Food chain dynamics in the chemostat

Food chain dynamics in the chemostat

M.P. Boer *, B.W. Kooi, S.A.L.M. Kooijman

Department of Theoretical Biology, Vrije Universiteit Amsterdam, De Boelelaan 1087, 1081 HV

Amsterdam, The Netherlands

Received 10 April 1997; received in revised form 5 January 1998

Abstract

The asymptotic behavior of a tri-trophic food chain model in the chemostat is stud-

ied. The Monod±Herbert growth model is used for all trophic levels. The analysis is car-

ried out numerically, by ®nding both local and global bifurcations of equilibria and of

limit cycles with respect to two chemostat control parameters: the dilution rate of the

chemostat and the concentration of input substrate. It is shown that the bifurcation

structure of the food chain model has much in common with the bifurcation structure

of a one-dimensional map with two turning points. This map is used to explain how at-

tractors are created and destroyed under variation of the bifurcation parameters. It is

shown that low as well as high concentration of input substrate can lead to extinction

of the highest trophic level. Ó 1998 Elsevier Science Inc. All rights reserved.

Keywords: Food chain; Chemostat; Global bifurcation; Boundary crisis; Heteroclinic

tangency; Homoclinic tangency; One-dimensional map; Escape mechanism

1. Introduction

In this paper we study the dynamics of a three species food chain model.These dynamics have been the subject of numerous studies, e.g. [1±11]. Recently,several authors have shown that these models exhibit a rich set of dynamicalbehaviors, such as `tea-cup' limit cycles and chaos, phenomena that seem tobe related with period doublings of limit cycles and global bifurcations [6±9].

* Corresponding author. Tel.: +31-20 444 7129; fax: +31-20 444 7123; e-mail: [email protected].

0025-5564/98/$19.00 Ó 1998 Elsevier Science Inc. All rights reserved.

PII: S 0 0 2 5 - 5 5 6 4 ( 9 8 ) 0 0 0 1 0 - 8

Mathematical Biosciences 150 (1998) 43±62

We will study the tri-trophic food chain in the chemostat. The species atthe bottom of the trophic is called the prey population. The growth rate ofthe prey population depends on the concentration of nutrient substrate inthe chemostat. In most of the literature on the dynamics of tri-trophic ecosys-tems it is assumed that the prey population, grows logistically in the absenceof predators [1±9]; for a comparison between logistic prey and chemostatmodels see [12].

The Monod±Herbert growth model [13,14] is often used in microbiology.We use this model for all trophic levels. In this model, it is assumed that speci®cingestion rate depends hyperbolically on food density. The speci®c growth rateis assumed to be a linear function of the speci®c ingestion rate. We will use theconcentration of input substrate and the dilution rate of the chemostat as bi-furcation parameters, since these parameters may be set by the experimenter.

The aim of this paper is to ®nd out for which values of the bifurcation pa-rameters stable coexistence of all trophic levels is possible. To this end we applybifurcation analysis and numerical simulation. We will interpret the dynamicalbehavior of the food chain model in biological terms.

The paper is organized as follows. In Section 2 we describe the model of thefood chain. In Section 3 we discuss the dynamical behavior of the bi-trophicfood chain, where the highest trophic level is absent. In Section 4 we analysethe local bifurcations of equilibria and limit cycles of the tri-trophic food chain.In Section 5 we explain how a chaotic attractor and its basin of attraction aredestroyed by a global bifurcation. Further we describe how the global bifurca-tions can be continued through the parameter plane. Finally, in Section 6 wesummarize the results of the local and global bifurcation analysis and we givea classi®cation of the stable modes of behavior.

2. The model

The food chain we analyse in this paper consists of substrate, prey (trophiclevel 1), predator (trophic level 2) and top-predator (trophic level 3). The den-sity of substrate in the chemostat will be denoted by X0. We use Xi to denote themass density of trophic level i, i � 1; 2; 3. We will assume that the speci®c in-gestion rate of trophic level i depends hyperbolically on food density Xiÿ1

AiXiÿ1

Ki � Xiÿ1

;

where Ai and Ki are the maximum speci®c ingestion rate and the half-saturationconstant of trophic level i. Monod [14] assumed that food is converted to bio-mass at a ®xed e�ciency. The conversion factor from trophic level iÿ 1 totrophic level i will be denoted by Ci. Herbert [13] accounted for endogenousmetabolism by introducing a constant maintenance rate coe�cient. The main-

44 M.P. Boer et al. / Mathematical Biosciences 150 (1998) 43±62

tenance rate coe�cient of trophic level i will be denoted by Mi. The speci®cgrowth rate of trophic level i is then given by

