Crisis-limited chaotic dynamics in ecological systems

Crisis-limited chaotic dynamics in ecological systems

Ranjit Kumar Upadhyay a,*, Vikas Rai b

a Department of Applied Mathematics, Indian School of Mines, Dhanbad 826 004, Indiab Structural Biology Unit, National Institute of Immunology, Aruna Asaf Ali Marg, New Delhi 110 067, India

Abstract

We review our recent e�orts to understand why chaotic dynamics is rarely observed in natural populations. The study of two-model

ecosystems considered in this paper suggests that chaos exists in narrow parameter ranges. This dynamical behaviour is caused by the

crisis-induced sudden death of chaotic attractors. The computed bifurcation diagrams and basin boundary calculations reinforce our

earlier conclusion [Chaos, Solitons & Fractals 8 (12) (1997) 1933; Int J Bifurc Chaos 8 (6) (1998) 1325] that the reason why chaos is

rarely observed in natural populations is hidden within the mathematical structure of the ecological interactions and not with the

problem associated with the data (insu�cient length, precision, noise, etc.) and its analysis. We also argue that crisis-limited chaotic

dynamics can be commonly found in model terrestrial ecosystems. Ó 2000 Elsevier Science Ltd. All rights reserved.

1. Introduction

Since the work of May [1,2], the possible existence of chaos in biological populations has been studied bymany theoretical ecologists. May's [3] research mainly concentrated on the existence of chaos in systemsgoverned by one-dimensional di�erence equations. Number of authors [4±8] have studied models of thepopulation dynamics of two species with non-overlapping generations and found that the dynamics can bedescribed either by long period limit cycles or chaos. May [2] reviewed the literature and concluded that thestudy of non-linear systems are indispensable as far as understanding about nature is concerned.

Later on, Gilpin [4] modelled a three-species (two-prey, one-predator) system by a set of three ordinarynon-linear coupled di�erential equations. He found that the model is capable of exhibiting strange at-tractors in which spiralling trajectories stretch and fold repeatedly to generate the geometry of this at-tractor. Scha�er and Kot [7,8] studied the Gilpin's model further and analysed its dynamical behaviourusing the numerical tools from non-linear dynamics. The studies of Scha�er and his collaborators led to theconclusion that chaotic dynamics could be expected to exist in nature.

Interaction networks in natural ecosystems consist of simple units known as food-chains. Hastings andPowell [5] considered a model consisting of a resource, a consumer and a predator and showed that chaosexists in a large parameter range of the parameter yc, which measures the ingestion rate per unit metabolicrate of consumer species. They had also observed that there exists an incubation period for chaos, i.e., oneis supposed to wait for some time so that the trajectories start evolving on a strange chaotic attractor.McCann and Yodzis [9] disagreed with these results of Hastings and Powell on the ground that the pa-rameter ranges reported were on an extreme and, therefore, not realistic. They had obtained biologicallyrealistic conditions for chaos to exist in this model. Speci®cally, they sought for chaotic solutions in whichall the three densities are non-zero. Such a situation was termed as persistent chaos. They had also made the

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Chaos, Solitons and Fractals 12 (2001) 205±218

* Corresponding author.

E-mail address: [email protected] (R.K. Upadhyay).

0960-0779/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 1 4 1 - 7

following observations: (i) chaotic dynamics which exist in biologically plausible regions in the parameterspace are not very common, and (ii) productive environment is a pre-requisite for a system to support adynamical behaviour such as chaos. Ruxton [10] considered the e�ect of inclusion of a population ¯oor(minimum viable population) on the occurrence of chaos in the model considered by McCann and Yodzis.He found that the inclusion of this key assumption, which renders the Hastings and Powell model a bitcloser to reality, suppresses chaos. Ruxton further examined the possibility whether suppression of chaoscan be counteracted by increasing the primary productivity of the food-chain. He found that any reductionin chaos which is caused by immigration or refuge can be compensated by a su�cient increase in the re-source renewal rate. He concluded with the observation that highly enriched systems are the most pro-spective candidates for chaotic dynamics to exist. These observations make it tempting to study theexistence of chaos in natural populations.

In recent years, many attempts have been made [2,11] to observe chaotic dynamics in natural popula-tions. To the best of our knowledge, no unambiguous evidence of chaos in these time-series has emerged.May [1] argued that the reason why ®eld ecologists have not been able to get any reliable proof of existenceof chaos is that they have pre-occupied notions and that they are observing what they want. It was arguedby many authors that poor data quality (short and noisy character of the time-series) makes the techniquesfrom non-linear dynamics an unsuitable tool for the analysis of ecological data, and thus, leads to failure ofsuch attempts. One common hindrance (in the study of natural populations) to understand the underlyingdynamical process has been the non-availability of data of suitable length and precision. Another di�cultyis that there is no method which can ®x these parameters a priori. On the other hand, the existence of chaosin almost all the physical systems [12,15,16] motivates one to critically study the same in natural popula-tions.

