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Mathematical Biosciences 162 (1999) 69±84
www.elsevier.com/locate/mathbio

Chaotic dynamics of a food web in a
chemostat
D.V. Vayenas, Stavros Pavlou *
Department of Chemical Engineering, University of Patras, Institute of Chemical Engineering and
High Temperature Chemical Processes, FORTH, GR-26500 Patras, Greece
Received 19 April 1999; received in revised form 21 July 1999; accepted 19 August 1999

Abstract
We analyze a mathematical model of a simple food web consisting of one predator
and two prey populations in a chemostat. MonodÕs model is employed for the dependence of the speci®c growth rates of the two prey populations on the concentration of
the rate-limiting substrate and a generalization of MonodÕs model for the dependence of
the speci®c growth rate of the predator on the concentrations of the prey populations.
We use numerical bifurcation techniques to determine the e€ect of the operating conditions of the chemostat on the dynamics of the system and construct its operating
diagram. Chaotic behavior resulting from successive period doublings is observed.
Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found. Ó 1999 Elsevier Science Inc. All rights reserved.
Keywords: Population dynamics; Chemostat; Operating diagram; Food web; Chaos

1. Introduction

Predation and competition are the two most common interactions between
two microbial populations inhabiting the same environment. Predation is a
direct interaction which occurs when individuals from one population derive
their nourishment by capturing and ingesting individuals from another

*

Corresponding author. Tel.: +30-61 997 640; fax: +30-61 993 255.
E-mail address: sp@chemeng.upatras.gr (S. Pavlou)

0025-5564/99/$ - see front matter Ó 1999 Elsevier Science Inc. All rights reserved.
PII: S 0 0 2 5 - 5 5 6 4 ( 9 9 ) 0 0 0 4 4 - 9

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

population. Competition is an indirect interaction which occurs when two
microbial populations compete for common resources. The simplest scheme
combining the two interactions is a food web consisting of two saprotrophs

(bacteria) competing for one rate-limiting substrate and one phagotroph
(protozoan) preying upon both saprotrophs. Such a system can be studied in
the laboratory with the help of the chemostat, which is a well-stirred vessel
where all microbial species grow together and which is fed with the rate-limiting nutrient for growth of the bacteria. With such an arrangement one can
study many di€erent microbial interactions which occur in large-scale systems.
The simple food web consisting of one predator and two prey populations is
just one step more complicated than the simple predator±prey system or the
system of competition for a single nutrient. It can be considered as resulting
from the predator±prey system by adding a second prey population or from the
competition system by adding a population preying upon both competing
populations. With respect to the dynamics of the system the question is how
they di€er from the dynamics of the simple competition or the simple predator±
prey system.
Speci®cally, it has been shown both theoretically [1±6] and experimentally
[7±11] that coexistence of two microbial populations competing for a single
nutrient in a chemostat is practically impossible when competition is the only
interaction between the populations. Then the question is whether presence of
a predator feeding on both competing populations makes their coexistence
possible. On the other hand, it is well established both theoretically [12±15] and
experimentally [10,16±19] that predator±prey systems exhibit sustained oscillations under a wide range of operating conditions of the chemostat. In this

case the question is whether presence of a second prey population leads to
more complicated dynamics.
Analysis of Lotka±Volterra type models of the one-predator, two-prey food
web [20±22] indicates that the answer to both questions is armative. Coexistence of the two competing prey populations is indeed observed and also
chaotic behavior is exhibited by the system. It should be examined, however,
whether the same conclusions are reached with more realistic chemostat models
and Monod-type kinetics for microbial growth. A model of a chemostat with a
one-predator, two-prey food web has been studied by Jost et al. [23]. They
employed MonodÕs model for the dependence of the speci®c growth rates of the
two prey populations on the concentration of the rate-limiting substrate and a
generalization of their so-called multiple saturation model for the dependence
of the speci®c growth rate of the predator on the concentrations of the prey
populations. They showed that their model predicts coexistence of all populations in the chemostat either at equilibrium or in a periodic state for a wide
range of the operating conditions. However, at that time they were not able to
do a complete bifurcation analysis and thus they could not ®nd any complex
dynamic behavior. The same workers [10] studied such a system experimentally