CiAiXiÿ1

Ki � Xiÿ1

ÿMi:

The dynamics of the tri-trophic food chain in the chemostat is now describedby the following system of ordinary di�erential equations:

dX0

dT� �Xr ÿ X0�Dÿ A1X0

K1 � X0

X1; �1a�dX1

dT� C1

A1X0

K1 � X0

X1 ÿM1X1 ÿ DX1 ÿ A2X1

K2 � X1

X2; �1b�dX2

dT� C2

A2X1

K2 � X1

X2 ÿM2X2 ÿ DX2 ÿ A3X2

K3 � X2

X3; �1c�dX3

dT� C3

A3X2

K3 � X2

X3 ÿM3X3 ÿ DX3; �1d�with T time, D the dilution rate of the chemostat, and Xr the concentration ofthe input substrate.

All parameters have positive values. The parameters D and Xr serve as bifur-cation parameters, since these parameters may be set by the experimenter. Thevalues of the other parameters are ®xed; they are given in Table 1. These pa-rameter values are relevant for a tri-trophic food chain in the chemostat

Table 1

Parameter values of the tri-trophic food chain model, after [12,15]

System (1) System (2)

Parameter Value Parameter Value

Xr ) xr Xr=Xref

D ) d D=Dref

C1 0.4 ) )C2 0.6 ) )C3 0.6 ) )A1 1:250 hÿ1 a1 C1A1=Dref � 5:0

A2 0:333 hÿ1 a2 C2A2=Dref � 2:0

A3 0:250 hÿ1 a3 C3A3=Dref � 1:5K1 8:0 mg dmÿ3 k1 K1=Xref � 0:16

K2 9:0 mg dmÿ3 k2 K2=�C1Xref� � 0:45

K3 10:0 mg dmÿ3 k3 K3=�C1C2Xref� � 0:833

M1 0:0250 hÿ1 m1 M1=Dref � 0:250

M2 0:0100 hÿ1 m2 M2=Dref � 0:100

M3 0:0075 hÿ1 m3 M3=Dref � 0:075

The concentration of substrate in the reservoir, Xr, and the dilution rate D are bifurcation pa-

rameters. Dref � 0:1 hÿ1 and Xref � 50 mg dmÿ3 are reference parameters.

M.P. Boer et al. / Mathematical Biosciences 150 (1998) 43±62 45

consisting of bacteria living on glucose, ciliates such as Tetrahymena sp. or Par-amecium, and carnivorous ciliates such as Didinium nasutum [11,15].

System (1) can be simpli®ed by choosing the dimensionless variables

x0 � X0

Xref

; x1 � X1

C1Xref

; x2 � X2

C1C2Xref

; x3 � X3

C1C2C3Xref

;

t � TDref ;

where Dref � 0:1 hÿ1 and Xref � 50 mg dmÿ3 are reference parameters. Then weobtain

dx0

dt� �xr ÿ x0�d ÿ a1x0

k1 � x0

x1; �2a�dx1

dt� a1x0

k1 � x0

x1 ÿ m1x1 ÿ dx1 ÿ a2x1

k2 � x1

x2; �2b�dx2

dt� a2x1

k2 � x1


k3 � x2

x3; �2c�dx3

dt� a3x2

k3 � x2

x3 ÿ m3x3 ÿ dx3; �2d�where the dimensionless parameters can be de®ned in terms of the original pa-rameters as shown in Table 1. The state space of system (2) is the non-negativecone

R4� � fx � �x0; x1; x2; x3� 2 R4: x0 P 0; x1 P 0; x2 P 0; x3 P 0g:

Furthermore, it can be shown that all solutions of system (2) with x�0� 2 R4�

eventually lie in a set

X � fx � �x0; x1; x2; x3� 2 R4�: x0 � x1 � x2 � x36 xrg:

So, the asymptotic behavior of system (2) is bounded by xr, the concentrationof substrate in the reservoir.

3. Dynamics in absence of the top-predator

The dynamics in absence of the top-predator is described by Kooi and Ko-oijman [16] and Kooijman [17]. This bi-trophic food chain has at most threeequilibria lying in R4

�. First, there is always a trivial solution �xr; 0; 0; 0�. Sec-ond, the equilibrium point

�xH0 ; x

H1 ; 0; 0� �

�d � m1�k1

a1 ÿ �d � m1� ;d�xra1 ÿ �d � m1��xr � k1��d � m1��a1 ÿ �d � m1�� ; 0; 0

� �lies in R4

� if and only if

a1xr

k1 � xrÿ m1 > d : �3�


Finally there is an equilibrium point E0 � ��x0;�x1;�x2; 0� which lies in R4� if and

only if

a2xH1

k2 � xH1

ÿ m2 > d : �4�Stable coexistence of prey and predator population obtains if condition (4)

is ful®lled. In region E of Fig. 1 the equilibrium E0 is the unique attractor of thebi-trophic food chain. Crossing the supercritical Hopf bifurcation curve H0

from left to right the equilibrium E0 becomes unstable and the stable limit cycle

L0 � �~x0�t�;~x1�t�;~x2�t�; 0�appears, whose period will be denoted as T0. Kooi and Kooijman [16] foundnumerically that this limit cycle L0 is the unique attractor of the bi-trophic foodchain in region L of Fig. 1.

