# 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

)