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

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 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 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 .

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.

The dynamic equations for the prey-predator interaction in a chemostat are as follows

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).

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

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.

[1] Saunders, P. T. , Bazin, M. J. “On the Stability of Food Chains, ” J. Theor. Biol., Vol.52, pp.121-142. (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).

Fig.1 s versus t ； D=0.01hr– １ Limit Cycle

Fig.2 b versus p ； D=0.01hr– １

t = 1000 - 2000 hr ； Limit Cycle

Introduction

Abstract

Research Methods

Results and Discussion

Conclusions

References

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.

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

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.3 s versus t ； D=0.0799hr– １ Third Type of Stable Steady State

Fig.4 b versus p ； D=0.0799hr– １ t = 0 - 2000 hr ； Third Type of Stable Steady State

Fig.5 s versus t ； D=0.2hr– １ Second Type of Stable Steady State

Fig.7 s versus t ； D=0.6hr– １ First Type of Stable Steady State

Fig.6 b versus p ； D=0.2hr– １ t = 0 - 2000 hr ； Second Type of Stable Steady State

Fig.8 b versus p ； D=0.6hr– １ t = 0 - 2000 hr ； First Type of Stable Steady State