Fig. 1 shows that increasing the concentration of input substrate xr cancause destabilization of the otherwise stable equilibrium E0. This phenomenonwas ®rst described in Rosenzweigs's famous paper [18] on `the paradox of en-richment'. For our model it is possible to give an explanation of this phenom-enon in biological terms. Let us assume for the moment that loss rates throughendogenous metabolism can be neglected (mi � 0 for i � 1; 2; 3). Then it can beshown that the equilibrium E0 is unstable if and only if

g1��x0� ÿ d > g01��x0��x1 � g02��x1��x2; �5�where

gi�x� � aixki � x

and g0i�x� �d

dxgi�x� � aiki

�ki � x�2 :

Fig. 1. Dynamics in absence of the top-predator. In region E there exists a globally asymptotic sta-

ble equilibrium E0. In region L the unique attractor is the limit cycle L0. Regions E and L are sep-

arated by the supercritical Hopf bifurcation curve H0.


Condition (5) can be interpreted as follows. The prey population size is kept incheck by both competition for resources and predatory pressure, which corres-pond, respectively, to the ®rst and second terms at the right-hand side of con-dition (5). If these control mechanisms are absent, then the net speci®c growthrate of the prey population is given by the left-hand side of condition (5). In-stability of the equilibrium E0 therefore obtains only if the prey population canovercome the control imposed by both the competition for resources and thepredatory pressure. De Roos et al. [19] called this destabilizing mechanismthe prey escape mechanism.

We will now describe how the terms in condition (5) are a�ected by an in-crease of xr. The term which corresponds to the competition for resources isa monotonically decreasing function of xr:

ooxr�g01��x0��x1� < 0:

Further, both the left-hand side and the second term on the right-hand side ofcondition (5) are monotonically increasing functions of xr:

ooxr�g1��x0� ÿ d� > 0 and

ooxr�g02��x1��x2� > 0:

Finally, it can be shown that

ooxr�g1��x0� ÿ d� > o

oxr�g01��x0��x1 � g02��x1��x2�;

which implies that an unstable equilibrium E0 cannot be stabilized by increas-ing the concentration of input substrate xr.

If the loss rate through endogenous metabolism cannot be neglected we arenot able to derive a condition for stability of E0 that can be translated directlyinto biological terms. However, numerical simulations suggest that the prey es-cape mechanism is still the driving force behind the oscillatory behavior of thebi-trophic food chain for high values of xr. Indeed, if there is an abundance ofresources, the prey population can grow very quickly from low to high densi-ties. This increase of prey density is followed by an increase of the predatordensity, which ultimately leads to a sharp decline in the number of prey ani-mals, by both exhausting the resources and high predator densities. This de-crease in prey densities leads to a decrease in predator densities and aftersome time the prey population again escapes the control imposed by the pre-dator population.

4. Local bifurcation analysis

In this section we describe the local bifurcations of equilibria and of limitcycles of system (2). Some of the bifurcation curves have been computed by


means of LOCBIF [20]. The other local bifurcation curves as well as the globalbifurcation curves (see Section 5) have been computed by means of a predic-tor±corrector continuation method with step-size control [21±23]. This contin-uation method was implemented in the C++ programming language [24].

Linearization of system (2) around the equilibrium E0 shows that the condi-tion

a3�x2

k3 � �x2

ÿ m3 � d �6�

de®nes the transcritical bifurcation curve of equilibria, TCe. Eq. (6) states thata transcritical bifurcation of E0 occurs if the speci®c growth rate of the top-pre-dator in E0 is equal to the dilution rate d. If the speci®c growth rate of the top-predator is higher than the dilution rate, the top-predator can invade, other-wise invasion is not possible. To the left of the Hopf bifurcation curve H0,the x3 � 0 hyperplane is attracting above the curve TCe and repelling belowTCe (Fig. 2).