D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

71

as well. Speci®cally, the predator was the ciliate Tetrahymena pyriformis, the
two prey populations were the bacteria Escherichia coli and Azotobacter vinelandii and the rate-limiting substrate for bacterial growth was glucose. Their
results indicate that all microbial populations can coexist in the chemostat in a
state of sustained oscillations.
Chaotic behavior results if we add to the predator±prey system another
population preying upon the predator, that is, introducing an additional
trophic level. This has been shown by several workers [24±28], who analyzed
the three-species food chain. However, it would be interesting to know whether
chaotic behavior results also if, instead of introducing an extra trophic level, we
add a second prey population resulting in the simplest possible food web.
In this work we do a detailed computational study of a model of a onepredator, two-prey food web in a chemostat. We use MonodÕs model for the
dependence of the speci®c growth rates of the two prey populations on the
concentration of the rate-limiting substrate and a generalization of MonodÕs
model for the dependence of the speci®c growth rate of the predator on the
concentrations of the prey populations. With the aid of numerical bifurcation
techniques we analyze the model equations and determine the e€ect of the
operating parameters of the chemostat on its dynamics.

2. Description of the system

We consider a chemostat in which all three microbial populations grow
together and which is fed with medium containing the limiting nutrient for
growth of the two competing bacterial populations. The food web can be
represented schematically as follows:

Scheme 1.

In this Scheme 1, S is the rate-limiting substrate for growth of the two bacterial
populations B1 and B2 upon which feeds the protozoan population P. The
balance equations for the three microbial populations and for the rate-limiting
substrate in the chemostat are
ds
1
1
ˆ D…sf ÿ s† ÿ l1 …s†b1 ÿ l2 …s†b2 ;
0
dt
Y1
Y2

…1a†

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

db1
ˆ ÿDb1 ‡ l1 …s†b1 ÿ /1 …b1 ; b2 †p;
dt0

…1b†

db2
ˆ ÿDb2 ‡ l2 …s†b2 ÿ /2 …b1 ; b2 †p;
dt0

…1c†

dp
ˆ ÿDp ‡ m…b1 ; b2 †p;

dt0

…1d†

where b1 , b2 and p are the concentrations of the two bacterial and the protozoan populations, respectively, in the chemostat, s is the concentration of the
rate-limiting substrate in the chemostat, sf the concentration of the rate-limiting substrate in the feed, D the dilution rate of the chemostat, Y1 and Y2 are
the yield coecients for biomass production of the bacterial populations on the
rate-limiting substrate, l1 (s), and l2 (s) are the speci®c growth rates of the
bacterial populations, m(b1 , b2 ) is the speci®c growth rate of the protozoan
population, and /1 (b1 , b2 ) and /2 (b1 , b2 ) are the speci®c feeding rates of the
protozoan population upon the two bacterial populations. The speci®c growth
rates of the bacterial populations are functions of the concentration of the ratelimiting substrate and are assumed to follow MonodÕs model:
li …s† ˆ

lmi s
;
Ki ‡ s

i ˆ 1; 2:

…2†

In Eq. (2) lmi are the maximum speci®c growth rates and Ki are the saturation
constants. The speci®c growth rate and the speci®c feeding rates of the protozoan population are in general functions of the concentrations of both
bacterial populations. We assume the speci®c growth rate of the protozoan
population to have a Monod type dependence on the weighted sum of the
concentrations of the two bacterial populations:
m…b1 ; b2 † ˆ

mm …X1 b1 ‡ X2 b2 †
:
L ‡ …X1 b1 ‡ X2 b2 †

…3†

In this expression mm is the maximum speci®c growth rate of the protozoan
population and L is an equivalent saturation constant. Also, X1 and X2 are the
yield coecients for protozoan production (protozoan mass produced per
bacterial mass consumed). By including these coecients in the speci®c growth
rate expression we consider the equivalent bacterial biomass concentration to

a€ect protozoan growth. Considering the de®nition of the protozoan speci®c
growth rate and of the yield coecients we see that it is necessary for the
speci®c feeding rates /1 (b1 , b2 ) and /2 (b1 , b2 ) to have expressions such that
m…b1 ; b2 † ˆ X1 /1 …b1 ; b2 † ‡ X2 /2 …b1 ; b2 †:

…4†

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

We assume the following expressions:
/i …b1 ; b2 † ˆ

m m bi
;
L ‡ …X1 b1 ‡ X2 b2 †

i ˆ 1; 2:

…5†

The expressions in Eqs. (3) and (5) are similar to the ones used by Kretzschmar
et al. [29] for a system of zooplankton grazing on two algal populations.
In order to write the system of Eqs. (1a)±(1d) in dimensionless form we
de®ne the following dimensionless quantities:
t ˆ t0 lm1 ;
uˆ

D
;
lm1

bi
p
s
sf
; yˆ
; w ˆ ; wf ˆ ;
Yi K1

K1
X1 Y1 K1
K1
l
mm
K2
L
; bx ˆ ; by ˆ
ax ˆ m2 ; ay ˆ
;
lm1
lm1
K1
X1 Y1 K1
xi ˆ

cˆ

X2 Y 2
:
X1 Y 1

Then, the system of Eqs. (1a)±(1d) becomes
dw
ˆ u…wf ÿ w† ÿ f1 …w†x1 ÿ f2 …w†x2 ;
dt

…6a†

dx1
ˆ ÿux1 ‡ f1 …w†x1 ÿ h1 …x1 ; x2 †y;
dt

…6b†

dx2
ˆ ÿux2 ‡ f2 …w†x2 ÿ h2 …x1 ; x2 †y;
dt

…6c†

dy
ˆ ÿuy ‡ g…x1 ; x2 †y;
dt

…6d†

where
f1 …w† ˆ

w
;
1‡w

…7a†

f2 …w† ˆ

ax w
;
bx ‡ w

…7b†

g…x1 ; x2 † ˆ

ay …x1 ‡ cx2 †
by ‡ …x1 ‡ cx2 †

…7c†

are the dimensionless speci®c growth rate expressions and
hi …x1 ; x2 † ˆ

ay x i
;
by ‡ …x1 ‡ cx2 †

i ˆ 1; 2

are the dimensionless speci®c feeding rate expressions.

…8†

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

3. Theory and methods
Depending on the values of the operating parameters, i.e., the chemostat
dilution rate u and the substrate concentration in the feed wf , the system may
exhibit several di€erent long-term dynamics. One would like to know the dependence of the system dynamics on the operating conditions. This dependence
can be summarized in an e€ective way with the help of the operating diagram.
It is a diagram which has u and wf as its coordinates and in which various
regions are de®ned representing ranges of the operating parameters for which
the system exhibits qualitatively di€erent dynamics. In order to construct the
operating diagram one must trace the boundaries of these regions in the operating parameter space. On these curves the system undergoes bifurcations of
steady states or periodic solutions and qualitative changes in its dynamic behavior occur.
Steady-state bifurcations occur at parameter values for which one real eigenvalue or a pair of complex conjugate eigenvalues of the Jacobian matrix
cross the imaginary axis in the complex plane and are accompanied by change
in the character of the steady states. In the system studied here the following
two types of steady-state bifurcations have been observed:
1. Transcritical bifurcation, when two steady states come together and
exchange their stability characteristics. In this case one real eigenvalue
becomes zero.
2. Hopf bifurcation, when a periodic solution (limit cycle) is born around a
steady state. In this case, the real part of a pair of complex conjugate eigenvalues vanishes.
Periodic solutions undergo bifurcations when one or more of their characteristic multipliers cross the unit circle in the complex plane. In the system
studied here the following limit-cycle bifurcations have been found:
1. Limit-point bifurcation, when two limit cycles collide and disappear. In this
case, one characteristic multiplier of the limit cycles crosses the unit circle
at 1.
2. Transcritical bifurcation, when two limit cycles come together and exchange
their stability characteristics. In this case also, one characteristic multiplier
of the limit cycles crosses the unit circle at 1.
3. Period-doubling bifurcation, when from one limit cycle a second limit cycle
of double period is born. In this case, one characteristic multiplier of the
limit cycle crosses the unit circle at ÿ1.
To compute a curve of the operating diagram one must ®rst locate a point
on the curve, i.e., a bifurcation point of the system. This can be accomplished
by the continuation algorithm AUTO [30], which computes the branches of
steady and periodic states as one of the operating parameters is changed and
also locates the bifurcation points. Then a curve of the operating diagram is
traced through two-parameter continuation of a bifurcation point. AUTO does

