Ind. Eng. Chem. Res. 2005, 44, 2525-2531
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Experimental Investigation and Confirmation of Static/Dynamic Bifurcation Behavior in a Continuous Ethanol Fermentor. Practical Relevance of Bifurcation and the Contribution of Harmon Ray Parag Garhyan and Said S. E. H. Elnashaie* Department of Chemical Engineering, Auburn University, 230 Ross Hall, Auburn, Alabama 36849
Professor Harmon Ray is my professor, and I am proud of that. I met him first in 1969 (35 years ago!) at the Chemical Engineering Department, University of Waterloo, Ontario, Canada, and since then, we have become friends and exchanged visits across thousands of miles. In June 1969 (just one year after graduating from Cairo University, Egypt), I was visiting some friends at Waterloo when I met Harmon; after a short discussion, he offered me the opportunity to continue my postgraduate studies with him, I accepted, changed my trajectory from studying in the U.K., and stayed in Waterloo to work with him. It was one of the most fruitful periods of my life. He taught me a lot about modeling, multiplicity, and catalytic processes. We were also lucky to be joined in Waterloo by Professor Milos Marek from Czechoslovakia and Professor Horn from Vienna. After I finished my M.Sc. degree, I was supposed to go with Harmon to SUNY-Buffalo to continue for my Ph.D., but for family reasons, I had to go to the U.K. to continue my Ph.D. first at University College London with Professors Peter Rowe and John Yates and then at University of Edinburgh with the late Professor Philip Calderbank and Professor David Cresswell. Harmon visited us in Edinburgh, stayed a few days, and gave very stimulating lectures. After I finished my Ph.D. in 1973, I returned to Egypt in 1974, and Harmon and I remained in continuous contact. When I had a brilliant B.Sc./M.Sc. student (Fouad Teymour, now Chairman of the Chemical and Environmental Engineering Department, IIT, Chicago) who wanted to continue for his Ph.D. in the U.S., I immediately advised him to join the group of Harmon at University of Wisconsin, Madison. Fouad joined Harmon’s group and did excellent work on the flourishing field of bifurcation and chaotic behavior of polymerization reactors. Harmon and his group were and are still doing pioneering work in this field. After that, in the 1980s and 1990s, we exchanged visits: Harmon visited me in King Saud University in Riyadh and invited me to Madison, both of which were excellent visits. Harmon is one of the excellent students from the Minnesota group of Professors Rutherford Aris and Neal Amundson. Since he finished his Ph.D. in Minnesota in the late 1960s after completion of his bachelor at Rice University, he has pioneered very advanced work on modeling, bifurcation and chaos, and optimization and control for catalytic and polymeric processes. His cooperation with mathematicians and considerable contribution to the fundamentals of the field are also of great value. He has published hundreds of papers in top international journals, as well as a number of important books, made numerous presentations at a variety of conferences, and supervised a large number of well-known researchers in academia and industry. He has also won a large number of prestigious awards. I still remember the excellent party we had in Madison for his 60th birthday. Time is flying and he is now 65, but still as productive and innovative as ever. On the happy occasion of his 65th birthday, we congratulate him and wish him and his family the best of everything. My Ph.D. student Parag Garhyan and I thank the editorial board of Ind. Eng. Chem. Res. for giving us the opportunity to honor this outstanding academician. We take this opportunity to present some of the initial experimental work being done in our laboratory at Auburn University to verify static and dynamic bifurcation in a continuous fermentor producing ethanol from glucose. sSaid S. E. H. Elnashaie
A model for ethanol production using pure glucose in a continuous perfectly mixed fermentor was developed, and extensive nonlinear analysis was carried out to explore the possibility of increasing the sugar conversion and ethanol productivity [Garhyan et al. Chem. Eng. Sci. 2003, 58 (8), 1479. Garhyan, P.; Elnashaie, S. S. E. H. Ind. Eng. Chem. Res. 2004, 43 (5), 1260]. An interesting mix of static and dynamic properties of the system was uncovered and analyzed. In the present work, laboratory-scale fermentation experiments were carried out to verify the existence of complex static and dynamic behavior such as multiplicity of steady states and periodic behavior. Introduction This paper is an extension of our bifurcation studies on a structured-unsegregated model for continuous * To whom correspondence should be addressed. nashaie@ eng.auburn.edu. Tel.: ++1-334-844-2060. Fax: ++1-334-8873905.