The curve TCe intersects the Hopf bifurcation curve H0 at the codimensiontwo point M . Linearization of system (2) around the limit cycle L0 reveals thatthe condition

1

T0

ZT0

0

a3~x2�s�k3 � ~x2�s� ÿ m3 ds � d �7�

de®nes a transcritical bifurcation curve of limit cycles, TCc, that originates atthe point M . Eq. (7) states that a transcritical bifurcation of L0 occurs if themean speci®c growth rate of the top-predator along L0 is equal to the dilutionrate d. If the mean speci®c growth rate of the top-predator along L0 is higherthan the dilution rate, the top-predator can invade, otherwise invasion is notpossible. To the right of the Hopf bifurcation curve H0, the x3 � 0 hyperplaneis attracting above the curve TCc and repelling below TCc (Fig. 2).

A subcritical Hopf bifurcation curve H� starts at the point M . For system(2), there are no other bifurcation curves through point M . In models with alogistic prey there is also a tangent bifurcation curve for equilibria that entersthe positive cone at point M [7,8].

Continuation of the bifurcation curves which emanate from point M (thecurves TCe, TCc and H�) shows that other codimension-two points exist. Atcodimension-two point K, lying on TCe, a tangent bifurcation curve Te entersthe positive cone. A second positive equilibrium exists in the region betweenthe curves TCe and Te. The Hopf bifurcation curve H consists of a subcriticaland a supercritical part, denoted by H� and Hÿ, respectively. These two partsare connected and thus a Bautin bifurcation point lies on H , where the ®rstLyapunov coe�cient vanishes [21]. This Bautin bifurcation point lies far be-yond the parameter range shown in Fig. 2. We were not able to calculate the


exact position of this point. Continuation of the curve TCc shows that thiscurve has a codimension-two singularity at point L. At this point L a tangentbifurcation curve of limit cycles Tc starts. Following the curve Tc another co-dimension-two point N can be found, which is a cusp where three limit cyclescollide simultaneously.

The dynamical behavior of system (2) is summarized in Fig. 3, where sevenregions in the parameter space with di�erent asymptotic behavior are shown.We now give a short description of the dynamical behavior in these regions.This description is based upon the analysis of the local bifurcations as wellas numerical evidence obtained by simulations of system (2).

In region 1 there is a unique attractor in the hyperplane x3 � 0. This attrac-tor is the equilibrium E0 (to the left of H0) or the limit cycle L0 (to the right of

Fig. 3. Two parameter bifurcation diagram of system (2). For an explanation of the dynamical be-

havior within a region see the text. The one-parameter bifurcation diagram for xr � 4:7 is shown in

Fig. 5.

Fig. 2. Bifurcation curves of equilibria and of limit cycles of system (2). For a description see the

text. The one-parameter bifurcation diagram for xr � 4:7 is shown in Fig. 5.


H0). For some subregions of region 1 there are either one or two strictly pos-itive equilibria. These equilibria are unstable. In region 1 stable coexistence ofall trophic levels is impossible because the dilution rate d is relatively high andthe speci®c growth rate of the top-predators is low, inasmuch as the number ofpredators is low.

Crossing TCe, the boundary between region 1 and 2, an equilibrium Ep en-ters the positive cone. In region 2 the equilibrium Ep is the unique attractor ofsystem (2). In this region we observe that the introduction of a top-predatorcan have a stabilizing e�ect: to the right of the Hopf bifurcation curve H0

the bi-trophic food chain has a non-point attractor, while the tri-trophic foodchain does have a point attractor.

As noted in Section 3, the oscillations of the bi-trophic food chain to theright of H0 are driven by the prey escape mechanism: the prey populationcan reach high densities, since the control imposed by the predatory pressureis not strong enough. On the other hand, high prey densities will ultimatelylead to a sharp decrease in the numbers of prey, induced by the growth ofthe predator population. The tri-trophic food chain attains the stable equilib-rium Ep in region 2, since the predator population is itself controlled by preda-tory pressure. This means that high prey densities will not lead to high predatordensities. As a result, there is no longer a strong force which drives back theprey population from high to low densities. Stabilization by the addition ofa third trophic level can also be observed in the models with logistic prey,see for instance [2].

In region 3 there are two attractors, the limit cycle L0 and the strictly pos-itive equilibrium Ep. There is a saddle cycle Lp which supports a separatrixdividing the two basins of attraction. (The term separatrix will be de®nedmore precisely in Section 5.) Simulations suggest that the initial size of thetop-predator populations has to be big enough to control the predator pop-ulation (Fig. 4).

Region 4 is reached from region 3 by moving through the supercritical bifur-cation curve Hÿ. The equilibrium Ep has become a repellor. This seems to berelated to some kind of predator escape mechanism: the predator can escape thepredatory pressure imposed by the top-predator population. The limit cycle L0

is stable. For some subregions within this region stable coexistence of all troph-ic levels is possible.

