Dynamics of the chemostat : a bifurcation theory approach

366 Pages·2011·9.789 MB·English

by  Abdelhamid Ajbar, K Alhumaizi

Mathematics/Mathematical Biology/Chemical Engineering D
y
Based on the authors’ extensive work in this field, Dynamics of the n
Chemostat: A Bifurcation Theory Approach explores the use of a
bifurcation theory to analyze the static and dynamic behavior of the m
chemostat.
i
c
The authors first survey the major work that has been carried out
on the stability of continuous bioreactors. They next present the s Dynamics of the

modeling approaches used for bioreactive systems, the different o
kinetic expressions for growth rates, and tools, such as multiplicity, f

bifurcation, and singularity theory, for analyzing nonlinear systems. t Chemostat
h
The text moves on to the static and dynamic behavior of the basic e
unstructured model of the chemostat for constant and variable
C A Bifurcation Theory
yield coefficients as well as in the presence of wall attachment. It
h
then covers the dynamics of interacting species, including pure and Approach
e
simple microbial competition, biodegradation of mixed substrates,
m
dynamics of plasmid-bearing and plasmid-free recombinant
cultures, and dynamics of predator–prey interactions. The authors o
also examine dynamics of the chemostat with product formation s
for various growth models, provide examples of bifurcation theory t
a
for studying the operability and dynamics of continuous bioreactor
t
models, and apply elementary concepts of bifurcation theory to
analyze the dynamics of a periodically forced bioreactor.
Using singularity theory and bifurcation techniques, this book
Abdelhamid Ajbar
A
presents a cohesive mathematical framework for analyzing and j
b
a Khalid Alhumaizi
modeling the macro- and microscopic interactions occurring in
r