In this contribution, we review our recent e�orts [13,14] to understand the failure of attempts to capturechaos in the real world, and propose a possible alternative line of reasoning to understand the failure ofattempts to observe chaos in ecological systems. The study suggests that the cause of failure might not bepoor data quality, as pointed by earlier authors, but an ecological reality. Two model food-chains havebeen considered to examine whether there is a biological basis for the crisis. Pseudo-prey method earlierintroduced in Ref. [13] is now presented in a more expository way for the bene®t of the reader. This helps usto select the biologically realistic parametric values to perform simulation experiments. An attempt tounravel the mechanism responsible for the existence of chaotic dynamics in narrow ranges of the parametricvalues in both the model systems is presented in Section 4. The last section discusses various implications ofthis study.

2. Model systems

Consider a situation where a prey population X is predated by individuals of population Y. This pop-ulation, in turn, serves as a favourite food for individuals of population Z. The rate equations for the threecomponents of the chain populations can be written as follows:

dXdt� a1X ÿ b1X 2 ÿ wYX

�X � D� ; �1a�

dYdt� ÿa2Y � w1YX

�X � D1� ÿw2YZ�Y � D2� ; �1b�

dZdt� cZ2 ÿ w3Z2

�Y � D3� ; �1c�

where a1; a2; b1;w;w1;w2;w3;D;D1;D2;D3; and c are positive constants, a1 the intrinsic growth rate of theprey population X, a2 the intrinsic death rate of the specialist predator Y in the absence of the only food X,c measures the rate of self-reproduction and the square term signi®es the fact that mating frequency isdirectly proportional to the number of males as well as females, D3 normalises the residual reduction in the

206 R.K. Upadhyay, V. Rai / Chaos, Solitons and Fractals 12 (2001) 205±218

predator population because of severe scarcity of the favourite food, b1 measures the e�ect of intra-speci®ccompetition, and D measures the e�ect of the prey in evading a predator's attack. It depends on theprotection a�orded by the environment to the prey. The larger the value of D, more elusive is the hostagainst any attack by parasites.

Eqs. (1a)±(1c) represent the Model I. Governing equations for populations X and Y are modelled by theVolterra scheme and for Z by the Leslie±Gower scheme. The last predator of the food-chain, Z, is ageneralist one, i.e, it has other options for food. We have also assumed that this predator belongs to asexually reproducing species. The second term on the right-hand side of Eq. (1c) is the Leslie±Gower term[17]. It measures the loss in the predator population due to rarity (per capita, Z=Y ) of its favourite food. TheLeslie±Gower formulation is based on the assumption that reduction in a predator population has a re-ciprocal relationship with per capita availability of its preferred food. In the case of severe scarcity, Z canswitch over to other populations but its growth will be limited by the fact that its most favorite food (Y) isnot available in abundance. This situation can be taken care by adding a positive constant D3 to the de-nominator.

Consider now the case when the predator Z is a vertebrate [26]. Then, Eq. (1b) is modi®ed as

dYdt� ÿa2Y � w1YX

�X � D1� ÿw2Y 2Z�Y 2 � D2� : �2�

Eqs. (1a), (2) and (1c) represent the Model II. The coe�cient w=�X � D� of the third term on the right-handside of Eq. (1a) is obtained by considering the probable e�ect of the density of the prey's (or host's)population on predators (parasites) attack rate. If this coe�cient is multiplied by X (the prey population atany instant of time), it gives the attack rate on the prey per predator. Denote wX=�X � D� by f �X �. It isreasonable to assume that this attack rate would be a function of the parasite's ability to attack the hosts.When X !1; f �X � ! w, which is the maximum that it can reach. The third term �w2Y 2�=�Y 2 � D2� on theright-hand side of Eq. (2) represents the per capita functional response of the vertebrate predator Z and was®rst introduced by Takahashi (see [20]). The ecological role of per capita functional response was welldescribed by May [20]. The interaction terms appearing in the rate equations (Eqs. (1a) and (1b); andEqs. (1a) and (2)) restore to some extent the symmetry which characterises the Lotka±Volterra model. Thebehaviour as X !1, leading to exponential growth of the parasite, is common to all the models. But thispossibility is always excluded due to the presence of the self-interaction terms in the rate equation for thehost. The typical situations represented by these models are presented in Figs. 1 and 2.