In region 5 the limit cycle L0 is unstable and there exists a stable limit cyclejust above the x3 � 0 hyperplane. For some subregions within this region thereis another strictly positive non-point attractor. The equilibrium Ep is unstableand the saddle cycle Lp exists.

Moving from region 5 to 6, through the curve Tc, we see that the saddle cy-cle Lp collides with another limit cycle and disappears. There is stable coexis-tence of all trophic levels in region 6. This stable coexistence can be cyclic orchaotic.


Moving from region 6 to 7, through the curve TCc, the limit cycle L0 be-comes stable. It is the unique attractor of system (2). So, in this region, at rel-atively high values of xr, stable coexistence of all trophic levels is not possible.In biological terms, we might say that the control mechanisms of predator toprey and of top-predator to predator are both not strong enough. Especiallythe prey population can break out very fast, since xr is high. This will leadto strong oscillations between prey and predator, which is unfavorable forthe top-predator population. The number of top-predators will furtherdecrease, leading to even stronger prey±predator oscillations. Ultimately, thetop-predator population will go extinct.

5. Global bifurcations

In this section we discuss the global bifurcations of system (2). Fig. 5 showsthat, for xr � 4:7, there are two global bifurcations, at d � 0:658 andd � 0:798. At these bifurcation points a chaotic attractor and its basin of at-traction are destroyed by the collision between the chaotic attractor and thesaddle cycle Lp. Grebogi and Ott [25] called this a boundary crisis.

5.1. Next-minimum map

By the construction of a next-minimum map of x3 we can partly explain howthe chaotic attractor existing for xr � 4:7 and d � 0:7980 is destroyed if the di-lution rate is decreased to d � 0:7979 (Fig. 6). The local minima in the graphsof Fig. 6 seem to lie on a single curve. So, the chaotic attractor is almost one-

Fig. 4. Dynamical behavior of system (2) for xr � 4:7 and d � 0:9. The projections of two orbits on

the �x2; x3�-plane are shown. One orbit lies in the basin of attraction of the limit cycle L0 in the

hyperplane x3 � 0, the other one lies in the basin of attraction of the interior equilibrium Ep.

The interior saddle cycle Lp supports a separatrix dividing the two basins of attraction.


dimensional. The attractor has two turning points, i.e. the map has a localmaximum and a local minimum. This suggests that the bifurcation structureof system (2) can be partially explained by considering a one-dimensional sys-tem

x 7! fa;b�x�; x 2 R; a 2 R; b 2 R; �8�

with two turning points. In the following we will often write f instead of fa;b, tosimplify notation.

We will assume that f has a negative Schwarzian derivative [26±28]. It canbe shown that two maps, both with a negative Schwarzian derivative and the

Fig. 6. Next-minimum maps of system (2) for xr � 4:7. In the left ®gure, with d � 0:7980, there is a

chaotic attractor. In the right ®gure, with d � 0:7979, the chaotic attractor is destroyed by the col-

lision between the chaotic attractor and the saddle limit cycle Lp. The last part of the trajectory to

extinction is indicated by the symbol �.

Fig. 5. One-parameter bifurcation of system (2) for xr � 4:7. Plotted as points are the local minima

or constant values of x3 of the attractors versus dilution rate d. The dashed curves indicate unstable

limit cycles or unstable equilibria. There are two global bifurcations, indicated with G. The num-

bers in the vertical strips refer to the regions in Fig. 3.


same number of turning points, will have basically the same bifurcation struc-ture [26,29,30]. Therefore we modeled the chaotic attractor of system (2) asshown in Fig. 6 simply as a cubic map. We could have used the normal formfa;b�x� � �x3 � ax� b, but instead we used the map

fa;b�x� � 16bx3 ÿ 24bx2 � 9bx� aÿ b;

which turns out to be more suitable to present the relationship between the bi-furcation structures of systems (2) and (8). The map f has two turning points,c1 � 1

4and c2 � 3

4, where f �c1� � a is a local maximum and f �c2� � aÿ b is a

local minimum (Fig. 7).We will analyse the bifurcation structure of system (8) in the �a; b�-plane,

with 0 < a < 1 and 0 < b < 1 (Fig. 8). System (8) always has a repelling ®xedpoint p3 > c2. To the left of the tangent bifurcation curve T there are no other®xed points. To the right of curve T there are two other ®xed points, p1 and p2,with p1 < p2 < p3 (Fig. 7). Point p1 is a hyperbolic repellor. The local stabilityof point p2 is not considered here, since this is unimportant for our present pur-poses.

