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Dynamics of the Prey-Predator Interaction for Two Continuous Cultures in Series
Hsiang-Yun Wang (王祥雲), 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
Because the concept of the importance of the environmental protection and the techniques of the biochemical waste treatment have
received more attention in recent years, the cultivation of the microorganism has become an important research subject.
A study is conducted to analyze the steady states and dynamic behavior of the reactions of the prey-predator interaction
in two continuous cultures in series. For the prey-predator interaction, 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). If the specific growth rates of Substrate Inhibition model and Monod model are used for the
prey and predator respectively, there are six types of steady states for the two continuous cultures in series reaction
system and the six types of steady states are analyzed in detail in this research. The dynamic equations of this system are
solved by the numerical method and the dynamic analysis is performed by the computer graphs. The results show that
the dynamic behavior of this system consists of stable steady states and limit cycles. The concentrations of predator, prey
and substrate versus time and 3D graphs are used for analysis of dynamic behavior.
b1a 
The techniques of the biochemical waste treatment have received more attention in recent years, and therefore the
cultivation of the microorganism has become an 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 was
conducted to analyze the steady state and dynamic behavior of the prey-predator interaction in two continuous cultures in
series. If the specific growth rates of Substrate Inhibition model and Monod model are used for the prey and predator
respectively, there are six types of steady states for this reaction system and the six 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 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.
Results and Discussion

m

where A1  K p 1 
 D k
1
b

b1a 
D1  k b

 ；B  K i K p
where A  K p 1 
D2  k b 


(10 a ) ；
s 2b
(9b)
and A < 0，A2-4B > 0

YD2

s f  s 2b
D2  k b

(10 b)
The values for the third steady state are computed from the steady state equations of equations (4)-(6).

D2  k a L
b2 
 m  D2  k a 
s23  C1s22  C2 s2  C3  0
(11) ；
b1b 
(19 a ) ；
0   D2 p2  b2  p2  k a p2
YD2 ( s f  s1b )
(19 b)
D1  k b
(20)
1
0  D2 b1  b2    s 2 b2   b2  p 2  k b b2 (21)
X
1
0  D2 s1  s2    s2 b2
(22 )
Y

D2  k a L
From the equations (20)-(22), we have
b2 
 m  D2  k a 
where C  K  s
p
(23)
；
s23  C4 s22  C5 s2  C6  0
C5  K i K p  K p s1 
m K p
YD2
b2
(25)
s13  C1s12  C2 s1  C3  0

D1  k a L
b1 
 m  D1  k a 
(27)
C1  K p  s f
m K p
YD1
b1
C3   K i K p s f ，There are three steady states for s1 at most.
X  s1   D1  kb b1
p1 
From the steady state equations of equation (2), we have
D1  k a
(28)
The steady state equations of equations (4)-(6) are as follows.
0  D2  p1  p2   b2  p2  k a p2
(29)
1
0  D2 b1  b2    s 2 b2   b2  p 2  k b b2 (30 )
X
1
0  D2 s1  s2    s2 b2
(31)
Y
The values of p2 , b2 and s2 are computed from equations (29)-(31)
For dynamic analysis, because of the complexity of the dynamic equations (1)-(6), numerical analysis method was used to solve the
values p1, b1, s1, p2, b2 and s2 of dynamic equations(1)-(6), and the dynamic analysis was performed by computer graphs. The initial
conditions for Fig.1- 4 are p1=5 mg/L, b1=25 mg/L, s1=10 mg/L, p2=5 mg/L, b2=25 mg/L, s2=10 mg/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, sf=250mg/L , ka=0hr-1, kb=0hr-1. For Fig.1 and Fig.2
D1=0.0595hr–1, D2=0.595hr–1 and for Fig.3 and Fig.4 D1=0.0825hr–1, D2=0.0825hr–1.Fig.1 and Fig.2 show the dynamic behavior of limit
cycle and Fig.3 and Fig.4 show the dynamic behavior of the sixth type stable steady state.
Conclusions
A study is conducted to analyze the steady states and dynamic behavior of the reactions of the prey-predator interaction in two continuous
cultures in series. If the specific growth rates of Substrate Inhibition model and Monod model are used for the prey and predator respectively,
there are six types of steady states for the two continuous cultures in series reaction system and the six types of steady states are analyzed in
detail in this research. The six types of steady states are computed from the steady state equations. For dynamic analysis, because of the
complexity of the dynamic equations (1)-(6), Runge-Kutta numerical analysis method is used to solve the dynamic equations(1)-(6) and the
dynamic analysis was performed by computer graphs. The results of dynamic analysis show that the dynamic behavior of this system
consists of limit cycles and stable steady states .
References
[1] Canale, R.P., Lustig, T.D., Kehrberger, P.M. and Salo, J.E. , “Experimental and Mathematical Modeling Studies of
Protozoan Predation on Bacteria,” , Biotechnol. and Bioeng., Vol.15, pp.707-728 (1973).
[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]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).
[4]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).
(12)
where C1  K p  s f
C2  K i K p  K p s f 
m K p
YD 2
b2
Fig.1
p2, b2 and s2 versus Time ；
D1=0.0595hr–1, D2=0.0595hr–1 ； limit
cycle
C 3   K i K p s f ， There are three steady states for s2 at most.
X  s2   D2  kb b2
p2 
D2  k a 
(13)
Fig.2 The dynamic graph for the second
continuous culture, 3 D Plot；D1=0.0595
hr–1, D2=0.0595hr–1； t = 1000-2000hr；
limit cycle
The values of the fourth steady state are computed from the steady state equations of
equations (2) and (3).
s1a 
 A1  A12  4B
2

