# Multiple Steady States in Continuous Culture with Substrate Inhibition

Multiple Steady States in Continuous Culture with Substrate Inhibition

The dynamic behavior of a chemostat can be described with the following system of ordinary differential equations:=(μ-D)X=-+D(-S),where is the dilution rate in , is the biomass concentration in g/l, is the substrate concentration in g/l, is the feed substrate concentration, is the yield, and is the specific growth rate (where is the half-saturation constant, is the inhibition constant, and is the maximum value of the specific growth rate, with =1 here).

dX

dt

dS

dt

μX

Y

S

in

D

-1

s

X

S

S

in

Y=-

dX

dS

μ=S+S+

μ

max

K

S

2

S

K

I

K

S

K

I

μ

max

μ

max

This Demonstration lets you plot the substrate (magenta curve) and biomass (blue curve) concentrations versus time. A phase plane analysis shows that there can be up to three steady states (SS) depending on the values of the dilution rate and inhibition coefficient. With the help of the specific growth rate curve, you can easily see when there are only three or one SS (in the latter case we have for all ). One SS always corresponds to the wash out case (), while the other two, if present, correspond to the case where . Indeed, as can be seen in the snapshot, the red dots are stable SS and the green dot is unstable SS. You can drag the locator to change the initial conditions. The cyan curve is a parametric plot of substrate versus biomass.

μ<D

S

X=0

D=μ