There are two global bifurcation curves associated with the repellors p1 andp3. First we will need some de®nitions. The map f n is inductively de®ned byf 0�x� � x and f n�x� � f �f nÿ1�x�� for n P 1. A point q 2 �p1; c1� is called homo-clinic to p1 if there exists n > 0 such that f n�q� � p1 [26]. A point q 2 �c2; p3� iscalled heteroclinic from p3 to p1 if there exists n > 0 such that f n�q� � p1 [26]. Aheteroclinic bifurcation curve is de®ned by

G1 � �a; b�: fa;b�c2� � p1

� � �a; b�: b � a ; aP1

9

� �:

Fig. 7. The map fa;b. The points p1, p2, and p3 are ®xed points of system (8). The point c1 is a max-

imum, with f �c1� � a. The point c2 is a minimum, with f �c2� � aÿ b.


Above the curve G1 (Fig. 8) there are in®nitely many heteroclinic points, be-low the curve there are no heteroclinic points. Point A, located at �1

9; 1

9�, is the

root of the bifurcation curves T and G1. A homoclinic bifurcation curve isgiven by

G2 � f�a; b�: fa;b�min�fa;b�c1�; c2�� p1g

� �a; b�: b � a if aP3

4; f 2

a;b�c1� � p1 if a <3

4

� �:

Fig. 8. Two-parameter bifurcation diagram of system (8). The curve T is a tangent bifurcation

curve of the ®xed points p1 and p2. There are two global bifurcation curves, G1 and G2; G1 is a he-

teroclinic bifurcation curve and G2 is a homoclinic bifurcation curve; G1 � G2 is both a homoclinic

and a heteroclinic bifurcation curve. In the dotted regions, the repellor p1 separates basins of attrac-

tion. For a further description see the text.


Above the curve G2 (Fig. 8) there are in®nitely many homoclinic points, whilebelow the curve there are no homoclinic points. Note that

G1 \ G2 � �a; b�: b � a ; aP3

4

� �:

Thus, crossing G1 \ G2 from above to below, both heteroclinic and homoclinicpoints disappear.

The bifurcation curves T , G1, and G2 divide the �a; b�-plane into four regions(Fig. 8). In region a all orbits starting below repellor p3 go to ÿ1. In region b

the interval �p1; p3� is invariant for system (8). The function f is called bimodalfor this region, since f has two turning points in �p1; p3�. A complex cascade ofintersecting supercritical period doubling and tangent bifurcation curves can befound [29]. Furthermore it can be shown, using Singer's theorem [26±29], thatsystem (2) has at most two attracting periodic orbits in this region. In region cthere is a closed invariant interval I � �p1; r� for system (8), where p2 < r < c2

and f �r� � p1. On this interval I the function f is called unimodal, since f hasone turning point in I . In this region a cascade of supercritical period doublingsthat leads to chaos can be found. System (8) has at most one attracting periodorbit in this region [26]. In region d the map f has no invariant subintervals.

Summarizing the dynamical behavior in Fig. 8, we can state that in regions b

and c the repellor p1 is a separating point: the set

fx: x � f n�q�; n 2 N; q < p1g \ fx: x � f n�q�; n 2 N; p1 < q6 c1gis empty. In region d the repellor p1 is not a separating point, while in region athe ®xed point p1 does not exist.

Returning to system (2), we can now give a partial explanation based onone-dimensional evidence of what happens at the global bifurcation pointxr � 4:7 and d � 0:798. Just before the chaotic attractor is destroyed, all localminimum values of x3 of the chaotic attractor lie between the minimum valueof x3 of Lp and the x3 value of Ep (Figs. 5 and 6). In terms of system (8), thechaotic attractor is destroyed by moving through the boundary between the re-gions b and d (Fig. 8).

An important di�erence between systems (2) and (8) is that (2) is invertible,while (8) is not invertible. Therefore, system (2) could be modeled more accu-rately by the system

x

y

� �7! fa;b�x� ÿ �y

x

� �;

x

y

� �2 R2; a; b; � 2 R; �9�

which is invertible if � 6� 0. System (9) reduces to system (8) if � � 0. If0 < �� 1, the bifurcation structures of systems (8) and (9) are basically thesame [31]. However, there is one important di�erence. The distance betweenthe curves G1 and G2 is zero for system (8) if aP 3

4. For system (9), the distance

between G1 and G2 is small and positive for aP 34

and � positive and small.Thus, the heteroclinic and homoclinic bifurcation curves do not coincide.


