condition $D<{\mu }_{\max }$ only makes the denominator positive; it does not ensure $s_{\ast }<s_{\mathrm{in}}$ or $x_{\ast }>0$. Because the Monod equation is increasing, the feed must permit a specific growth rate greater than $D$. At a positive chemostat steady state, increasing feed nutrient concentration at fixed $D$ increases the biomass concentration without changing $s_{\ast }$, within the one-nutrient assumptions.

For a concrete chemostat, let ${\mu }_{\max }=1{\mathrm{h}}^{-1}$, $K_s=1\mathrm{g}{\mathrm{L}}^{-1}$, $s_{\mathrm{in}}=9\mathrm{g}{\mathrm{L}}^{-1}$ and $Y=0.5\mathrm{g}\text{ biomass}/(\mathrm{g}\text{ nutrient})$. At $D=0.5{\mathrm{h}}^{-1}$,The nutrient supply minus outflow is $D(s_{\mathrm{in}}-s_{\ast })=4\mathrm{g}{\mathrm{L}}^{-1}{\mathrm{h}}^{-1}$, matching nutrient consumption $\mu (s_{\ast })x_{\ast }/Y=4\mathrm{g}{\mathrm{L}}^{-1}{\mathrm{h}}^{-1}$. Biomass production and removal both equal $2\mathrm{g}{\mathrm{L}}^{-1}{\mathrm{h}}^{-1}$. The chemostat washout threshold is $D_c=\mu (9)=0.9{\mathrm{h}}^{-1}$. For example, $D=0.95{\mathrm{h}}^{-1}<{\mu }_{\max }$ formally gives $s_{\ast }=19\mathrm{g}{\mathrm{L}}^{-1}$ and $x_{\ast }=-5\mathrm{g}{\mathrm{L}}^{-1}$, which cannot describe a physical culture.

There is a useful exact simplification: the nutrient-equivalent concentration $z=s+x/Y$ satisfies $\dot{z}=D(s_{\mathrm{in}}-z)$, so $z(t)=s_{\mathrm{in}}+[z(0)-s_{\mathrm{in}}]e^{-Dt}$. This combines dissolved nutrient and the nutrient equivalent of biomass; it does not assert that these two variables account for every chemical constituent. In coordinates $(z,x)$, the Jacobian matrix at the positive chemostat steady state has eigenvalues $-D$ and $-{\mu }^{\prime }(s_{\ast })x_{\ast }/Y$, both negative. Hence the positive chemostat steady state is locally asymptotically stable. In the worked example the eigenvalues are $-0.5{\mathrm{h}}^{-1}$ and $-2{\mathrm{h}}^{-1}$.

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 $D_c$.

The dilution rate is the fraction of a constant-volume chemostat replaced per unit time: volumetric flow $Q$ divided by culture volume $V$. For $Q=1\mathrm{L}{\mathrm{h}}^{-1}$ and $V=2\mathrm{L}$, $D=0.5{\mathrm{h}}^{-1}$. A well-mixed constituent with no source and absent from the feed has concentration $a(t)=a(0)e^{-Dt}$, since $\dot{a}=-Da$. Thus $1/D=2\mathrm{h}$ is the time for its concentration to fall by a factor $e$, not the time by which all original material has left.

At a positive chemostat steady state, the specific growth rate equals the dilution rate; the biomass concentration stays constant because production balances outflow. For $D=0.5{\mathrm{h}}^{-1}$, growth alone would have doubling time $\ln 2/D\approx 1.39\mathrm{h}$, even though the vessel's biomass concentration does not double. Raising the dilution rate beyond $\mu (s_{\mathrm{in}})$ causes chemostat washout; comparing it only with ${\mu }_{\max }$ misses the nutrient limitation imposed by the feed.

The Monod equation is an empirical saturating law for the specific growth rate of a culture limited by one nutrient. Write $s\ge 0$ for its residual nutrient concentration, ${\mu }_{\max }>0$ for the limiting specific growth rate, and $K_s>0$ for the nutrient concentration at half that rate:Here $\mu$ and ${\mu }_{\max }$ have units of inverse time, whereas $K_s$ has the same units as $s$. The notation $K(C)$ in the title denotes the same growth function, with $C=s$, $K_{\max }={\mu }_{\max }$ and $K_0=K_s$. For $s\ll K_s$, $\mu (s)\approx ({\mu }_{\max }/K_s)s$; for $s\gg K_s$, $\mu (s)$ approaches ${\mu }_{\max }$. Its derivative ${\mu }^{\prime }(s)={\mu }_{\max }{K}_s/(K_s+s{)}^2$ is positive: more nutrient increases the specific growth rate, with progressively smaller increments.

For example, take ${\mu }_{\max }=1{\mathrm{h}}^{-1}$ and $K_s=1\mathrm{g}{\mathrm{L}}^{-1}$. Nutrient concentrations of $0$, $1$ and $9\mathrm{g}{\mathrm{L}}^{-1}$ give specific growth rates $0$, $0.5$ and $0.9{\mathrm{h}}^{-1}$. In a chemostat with dilution rate $D=0.5{\mathrm{h}}^{-1}$, the intersection $\mu (s)=D$ selects the residual nutrient concentration $s=1\mathrm{g}{\mathrm{L}}^{-1}$, provided the feed can supply enough nutrient for a positive biomass concentration.

The Monod equation is useful because two measurable parameters describe both nutrient scarcity and saturation. It assumes the other conditions affecting growth remain controlled and does not represent inhibition at high nutrient concentration. Although its formula resembles the Michaelis-Menten equation, a whole culture's specific growth rate combines many processes; the resemblance alone does not derive it from Michaelis-Menten kinetics or identify $K_s$ with a Michaelis constant.

In his 1949 review, Monod presents the hyperbolic rate law as an empirical fit in equation (2), pp. 383-384, and discusses the difficulty of identifying a single growth-controlling reaction. His reported division rates use doublings per unit time; the natural-exponential specific growth rate used here is that division rate multiplied by $\ln 2$. ^[ref].

The specific growth rate is biomass production per unit existing biomass per unit time. For biomass concentration $x>0$ with growth alone, $\dot{x}=\mu x$ and $\mu =\mathrm{d}(\ln x)/\mathrm{d}t$. A constant positive specific growth rate produces exponential growth, $x(t)=x(0)e^{\mu t}$, with doubling time $\ln 2/\mu$. In a chemostat, removal makes the net rate $\dot{x}/x=\mu (s)-D$, rather than $\mu (s)$ itself.