m
where A1  K p 1 
 D k
1
b

(14 a ) ；
(24)
1
C 2  K i K p  K p s f 
 A  A2  4 B

2
b2b
2
From the steady state equations of equations (4)-(6), we have
where
The values for the second steady state are computed from equations (5) and (6).
(9a ) ；
(18 b)

 ；B  K i K p ， and A1 < 0，A12-4B > 0


From the steady state equations of equation (3), we have
first continuous culture (mg/L), s1 = concentration of the substrate for the first continuous culture (mg/L), p2＝
concentration of the predator for the second continuous culture (mg/L), b2 = concentration of the prey for the second
continuous culture (mg/L), s2 = concentration of the substrate for the second continuous culture (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), V1 = reactor volume for the first continuous culture (L), V2 = reactor volume
for the second continuous culture (L), D1 =F/V1 = dilution rate for the first continuous culture (hr–1), D2 =F/V2 = dilution
rate for the second continuous culture (hr–1). ka = death rate coefficient for the predator (hr–1)，kb = death rate
coefficient for the prey (hr–1).The specific growth rates of Substrate Inhibition model and Monod model are used for the
prey and predator respectively.
μm s
νmb
 ( s) 
(7 )
2
ν(b) 
(8)
s
；
Lb
Ki  s 
KP
μm , Ki, Kp = constants ; νm = maximum specific growth rate (hr–1) ; L= saturation constant
(mg/L).
For the situation of this system, six types of steady state solutions are possible for equations (1)-(6)
(1) p1=0 , b1=0 , s1=sf , p2=0 , b2=0 , s2=sf
(2) p1=0, b1=0 , s1=sf , p2=0 , b2＞0 , s2＞0
(3) p1=0 , b1=0 , s1=sf , p2＞0 , b2＞0 , s2＞0
(4) p1=0, b1＞0 , s1＞0 , p2=0 , b2＞0 , s2＞0
(5) p1=0, b1＞0 , s1＞0 , p2＞0 , b2＞0 , s2＞0
(6) p1＞0, b1＞0 , s1＞0 , p2＞0 , b2＞0 , s2＞0

YD1 ( s f  s1a )
s1b 
；
 A1  A12  4B
The values of the sixth steady state are obtained from steady state equations of equation (1)
where p1＝concentration of the predator for the first continuous culture (mg/L), b1 = concentration of the prey for the
b2a
2
(18a )
X D2 b1  b2    s2 b2  kb b2 
p2 
D2  k a

YD2

s f  s2a
D2  k b
 A1  A12  4 B
C6   K i K p s1 ，There are three steady states for s2 at most.
dp1
  D1 p1   b1  p1  k a p1
(1)
dt
db1
1
  D1b1   s1 b1   b1  p1  k b b1 (2)
dt
X
ds1
1
 D1 s f  s1   s1 b1
(3)
dt
Y
dp 2
 D 2  p1  p 2    b2  p 2  k a p 2 (4)
dt
db2
1
 D2 b1  b2    s 2 b2   b2  p 2  k b b2 (5)
dt
X
ds2
1
 D2 s1  s 2    s 2 b2
(6)
dt
Y
s2a
D1  k b
The values of b2 and s2 can be obtained from equations (16) and (17).
The values of the fifth steady state are computed from the steady state equations of equations (2) and (3).
4
 A  A2  4B

2

m
(15 b)
0  D2 b1  b2    s2 b2  kbb2
(16)
1
0  D2 s1  s2    s2 b2
(17 )
Y
The dynamic equations for the prey-predator interaction in two continuous cultures in series
are as follows:

D1  k b
b1b 
(15 b) ；
YD1 ( s f  s1b )
Because p2=0, the steady state equations of equations (5) and (6) are as follows.
s1a 
Introduction
YD1 ( s f  s1a )
s1b 
 A1  A12  4B
2

 ；B  K i K p and A1 < 0，A12 - 4B > 0


(14 b)
Fig.3 p2, b2 and s2 versus Time；
D1=0.0825hr–1, D2=0.0825hr–1 ；
the sixth type stable steady state
Fig.4 The dynamic graph for the second
continuous culture, 3 D Plot；D1=0.0825
hr–1, D2=0.0825hr–1； t = 0-2000hr；the
sixth type stable steady state
(26 )