3. The methodology

We present a method for the dynamical study of three-species ecosystems. The species are connectedthrough a food-chain relationship. The top prey (X) and the middle predator (Y) give a biologicallymeaningful subsystem (subsystem A). In order to be a biologically meaningful system, a subsystem should

Fig. 1. Typical ecological situation represented by model I.

Fig. 2. Typical ecological situation represented by model II.

R.K. Upadhyay, V. Rai / Chaos, Solitons and Fractals 12 (2001) 205±218 207

qualify as a Kolmogorov system (see [20]). The last term in Eq. (1b) is omitted to get this subsystem. Thesubsystem is

dXdt� a1X ÿ b1X 2 ÿ �wXY �=�X � D�: �3a�

(Subsystem A)

dYdt� ÿa2Y � �w1XY �=�X � D1�: �3b�

3.1. Kolmogorov analysis

For the above subsystem F �X ; Y � and G�X ; Y � are given by

F �X ; Y � � a1 ÿ b1X ÿ �wY �=�X � D�;

G�X ; Y � � ÿa2 � �w1X �=�X � D1�:

Applying the conditions of the Kolmogorov theorem, we obtain the following:(i)

oFoY

< 0 ) ÿw=�X � D� < 0:

This condition is satis®ed as w and D are positive constants.(ii)

XoFoX� Y

oFoY

< 0 ) b1X �X � D�2 � wDY > 0:

This condition holds in the domain X ; Y > 0 as b1;D;w are positive constants.(iii) oG=oY � 0 is automatically satis®ed.(iv)

XoGoX� Y

oGoY

> 0 ) w1D1X > 0 ) D1 > 0:

(v) F �0; 0� > 0 gives a1 > 0. This condition is automatically satis®ed as a1 > 0.(vi) F �0;A� � 0 gives A � a1D=w. Since a1;D;w are positive constants, this implies A > 0, which is true.(vii) F �B; 0� � 0 gives B � a1=b1. Since a1; b1 are positive constants this implies B > 0, which is true.(viii) G�C; 0� � 0 gives C � �a2D1�=�w1 ÿ a2�. Since C > 0, we get a constraint w1 > a2:(ix) The condition B > C gives the constraint

a1=b1 > �a2D1�=�w1 ÿ a2�:Thus, Kolmogorov theorem is satis®ed when

w1 > a2; D1 > 0; �4a�

a1=b1 > �a2D1�=�w1 ÿ a2�: �4b�

3.2. Linear stability analysis

For X -isocline

dXdt� 0 ) a1 ÿ b1X ÿ �wY �=�X � D� � 0: �4c�


For Y -isocline

dYdt� 0 ) a2 � w1X

X � D1

: �4d�

The intersection of the two isoclines is the equilibrium point �X �; Y ��, where X � � �a2D1�=�w1 ÿ a2� andY � � �a1 ÿ b1X ��D� X ��=w: Writing N1 for the host and N2 for the parasite in the subsystem, we have

F1�N1;N2� � a1N1 ÿ b1N 21 ÿ

wN1N2

N1 � D;

F2�N1;N2� � ÿa2N2 � w1N1N2

N1 � D1

:

Now, the elements of the community matrix are given by

a11 � oF1

oN1

� �� a1 ÿ 2b1N �1 ÿ

wDN �2�N �1 � D�2 ; a12 � oF1

oN2

� �� ÿ wN �1

N �1 � D;

a21 � oF2

oN1

� �� w1D1N �2�N �1 � D1�2

; a22 � oF2

oN2

� �� ÿa2 � w1N �1

�N �1 � D1� � 0:

(using Eq. (4d))Therefore, the community matrix is given by

M �a1 ÿ 2b1N �1 ÿ

wDN �2�N �1 � D�2 ÿ wN �1

N �1 � Dw1D1N �2�N �1 � D1�2

0

26643775:

The eigenvalues of the matrix M are the roots of

k2 � ak� b � 0; �5a�with

a � ÿ�a11 � a22� �5b�and

b � a11a22 ÿ a12a21: �5c�The subsystem is locally stable, if the eigenvalues are negative or have negative real parts. A necessary andsu�cient condition is a > 0 and b > 0, where a and b are de®ned in Eqs. (5b) and (5c), respectively. Wehave, at the equilibrium point

a > 0 ) 2b1N �1 �wDN �2�N �1 � D�2 ÿ a1 > 0:

Substituting the values of N �1 ;N�2 and simplifying we obtain

2b1

a2D1

w1 ÿ a2

� �� b1Dÿ a1 > 0: �6a�

We have b > 0 ) ww1N �1 N �2 D1 > 0 which gives

D1 > 0: �6b�

Suitable choices of the parameters satisfying Eqs. (6a) and (6b) will lead to a solution with stable equi-librium point and choices violating these conditions will lead to limit cycles.