sugar fermentation to ethanol using Zymomonas mobilis.1,2 The present work utilizes bifurcation analysis as a tool for evaluating the transient model of the continuous fermentation process for the production of ethanol. Bifurcation analysis utilizing the model equations is used to locate steady-state solutions, periodic solutions, and bifurcation points where the static and
10.1021/ie049679w CCC: $30.25 © 2005 American Chemical Society Published on Web 10/07/2004
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tor showing all of the concentrations and flow rates. The model is expressed in the form of the following four coupled nonlinear differential equations
(
Figure 1. Schematic of fermentor.
dynamic behavior changes drastically. Qualitative and quantitative changes are represented in the form of bifurcation diagrams. These diagrams are used to determine the static and dynamic accuracy of the model as compared to experimental results. It is also useful for providing deep insights into the design of experiments aimed at validating exotic model predictions. The qualitative properties of a nonlinear dynamical system can change significantly as a result of small variations in model parameters, unlike the behavior in a linear dynamical system. Multiplicity of steady states, stability of steady states, onset and existence of periodic or oscillatory states, and more complex strange nonchaotic or chaotic attractors are some examples of these complex nonlinear qualitative properties.3 In a simple manner, bifurcation analysis can be defined as the study of the change in the qualitative properties of a nonlinear dynamical system when some key parameters are varied. Detailed and mathematically oriented explanations of static/dynamic bifurcation and chaos for catalytic systems can be found in the textbook by the author.3 A number of other investigators have applied bifurcation analysis to continuous bioreactor models.4-9 In most of these studies, the investigators presented detailed characterizations of the model behavior with negligible experimental verification. The aim of the present work is different: we are utilizing our previous nonlinear investigations1,2 to design and run continuous experiments to validate the dynamical model and its bifurcation characteristics. Model Description Biochemical reactors can be viewed as highly complex dynamical systems in which various chemical components are present in the intracellular and extracellular spaces and each cell has unique properties. A rigorous model accounting for the above complexities is called a structured-segregated model,10-12 and such models are difficult to formulate and analyze. The presence of oscillations within cellular systems has been observed and mathematically described, such as glycolytic oscillations and oscillations in intracellular calcium concentrations in different types of cells.13 Obviously, oscillations can be observed in a cell suspension only if the behavior of the single cells is synchronized.14 More simplified models can be formulated either by neglecting the intracellular chemical variations (unstructured models)15-17 or by neglecting the heterogeneity of the cell population (unsegregated models).18-20 The simplest models are unstructured-unsegregated models.21 The structured-unsegregated biokinetic model describing the continuous fermentation of glucose to ethanol in a continuously stirred fermentor can be described by a set of four differential equations.1 This model has four state variables, namely, the concentrations of sugar (CS), ethanol (CP), microorganism (CX), and an internal key component (Ce). The term E is the ratio of the internal key component concentration to the microorganism concentration (E ) Ce/CX). Figure 1 presents a simplified schematic diagram of the fermen-
k2 k3 k1 dE ) C + C 2 µ - µE dt µmax µmax P µmax P
)
(1)
dCX ) µCX + D(CXO - CX) dt
(2)
(
) )
dCS 1 )µ + mS CX + D(CSO - CS) dt YSX
(
dCP 1 ) µ + mP CX + D(CPO - CP) dt YPX
(3)
(4)
where
µ)
CSEµmax KS + CS
and
D)
q V
