chaos in natural populations: edge or wedge?

Ecological Complexity 1 (2004) 127–138

Chaos in natural populations: edge or wedge?

Vikas Rai∗

Department of Applied Mathematics, Delhi College of Engineering, Bawana Road, Delhi 110042, India

Received 18 June 2003; received in revised form 21 December 2003; accepted 11 February 2004

Available online 10 April 2004

Abstract

A theory for the ‘edge of chaos’ (EOC) in ecological systems is presented. The theory envisions that the essential ecologicalinteractions can be described as a coupling of two basic predator–prey oscillators: one involving a specialist predator and theother a generalist predator. Three population dynamic models, which are obtained by coupling these oscillators, display ‘edge ofchaos’ in parameter space. Additionally, one of the models helps us understand an interesting population dynamic phenomenon(short-term recurrent chaos) recently observed in vole populations of northern Fennoscandia.© 2004 Elsevier B.V. All rights reserved.

Keywords: Predator–prey interaction; Specialist and generalist predators; Natural populations; Model systems; Pseudo-prey method;Simulation experiments; Edge of chaos; Short-term recurrent chaos

1. Introduction

There does not exist an unequivocal example ofchaos in a natural population till today. Given the shortand noisy character of the ecological time series, thisis not surprising as underlying theory of the methodsof non-linear time series analysis assumes that the ex-ogenous forces are absent.Ellner and Turchin (1995)developed a methodology and tools to detect deter-ministic chaos in short and noisy time series. Theseauthors analyzed laboratory and field data extractedfrom published records and found that deterministicchaos is not a dominant explanation for the variabil-ity exhibited by these records, although many of thempossessed potential to support the phenomenon. The

∗ Fax: +91-11-27871023.E-mail address: [email protected] (V. Rai).

authors concluded that their comprehensive analysissuggested that these population systems were evolv-ing to the ‘edge of chaos’ (EOC) (Kaufmann andJonsen, 1990; Kaufmann, 1993; Ikegami and Kaneko,1992; Kaneko and Ikegami, 1992). In the same year,Jorgensen (Jorgensen, 1995) produced simulation re-sults from a realistic lake model, which confirmed theearlier suggestion that the natural systems evolve toEOC. Later on,Rinaldi and De Feo (1999)proved thatecosystems enjoy special benefits being at the EOC.

One of the reasons why chaos in nature has rarelybeen detected is that the phenomenon really is a rarityand that this reflects something about the organiza-tion of ecological systems. One possibility is that thereality of ecological interactions, their non-linearcharacter notwithstanding, necessitates high level ofdissipation which, in turn, precludes the possibility ofchaotic motion in the absence of noise. Another is that

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128 V. Rai / Ecological Complexity 1 (2004) 127–138

the chaotically fluctuating populations are prone to ex-tinction with the consequence that group selection actsto eliminate species, which would otherwise evolveinto chaos. The argument depend on the observationthat in simple models, such as the logistic and Rickermaps (Ricker, 1954), minimum population sizes de-cline as one enters, and moves farther into, the chaoticregion. The present author has discovered that this is anartifact of these simple maps, which, at the best, can beregarded as caricatures of reality. We (Upadhyay andRai, 1997) proposed a model to understand why thereis so little empirical evidence supporting chaoticdynamics in natural population systems. This newclass of models executed chaotic motion, whereinno constituent population assumes extinction-sizeddensities.

In what follows, I describe the genesis of thesenew classes of model systems. It is shown that thesemodel systems evolve to the ‘edge of chaos’ in param-eter space.Section 2describes methodology for per-forming simulation experiments. The last two sectionspresent and discuss the results obtained there from.