This subsystem would admit limit cycle solutions when the Kolmogorov conditions (4a) and (4b) aresatis®ed and one or more of conditions (6a) and (6b) from the linear stability analysis are violated.

A set of parameter values satisfying these conditions are:

a1 � 2; b1 � 0:05; D � 10; a2 � 1; w1 � 2; D1 � 10:

We choose w � 1 as it is the maximum per capita removal rate of prey population.There may exist other sets of parameter values which satisfy the above criteria. However, we have chosen

one set for analysis.Suppose that there exists a prey ~X for predator Z other than Y. Assume that A is the rate of self-

reproduction for this prey and K is the carrying capacity of its environment. In the Leslie±Gower formu-lation, the growth rate equations for the two populations are

d ~Xdt� A ~X 1

ÿ

~XK

!ÿ B ~X Z

� ~X � E� ; �7a�

(Subsystem B)

dZdt� cZ2 ÿ w3Z2

� ~X � D3�: �7b�

Let us now analyse this subsystem.

3.3. Kolmogorov analysis

In this case, F � ~X ; Z� and G� ~X ; Z� are

F � ~X ; Z� � A 1

ÿ

~XK

!ÿ BZ

� ~X � E� ;

G� ~X ; Z� � Z c

ÿ w3

� ~X � D3�

!:

(i)

oFoZ

< 0 ) ÿB=� ~X � E� < 0:

This condition is satis®ed as B and E are positive constants.(ii)

~XoF

o ~X� Z

oFoZ

< 0 ) ~X

ÿ A

K� BZ

� ~X � E�2!ÿ BZ

� ~X � E� < 0; or ÿ BEZ

� ~X � E�2

� A ~XK

!< 0:

This condition holds in the domain ~X ; Z > 0, as B;E;A;K are positive constants.(iii)

oGoZ� 0 ) cÿ w3=� ~X � D3� � 0 ) c � w3=� ~X � D3�: �7c�

Since ~X > 0, this implies c < w3=D3.(iv)

~XoG

o ~X� Z

oGoZ

> 0 ) ~X Zw3

� ~X � D3�2" #

> 0: �7d�

This condition is always satis®ed.


(v) F �0; 0� > 0 gives A > 0. This condition is automatically satis®ed as A > 0.(vi) F �0;A�� 0 gives A� � AE=B. Since A;E;B are positive constants, this implies A� > 0, which is true.(vii) F �B�; 0� � 0 gives ) B� � K: Since K is a positive constant, this implies B� > 0, which is true.(viii) G�C�; 0� � 0 gives �cÿ w3=�C� � D3��Z � 0 or c � w3=�c� � D3�. Since C� > 0, we again obtaincondition (2.3.5d). We also obtain C� � �w3 ÿ cD3�=c.(ix)

The condition B� > C� gives K > �w3 ÿ cD3�=c: �7e�Thus, for this subsystem to be a system in the Kolmogorov sense, the following constraints should besatis®ed:

c � w3=� ~X � D3�; or c < w3=D3; �8a�

K > �w3 ÿ cD3�=c: �8b�A Kolmogorov system can either admit a stable equilibrium or a stable limit cycle solution. Following thelinear stability analysis discussed earlier, we ®nd that the subsystem given by (7a) and (7b) would be locallystable if

A2 ~X �

K

ÿ 1

!� BEZ�

� ~X � � E�2 > 0; �9a�

where we have used (7c), and

2 A

(ÿ 2A ~X �

Kÿ BEZ�

� ~X � � E�2)

c

(ÿ w3

� ~X � � D3�

)� Bw3

~X �Z�

� ~X � � E�� ~X � � D3�2> 0:

Using (7c), we obtain

Bw3~X �Z�

� ~X � � E�� ~X � � D3�2> 0; �9b�

which is always true, where ~X � and Z� are the equilibrium points

~X � � w3

c

�ÿ D3

�; Z� � � ~X � � E� 1

ÿ

~X �

K

!AB

� �:

When the values of the parameters of this subsystem are chosen in such a way that the constraints (8a) and(8b) are satis®ed and one or more of inequalities (9a) and (9b) are violated, the subsystem admits a limitcycle solution. We ®nd that for the following values of the parameters the subsystem has a limit cyclesolution:

A � 1; K � 50; B � 1; E � 20; c � 0:0062; w3 � 0:2; D3 � 20:

For this set of values, we obtain ~X � � 12:26; Z� � 24:20: In this case, Eq. (9a) is not satis®ed.We now try to link the two subsystems. One possible way which is biologically sound is shown in Fig. 3.

