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The Effect of the Dilution Rate on the Dynamics of a Chemostat
Chung-Te Liu (劉崇德) , Chung-Min Lien (連崇閔) , Hau Lin (林浩)
Department of Chemical and Materials Engineering, Southern Taiwan University
南台科技大學化學工程與材料工程系
Abstract
The technique of the cultivation of the microorganism is a very important research subject. The prey-predator interaction exists in
the rivers frequently and a common interaction between two organisms inhabiting the same environments involves one
organism(predator) deriving its nourishment by capturing and ingesting the other organism(prey). A study is conducted to analyze the
steady states and dynamic behavior of the prey-predator interaction chemostat system. The specific growth rates of Monod model and
Multiple Saturation model are used for the prey and predator respectively. The dynamic equations of this system are solved by the
numerical method and the dynamic behavior is analyzed by computer graphs. The graphs of concentrations of predator, prey and
substrate versus reaction time and the graphs of the concentration of prey versus concentration of predator, concentration of predator
versus concentration of substrate and concentration of substrate versus concentration of prey are plotted for dynamic analysis. The
results show that the dynamic behavior of this system consists of stable steady states and limit cycles. When the parameters μm =
0.56hr–1, νm = 0.1hr–1, K = 16.0 mg/L, L1 = 6.1 mg/L, L2 = 0 mg/L, X = 0.73, Y = 0.428, sf = 215.0 mg/L, k1 = 0hr–1, k2 = 0hr–1, the
dynamic behavior of the dilution rates D=0.01 hr–1, D=0.0799 hr–1, D=0.2 hr–1and D=0.6 hr–1shows the limit cycle, the third type of
stable steady state, the second type of stable steady state and the first type of stable steady state respectively.
Introduction
Because of the contamination of the biochemical waste , the techniques of waste treatment have been applied frequently
and the techniques of the cultivation of the microorganism have received more attention in recent years. The preypredator interaction exists in the rivers frequently and a common interaction between two organisms inhabiting the same
environments involves one organism (predator) deriving its nourishment by capturing and ingesting the other organism
(prey). A study was conducted to analyze the steady state and dynamics of the prey-predator interaction in a chemostat.
There are three types of steady states for this system and the three types of steady states are analyzed in detail. The
dynamic equations of this system are solved by the numerical method and the dynamic analysis is performed by computer
graphs. In this study, the graphs of concentrations of predator, prey and substrate versus reaction time and the graphs of
the concentration of prey versus concentration of predator, concentration of predator versus concentration of substrate
and concentration of substrate versus concentration of prey are plotted for dynamic analysis .
Fig.4 b versus p ； D=0.0799hr–１
Fig.3 s versus t ； D=0.0799hr– １
Third Type of Stable Steady State
t = 0 - 2000 hr； Third Type of Stable
Steady State
Research Methods
The mathematical methods and numerical method are used in this study. For this chemostat system, steady state
equations are solved by mathematical methods. For dynamic analysis, because of the complexity of the dynamic
equations, Runge-Kutta numerical analysis method is used to solve the dynamic equations, and the dynamic analysis
was performed by computer graphs.
Results and Discussion
The dynamic equations for the prey-predator interaction in a chemostat are as follows
Fig.5
s
versus
t ；
D=0.2hr– １
Second Type of Stable Steady State
Fig.6 b versus p ； D=0.2hr–１
t = 0 - 2000 hr； Second Type of Stable
Steady State
where p = concentration of the predator (mg/L), b = concentration of the prey (mg/L), s =
concentration of the substrate (mg/L), sf = feed concentration of the substrate (mg/L), X= yield
coefficient for production of the predator, Y = yield coefficient for production of the prey, F = flow
rate(L/hr), V = reactor volume (L), D = F / V = dilution rate (hr–1), k1＝death rate coefficient for the
predator (hr–1)，k2＝death rate coefficient for the prey (hr–1).
Monod’s model is used for the specific growth rate of the prey and Multiple Saturation model is used
for the specific growth rate of the predator
Fig.7 s versus t；D=0.6hr–１
Fig.8 b versus p ； D=0.6hr–１
First Type of Stable Steady State
t = 0 - 2000 hr； First Type of Stable Steady
State
Conclusions
For the situation of this this system, three types of steady state solutions are possible
(1) Washout of both prey and predator：
p＝0，b＝0，s＝sf
(2) Washout of the predator only：
p＝0，b＞0，sf＞s＞0
(3) Coexistance of both prey and predator：
p＞0，b＞0，sf＞s＞0
For dynamic analysis, because of the complexity of the dynamic equations , Runge-Kutta numerical analysis
method is used to solve the dynamic equations, and the dynamic analysis is performed by computer graphs. Fig.1,
Fig.3, Fig.5, and Fig.7 show the graphs of s versus time for different parameters. Fig.2, Fig.4, Fig.6 and Fig.8 show b
versus p . The initial conditions for Fig.1- Fig.8 are p = 5.0 mg/L, b = 25.0 mg/L, s = 10.0 mg/L . The parameters for
Fig.1- 8 are μm=0.56hr–1, νm =0.1hr–1，K=16mg/L, L1=6.1mg/L, L2= 0 mg/L, X=0.73, Y=0.428, sf=215mg/L , k1=0
hr-1, k2=0 hr-1 . Fig.1 and Fig.2 show the Limit Cycle behavior ; Fig.3 and Fig.4 show the Third Type of Stable Steady
State behavior ; Fig.5 and Fig.6 show the Second Type of Stable Steady State behavior and Fig.7 and Fig.8 show the
First Type of Stable Steady State behavior.
Fig.1 s versus t ； D=0.01hr–
Limit Cycle
１
Fig.2 b versus p ； D=0.01hr–１
t = 1000 - 2000 hr；Limit Cycle
In this study, for the prey-predator interaction chemostat system , there are three types of
steady state solutions . The steady state equations are solved by mathematical methods. The
dynamic equations of this system are solved by the numerical method and the dynamic
analysis is performed by computer graphs. The graphs of concentrations of predator, prey
and substrate versus reaction time and the graphs of the concentration of prey versus
concentration of predator, concentration of predator versus concentration of substrate and
concentration of substrate versus concentration of prey are plotted for dynamic analysis.
The results show that the dynamic behavior of this system consists of stable steady states
and limit cycles.
References
[1] Saunders, P. T. , Bazin, M. J. “On the Stability of Food Chains, ” J. Theor. Biol., Vol.52, pp.121142. (1975).
[2] Hastings, A. , “Multiple Limit Cycles in Predator-Prey Models,” J. Math. Biol., Vol.11, pp.51-63
(1981).
[3] Pavlou, S., “Dynamics of a Chemostat in Which One Microbial Population Feeds on Another,”
Biotechnol. and Bioeng., Vol. 27, pp.1525-1532 (1985).
[4] 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.
and Bioeng. , Vol.57, pp.251-261. (1998).
[5]Ajbar, A., “Classification of Static and Dynamic Behavior in Chemostat for Plasmid-Bearing,
Plasmid-Free Mixed Recombinant Cultures,” Chem. Eng. Comm., Vol.189, pp.1130-1154 (2002).
[6]Kim, H and Pagilla, K.R., “Competitive Growth of Gordonia and Acinetobacter in Continuous
Flow Aerobic and Anaerobic/Aerobic Reactors,” J. Bioscience and Bioeng., Vol.95, pp.577-582
(2003).