chemostats. The text includes models that describe the intracellular •
A
and operating elements of the bioreactive system. It also explains l
h
u
the mathematical theory behind the models.
m
a
i
z
i
K13016
K13016_Cover.indd 1 6/29/11 12:33 PM
Dynamics of the
Chemostat
A Bifurcation Theory
Approach
K13016_FM.indd 1 6/29/11 3:35 PM
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Dynamics of the
Chemostat
A Bifurcation Theory
Approach
Abdelhamid Ajbar
Khalid Alhumaizi
K13016_FM.indd 3 6/29/11 3:35 PM
CRC Press
Taylor & Francis Group
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Boca Raton, FL 33487-2742
© 2012 by Taylor & Francis Group, LLC
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Version Date: 20110613
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We thank God Almighty for the blessings bestowed during the writing of this
book. Our thanks also go to our parents and familiesfortheir endless support.
For Dr. Ajbar: to mother Khadija, wife Saida and kids, Ayman, Sami, and
Ziyad. For Dr. Alhumaizi: to mother Sara, wife Muneera and kids, Shatha,
Alaa, and Ibrahem.
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
Contents
Preface xiii
1 INTRODUCTION TO STABILITY OF BIOREACTORS 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 StabilityStudies of Continuous Bioreactors . . . . . . . . . . 3
1.3 Methodologies for StabilityAnalysis . . . . . . . . . . . . . . 6
2 INTRODUCTION TO BIOREACTORS MODELS 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Continuous Bioreactors . . . . . . . . . . . . . . . . . . . . . 9
2.3 Modeling Bioreactors . . . . . . . . . . . . . . . . . . . . . . 11
2.3.1 Theoretical, Empirical, and Semiempirical Models . . 11
2.3.2 Unstructured vs. Structured Models . . . . . . . . . . 12
2.3.3 Nonsegregated vs. Segregated Models . . . . . . . . . 13
2.4 Kinetic Models for Cell Growth . . . . . . . . . . . . . . . . 14
2.4.1 Substrate-Limited Growth . . . . . . . . . . . . . . . 14
2.4.2 Growth Models with Inhibition . . . . . . . . . . . . . 15
2.4.2.1 Substrate Inhibition . . . . . . . . . . . . . . 15
2.4.2.2 Product Inhibition . . . . . . . . . . . . . . 16
2.4.2.3 Inhibitionby Toxic Compounds . . . . . . . 16
2.4.2.4 Multiple Substrates . . . . . . . . . . . . . . 16
2.5 Product Formation . . . . . . . . . . . . . . . . . . . . . . . 17
3 INTRODUCTION TO STABILITY AND BIFURCATION
THEORY 19
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Local Stability of Steady States . . . . . . . . . . . . . . . . 20
3.3 Steady-State Multiplicity . . . . . . . . . . . . . . . . . . . . 23
3.4 Dynamic Bifurcation . . . . . . . . . . . . . . . . . . . . . . 25
3.5 Numerical Techniques . . . . . . . . . . . . . . . . . . . . . . 27
3.6 SingularityTheory . . . . . . . . . . . . . . . . . . . . . . . . 28
3.6.1 Codimension-0Singularities . . . . . . . . . . . . . . . 33
3.6.2 Codimension-1Singularities . . . . . . . . . . . . . . . 33
vii
viii Contents
3.6.2.1 Hysteresis . . . . . . . . . . . . . . . . . . . . 33
3.6.2.2 Isola and Mushroom . . . . . . . . . . . . . . 33
3.6.3 Codimension-2Singularities . . . . . . . . . . . . . . . 34
3.6.4 Codimension-3Singularities . . . . . . . . . . . . . . . 34
3.6.5 Hopf Degeneracies . . . . . . . . . . . . . . . . . . . . 34
3.6.6 Type I Degeneracies . . . . . . . . . . . . . . . . . . . 35
3.6.6.1 F Degeneracy . . . . . . . . . . . . . . . . . 36
1
3.6.6.2 F Degeneracy . . . . . . . . . . . . . . . . . 36
2
3.6.6.3 G Degeneracy . . . . . . . . . . . . . . . . . 37
1
3.6.7 Type II and III Degeneracies . . . . . . . . . . . . . . 37
3.6.8 Examples of Type II Degeneracies . . . . . . . . . . . 40
3.6.9 Examples of Type III Degeneracies . . . . . . . . . . . 41
4 THE BASIC MODEL OF IDEAL CHEMOSTAT 43
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2 Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3 Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Codimension-1Singularity . . . . . . . . . . . . . . . 46
4.3.2 Codimension-2Singularity . . . . . . . . . . . . . . . 50
4.3.3 Monod Kinetic Model . . . . . . . . . . . . . . . . . . 52
4.4 Dynamic Behavior for Constant Yield Coefficient . . . . . . 53
4.5 Dynamic Behavior for Variable Yield Coefficient . . . . . . . 55
4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 61
5 THE CHEMOSTAT WITH WALL ATTACHMENT 63
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2 Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Static Analysis for InhibitionKinetics . . . . . . . . . . . . . 67
5.4 Static Analysis for Monod Growth . . . . . . . . . . . . . . . 72
5.5 Quantification of the StabilizingEffect of WallsAttachment 76
5.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 79
6 PURE AND SIMPLE MICROBIAL COMPETITION 81
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3 Static Bifurcation for Substrate Inhibition . . . . . . . . . . 85
6.3.1 Hysteresis Singularity . . . . . . . . . . . . . . . . . . 85
6.3.2 Double Limit Singularity . . . . . . . . . . . . . . . . 88
6.3.3 Isola and Mushroom . . . . . . . . . . . . . . . . . . . 89
6.3.4 Pitchfork Singularity . . . . . . . . . . . . . . . . . . . 90
6.4 Existence of Periodic Solutions . . . . . . . . . . . . . . . . . 91
6.5 Monod Kinetics Model . . . . . . . . . . . . . . . . . . . . . 93
Contents ix
6.5.1 Periodic Solutions for Monod Kinetics . . . . . . . . . 94
6.6 Case of Sterile Feed . . . . . . . . . . . . . . . . . . . . . . . 95
6.6.1 Periodic Solutions for Sterile Feed Conditions . . . . . 96
6.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 97
7 STABILITY OF CONTINUOUS RECOMBINANT DNA
CULTURES 99
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
7.2 Process Model . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Dynamic Bifurcation . . . . . . . . . . . . . . . . . . . . . . 104
7.3.1 F Degeneracy . . . . . . . . . . . . . . . . . . . . . . 106
1
7.3.2 H Singularity . . . . . . . . . . . . . . . . . . . . . . 108
01
7.4 Applications to Monod/Haldane Substrate-Inhibited Kinetics 109
7.4.1 Monod-Monod Case . . . . . . . . . . . . . . . . . . . 113
7.4.2 Inhibition-MonodCase. . . . . . . . . . . . . . . . . . 116
7.4.3 Monod-InhibitionCase. . . . . . . . . . . . . . . . . . 120
7.4.4 Inhibition-InhibitionCase . . . . . . . . . . . . . . . . 122
7.5 Implicationof Resulting Dynamics . . . . . . . . . . . . . . . 125
7.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 128
8 BIODEGRADATION OF MIXED SUBSTRATES 131
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
8.2 Bioreactor Model . . . . . . . . . . . . . . . . . . . . . . . . 132
8.3 Static Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 134
8.3.1 Hysteresis . . . . . . . . . . . . . . . . . . . . . . . . . 135
8.3.2 Isola and Mushroom . . . . . . . . . . . . . . . . . . . 138
8.3.3 Pitchfork Singularity . . . . . . . . . . . . . . . . . . . 140
8.3.4 Winged-Cusp Singularity . . . . . . . . . . . . . . . . 143
8.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 144
9 PREDATOR–PREY INTERACTIONS 145
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
9.2 Bioreactor Model . . . . . . . . . . . . . . . . . . . . . . . . 146
9.3 Existence of Oscillatory Behavior . . . . . . . . . . . . . . . 147
9.4 Construction of Operating Diagrams . . . . . . . . . . . . . . 148
9.5 Application to the Saturation Model . . . . . . . . . . . . . . 149
9.6 Application to the Multiple Saturation Model . . . . . . . . 153
9.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . 157