The linking scheme would depend on how the individual populations of the two subsystems are related witheach other.

This link mechanism can be mathematically represented by adding a term ÿ�w2YZ�=�Y � D2� to thesecond equation of the ®rst subsystem. This gives us the original equation (1b). Since the meaning of D1

and D2 are the same and D2 is not an important parameter as far as the asymptotic dynamics of thecomplete system is concerned, D2 can be given the same numerical value as that of D1. On the other hand,w2 can be varied from 0:1 to 1 since w2 plays a role similar to that of w. Physically, the parameter w2 acts asa coupling constant for the two non-linear oscillators de®ned by subsystems A and B.

The most crucial part of the present methodology is the following conjecture:


A model ecological system would either exhibit limit cycle solutions or chaos when both the correspondingsubsystems are in oscillatory mode.

We shall call this method as the pseudo-prey method.The selection of the parameter values for the original system is now restricted by omitting those ap-

pearing in the growth equation of the pseudo-prey. In the present case, the set of parameter values forwhich the system admits limit cycle solution is found to be

a1 � 2; b1 � 0:05; w � 1; D � 10; a2 � 1; w1 � 2; D1 � 10; w2 � 0:3; D2 � 10;

c � 0:0062; w3 � 1; D3 � 20:

4. Results

The parametric values chosen in accordance with the pseudo-prey method were used to run the ODEsoftware package. The models were numerically integrated to get the time-series corresponding to thevariables of the model systems. Since every non-linear system has a ®nite amount of transients, the datapoints representing transient behaviour were discarded. Phase portraits were drawn using these data toobtain the geometry of the attractor. The geometrical object (phase portrait) with zero phase space volumeand represented by a point on the phase plane is called a stable focus. If the system trajectory evolvesstrictly on a closed path, then the attractor is said to be a limit cycle attractor. On the other hand, if thetrajectory meanders in a bounded phase space of ®nite volume, then the corresponding attractor is knownas a strange chaotic attractor.

The parametric values which are common for all the simulation experiments are as follows:

Model I : w � 1; D � 1; a2 � 1; w1 � 2; D1 � 10; D2 � 10; D3 � 20; w3 � 1:

The parameters used in the 2D scans are a1; b1;w2 and c which represent the rate of self-reproduction forthe prey X, the strength of intra-speci®c competition, the maximum value (or limiting value) which percapita reduction rate w2Y =�Y � D2� can attain and the growth rate of the generalist predator Z due tosexual reproduction, respectively.

Model II : w � 1; D � 1; a2 � 1; w1 � 2; D1 � 10; D2 � 100; D3 � 20; w3 � 1:

The parameters used in the 2D scans are a1; b1;w2 and c which represent the rate of self-reproduction forthe prey X, the strength of competition among individuals of this species, maximum value at which percapita reduction rate can attain and the rate of self-reproduction of species Z, respectively.

It may be noted that the parameters chosen for the 2D scans are the parameters which control thedynamics of the model systems. The basis for performing the 2D scans is the belief that changes in physicalconditions may bring corresponding changes in, at least, two parameters at a time. The change in the natureof dynamics is monitored. The computed results are given in graphical form. Results pertaining to Model Iare given in Figs. 4±7. The results are summarized as follows.

Fig. 3. Relationship between food-chain species and pseudo-prey.


4.1. Model I

(A) w2 � 0:55, c � 0:0257 and 1:56 a16 2:55, 0:0356 b16 0:06 (see Fig. 4).(i) Chaos exists at the discrete points �a1; b1� � �1:95; 0:05�; �2; 0:05�.(ii) For b1 � 0:06; chaos does not exist in the range 2:07 < a1 < 2:09:(iii) For 2:1 < a1 < 2:5, chaos exists along the line b1 � 0:06 except at the points �2:14; 0:06�,�2:15; 0:06�, �2:25; 0:06�, �2:27; 0:06�, �2:33; 0:06�, �2:37; 0:06�, �2:41; 0:06�, and �2:42; 0:06�.

Fig. 4. Model system I. 2D scan diagram between �a1; b1� parameter space with w2 � 0:55, c � 0:0257, w � 1, D � 10, a2 � 1, w1 � 2,

D1 � 10, D2 � 10, D3 � 20 and w3 � 1.