D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

75

two-parameter continuation of certain steady-state and limit-cycle bifurcations, like limit-point bifurcations, Hopf bifurcations and period-doubling
bifurcations. However it cannot do two-parameter continuation of transcritical
bifurcations of steady and periodic states, but it can only locate them in the
one-parameter bifurcation diagram. In order to trace the transcritical bifurcation curves in the two-parameter space one may use one of the two parameters as variable and do one-parameter continuation by adding an extra
equation. This extra equation results in the case of transcritical bifurcation of
steady states by setting one eigenvalue equal to zero or equivalently by setting
the determinant of the Jacobian matrix of the system equal to zero. In the case
of transcritical bifurcation of limit cycles the extra equation results by setting a
characteristic multiplier equal to one. In this way, all the bifurcation curves can
be traced in the operating parameter space and thus the operating diagram can
be constructed. A detailed description of all these techniques has been given by
Pavlou [31].

4. Results and discussion
The system of equations (6a)±(6d) has seven possible steady states:
Extinction of all populations: x1 ˆ x2 ˆ y ˆ 0 (washout state).
Survival of population X1 only: x1 > 0; x2 ˆ y ˆ 0 (X1 state).
Survival of population X2 only: x2 > 0; x1 ˆ y ˆ 0 (X2 state).
Survival of populations X1 and X2 only: x1 ; x2 > 0; y ˆ 0 (X12 state).
Survival of populations X1 and Y only: x1 ; y > 0; x2 ˆ 0 (YX1 state).
Survival of populations X2 and Y only: x2 ; y > 0; x1 ˆ 0 (YX2 state).
Survival of all three populations: x1 ; x2 ; y > 0 (YX12 state).
From the theory of microbial competition [32] it is known that two microbial populations involved in pure and simple competition in a chemostat
with time-invariant operating conditions can coexist only when the speci®c
growth rate curves of the two populations cross and the dilution rate has exactly the value corresponding to the point of intersection. However, in that case
the system is structurally unstable and the coexistence state is not attainable in
practice. Thus, survival of populations X1 and X2 alone (X12 state) is realizable
only at a speci®c value of the dilution rate u ˆ …ax ÿ bx †=…1 ÿ bx †. An important question is whether the presence of the predator population Y can lead to
coexistence of populations X1 and X2 in a practically attainable state. Namely,
whether there exists a region in the operating diagram of the system where the
state of coexistence of all three populations (YX12 state) is stable.
An operating diagram of the system is shown in Fig. 1. The parameter
values that we used in the kinetic expressions ((7a)±(7c) and (8)) for the construction of this diagram are listed in Table 1. These values were chosen so that
we observe all the interesting dynamics of the system. The character of the

1.
2.
3.
4.
5.
6.
7.

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

Fig. 1. (a) Operating diagram for the system of equations (6a)±(6d) and (b) enlargement of the
region marked in (a). Labeling of curves: TXi , transcritical bifurcation between washout state and
Xi state; TYXi , transcritical bifurcation between Xi state and YXi state; TYXij , transcritical bifurcation between YXi state and YX12 state; Hk , Hopf bifurcation of k state; TPi , transcritical bifurcation between periodic YXi state and periodic YX12 state; Pi , period-doubling bifurcation of
limit cycle of period i; Li , limit-point bifurcation of limit cycle of period i. Values of kinetic parameters in Table 1. Character of steady states in Table 2. Character of periodic states in Table 3.