A detailed description of this model and the values of parameters used for this basic analysis are given in our previous work.1 Nonlinear Investigation A detailed nonlinear investigation of this continuous fermentor was carried out. The bifurcation diagrams were obtained using the software package AUTO97.22 This package is able to perform both steady-state and dynamic bifurcation analysis, including the determination of entire periodic solution branches using efficient continuation techniques.23 The DIVPAG subroutine available with IMSL Libraries for FORTRAN with an automatic step size to ensure accuracy for stiff differential equations was used for numerical dynamic simulations. Some of the mathematical results are presented in this section; they are then verified by laboratory experiments in the next section of this paper. Case 1: CSO ) 140 g/L and Dilution Rate D is Used as the Bifurcation Parameter. Details of the static and dynamic bifurcation behavior for this case are shown in Figure 2, with dilution rate D as the bifurcation parameter for product concentration (CP). Dotted vertical lines show the locations of the dilution rates at which the experiments were performed. It is clear that the static bifurcation diagram is an incomplete S-shaped hysteresis type with a static limit point (SLP) at the very low value of DSLP ) 0.0035 h-1. The dynamic bifurcation shows a Hopf bifurcation (HB) at DHB ) 0.05 h-1. The periodic branch emanating from the HB terminates homoclinically (i.e., reaches a homoclinic termination, HT) when it touches the saddle point very close to the SLP at DHT ) 0.0035 h-1. The region of interest in this case is DHB > D > DHT, which is characterized by a unique periodic attractor (surrounding the unique unstable steady state). It is clear that, in this region, the average of the oscillations for the periodic attractor gives a higher ethanol concentration than that of the corresponding steady states. The average concentrations are calculated by taking the average of the concentrations over one period of oscillation. The operation of the fermentor under periodic
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Figure 2. Bifurcation diagram for CSO ) 140 g/L and D as the bifurcation parameter.
conditions is productive and also gives higher ethanol concentrations. The best production policy in terms of the ethanol concentration, yield, and productivity for this case is a periodic attractor. In general, there is a tradeoff between concentration and productivity, which requires an economic optimization study to determine the optimum value of D. Case 2: CSO ) 200 g/L and Dilution Rate D is Used as the Bifurcation Parameter. This is a case with a very high feed sugar concentration. Figure 3 shows the static and dynamic bifurcation diagrams for this case. This case is characterized by the existence of fully developed chaos because of the period doubling to fully developed chaos24 (Figure 3B); the sequence is P1 f P2 f P4 f P8 f ‚‚‚ f fully developed chaos (Figure 4), which terminates homoclinically at DHT ) 0.045835 h-1.1 Experiments were carried out at two dilution rates (0.25 and 0.045 h-1), identified by dotted vertical lines in Figure 3A,B. In the region that includes the range of DSLP > D > DHB (i.e., 2.25 > D > 0.054), bistability exists, with a high-conversion stable static branch (conversion values in the range 0.975-1.0), as well as a low-conversion (conversion values in the range 0.42-0.595) stable static branch (Figure 3A). A comparison between the values of the low- and high-conversion stable static branches at D ) 1.5 h - 1 shows that the high-conversion branch achieves an improvement of 120.26% in ethanol productivity over the low-conversion branch. In the range of DHB > D > DPD (i.e., 0.054 > D > 0.04604), bistability again exists, with a high-conversion stable static branch as well as a stable periodic attractor with periodicity 1 (Figure 3B). Experimental Setup Batch and continuous runs were conducted to experimentally verify some of the characteristics of the fermentation processes at different parameter values. Most of the experimental runs were conducted in continuous mode because the prime objective of the
Figure 3. Bifurcation diagrams for CSO ) 200 g/L and D as the bifurcation parameter.
