WOLFRAM|DEMONSTRATIONS PROJECT

Bifurcation in a Biochemical Reactor

​
Monod
substrate inhibition
bifurcation diagram
nontrivial steady state
x
2f
4.
μ
max
0.6
k
m
0.12
k
1
0.4545
D
s
0.6
In 1942, Monod proposed the following form of the specific growth coefficient:
μ
Monod
=
μ
max
x
2
k
m
+
x
2
The specific growth coefficient with the substrate inhibition
SI
is given by
μ
SI
=
μ
max
x
2
k
m
+
x
2
+
k
1
2
x
2
.
The biochemical reactor is governed by two coupled equations
d
x
1
dt
=(μ-D)
x
1
,
d
x
2
dt
=D(
x
2,f
-
x
2
)-μ
x
1
/Y
,
where
x
1
is the biomass concentration,
x
2
is the substrate concentration,
D
is the dilution rate,
Y
is the yield,
x
2,f
is the feed substrate concentration, and
μ
is the specific growth coefficient.
The steady states are the solutions of the following system of equations:
(μ-D)
x
1
=0
,
D(
x
2,f
-
x
2
)-μ
x
1
/Y=0
.
The trivial solution is obtained for
x
1
=0
and
x
2
=
x
2,f
. This corresponds to a situation where there are no cells left in the reactor, a phenomena called wash out.
The nontrivial solution is obtained if
μ=D
and
x
1
=Y(
x
2,f
-
x
2
)
.
This Demonstration finds the nontrivial steady states and shows the bifurcation diagram (
x
2
versus the bifurcation parameter
D
).
For the Monod case there is a single nontrivial steady state if
D<
μ
max
. This steady state is stable. On the other hand, the trivial steady state is either stable (when
D>
μ
max
) or unstable (
D<
μ
max
).
For the
SI
model, there are two nontrivial steady states if the value of
D
is in the pink region (click "nontrivial steady state"). In that case, the following inequalities hold:
μ
max
x
2,f
k
m
+
x
2,f
+
k
1
2
x
2,f
<D<
μmax
(1+2
k1km
)
. The magenta dot (low value of
x
2
) is stable because
dμ
d
x
2
>0
. The cyan dot (intermediate value of
x
2
) is unstable (a saddle point) because
dμ
d
x
2
<0
. The trivial solution is either stable (
D>
μ
max
x
2,f
k
m
+
x
2,f
+
k
1
2
x
2,f
) or unstable (
D<
μ
max
x
2,f
k
m
+
x
2,f
+
k
1
2
x
2,f
). If
D<
μ
max
x
2,f
k
m
+
x
2,f
+
k
1
2
x
2,f
, there is only one nontrivial steady state indicated by the magenta dot (low value of
x
2
). This steady state is stable because
dμ
d
x
2
>0
. The other value of
x
2
verifies
x
2
>
x
2,f
, thus
x
1
<0
(i.e., this solution is not feasible). Finally, if
D>
μmax
(1+2
k1km
)
, nontrivial solutions are not possible.