Table 1
Values of kinetic parameters in Eqs. (7a)±(7c) and (8) for construction of the operating diagrams
ax
bx
ay
by
c

Fig. 1

Fig. 4

Fig. 5

0.7
0.3
2.8
17.8
20.3

0.7
0.3
1
2
3

0.7
0.3
1
2
1

steady states in each region of the operating diagram is listed in Table 2. Also,
in some regions of the diagram there exist stable or unstable periodic states.
These are shown in Table 3. As mentioned above, coexistence of populations
X1 and X2 without the presence of population Y is possible only on the horizontal line separating regions 3 and 5, 9 and 10, 15 and 22, and 14 and 16, i.e.,
for u ˆ …ax ÿ bx †=…1 ÿ bx † ˆ 0:57143. On the other hand, coexistence of all
three populations in a steady state is observed for a wide range of operating
conditions, and speci®cally in regions 9, 22 and 23. Thus, the presence of the
predator Y makes possible the coexistence of the two competitors X1 and X2 .
This coexistence steady state undergoes a Hopf bifurcation on the curve
marked HYX12 resulting in a stable coexistence limit cycle. Thus, coexistence
of all three populations is observed also in regions 21 and 24, but in a state of
sustained oscillations. The coexistence limit cycle undergoes a sequence of

77

D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

Table 2
Character of each steady state in the various regions of the operating diagrams shown in Figs. 1, 4,
and 5a
Region

Washout

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17,
18
19
20,
22,
23,
24,
31
34
36
37
38

S
D1
D2
D1
D2
D1
D2
D2
D2
D2
D1
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2
D2

35

21
32
26, 27, 33
25, 28±30

X1 state
S
S
D1
D2
D1
D1
S
D1
D1
D1
D1
S
D2
D2
D2
D1
D1
D1
D2
D2
D1
D2
D1
D2
D2

X2 state

D1
S
S
D1
D1
D1
D1
D2
D1
D2
D2
D2
D1
D1
D1
D1
D1
D1
D1
D1
D2
D1
D2
D1
D1

YX1 state

D1

S
S
S
S
S
D2
D1

D1
D1
D2
D3
D2
S
D3

YX2 state

S
S
S
D1
D1

D1
D3
D3
D3
D3
D2
D2
D3
D3
D3
D3
D3
D3
D1
D1
D2

YX12 state

S

D2
S
S
D2
D2

a

S, stable; D1 , saddle with one positive eigenvalue; D2 , saddle with two eigenvalues with positive
real parts; D3 , saddle with three eigenvalues with positive real parts.

period doubling bifurcations on the curves marked P1 , P2 , P4 . This is illustrated
in Fig. 2, where limit cycles of periods 1, 2, 4 and 8 are shown. To obtain these
periodic solutions, the operating parameter wf was kept constant and the other
operating parameter u was changed in order to cross successively the period
doubling curves. The period doublings continue up to the point where the
system exhibits chaotic behavior. The curves on which period doublings occur
lie closer together as the period increases, in accordance with FeigenbaumÕs
scenario [33]. This makes computation of the curves on which higher period
doublings occur very dicult. An example of chaotic behavior of the system,
which is observed when the operating conditions fall in region 30, is shown in
Fig. 3. A way of certifying chaotic behavior is through the Lyapunov exponents. A chaotic system must contain at least one positive Lyapunov exponent.

78

D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

Table 3
Character of each periodic state in the various regions of the operating diagrams shown in Figs. 1,
4, and 5a
Region
14±16, 22, 23
17, 31
18±20, 25
21, 24
26
27
28
29
30
32, 33
34, 35
36
38

YX1 state
S

U
S
U

YX2 state

YX12 state

U
U
S
U
U
U
U
U
U
S
U

S
S, U
U, U, S(2)
U, S(2)
U, U(2), S(4)
U, U(2), U(4). . ...chaos
U
S

S

a

S, stable; U, unstable. Number in parentheses denotes period of limit cycle whenever di€erent
from 1.