2. Origin of the model systems

Among four basic interactions (predator–prey, com-petition, interference, and mutualism), predator–preyinteraction is the most common. There are two typesof predators: (i) specialists and (ii) generalists. A spe-cialist predator is the one, which dies out exponen-tially fast when its favorite food is absent or is inshort supply. The latter predators switch over to al-ternative food options when its most preferred foodis in short supply. The Rosenzweig–MacArthur (RM)model (Rosenzweig and MacArthur, 1963) is the one,which describes the dynamics of a specialist predatorand its prey:

dX

dt= a1X − b1X

2 − wYX

X + D, (1a)

dY

dt= −a2Y + w1YX

X + D1, (1b)

where X is prey for specialist predator,Y withHolling type II functional response,a1 is the rate ofself-reproduction for the prey,b1 measures the inten-sity of competition between the individuals of prey

X, w the maximum rate of per capita removal of preyspeciesX due to predation by its predatorY, D is thatvalue of population density ofX at which per capitaremoval rate is half ofw, the parametera2 measureshow fast the predatorY will die when there is no preyto capture, kill, and eat,w1X/(X + D1) is the percapita rate of gain in predatorY, w1 is its maximumvalue andD1 denotes the population density of theprey at which per capita gain per unit time inY is halfits maximum value (w1).

A model given by Holling and Tanner (HT) (Pielou,1977) describes the dynamics of a generalist predatorand its prey:

dZ

dt= AZ

(1 − Z

K

)− w3UZ

Z + D3(2a)

dU

dt= cU − w4U

2

Z, (2b)

whereZ is the most favorite food for the generalistpredatorU. In this model, prey and predator both growlogistically.A is the per capita rate of self-reproductionof the predatorZ and K the carrying capacity of itsenvironment. The parameterw3 measures the max-imum of the per capita rate of removal of the preypopulation by its predatorU. D3 is the half- saturationconstant for preyZ. The parameterc represents the percapita rate of self-reproduction of the predatorU andw4 measures severity of the limitation put to growthof predator population by per capita availability ofits prey.

Eq. (2a) and (2b)together represent this model. Inwhat follows, I attempt to arrive at a graphical repre-sentation, which gives us an idea about the parameterregions displaying distinct dynamical possibilities.

A predator–prey system qualifies as a Kolmogorov(K) system if it can be cast in the following form:

dX

dt= XF(X, Y)

dY

dt= YG(X, Y), (3)

whereF andG are continuous and analytic functionsin the domain,X ≥ 0, Y ≥ 0.

It can be easily seen that both of the models qualifyas a K-system. Kolmogorov analysis of RM modelyields following conditions:

w1 > a2,a1

b1>

D1a2

w1 − a2.

V. Rai / Ecological Complexity 1 (2004) 127–138 129

Fig. 1. Two stability regions are separated by a straight line withslope 2D1/(w1−a2) and meets the abscissa atD.

The local stability analysis puts the additional con-straint:

2b1

(a2D1

w1 − a2

)+ b1D − a1 > 0. (4)

One obtains a stable limit cycle in the phase space, ifinequality (4) is violated and those from Kolmogorovanalysis (Yodzis, 1989; May, 2001)are honored whilemaking the choices of the parameter values. The twoconditions can be combined to give a straight linewith slope 2D1/(w1 − a2). This line is located at adistanceD from the origin of the coordinate system.a1/b1 anda2 are the ordinate and abscissa. The straightline divides the space between two regions: the regionabove presents parameter values, which correspondto stable limit cycle solutions. The region defined bythe dividing line and the straight linea1/b1 = D ispopulated by stable equilibrium solutions (cf.Fig. 1).

Local stability analysis of HT model (May, 2001)provides a criterion:

c

A>

2(α − R)

1 + α + β + R, (5)

where

α = w3c

w4A, β = D3

K, and

R = [(1 − α − β)2 + 4β]1/2. (6)

A graphical representation of the criterion (5) forthe caseA/c = w3/w4 is given in May’s book (May,

2001, p. 192, Figure A.1.). The unstable region signi-fies stable limit cycle solutions, which emanate fromstable ones through super critical Hopf bifurcations.The assumed relationship is biologically realistic andwas chosen for the shake of simplicity. Similar stabil-ity boundaries exist for other relationships. It shouldbe noted that a Kolmogorov analysis of this systemdoes not put any constraint on the parameter values ofthe system (Eq. (2a) and (2b)).

