An Analysis of Dynamics of the Prey-Predator Interaction in a Chemostat

Wen-Ke Su (蘇文柯 ) , 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

The dynamics and steady state behavior of biochemical reaction systems have been studied for many years . 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 dynamic behavior of the prey-predator interaction in a continuous stirred tank reactor (chemostat). There were three types of steady states for this system and the three types of steady states were analyzed in detail. The dynamic equations of this system were solved by the numerical method and the dynamic analysis was performed by computer graphs. The graphs of predator, prey and substrate versus reaction time and three-dimensional( 3D ) graphs for different parameters were plotted for dynamic analysis. The graphs of predator, prey and substrate versus reaction time and three-dimensional graphs were plotted for dynamic analysis.

In this study, the mathematical methods and numerical analysis were used. For this biochemical reaction system, steady state equations were solved by mathematical methods. For dynamic analysis, due to the complexity of the dynamic equations, Runge-Kutta numerical analysis method was used to solve the dynamic equations, and the dynamic analysis was performed by computer graphs. The mathematical methods and numerical analysis were used in this research. For this biochemical reaction system, steady state equations were solved by mathematical methods. For dynamic analysis, due to the complexity of the dynamic equations, Runge-Kutta numerical analysis method was 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

The solutions of the third type of steady states for this system are :

For dynamic analysis, because of the complexity of the dynamic equations, Runge-Kutta numerical analysis method was used to solve the dynamic equations, and the dynamic analysis was performed by computer graphs. Fig.1 and Fig.3 show the graphs of s versus time for different parameters. Fig.2 and Fig.4 show 3 D Plot for different parameters. The initial conditions for Fig.1-4 are p = 5.0 mg/L, b = 25.0mg/L, s = 10.0mg/L . The parameters for Fig.1- 4 are μm=0.56hr–1, νm =0.1hr–1 ， Ki=16mg/L, Kp=10000mg/L, L=6.1mg/L, X=0.73, Y=0.428, D=0.0715hr– １ , k1=0 hr-1, k2=0 hr-1 . Fig.1 and Fig.2 show the Stable Steady State behavior and Fig.3 and Fig.4 show the Limit Cycle behavior.

For the prey-predator interaction chemostat system , three types of steady state solutions are possible. The steady state equations were solved by mathematical methods. The dynamic equations of this system were solved by the numerical method and the dynamic analysis was performed by computer graphs. The graphs of predator, prey and substrate versus reaction time and three-dimensional graphs were plotted for dynamic analysis. The results show that the dynamic behavior of this system consists of stable steady states and limit cycles which were shown by concentration of s (substrate) versus t (time) and 3 D graphs.

[1] Hastings, A. “ Multiple Limit Cycles in Predator-Prey Models , ” J. Math. Biol. Vol.11, pp.51-63 (1981).[2] Pavlou, S., “Dynamics of a Chemostat in Which One Microbial Population Feeds on Another,” Biotechnol. and Bioeng., Vol. 27, pp.1525-1532 (1985).[3] 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). [4] Jones K. D., Kompala D. S., “ Cybernetic Model of the Growth Dynamics of Saccharomyces Cerevisiae in Batch and Continuous Cultures,” J. of Biotechnology , Vol.71, pp.105-131 (1999).[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).

Fig.1 s versus Time ； sf=190mg/L

Stable Steady State

Fig.2 3 D Plot ； sf=190mg/L

t = 0 - 2000 hr ； Stable Steady State

Introduction

Abstract

Research Methods

Results and Discussion

Conclusions

References

The study of steady state behavior and dynamics of biochemical reaction system is a very important topic in the research field of biochemical engineering. A study was conducted to analyze the steady state and dynamics of the prey-predator interaction in a chemostat. The specific growth rates of Substrate Inhibition model and Monod model were used for the prey and predator respectively. The dynamic equations were derived by assuming that the reaction was occurring in a perfectly mixed flow reactor (chemostat). It was assumed that the prey (such as the bacterium) was only fed withthe substrate (such as the glucose) and the predator (such as the ciliate )was only fed with the prey and no other substance exchanged between the system and environment. There were three types of steady states for this system and the three types of steady states were analyzed in detail. The dynamic equations of this system were solved by the numerical method and the dynamic analysis was performed by computer graphs. The graphs of predator, prey and substrate versus reaction time and three-dimensional graphs were plotted for dynamic analysis. The results showed that the dynamic behavior of this system consisted of stable steady states and limit cycles which were shown by concentration of s (substrate) versus t (time) and 3 D graphs.

Substrate Inhibition model model is used for the specific growth rate of the prey and Monod model is usedfor the specific growth rate of the predator

The solutions of the second type of steady states for this system are :

Fig.3 s versus Time ； sf=205mg/L Limit Cycle

Fig.4 3 D Plot ； sf=205mg/L

t = 1000 - 2000 hr ； Limit Cycle

pkp)b(νDpdt

dp1

bkp)b(νX

1b)s(μDb

dt

db2

b)s(μY

1)ss(D

dt

dsf

p2

i

m

K/ssK

sμ)s(μ

bL

ν)b(ν m

b

whereμm = constant (hr–1) ; Ki, Kp = constants (mg/L) ; m = maximum specific growth rates (hr–1) ; L = saturation constant (mg/L).

2

B4AAs

2

1

2

B4AAs

2

2

where A = Kp [ 1 – μm / ( D + k2 ) ] , B = Ki Kp

)s(skD

YDb 1f

21

)s(s

kD

YDb 2f

22

sf ＞ s1 ＞ 0 , sf ＞ s2 ＞ 0

)kD(ν

L)k(Db

1m

1

s3 + αs2 + βs + γ = 0 where α = Kp – sf

bYD

KsKKK pm

fppi

γ = – Ki Kp sf

Therefore there are three solutions for s at most.

)kD(

)]bkD()s(μ[Xp

1

2