The Lyapunov exponents for the attractor shown in Fig. 3 were computed by a
method described by Wolf et al. [34] and were found to be 0.0037, 0, ÿ0.0014,
ÿ0.018.
Another interesting feature of the system is multistability. Speci®cally, there
exist operating conditions for which the system exhibits both stable steadystate coexistence and stable periodic coexistence. Which of the two states will
be reached by the system depends on the initial conditions. This type of behavior is observed in regions 26 and 27.
The operating diagram shown in Fig. 1 was constructed by proper choice of
the kinetic parameters so that all the interesting behaviors of the system are
observed. However, for other parameter values some of the features of the
system may be lost. Considering the values with which the operating diagram
in Fig. 1 was constructed, it is interesting to see, how much we can change the
value of any of the parameters and still observe chaotic behavior. Thus,
changing the value of one parameter at a time, keeping all the others at the
values of Fig. 1, we ®nd the following ranges for which the system can exhibit
chaotic dynamics: 0:695 < ax < 0:71; 0:281 < bx < 0:302; 2:15 < ay < 2:92;
10:0 < by < 35:0; 12:5 < c < 48:0. Of course, if we change the values of two or
more parameters simultaneously we will be able to ®nd other ranges of parameter values.
From a practical point of view, it is important to examine what types of
behavior are observed for biologically realistic parameter values. In Table 4 we
have listed values of kinetic parameters for some experimental systems that
have been reported in the literature or are estimated from data reported in the
literature. From this list we can derive the following rough ranges for the

D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

79

Fig. 2. Stable limit cycles of the system of Eqs. (6a)±(6d). Operating parameter values: wf ˆ 1:85
and (a) u ˆ 0:04 (period 1), (b) u ˆ 0:03 (period 2), (c) u ˆ 0:0225 (period 4), (d) u ˆ 0:021 (period
8). Kinetic parameter values as in Fig. 1.

Fig. 3. Chaotic attractor of the system of equations (6a)±(6d). Operating parameter values:
u ˆ 0:01 and wf ˆ 1:85. Kinetic parameter values as in Fig. 1.

dimensionless kinetic parameters: 0:4 < ax < 2:5; 8  10ÿ6 < bx < 1:2  105 ;
0:2 < ay < 1:5; 0:25 < by < 1  105 ; 0:2 < c < 5. We see that from the values

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

Table 4
Values of kinetic parameters for some experimental systems
Escherichia coli/glucose
Escherichia coli/BC medium
Azotobacter vinelandi/BC medium
Aerobacter aerogenes/sucrose
Alcaligenes faecalis/asparagine
Propionibacterium shermanii/glucose
Propionibacterium shermanii/lactose
Saccharomyces cerevisiae/glucose
Saccharomyces cerevisiae/ethanol
Bacillus cereus/glucose
Bacillus cereus/fructose
Candida tropicalis/glucose
Candida tropicalis/fructose
Enterobacter aerogenes/organic carbon
Escherichia coli/BC medium
Escherichia coli/glucose

Colpoda steinii/Escherichia coli
Tetrahymena pyriformis/Klebsiella aerogenes
Tetrahymena pyriformis/Klebsiella aerogenes
Dictyostelium discoedium/Escherichia coli
Tetrahymena pyriformis/Aerobacter aerogenes
Colpidium campylum/Alcalegenes faecalis
Colpoda steinii/Escherichia coli
Paramecium primaurelia/Enterobacter aerogenes
Tetrahymena pyriformis/Escherichia coli
Tetrahymena pyriformis/Escherichia coli
a

lmi (hÿ1 )

Ki (mg/ml)

Yi

Ref.

0.25
0.32
0.23
0.56
0.114
0.14
0.14
0.33
0.22
0.52
0.30
0.74
0.50
0.621
0.31±0.33
0.25

5  10ÿ4
1  10ÿ7
1:2  10ÿ2
1:6  10ÿ4
1:07  10ÿ4
1  10ÿ2
2  102
4  10ÿ2
5  10ÿ2
0.15
3  10ÿ2
1:2  10ÿ2
1  10ÿ2
2:43  10ÿ4
1  10ÿ7
5  10ÿ4

1.0a

[16]
[10]
[10]
[38]
[17]
[40]
[40]
[40]
[40]
[40]
[40]
[40]
[40]
[41]
[19]
[42]

mm (hÿ 1)

L=Xi (mg/ml)

Xi

Ref.