experiments was to verify the continuous system modeled, investigated, and analyzed in the previous section and earlier work.1,2 Microorganism and Fermentation Medium. Zymomonas mobilis strain ATCC 10988 obtained from ATCC was used for the experimental runs. The strain was kept on agar dishes containing 20 g/L glucose and 10 g/L yeast extract in a refrigerator and was transferred every 2-4 weeks. The strain was also preserved at -20 °C in Eppendoff tubes containing 15% (w/v) glycerol. The cultivation medium consisted of 50 g/L glucose, 1 g/L KH2PO4, 2 g/L NH4Cl, 0.49 g/L MgSO4‚ 7H2O, 5 mg/L calcium panthothenate, 5 mg/L FeSO4‚ 7H2O, 7.2 mg/L ZnSO4‚7H2O, 1.5 mg/L CaCl2‚2H2O, 4.2 mg/L MnSO4‚H2O, 2.0 mg/L CuSO4‚5H2O, 1.6 mg/L CoSO4‚7H2O, 50 mg/L NaCl, and 50 mg/L KCl. Experimental Setup and Operation. The medium of inoculum consisted of 20 g/L glucose and 10 g/L yeast extract. The cultures were seeded with 150 mL of
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Figure 5. Simplified schematic of the experimental setup.
Figure 6. Batch experiment of glucose fermentation with Zymomonas mobilis.
Experimental Results and Discussion Figure 4. One-dimensional Poincare´ bifurcation diagrams showing period-doubling route to chaos for CSO ) 200 g/L.
inoculum, and the culture pH was kept at 5.0 by an automatic pH controller using 1 M NaOH. Steady states in continuous cultures were assumed to be established after 6-8 times the residence time. Samples were taken at an interval of 3 or 6 h for the continuous mode of operation. A schematic diagram of the experimental fermentor with a working volume of 2.8 L operating in continuous mode is shown in Figure 5. Analytical Methods. Glucose and ethanol were determined by HPLC using a Bio-Rad Aminex HPX87H column. Glucose was also monitored with a YSI 2300 glucose/lactate analyzer (YSI Co., Yellow Springs, OH). The optical density of the fermentation broth was noted at a wavelength of 600 nm using a Gilford 250 spectrophotometer. The dry weight of the biomass (dry cell mass, DCM) was determined by centrifugation. The biomass was washed first with saline water and then mixed, centrifuged, washed twice with deionized water, and dried at 85°C until reaching a constant weight.
Batch Experiments. A few batch experiments were conducted for different initial sugar concentrations. The purpose of conducting the batch experiments was to formulate a growth curve that could be used to predict the inoculation time for the continuous experiments. Typical results from one of the batch fermentation runs are shown in Figure 6. The initial glucose concentration in this batch experiment was 48.8 g/L, and the ethanol concentration was 0.002 g/L. It can be seen in this figure that the glucose, ethanol, and biomass concentrations remain almost constant during the first 6-8 h of the batch operation. After this lag phase, an exponential phase is observed in which the biomass concentration increases sharply and, in doing so, consumes a great deal of glucose to produce ethanol. Thus, an exponential increase in the biomass and ethanol concentrations is observed while the glucose concentration drops to almost zero. After the exponential phase, a stationary phase is observed, and it is seen that, after some time, the biomass concentration starts to decrease slightly, as no more glucose is available for consumption and growth of active microorganism stops. Continuous Experiments. Several runs of the continuous fermentation experiments (initially starting in batch fashion and then switching to continuous mode)
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Figure 7. Comparison of simulated and experimental ethanol concentrations in continuous operation mode for CSO ) 140 g/L at D ) 0.022 h-1 (corresponding to case 1).
Figure 8. Comparison of simulated and experimental ethanol concentrations in continuous operation mode for CSO ) 140 g/L at D ) 0.04 h-1 (corresponding to case 1).