When parameters are chosen in such a way that bothRM and HT systems display regular persistent periodicbehavior, these systems act as oscillators. Let us callRM oscillator as LCO (1) and HT oscillator as LCO(2). When suitably coupled, these systems will forceeach to generate chaos. An ecologically sound cou-pling of these systems gives us the following model:

2.1. Model 1

dX

dt= a1X − b1X

2 − wYX

X + D, (7a)

dY

dt= −a2Y + w1YX

X + D1− w2Y

2U

Y2 + D22

, (7b)

dZ

dt= AZ

(1 − Z

K

)− w3UZ

Z + D3, (7c)

dU

dt= cU − w4U

2

Y + Z, (7d)

where parametera1 is the rate of self-reproductionof speciesX, b1 measures the intensity of competi-tion among individuals of speciesX for space, food,etc. w/(X + D) is the per capita rate of removal ofspeciesX by Y. D is a measure of protection providedby the environment to preyX. Y a specialist predator,i.e. X is the only food for it. Therefore,Y dies outexponentially in the absence ofX. w is the maximumvalue that functionwX/(X + D) can attain.a2 is therate at whichY dies out exponentially in the absenceof its prey X. w1X/(X + D1) denotes the per capitagain in the specialist predator population due to pro-portionate loss in its prey.W1 is the maximum valuethat this function can take.D1 is the half-saturationconstant forY. A and K are respectively, the rate ofself-reproduction and carrying capacity for preyZ.The last term inEq. (7b)represents the functional re-sponse of the predatorU, which is a generalist preda-


tor. It switches its prey whenever its favorite foodoption Z is in short supply. The last term inEq. (7d)describes how loss in speciesU depends on per capitaavailability of its preys (Z andY). This gives us model1. W2 serves as the coupling parameter.

2.2. Model 2

Let us imagine a situation that arises when one ofthe prey species (Z) leaves out the patch for a betterhabitat. The resulting system is described by the fol-lowing equations:

dX

dt= a1X − b1X

2 − wYX

X + D, (8a)

dY

dt= −a2Y + w1YX

X + D1− w2Y

2Z

Y2 + D22

, (8b)

dZ

dt= cZ − w3Z

2

Y. (8c)

It should be noted that generalist predator is de-noted byZ, instead ofU. This is for the shake of thecontinuity of the notations for different species in themodel.

The generalist predatorZ (Eq. (8c)) is a sexualspecies. For most sexual species over most popula-tion densities, reproduction is determined primarilyby female numbers. Because of thefemale-biased de-mography of the sexual species the population growthrate is linear in population size, which is well belowthe carrying capacity. However, the growth of a sex-ually reproducing population is proportional to thesquare of the number of individuals present, in situ-ations when the population is at very low densities.This is known as the Allee effect (Allee et al., 1949).In such species, the decline of the population to lowdensities occurs due to a variety of causes (Lande,1988). Most important of these isinbreeding depres-sion, which results from physical or chemical mod-ification of the environment of organisms by socialinteraction or by density-dependent mating success.For example, some aquatic organisms condition theirmedium by releasing substances that stimulate growthof species, which have similar genetic make-up. Sparsepopulations rarely provide sufficient opportunitiesfor social interaction necessary for reproduction.In this case, the last equation of model 2 is modi-

fied to:

dZ

dt= cZ2 − w3Z

2

Y + D3. (9)

Some insect top predators very often switch to al-ternative prey in situations when their favorite food isin short supply. This fact can be accommodated by re-placingY2 with Y in the last term ofEq. (8b)as theirfunctional response is of Holling type II.

Eq. (8a)–(8c)define the linear phase of model 2.The non-linear phase is described byEqs. (8a) and(8b) and (9). The invertebrate version is obtained byreplacingY2 with Y in the last term ofEq. (8b).

