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 depen-dence 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 eect of the operating con-ditions 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 op-erating 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 www.elsevier.com/locate/mathbio
*
Corresponding author. Tel.: +30-61 997 640; fax: +30-61 993 255. E-mail address:[email protected] (S. Pavlou)
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-lim-iting nutrient for growth of the bacteria. With such an arrangement one can study many dierent 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 dier 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 oscil-lations 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.
as well. Speci®cally, the predator was the ciliate Tetrahymena pyriformis,the two prey populations were the bacteriaEscherichia coliandAzotobacter vine-landiiand 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 one-predator, 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 eect 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:
In this Scheme 1,Sis 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
dt0D sfÿs ÿ
1
Y1
l1 sb1ÿ
1
Y2
l2 sb2; 1a
db1
dt0 ÿDb1l1 sb1ÿ/1 b1;b2p; 1b
db2
dt0 ÿDb2l2 sb2ÿ/2 b1;b2p; 1c
dp
dt0 ÿDpm b1;b2p; 1d
whereb1,b2and pare the concentrations of the two bacterial and the
proto-zoan populations, respectively, in the chemostat,sis the concentration of the rate-limiting substrate in the chemostat, sf the concentration of the
rate-lim-iting substrate in the feed,Dthe dilution rate of the chemostat,Y1 andY2are
the yield coecients 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 rate-limiting substrate and are assumed to follow MonodÕs model:
li s lmis
Kis
; i1;2: 2
In Eq. (2)lmiare the maximum speci®c growth rates andKiare the saturation
constants. The speci®c growth rate and the speci®c feeding rates of the pro-tozoan 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
mm X1b1X2b2
L X1b1X2b2
: 3
In this expression mm is the maximum speci®c growth rate of the protozoan
population andLis an equivalent saturation constant. Also,X1andX2are the
yield coecients for protozoan production (protozoan mass produced per bacterial mass consumed). By including these coecients in the speci®c growth rate expression we consider the equivalent bacterial biomass concentration to aect protozoan growth. Considering the de®nition of the protozoan speci®c growth rate and of the yield coecients we see that it is necessary for the speci®c feeding rates/1(b1,b2) and/2(b1,b2) to have expressions such that
We assume the following expressions:
/i b1;b2
mmbi
L X1b1X2b2
; i1;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:
Then, the system of Eqs. (1a)±(1d) becomes
dw
are the dimensionless speci®c growth rate expressions and
hi x1;x2
ayxi by x1cx2
; i1;2 8
3. Theory and methods
Depending on the values of the operating parameters, i.e., the chemostat dilution rateu and the substrate concentration in the feedwf, the system may
exhibit several dierent long-term dynamics. One would like to know the de-pendence of the system dynamics on the operating conditions. This dede-pendence can be summarized in an eective 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 dierent dynamics. In order to construct the operating diagram one must trace the boundaries of these regions in the op-erating parameter space. On these curves the system undergoes bifurcations of steady states or periodic solutions and qualitative changes in its dynamic be-havior occur.
Steady-state bifurcations occur at parameter values for which one real ei-genvalue or a pair of complex conjugate eiei-genvalues 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 eigen-values vanishes.
Periodic solutions undergo bifurcations when one or more of their charac-teristic 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.
two-parameter continuation of certain steady-state and limit-cycle bifurca-tions, like limit-point bifurcabifurca-tions, 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 bifur-cation curves in the two-parameter space one may use one of the two pa-rameters 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: 1. Extinction of all populations:x1x2y0 (washout state).
2. Survival of populationX1 only:x1>0;x2y0 (X1 state).
3. Survival of populationX2 only:x2>0;x1y0 (X2 state).
4. Survival of populationsX1andX2 only:x1;x2>0;y0 (X12 state).
5. Survival of populationsX1andYonly:x1;y>0;x20 (YX1 state).
6. Survival of populationsX2andYonly:x2;y>0;x10 (YX2 state).
7. Survival of all three populations:x1;x2;y>0 (YX12 state).
From the theory of microbial competition [32] it is known that two mi-crobial 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 ex-actly 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 populationsX1andX2alone (X12 state) is realizable
only at a speci®c value of the dilution rateu axÿbx= 1ÿbx. An
impor-tant question is whether the presence of the predator populationYcan lead to coexistence of populationsX1andX2in 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.
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 andX2 without the presence of populationYis possible only on the
hori-zontal line separating regions 3 and 5, 9 and 10, 15 and 22, and 14 and 16, i.e., for u axÿbx= 1ÿbx 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 predatorYmakes possible the coexistence of the two competitorsX1andX2.