Fig. 5. Model system I. 2D scan diagram between �a1;w2� parameter space with b1 � 0:05; c � 0:0257 (remaining parameter values are

same as in Fig. 4).

Fig. 6. Model system I. 2D scan diagram between �a1; c� parameter space with b1 � 0:05;w2 � 0:55 (remaining parameter values are

same as in Fig. 4).

Fig. 7. Model system I. 2D scan diagram between �w2; c� parameter space with a1 � 2 or 2:5; b1 � 0:05 (remaining parameter values

are same as in Fig. 4).


(iv) Chaos does not exist along the dotted line b1 � 0:2a1 ÿ 0:35 except at the points�a1; b1� � �2:0; 0:05�; �2:03; 0:056�, �2:035; 0:057�; �2:05; 0:06�.

(B) For b1 � 0:05; c � 0:0257 and 1:56 a16 2:5 , 0:16w26 2:5 (see Fig. 5).(i) For a1 � 1:95, chaos exists in the range 0:526w26 0:63 , 0:696w26 0:85 and 0:956w26 2:5. Nochaos exists in the range 0:646w26 0:68, 0:866w26 0:94.(ii) For a1 � 2, chaos exists for 0:556w26 1 except at the points �1:96; 0:65� , �1:97; 0:65� and�2:04; 0:5�.(iii) No chaos exists in the ranges 1:56 a16 1:89 and 2:16 a16 2:5 for all w2.

(C) b1 � 0:05;w2 � 0:55 and 1:56 a16 2:5, 0:0036 c6 0:05 (see Fig. 6).(i) For a1 � 2:05, chaos exists for 0:02556 c6 0:036:(ii) Chaos exists at some discrete points. For example, chaos exists for �a1; c� � �2:0; 0:026�,�2:0; 0:036�, �1:95; 0:026�, �1:95; 0:036�, �2:25; 0:045�, �2:5; 0:045� and �2:5; 0:049�.

(D) a1 � 2 or 2:5; b1 � 0:05 and 0:16w26 1, 0:0036 c6 0:05 (see Fig. 7).(i) For w2 � 0:3, chaos exists for all c in 0:0366 c6 0:045(ii) For c � 0:045, chaos exists for all w2 in 0:36w26 0:75.(iii) Besides the above range, chaos exists at the discrete points shown by circles. For example, chaosexists for �w2; c� � �0:1; 0:045�; �1:0; 0:026� and �1:0; 0:036�.

(E) a1 � 2:0; c � 0:0257 and 0:0356 b16 0:06, 0:16w26 1.(i) For b1 � 0:05, chaos exists for all w2 in 0:556w26 1:0.(ii) Chaos exists at some other discrete points. For example, chaos exists for �b1;w2� � �0:048; 1:0�.

(F) a1 � 2;w2 � 0:55 and 0:0356 b16 0:06, 0:0036 c6 0:05.(i) For b1 � 0:05, chaos exists for all c in 0:0266 c6 0:045.(ii) Chaos exists at some other discrete points. For example, chaos exists for �b1; c� ��0:035; 0:045�; �0:048; 0:045�.

The results of the dynamical study of the Model II are summarized as follows.

4.2. Model II

(A) w2 � 1:45; c � 0:0257 and 0:56 a16 2:5; 0:036 b16 0:1. Chaos exists only at the discrete point:�a1; b1� � �2; 0:05�.

(B) b1 � 0:05; c � 0:0257 and 0:56 a16 2:5, 0:0556w26 1:95. For a1 � 2:0, chaos exists for all w2 in1:456w26 1:6.

(C) b1 � 0:05;w2 � 1:45 and 1:56 a16 2:5, 0:0036 c6 0:05. Chaos exist only at the discrete points. Forexample, chaos exists for �a1; c� � �2:0; 0:0257�; �2:0; 0:026�.

(D) a1 � 2; c � 0:0257 and 0:016 b16 0:1, 0:0556w26 1:45: For b1 � 0:05, chaos exists for all w2 in1:356w26 1:61:

(E) w2 � 1:45; a1 � 2 and 0:016 b16 0:1, 0:0036 c6 0:05: Chaos exists only at discrete points. Forexample, chaos exists for �b1; c� � �0:05; 0:0257� and �0:05; 0:026�.

(F) a1 � 2; b1 � 0:05 and 0:0556w26 1:45, 0:0036 c6 0:045:Chaos exists only at discrete points. For example, chaos exists for �w2; c� � �1:5; 0:0257� and �1:5; 0:026�.