It can be noted that the dynamics of the otherpredator–prey community (generalist predator–prey)is modeled following a scheme given by Leslie andGower (Leslie, 1948; Leslie and Gower, 1960; Pielou,1977). This formulation of predator–prey dynamicshas received severe criticism byYodziz (1994)andAbrams (1994). In order to test how far the dynamicsof the system is independent of the chosen formulationof the predator–prey dynamics, Abrams (P. Abrams,personal communication, 2000) suggested an alter-native representation (seeFig. 3a) of the biologicalsystem described by model 2 (Eq. (8a)–(8c)).

2.3. Model 3

Fig. 2a describes the interaction of species rep-resented byEq. (8a)–(8c)(which describe model 2)pictorially. Fig 2b is the operational description ofthe same. The units (prey-specialist predator; pseudo-prey-generalist predator) represent two distinct non-linear oscillators. At suitable choices of parametervalues, two oscillators execute regular, persistent, andperiodic motion. The interaction of these two limitcycles governs dynamical behavior of model 2.

Limit cycle oscillator of the first kind (LCO(1)) isobtained when parameters of Rosenzweig–McArthurmodel (RM, first subsystem) (Rosenzweig andMacArthur, 1963) are set at limit cycle values de-rived from an application of pseudo-prey method(Upadhyay and Rai, 1997). Likewise, the other sub-system (obtained by cutting the link between the spe-cialist and generalist predators in the second diagramof Fig. 2a when set at appropriate parameter valuesserves as a limit cycle oscillator of the second kind.The predator–prey interaction in RM model is formu-


Fig. 2. (a) A representation of the biological system and (b) two coupled limit cycle oscillators.

lated in Volterra scheme, which assumes exponentialdecay of a predator population in the absence of itssole prey. In contrast to this, the second subsystemis a valid representation of predator–prey interactionwhen the predator has other food options in additionto its most favorite prey.

The dynamics of predator–prey interactions de-scribed inFig. 3acan be mathematically representedby the following equations:

dx

dt= Ax(1 − x) − kxy

x + d

dy

dt= −cy + w1xy

x + d− y2z

y2 + d1

du

dt= au(1 − u) − k1uv

u + d2

dv

dt= −bv + w4uv

u + d2− wv2z

v2 + d3

dz

dt= w6y

2z

y2 + d1+ w7v

2z

v2 + d3− cz

. (10)

The original model 2 (Eq. (8a)–(8c)) is cast intoa form described byEq. (10a)–(10e). The variablesappearing in these equations are related to originalones by the following relationships:

x = X

K, y = wY, d = D

K, k = 1

K, z = w2Z,

d2 = D2w2, u = U

K1, v = w3V, d3 = D3

K1,

k1 = 1

K1, w = w5

w2and d4 = D4w

23,

whereX is prey 1,Y is predator 1,U is prey 2,V ispredator 2 andZ is the generalist predator, which hasspecialist feeding on predator 1 and predator 2.

The alternative representation (model 3) differsfrom model 2 in one essential way: the generalistpredator has specialist feeding on two preys (preda-tor 1 and predator 2) in the former. Prey 2 is thepseudo-prey (cf.Figs. 2a and 3a). The specialist feed-ing is formulated under Volterra scheme. For the firstrepresentation (model 2), subsystemA is L.C.O. (1)

Fig. 3. (a) An alternative representation of the biological systemand (b) the four coupled limit cycle oscillations of type 1.


and subsystem B (with model parameters set at stablelimit cycle values) serves as L.C.O. (2). It should benoted that none of the subsystems for the alternativerepresentation (model 2) is described by a limit cycleoscillator of type 2.

3. Simulations

As discussed inSection 2, parameters of model 1were set at the following values:

a1 = 2, a2 = 1, w1 = 2, c = 0.2,

w4 = 0.0257, w = 1, D = 10, D1 = 10,

w2 = 0.05, D2 = 10, A = 1.5, K = 100,

w3 = 0.74, and D3 = 20.