5.2. Global bifurcation curves

The global bifurcation curves of system (2) can be approximated by the fol-lowing procedure. First we de®ne the Poincar�e section

R � x � �x0; x1; x2; x3� 2 R4�: x2 � �d � m3�k3

a3 ÿ �d � m3� ; G�x�P 0

� �;

where G : R4� ! R is de®ned by

G�x� � a2x1

k2 � x1


k3 � x2

x3;

which is equal to the right-hand side of Eq. (2c). The local minimum values ofx3 of trajectories of system (2) lie on R. Furthermore, the strictly positive equi-libria lie on R and R intersects the strictly positive non-point attractors at anon-zero angle.

Let us introduce some notation for the dynamics on R. We use the coordi-nates y � �x0; x1; x3� on R. The location of the equilibrium Ep on R will be de-noted by �y1. The point �y2 represents the intersection point of the saddle cycle Lp

with R. Let P : R! R be the Poincar�e or next-return map, with P��yi� � �yi fori � 1; 2. The Jacobian matrix of P evaluated at point �yi will be denoted by Ji fori � 1; 2. The map P n is inductively de®ned by P 0�y� � y and P n�y� � P �P nÿ1�y��for n P 1. The inverse of the map P n will be denoted by Pÿn. The stable andunstable manifold of a ®xed point �y of the map P are de®ned by

W s��y� � fy 2 R: P n�y� ! �y; n! �1gand

W u��y� � fy 2 R: P n�y� ! �y; n! ÿ1g;respectively [21]. In a small neighbourhood Ui � R of �yi, we use linear approx-imations of W s��yi� and W u��yi�, which will be denoted by T s��yi� and T u��yi�, res-pectively (i � 1; 2).

The analysis in Section 5.1 suggests that at xr � 4:7 and d � 0:798 (Fig. 5) aheteroclinic tangency crisis [32] can be found: the manifold W u��y1� becomestangent with W s��y2�. In addition, also a homoclinic tangency crisis [32] is foundat xr � 4:7 and d � 0:798: the manifold W u��y2� becomes tangent with W s��y2�.Another homoclinic tangency crisis of �y2 occurs at xr � 4:7 and d � 0:658(Fig. 5). In regions 4 and 5 of Fig. 3 we found numerically that the unstablemanifolds W u��y1� and W u��y2� are one-dimensional. The stable manifoldW s��y2� is two-dimensional.

The linear approximation of the unstable manifold of the ®xed point �yi isde®ned by

T u��yi� � fy 2 Ui: y � �yi � nui; n 2 Rg;


where ui is the normalized unstable eigenvector of Ji (i � 1; 2) [21,23,27]. Wede®ne the interval

Ji � fn 2 R: �yi � nui 2 Uigand introduce the function Hi : Ji ! R de®ned by

Hi�n� � P n��yi � nui�for i � 1; 2 and some n 2 N.

Using the Fredholm Alternative Theorem, it can be shown that the linearapproximation of the stable manifold of �y2 is given by

T s��y2� � fy 2 U2: hy ÿ �y2; vi � 0g;where v is the normalized unstable eigenvector of the transpose of J2 [21,23,33±35]. The function h�; �i computes the standard inner product in R3. It followsthat the heteroclinic bifurcation curve G1 and the homoclinic bifurcation curveG2 of system (2) can be approximated by

Gi � f�d; xr�: hHi�� ÿ �y2; vi � 0; hH 0i ��; vi � 0; � 2 Ji; Hi�� 2 U2g;where H 0i is the derivative of Hi (i � 1; 2). In Fig. 9 a portion of the global bi-furcation diagram is shown. Fig. 9 has much in common with the bifurcationdiagram of system (8) (Fig. 8). The heteroclinic bifurcation curve G1 starts at apoint A on the tangent bifurcation curve Tc. The homoclinic bifurcation curveG2 lies entirely above G1. A part of G2 lies very close to G1. In Fig. 9 these twocurves are indistinguishable for xrK4:3 and dJ0:6.

Fig. 9. Detail of the two-parameter bifurcation diagram of system (2). In the dotted regions the sta-

ble manifold W s�Lp� separates the basins of two attractors. The dotted region is bounded by (i) the

local bifurcation curve Tc; along this curve the saddle cycle Lp disappear by the collision with a sta-

ble limit cycle. (ii) The homoclinic bifurcation curve G2: along this curve a chaotic attractor is de-

stroyed. For a further description see the text.


In Section 4 we mentioned that in region 3 of Fig. 3 the saddle cycle Lp

supports a separatrix which divides the basins of two attractors, theequilibrium Ep and the limit cycle L0. Let us de®ne this separatrix more pre-cisely, using the notation introduced in this section. For regions 3, 4, and 5 ofFig. 3, a separatrix is de®ned by the stable manifold of the saddle cycle Lp,denoted by W s�Lp�. This manifold is three-dimensional, since the two-dimen-sional manifold W s��y2� is formed by the intersection of W s�Lp� with R, see forinstance [27].