were conducted to verify the complex nonlinear behavior of the fermentation process discovered and explained in the previous section and papers.1,2 Although, in one of the previous works,2 the effect of continuous ethanol removal using membranes on the fermentation process and the complex nonlinear dynamical behavior were investigated, no provision for ethanol removal by any means is incorporated in the experimental setup in this work. The continuous fermentation experiments were conducted with two different feed sugar concentrations: 140 and 200 g/L. These feed concentrations correspond to cases 1 and 2, respectively, discussed in previous section. Feed Sugar Concentration CSO ) 140 g/L. Results of the continuous runs are presented in Figures 7-9. These experimental runs were carried out with an inlet feed glucose concentration of 140 g/L. The aim of these experiments was the validation of the periodic behavior shown in Figure 2. Therefore, in the experiments, the stream coming out of the fermentor was continuously collected in a reservoir placed in an ice bath so that any further action of microorganism with the remaining
Figure 9. Comparison of simulated and experimental ethanol concentrations in continuous operation mode for CSO ) 140 g/L at D ) 0.06 h-1 (corresponding to case 1).
glucose is prevented. The outlet stream was collected for a certain period of time (84 h) over an ice bath and mixed well so that an analysis could be performed to determine the average concentrations of ethanol, glucose, and microorganisms. Figures 7-9 show a comparison of the simulated and experimental results at three different dilution rates: 0.022, 0.04, and 0.06 h-1. After the continuous-mode fermentation was started, the system was allowed to stabilize such that the initial transients were “washed out”, and then the samples were analyzed to record the data. A dilution rate of 0.022 h-1 was used for the results presented in Figure 7, where the dotted line shows the results of the dynamic simulation obtained from the model and the small circles are the experimental values. According to the nonlinear analysis in previous section, the average ethanol concentration should be equal to 65.3 g/L. The experimental ethanol concentration over time has a fair consistency with the simulated result, and the average experimental ethanol concentration was determined to be 63.89 g/L. The small difference between the average ethanol concentrations in the simulation and experimental run might be due to the fact that some of the ethanol might have escaped in vapor form from the fermentor, thus reducing the overall ethanol concentration in the fermentation broth. Moreover, the experimental average was calculated using a finite number of points, which can exclude the maxima and minima of the oscillations. Figure 8 shows the same data for a dilution rate of 0.04 h-1 and it also corresponds to a stable periodic attractor. Again, it is observed that the experimental and simulated concentrations closely match each other. The simulated average ethanol concentration in this case is 63.4 g/L, and the experimental average ethanol concentration is slightly lower at 61.93 g/L, which might be due to the same reason mentioned above. Figure 9 depicts the system behavior at a dilution rate value of 0.06 h-1. At this high dilution rate, the system has crossed the Hopf bifurcation point (Figure 2) and has only a unique stable attractor (point attractor), which means that the state variables do not change with time. It can be seen in Figure 9 that the ethanol concentration is fairly constant, with very little variation over time. The average value of the ethanol concentra-
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Figure 10. Experiments with CSO ) 200 g/L at D ) 0.25 h-1 (leading to the high-ethanol-concentration branch in case 2).
Figure 12. Experiments with CSO ) 200 g/L at D ) 0.045 h-1 (leading to the stable branch in case 2).
Figure 11. Experiments with CSO ) 200 g/L at D ) 0.25 h-1 (leading to the low-ethanol-concentration branch in case 2).
Figure 10 shows the trajectory of the ethanol concentration with time. The part on the left side of the bold vertical line depicts the batch mode of operation, and the part on right side of line is for the continuous mode. The experiment was run in batch mode to achieve an ethanol concentration close to the value corresponding to the high-ethanol-concentration branch of about 100 g/L, after which it was switched to continuous mode. It is seen that the ethanol concentration decreases slightly with time and finally reaches a stable value of about 89.6 g/L. The expected value of the stable steady state for this case from the model was 95 g/L. This discrepancy can be attributed to some loss of ethanol due to evaporation and calls for further improvement in the model parameters. Similarly to Figure 10, Figure 11 shows the trajectory of the ethanol concentration leading to the lowerethanol-concentration branch. This time, the continuous operation was started when the ethanol concentration was about 54 g/L during batch operation. The ethanol concentration finally settled at a value of 51.1 g/L, which is slightly lower than the model-expected value of 55 g/L. From Figures 10 and 11, the multiplicity phenomenon is confirmed, as the final ethanol concentration is dependent on the initial conditions of the process for identical parameter values. A lower dilution rate (0.045 h-1) was used for Figure 12, where the final ethanol concentration is 93.4 g/L whereas the simulated value is about 98 g/L. Despite the use of different initial conditions, the system could not settle at the periodic attractor as expected from the bifurcation diagram (Figure 3B). This might be due to the fact that the region of attraction for this stable periodic attractor is very small as compared with the region of attraction of the stable steady state. More work is being carried out to investigate the complex region of periodic behavior and period doubling to chaos.