Parameters of a model are intrinsic attributes ofa system, therefore, arbitrary choices of their valuesare dangerous and may lead to spurious results. Inorder to make suitable choices of parameter values,we (Upadhyay and Rai, 1997) proposed a methodto select biologically meaningful parameter values(Upadhyay et al., 1998) for simulation of differentialequation models (3D systems) commonly designedand studied in mathematical ecology. The crucialpart of the method is to obtain two subsystems (forexample, one obtains two subsystems by cuttingthe link between two predators in the second dia-gram of Fig. 2a). Both of these can be treated asKolmogorov system and can be subsequently ana-lyzed by using the theorem as mentioned inSection2. This analysis along with the usual local stabil-ity analysis gives us an idea about parameter valuesfor simulation experiments. The biological mean-ing of Kolmogorov conditions, which are part andparcel of Kolmogorov analysis, is described inMay(2001).

The values of parameters of model 2, thus obtained,are:

a1 = 2, b1 = 0.05, w = 1, D = 10, a2 = 1,

w1 = 2, D1 = 10, w2 = 1.45, D2 = 10,

c = 0.0257, w3 = 1 and D3 = 20.

Parameters of model 3 were set at the followingvalues:

A = 2, k = 1

K= 0.025, d = D

K= 0.25,

c = 1, w1 = 2, d1 = D2w2 = 100, a = 2,

k1 = 1

K1= 0.025, d2 = 0.25, b = 1, w4 = 2,

w = 1, 0.6, d3 = 100, c = 0.2, w6 = 0.1.

These values were derived from those for model 2 asthe two models represent the same biological system.

An important part of the methodology of selectionof parameter values for simulation experiments is thefollowing theorem:

Two coupled K-systems in the oscillatory modewould yield either cyclic (stable limit cycles andquasi-periodicity) or chaotic solutions depending onthe strength of coupling between the two.

If key parameters of a model system are known, onecan study the dynamics by varying these system pa-rameters on both sides of their basic value for whichboth subsystems yield a stable limit cycle in the phasespace. Rest of the parameters are kept constant. Forexample,a1 andc are two key parameters for model 2.The range for variation of these parameters in simula-tion experiments is decided on the available biologicalinformation.

It should be noted that the attractors with apprecia-ble sizes of basins of attraction are ecologically rele-vant. The models were simulated for different initialconditions at all the sets of parameter values, wherechaotic attractors were observed. This was done to en-sure that these attractors have appreciable basin sizes.

4. Results

The model systems were scanned for chaotic dy-namics in 2D parameter space. Results of simulationexperiments (Upadhyay et al., 2001) for the 4D sys-tem (model 1) are presented inFig. 4(a–c).

The parameter space for model 1 (first representa-tion) is constituted bya1 andc. The alternative repre-sentation was studied in a parameter space spanned byA andw6. It is easy to verify that the two choices ofparameter spaces are appropriate if one pays a goodlook to the two models. The values of model parame-


Fig. 4. (a) Shown are the points where chaos was observed inmodel 1. 0.2 ≤ a1 ≤ 3.5, 0.03 ≤ b1 ≤ 0.1. The values of otherparameters are given in the text (cf.Section 3); (b) 0.2 ≤ a2 ≤ 1.5,1 ≤ w1 ≤ 4, the values of other parameters to generate this figureare given in the text; and (c) 0.1 ≤ c ≤ 1.2, 0.01 ≤ w4 ≤ 0.5.The values of other parameters were the same as given in the text(cf. Section 3).

ters for simulation experiments were obtained from amethodology with pseudo-prey method in the center.The ranges of parameters were decided so as to be inagreement with those given in Jorgensen’s handbook(Jorgensen, 1979). For all the simulation experimentson the 5D system (model 3), it was assumed thatw6 =w7. The points in these parameter spaces represent val-ues where chaotic solutions were obtained. Distribu-tion of points (Fig. 5) for UR model corresponds to theanswer that the system displays short-term recurrentchaos. It is characterized by chaotic bursts repeated atunpredictable intervals, and the reason for terminationof chaotic behavior is existence of chaos at discreteparameter values, sometimes with narrow parameterranges. Model 3 was simulated under two differentassumptions: (1) when there is no prey preference inthe diet of the top predator, and (b) when there doesexist such a preference. The model displayed chaoticbehavior only in the former case.Fig. 6 shows pa-rameter values inA−w6 space, which yielded chaoticsolutions for model 3.