From the above summarized results, it can be observed that chaos exists in very narrow parameter regimes,i.e., chaotic dynamics is manifesting itself in very narrow regimes as the sensitive parameters of the systemare varied. The fact that such a dynamical behaviour has not been observed by the earlier investigatorssuggests that the biology of the bottom predator (whether it is sexually reproducing, generalist, etc.) may bea crucial factor in the governance of the dynamical consequences of any given food-chain. It may be notedthat in both the model systems, bottom predators are sexually reproducing and generalists. The presentstudy may partially provide reasons for the failure of the earlier attempts to observe chaos in naturalpopulations. We argue that the biological basis of the aforementioned crisis must not be overlooked al-though there could be other reasons, e.g., poor data quality (insu�cient length, precision, unsuitablesampling rate etc.). Since we have varied both the genetic (a1; c) and the environmental parameters (b1), thereported results have strong biological signi®cance.

The other objective of the present study is to identify the mechanism which may be responsible forchaos to exist in narrow parameter ranges in both the model systems. We have used basin boundary


calculations, bifurcation diagrams, etc. to investigate as to why chaos existed in very narrow parametricranges.

These calculations were performed using the software provided with the book entitled ``Dynamics:Numerical Explorations'' [21]. Fig. 8 shows the bifurcation diagram for the Model system I given byEqs. (1a)±(1c) for the set of parametric values a1 � 2; b1 � 0:05;w � 1; and D � 10: The magni®ed bifur-cation diagram for the range 1:656 a16 3:13 is presented in Fig. 9. Similar calculations for the companionsystem (Model II) are presented in Figs. 10 and 11. In Figs. 12 and 13, basin boundary calculations are

Fig. 8. Bifurcation diagram for model I at a1 � 2. Rest of the parameters are w2 � 0:55, c � 0:0257, w � 1, D � 10, a2 � 1, b1 � 0:05,

w1 � 2, D1 � 10, D2 � 10, D3 � 20 and w3 � 1.

Fig. 9. Magni®ed bifurcation diagram for model system I in the range a1 � 1:65±3:13.

Fig. 10. Bifurcation diagram for model II at a1 � 2. Rest of the parameters are w2 � 1:45, c � 0:0257, w � 1, D � 10, a2 � 1,

b1 � 0:05, w1 � 2, D1 � 10, D2 � 10, D3 � 20, and w3 � 1.

Fig. 11. Magni®ed bifurcation diagram for model system II in the range a1 � 2:85±3.


presented for the chaotic attractors in the Model systems I and II, respectively at a1 � 2, and a1 � 2:005.The values of b1;w and D are taken as the same as in the previous ®gures. As the value of the parameter a1

is increased, there is an increase in the size of the attractor (see Fig. 12). The increase in the attractor sizewas also observed in the companion system for the same set of parametric values. The encroachment intothe basin of chaotic attractor by the basin of attractor at1 can be observed in Fig. 12. The increase in sizeof a strange chaotic attractor as a system parameter is varied is considered to be the hallmark of the crisis(sudden destruction of a chaotic attractor) route to chaotic dynamics [22,23]. The crisis occurs precisely atthe point where the unstable period-3 orbit created at the original saddle-node bifurcation intersects withthe narrow chaotic region. The crisis occurs at a1 � 2:15 for the ®rst time in the companion system (seeFig. 13).

The occurrence of such phenomena in one-dimensional unimodal mappings was discussed by Grebogiet al. [22,23]. Their analysis helps us to conclude that the existence of chaos in Model systems I and II

Fig. 12. Basin boundary calculations for the model system I at a1 � 2 and at a1 � 2:005.

Fig. 13. Basin boundary calculations for the model system II at a1 � 2 and at a1 � 2:005.


in narrow parameter ranges is a result of crisis-limited chaotic dynamics. This conclusion is corroborated bythe basin boundary computations and attractor size comparisons for both the model systems. The chaoticbehaviour in both the systems is not continuing further, as the unstable period-3 orbits which originate atthe time of saddle-node bifurcation do not allow it to move further.