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 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
bifur-cation between YXi state and YX12 state; Hk, Hopf bifurcation ofkstate; TPi, transcritical
bi-furcation between periodic YXi state and periodic YX12 state;Pi, period-doubling bifurcation of
limit cycle of periodi;Li, limit-point bifurcation of limit cycle of periodi. Values of kinetic
pa-rameters 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
Fig. 1 Fig. 4 Fig. 5
ax 0.7 0.7 0.7
bx 0.3 0.3 0.3
ay 2.8 1 1
by 17.8 2 2
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 parameterwf 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 dicult. 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 expo-nents. A chaotic system must contain at least one positive Lyapunov exponent. Table 2
Character of each steady state in the various regions of the operating diagrams shown in Figs. 1, 4, and 5a
Region Washout X1 state X2 state YX1 state YX2 state YX12 state
1 S
S, stable; D1, saddle with one positive eigenvalue; D2, saddle with two eigenvalues with positive
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 steady-state coexistence and stable periodic coexistence. Which of the two steady-states will be reached by the system depends on the initial conditions. This type of be-havior 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 pa-rameter 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 Table 3
Character of each periodic state in the various regions of the operating diagrams shown in Figs. 1, 4, and 5a
Region YX1 state YX2 state YX12 state
14±16, 22, 23 U
dimensionless kinetic parameters: 0:4<ax<2:5;810ÿ6<bx<1:2105;
0:2<ay <1:5;0:25<by <1105;0:2<c<5. We see that from the values
Fig. 2. Stable limit cycles of the system of Eqs. (6a)±(6d). Operating parameter values:wf1:85
and (a)u0:04 (period 1), (b)u0:03 (period 2), (c)u0:0225 (period 4), (d)u0:021 (period 8). Kinetic parameter values as in Fig. 1.
used in the construction of the operating diagram in Fig. 1, the ones ofayandc
do not lie in these ranges. Although it might be possible to ®nd extreme ex-perimental cases where these values are observed, it is interesting to see what form takes the operating diagram for parameter values common in experi-mental 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 coexis-tence to be possible it is necessary that the speci®c growth rate curves of the Table 4
Values of kinetic parameters for some experimental systems
lmi(hÿ1) Ki(mg/ml) Yi Ref.
Escherichia coli/glucose 0.25 510ÿ4 1.0a [16]
Escherichia coli/BC medium 0.32 110ÿ7 [10]
Azotobacter vinelandi/BC medium 0.23 1:210ÿ2 [10]
Aerobacter aerogenes/sucrose 0.56 1:610ÿ4 0.428 [38]
Alcaligenes faecalis/asparagine 0.114 1:0710ÿ4 0.15 [17]
Propionibacterium shermanii/glucose 0.14 110ÿ2 [40]
Propionibacterium shermanii/lactose 0.14 2102 [40]
Saccharomyces cerevisiae/glucose 0.33 410ÿ2 0.12 [40]
Saccharomyces cerevisiae/ethanol 0.22 510ÿ2 1.10 [40]
Bacillus cereus/glucose 0.52 0.15 [40]
Bacillus cereus/fructose 0.30 310ÿ2 [40]
Candida tropicalis/glucose 0.74 1:210ÿ2 [40]
Candida tropicalis/fructose 0.50 110ÿ2 [40]
Enterobacter aerogenes/organic carbon 0.621 2:4310ÿ4 2.5a [41]
Escherichia coli/BC medium 0.31±0.33 110ÿ7 [19]
Escherichia coli/glucose 0.25 510ÿ4 0.9a [42]
mm(hÿ1) L=Xi(mg/ml) Xi Ref.
Colpoda steinii/Escherichia coli 0.23 610ÿ3 0.78 [35]
Tetrahymena pyriformis/Klebsiella aerogenes 0.22 1:1610ÿ4 0.5 [36]
Tetrahymena pyriformis/Klebsiella aerogenes 0.43 1:1710ÿ4 0.54 [37]
Dictyostelium discoedium/Escherichia coli 0.24 1:210ÿ4a [16]
Tetrahymena pyriformis/Aerobacter aerogenes 0.1 6:110ÿ3 0.73 [38]
Colpidium campylum/Alcalegenes faecalis 0.11 110ÿ4 0.5 [17]
Colpoda steinii/Escherichia coli 0.37 310ÿ3a 0.45 [39]
Paramecium primaurelia/Enterobacter aero-genes
0.132 910ÿ2a 0.968 [41]
Tetrahymena pyriformis/Escherichia coli 0.31 110ÿ2a [19]
Tetrahymena pyriformis/Escherichia coli 0.31 210ÿ2a 0.6a [42] a
two prey populations cross, i.e., the values of ax and bx must be such that axÿbx= 1ÿbx>0. The values of ay and by do not seem to play an
im-portant 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 dierent 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.
than unity. Whenc1 the two curves of the operating diagram marked TYX12
and TYX21coincide 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 eect 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 eect 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 three-species 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 biologi-cally 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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