Chaotic dynamics in model 2 was observed onlywhena2 was fixed at 1. For other values of this param-eter, no chaos was found in the model. This suggeststhat there does not exist a window in parameter spacefor chaotic dynamics.

Chaotic dynamics does not exist in model 2 withinvertebrate top predator for other values ofa2. Thisalong withFig. 7 suggests that there does not exist awindow in parameter space for chaotic dynamics.

5. Discussion

Let us try to understand what these simulation re-sults convey. One thing is sure that the chaotic behav-ior displayed by all these model systems is not of usualtype (“onset of chaos” (OOC)) scenario, wherein a dy-namical system suddenly passes from a state of regu-larity (constant, periodic or quasi-periodic motion) tochaotic behavior as some control parameter is modi-fied (May, 1976; Ott, 1993). In OOC scenario, thereexist ‘windows of periodicity’ in the chaotic region).On complete contrast to this, parameter values sup-porting chaotic dynamics in the present model sys-tems are very narrow and, in most systems, are dis-crete. Neither the ‘region of chaotic behavior’ nor the‘windows of periodicity’ in the chaotic region is visi-


Fig. 5. Distribution of points, where chaos was observed in model 2.a1 was varied in the interval (0.2–2.5), andc was varied in theinterval (0.005–0.03).

ble. Since well-mixed conditions are assumed, the pa-rameters of these population (reacting) systems wouldinherently fluctuate around a mean. These populationsystems are poised between order and chaos, thus, dis-playing ‘edge of chaos’. As chaos (exponential sensi-tivity of system’s trajectories on “initial conditions”)takes time (inversely proportional to max. Lyapunovexponent of the system) to develop, the system locks

Fig. 6. Distribution of points where chaos was observed in model 3.

itself onto a non-chaotic attractor before it settles intoa fully developed chaotic state. These dynamical tran-sitions are caused by changes in system parameters.Thus, chaotic dynamics is frequently interrupted bya purely deterministic cause, which is intrinsic to thesystem. This is why these systems appear to evolveto the ‘edge of chaos’ in the parameter space. At thisjuncture, it should be noted that manifestation EOC


Fig. 7. Distribution of points, where chaos was observed in model 2 with invertebrate top predator. The model equations are described byEq. (8a)–(8c), with Y2 replaced withY in the last term ofEq. (8b). a1 was varied in the range 0.5–2.5. The parameterc was varied inthe same range as it was done to generateFig. 4. a2 was set at 1. Values of other parameters are given in the text (cf.Section 3).

in this case is entirely different from how its pio-neers would have expected it to be (Kaufmann, 1993;Mitchell et al., 1993).

Simulations of model 2 present us an opportunity tounderstand phenomenon of short-term recurrent chaos(strc) (Turchin and Ellner, 2000) observed in voles innorthern Fennoscandia. (Turchin and Hanski, 1997). Iassume that the essential elements of vole dynamics(Turchin et al., 2000; Bjornstad and Grenfell, 2001)can be described by the following system:

Although weasel is a specialist predator, yetTurchinand Hanski (1997)have argued in favor of an inter-ferential form of predator–prey interaction leading tologistic growth of the predator. The thrust of their ar-gument is that weasel is a small mammalian predator,which spends most of its energy intake on generat-ing heat. The interferential form turns out to be themost suitable representation at high rodent densities,if one considers the fact that this animal is territorial.Eqs. (8a) and (8b) and (9)) describe the non-linearphase of the weasel’s population dynamics. The lin-ear phase of the animal’s dynamics is governed by