5. Discussion

We have attempted to study the reasons for the failure of attempts to observe chaos in the real ecologicalsystems. It was found that chaos existed in narrow parameter ranges in the model systems studied. Theexistence of chaos in narrow parameter ranges means that it will rarely be observed in the real world as theenvironmental noise can easily drag the dynamics away from chaotic regime. An attempt was also made toidentify the mechanism which terminates the chaotic dynamics abruptly. It was found that crisis is theunderlying factor which generates chaotic dynamics in the two-model systems considered. To the best ofour knowledge this is the ®rst instance of crisis-limited chaotic dynamics being observed in a model food-chain. Since food-chains are the basic constituents of the interaction networks, it may be argued thatterrestrial ecosystems may not be the prospective places where chaos should be looked for. Crisis-induceddeaths of strange chaotic attractors have been observed earlier by Huberman and Crutch®eld [24], Kaplanand Yorke [25] in systems described by ordinary di�erential equations, and by Russell and Ott [19] in a non-linear coupled plasma wave problem. Ott's book [16] provides a lucid description of these developmentsand the underlying theory. Although we chose to work with a1 (which measures the rate of self-reproduction for species X) as the control parameter, similar dynamical behaviour was observed when otherparameters of the model system are taken as control parameters. It is possible that crisis-induced de-struction of chaotic attractors may be a common phenomenon in model systems describing real worldecosystems, and, therefore, may explain why attempts to observe chaos in natural populations have failed.This failure, having a biological origin, may be attributed to the fact that interaction networks in real worldecosystems are not suitable for chaos to be supported. The analysis presented in this paper supports theview that the cause of failure of majority of the e�orts to observe chaos in natural populations may not bepoor quality of the data (insu�cient length, precision, unsuitable sampling rate, etc.), but may be anecological reality and hence can be described in terms of established biological principles. Moreover, weargue that crisis-limited chaotic dynamics may be common in model systems describing di�erent ecologicalsituations.

Fig. 11 shows an interesting phenomenon. The magni®ed bifurcation diagram for Model system II in therange a1 � �2:85; 3� shows that the system possesses a rich variety of dynamical behaviour. Closed curves inthis diagram correspond to invariant KAM tori in the phase space. Later on, these curves break and giverise to chaotic dynamics. Similar features are also present in Model system I (see Fig. 9) although in adevelopment stage. The interaction of the Hamiltonian dynamics with the processes of crisis-inducedsudden deaths of chaotic attractors in this parameter range can give rise to dynamical complexity [18].These systems may serve as good models to probe into the details of di�erent aspects of dynamical com-plexity.

References

[1] May RM. Proc Roy Soc Lond Ser A 1987;27:419.

[2] May RM. Proc Roy Soc Lond Ser A 1987;413:27.

[3] May RM. J Theore Biol 1975;51:511.

[4] Gilpin ME. Amer Naturalist 1979;107:306.

[5] Hastings A, Powell T. Ecology 1991;72(3):896.

[6] May RM. Nature 1976;261:459.

[7] Scha�er WM. Ecology 1985;66:93.

[8] Scha�er WM, Kot M. Bioscience 1985;35(6):342.

[9] McCann K, Yodzis P. Ecology 1994;75:561.

[10] Ruxton GD. Ecology 1996;77(1):317.

[11] Sugihara G, May RM. Nature 1990;344:734.


[12] Eckmann JP. Rev Mod Phys 1981;53:643.

[13] Upadhyay RK, Rai V. Chaos, Solitons & Fractals 1997;8(12):1933.

[14] Upadhyay RK, Iyengar SRK, Rai V. Int J Bifurc Chaos 1998;8(6):1325.

[15] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcation of vector ®elds. Applied Mathematical

Sciences, vol. 42. New York: Springer; 1983.

[16] Ott E. Chaos in dynamical systems. Cambridge,UK: Cambridge University Press; 1993.

[17] Leslie PH, Gower JC. Biometrika 1960;47:219.

[18] Poon L, Grebogi C. Phys Rev Lett 1995;75(22):4023.

[19] Russell DA, Ott E. Phys Fluids 1981;24:1976.

[20] May RM. Stability and complexity in model ecosystems. Princeton, NJ: Princeton University Press; 1973. p. 85.

[21] Nusse HE, Yorke JA. Dynamics: numerical explorations. New York: Springer; 1994. p. 269 (Chapter 7).

[22] Grebogi C. Z. Naturforsch 1994;49a:1207.

[23] Grebogi C, Ott E, Yorke JA. Phys Rev Lett 1982;48:1507.

[24] Huberman BA, Crutch®eld JP. Phys Rev Lett 1979;43:1743.

[25] Kaplan JL, Yorke JA. Commun Math Phys 1979;67:93.

[26] Hallam TG. Population dynamics in an inhomogeneous environment. In: Hallam TG, Levin SA, editors. Biomathematics 17,

Mathematical Ecology: Introduction. Berlin: Springer-Verlag; 1986. p. 61.


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Crisis-limited chaotic dynamics in ecological systems