In the dotted region of Fig. 9, the manifold W s�Lp� actually separates twobasins of attraction. This dotted region is enclosed by the homoclinic bifurca-tion curve G2 and the tangent bifurcation Tc. Crossing the global bifurcationcurve G2, the separatrix becomes virtual, which means that the separatrix doesno longer separate two basins of attraction [36]. Moving through the tangentbifurcation curve Tc, the saddle cycle Lp and its stable manifold W s�Lp� disap-pear by the collision with a stable limit cycle.

6. Summary and conclusions

Gathering together the information obtained from the local bifurcationanalysis in Section 4 and the global bifurcation analysis in Section 5, we cannow give a classi®cation of the stable modes of behavior of system (2). We dis-tinguish three types of asymptotic behavior (Fig. 10): stable coexistence for allpositive initial conditions, bistability, and extinction of the top-predator.

Fig. 10. The asymptotic behavior of system (2). In the gray regions there exists at least one strictly

positive attractor. In the dark gray region the hyperplane x3 � 0 is repelling, in the other regions the

hyperplane x3 � 0 is attracting. In the light gray regions the asymptotic behavior depends on initial

conditions. The one-parameter bifurcation diagram for xr � 4:7 is shown in Fig. 5.


If there is stable coexistence of all trophic levels for all positive initial condi-tions, system (2) is persistent [37]. This is the case below the transcritical bifur-cation curves TCe (to the left of H0) and TCc (to the right of H0) in Fig. 2. Thestable coexistence is stationary in region 2 and cyclic or chaotic in regions 5 and6 (Fig. 3). In the region enclosed by the curves H0, TCc, and Hÿ (Fig. 2) the ad-dition of a third trophic level has a stabilizing e�ect: the tri-trophic food chainhas a point attractor, while the bi-trophic food chain has a non-point attractor.

If the asymptotic behavior depends on initial conditions, there are at leasttwo attractors, a strictly positive attractor and an attractor in the x3 � 0 hyper-plane. The two domains of attraction are separated by W s�Lp�, the stable three-dimensional manifold of the saddle limit cycle Lp. The saddle limit cycle Lp andits stable manifold W s�Lp� exists in the regions 3, 4, and 5 (Fig. 3), enclosed bythe curves TCc, Tc, and H� (Fig. 2). In region 3 the manifold W s�Lp� separatesthe attraction domain of the strictly positive stable equilibrium Ep from the at-traction domain of the limit cycle L0. In a subregion of region 4, bounded bythe homoclinic bifurcation curve G2, the manifold W s�Lp� separates two non-point attractors.

The top-predator population will go extinct for almost all initial conditionsif the x3 � 0 hyperplane is attracting and there are no strictly positive attrac-tors. There are three regions in Fig. 10 for which the attractor of the bi-trophicfood chain (Fig. 1) is the unique attractor of the tri-trophic food chain:1. Region 1 of Fig. 3. This region is bounded by the transcritical bifurcation

curve TCe and the subcritical Hopf bifurcation curve H� (Fig. 2). In this re-gion stable coexistence of all trophic levels is impossible, since the value of dis relatively high and the value of xr is relatively low.

2. Region 7 of Fig. 3, which is enclosed by the transcritical bifurcation curveTCc and the tangent bifurcation curve Tc (Fig. 2). In this region, at relative-ly high values of xr, stable coexistence is not possible since the control mech-anism of predator to prey and top-predator to predator are both not strongenough.

3. The region enclosed by the homoclinic bifurcation curve G2 (Figs. 9 and 10).Entering this region by crossing the curve G2, a chaotic attractor is de-stroyed by the collision with the saddle limit cycle Lp (Fig. 6).As noted in Section 3, increasing the concentration of input substrate xr for

a bi-trophic food chain can cause destabilization of the otherwise stable equi-librium E0. For a tri-trophic food chain the increase of xr can even lead to ex-tinction of the highest trophic level, since an abundance of resources for thelowest trophic level leads to strong oscillations between prey and predator,which is unfavorable for the top-predator population. Therefore we supportRosenzweig's warning, although for di�erent reasons, that ``Man must be care-ful in attempting to enrich an ecosystem in order to increase its food yield.There is a real chance that such activity may result in a decimation of the foodspecies that are wanted in greater abundance''.


Acknowledgements

The authors thank Yuri Kuznetsov for valuable discussions and Hugo vanden Berg, Fleur Kelpin, and Cor Zonneveld for useful comments on the manu-script. The research of the ®rst author was supported by the Netherlands Or-ganization for Scienti®c Research (NWO).

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Food chain dynamics in the chemostat