0.23
0.22
0.43
0.24
0.1
0.11
0.37
0.132

6  10ÿ3
1:16  10ÿ4
1:17  10ÿ4
1:2  10ÿ4a
6:1  10ÿ3
1  10ÿ4
3  10ÿ3a
9  10ÿ2a

0.78
0.5
0.54
0.73
0.5
0.45
0.968

[35]
[36]
[37]
[16]
[38]
[17]
[39]
[41]

0.31
0.31

1  10ÿ2a
2  10ÿ2a

0.6a

[19]
[42]

0.428
0.15

0.12
1.10

2.5a
0.9a

Estimated value.

used in the construction of the operating diagram in Fig. 1, the ones of ay and c
do not lie in these ranges. Although it might be possible to ®nd extreme experimental cases where these values are observed, it is interesting to see what
form takes the operating diagram for parameter values common in experimental system. Such an operating diagram is shown in Fig. 4. The parameter
values used are again listed in Table 1. We see that most of the regions of the
operating diagram are preserved except the ones in which chaotic behavior is
observed. An important observation is that coexistence of all the microbial
populations is still possible for a wide range of operating conditions, thus
indicating that it must be common in experimental systems and, as we have
already mentioned, it has been observed in at least one case [10]. For coexistence to be possible it is necessary that the speci®c growth rate curves of the

D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

81

Fig. 4. Operating diagram for the system of equations (6a)±(6d). Labeling of curves as in Fig. 1.
Values of kinetic parameters in Table 1. Character of steady states in Table 2. Character of periodic
states in Table 3.

two prey populations cross, i.e., the values of ax and bx must be such that
…ax ÿ bx †=…1 ÿ bx † > 0. The values of ay and by do not seem to play an important role with regard to coexistence of the populations. As for the value of
c, it seems that the only condition for coexistence is that it must be di€erent

Fig. 5. Operating diagram for the system of equations (6a)±(6d). Labeling of curves as in Fig. 1.
Values of kinetic parameters in Table 1. Character of steady states in Table 2. Character of periodic
states in Table 3.

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D.V. Vayenas, S. Pavlou / Mathematical Biosciences 162 (1999) 69±84

than unity. When c ˆ 1 the two curves of the operating diagram marked TYX12
and TYX21 coincide and the regions where coexistence is observed vanish. This
is illustrated in the operating diagram in Fig. 5.

5. Conclusions
We performed a detailed computational analysis of a mathematical model of
a simple food web consisting of one predator and two prey populations in a
chemostat. The dynamic behavior of the system was studied with respect to the
e€ect of the operating conditions of the chemostat. Towards this end numerical
bifurcation techniques were used for the construction of the operating diagram,
which summarizes the e€ect of the operating parameters of the system on its
dynamics. The analysis shows that there exists a wide range of conditions for
which all three populations can coexist. This result is in accordance with earlier
observations that presence of a population preying upon two populations
competing for a single rate-limiting nutrient stabilizes their coexistence. An
important conclusion of the analysis is that there exist conditions for which the
system exhibits chaotic behavior. By changing an operating parameter of the
system, a transition from simple periodic to chaotic behavior takes place
through a sequence of period doublings. This observation is important from an
ecological point of view. It is known that a simple food chain with one predator
and one prey population exhibits at most periodic behavior, whereas a threespecies food chain can exhibit chaotic behavior. The present analysis shows
that addition of a second prey population instead of an extra trophic level can
also lead to chaotic behavior. Finally, the system studied here has another
interesting feature, which is common in many systems of interacting microbial
populations. Namely, it exhibits multistability, which means that there exist
certain operating conditions for which the system may reach either steady state
or periodic state depending on its initial conditions. However, both chaotic
behavior and multistability were found only for certain values of the kinetic
parameters which are not all common in experimental systems. For biologically common parameter values, coexistence of all microbial populations is
realized in a steady or periodic state, but not in a chaotic state.

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