tion over a long period of time for this dilution rate is equal to 57.33 g/L, whereas the simulated value was 57.9 g/L. Feed Sugar Concentration CSO ) 200 g/L. For the second case, continuous fermentation experiments were carried out with the feed sugar concentration of 200 g/L. The main purpose of these experiments was to validate experimentally the existence of the multiplicity phenomenon (Figure 3) in the discussed model. Three different experimental runs were completed, each starting in batch mode and then being switched to continuous mode. The experiments were started in batch mode and were run to achieve certain glucose, ethanol, and microorganism concentrations to simulate different initial conditions; later, the continuous feed of pure glucose and product removal at the same flow rate were started to switch to continuous operation mode. Two dilution rates were used: The first was D ) 0.25 h-1, for which two different initial conditions were tested. It is clear that the system settles to different steady states (Figures 10 and 11) because of the multiplicity phenomenon occurring at this dilution rate. The second dilution rate was D ) 0.045 h-1, for which another steady state was achieved (Figure 12).
Conclusions and Ongoing Work An extensive nonlinear investigation of the continuous fermentation process for producing ethanol from sugar was carried out in our previous works.1,2 Bifurcation analysis provided insight into the possible utilization of periodic attractors to enhance the conversion,
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yield, and productivity of the fermentation process. This work is an extension of the previous work in the form of experimental verification of the mathematical investigation reported earlier. It is seen in the continuous experiments that the experimental values of the state variables closely match the simulated values, thus confirming that the simplified structured-unsegregated model is suitable for the description of the present fermentation process. Experiments were carried out to show that a change in bifurcation parameter (dilution rate, D h-1) results in sustained oscillations. Moreover, when the dilution rate is above the Hopf bifurcation value, the oscillations disappear to give a steady-state value. Experiments were also carried out to show the existence of multiple steady states (multiplicity) by starting the experiments at different initial conditions. More continuous experiments are currently being performed for different inlet feed sugar concentrations and dilution rates with the aim of experimentally obtaining the complete bifurcation diagrams obtained in previous works.1,2 Acknowledgment This research was supported by Auburn University though Grant 2-12085. Our appreciation is also extended to Prof. Robert P. Chambers, Prof. Y. Y. Lee, and Haodi Dong for assistance with the experimental work. Literature Cited (1) Garhyan, P.; Elnashaie, S. S. E. H.; Al-Haddad, S. M.; Ibrahim, G.; Elshishini, S. S. Exploration and exploitation of bifurcation/chaotic behavior of a continuous fermentor for the production of ethanol. Chem. Eng. Sci. 2003, 58 (8), 1479. (2) Garhyan, P.; Elnashaie, S. S. E. H. Static/Dynamic Bifurcation and Chaotic Behavior of an Ethanol Fermentor. Ind. Eng. Chem. Res. 2004, 43 (5), 1260. (3) Elnashaie, S. S. E. H.; Elshishini, S. S. Dynamic Modelling, Bifurcation and Chaotic Behavior of Gas-Solid Catalytic Reactor; Gordon and Breach Publishers: London, 1996. (4) Ajbar, A. On the existence of oscillatory behavior in unstructured model of bioreactors. Chem. Eng. Sci. 2001, 56, 1991. (5) Lyberatos, G.; Kuszta, G.; Bailey, J. E. Bifurcation from the potential field analogue of some chemical reaction systems. Chem. Eng. Sci. 1985, 40, 1679. (6) Pavlou, S.; Kevrekidis, I. G. Microbial predation in a periodically operated chemostat: A global study of the interaction between natural and externally imposed frequencies. Math. Biosci. 