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