Eq. (8a)–(8c). The linear phase of model 2 does notdisplay any chaotic behavior. Only when weasel issufficiently low in number that the non-linear phase isswitched on and the system is locked on to a chaoticstate. As parameter values, which support chaotic be-havior, are discrete and have narrow range, the chaoticbehavior is frequently interrupted by non-chaotic be-havior as the system changes its parameters. As weaselgrows in number, exclusively non-chaotic attractorsgovern dynamics of this population system.

At this point, it should be noted that chaotic attrac-tors withoutextinction-sized densities could only beused to explain natural population dynamic phenom-ena (e.g. the short-term recurrent chaos (strc)). Thisis so because those with extinction-sized densitiessignify that populations will go extinct when actedupon by exogenous chance fluctuations. Two typicalchaotic attractors observed in model 2 are shown inFigs. 8 and 9.

It can be easily seen that these chaotic attractors arefree from extinction-sized densities. This also givesan idea about how misleading are the conclusionsdrawn from studies of logistic and Ricker maps, whichabound in such densities in the chaotic region. Thesechaotic attractors have appreciable sizes of basins ofattraction and, thus, render a firm foundation to the


Fig. 8. Chaotic attractor observed in model 2 fora1 = 2, b1 = 0.05, w = 1, D = 10, a2 = 1, w1 = 2, D1 = 10, w2 = 1.45, D2 = 10,c = 0.0257,w3 = 1, andD3 = 20.

theory of ‘edge of chaos’ presented in this paper and tothe associated dynamical phenomenon of short-termrecurrent chaos.

Alternatively, strc can also result because of thecoexistence of regular and chaotic attractors at thesame set of parametric values. It has been shown re-

Fig. 9. Chaotic attractor observed in model 2 (non-linear phase) with Holling type II functional response for the top predator. The systemparameters were fixed ata1 = 1.93, b1 = 0.06, w = 1, D = 10, a2 = 1, w1 = 2, D1 = 10, w2 = 0.405, D2 = 10, c = 0.0257,w3 = 1,andD3 = 20.

cently that the chaotic behavior might occur in a widerange of parameter values in host–parasite (Kaitalaet al., 2000) and in diffusively coupled predator–prey(Medvinsky et al., 2001) systems. These authors alsoshow that the boundary of basin of these regular andchaotic attractors can be fractal. In such a case, the


dynamical system under the influence of environmen-tal fluctuations would display strc. This mechanism ofstrc generation is difficult to test either in the labora-tory or in the field. On the other hand, the present the-ory makes testable predictions regarding the dynam-ics of these population systems, which can easily beverified by careful experimentation and well-plannedfield studies.

Thus, we see that there are two independent mecha-nisms of strc generation. A natural question that arisesat this stage is that which of the two is dominant andwhen. This will depend on the values of the crucialparameters of the system. Stochastic influences willdictate the dynamics when the basins of attractionof the coexisting attractors are intermixed. At othersets of parameter values, deterministic changes inthe system control parameters will guide the systemthrough its evolutionary course. The fact that all themodel systems studied displayed strc, suggests thatthe phenomenon should be common in natural andlaboratory populations. In fact, this turns out to be thecase. The phenomenon has not only been observed involes in northern Fennoscandia, but the characteris-tic property of this behavior, i.e. occasionally strongamplification of perturbations, has also been foundin Dungeness crab (Higgins et al., 1997), larval fish(Dixon et al., 1999), and laboratory populations offlour beetles (Costantino et al., 1997), and blowflies(Smith et al., 2000).

Acknowledgements

The author thanks Peter Abrams for invigoratingdiscussions and two anonymous reviewers for mean-ingful comments on an earlier draft of this manuscript.Madhur Anand and Chris Bauch provided some use-ful comments on the manuscript.

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