1992, 108, 1. (7) Zhang, Y.; Henson, M. A. Bifurcation Analysis of Continuous Biochemical Reactor Models. Biotechnol. Prog. 2001, 17, 647. (8) Xiu, Z. L.; Zeng, A. P.; Deckwer, W. D. Multiplicity and stability analysis of microorganisms in continuous culture: Effects
of metabolic overflow and growth inhibition. Biotechnol. Bioeng. 1998, 57 (3), 251. (9) Zamamiri, A. M.; Birol, G.; Hjortso, M. A. Multiple steady states and hysteresis in continuous, oscillating cultures of budding yeast. Biotechnol. Bioeng. 2001, 75 (3), 305. (10) Cazzador, L.; Alberghina, L.; Martegani, E.; Mariani, L. Bioreactor control and modeling: A simulation program based on a structured population model of budding yeast. In Bioreactors and Biotransformations; Moody, G. W., Baker, P. B., Eds.; Elsevier: London, 1987; p 64. (11) Eakman, J. M.; Fredrickson, A. G.; Tsuchiya, H. H. Statistics and dynamics of microbial cell populations. Chem. Eng. Prog. Symp. Ser. 1966, 62, 37. (12) Srienc, F.; Dien, B. S. Kinetics of the cell cycle of Saccharomyces cerevisiae. Ann. N.Y. Acad. Sci. 1992, 59. (13) Goldbeter, A. Biochemical Oscillations and Cellular Rhythms. The Molecular Bases of Periodic and Chaotic Behavior; Cambridge University Press: Cambridge, U.K., 1996. (14) Wolf, J.; Sohn, H. Y.; Heinrich, R.; Kuriyama, H. Mathematical Analysis of a Mechanism for Autonomous Metabolic Oscillations in Continuous Culture of Saccharomyces cerevisiae. FEBS Lett. 2001, 499, 230. (15) Bellgardt, K.-H. Analysis of synchronous growth of baker’s yeast: Part I: Development of a theoretical model for sustained oscillations. J. Biotechnol. 1994, 35, 19. (16) Duboc, P.; von Stockar, U. Modeling of oscillating cultivations of Saccharomyces cerevisiae: Identification of population structure and expansion kinetics based on on-line measurements. Chem. Eng. Sci. 2000, 55, 149. (17) Hjortso, M. A.; Nielsen, J. A conceptual model of autonomous oscillations in microbial cultures. Chem. Eng. Sci. 1994, 49, 1083. (18) Cazzador, L. Analysis of oscillations in yeast continuous cultures by a new simplified model. Bull. Math. Biol. 1991, 53, 685. (19) Jones, K. D.; Kompala, D. S. Cybernetic modeling of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures. J. Biotechnol. 1999, 71, 105. (20) Ramkrishna, D.; Kompala, D. S.; Tsao, G. T. Are microbes optimal strategists? Biotechnol. Prog. 1987, 3, 121. (21) Daugulis, A. J.; McLellan, P. J.; Li, J. Experimental investigation and modeling of oscillatory behavior in the continuous culture of Zymomonas mobilis. Biotechnol. Bioeng. 1997, 56 (1), 99. (22) Doedel, E. J.; Champneys, A. R.; Fairgrieve, T. F.; Kuznetsov, Y. A.; Sandstede, B.; Wang, X. J. AUTO97: Continuation and Bifurcation Software for Ordinary Differential Equations; Department of Computer Science, Concordia University: Montreal, Canada, 1997. (23) Kubicek, M. and Marek, M. Computational Methods in Bifurcation Theory and Dissipative Structures; Springer-Verlag: New York, 1983. (24) Feigenbaum. M. J. Universal behaviour in nonlinear systems. Los Alamos Sci. 1980, 1, 4.
Received for review April 20, 2004 Revised manuscript received July 18, 2004 Accepted July 23, 2004 IE049679W
