Evidence of peak-to-peak dynamics in ecology

R E P O R TEvidence of peak-to-peak dynamics in ecology

Sergio Rinaldi, Matteo Candaten

and Renato Casagrandi

CIRITA, Politecnico di Milano,

Via Ponzio 34/5, 20133 Milan,

Italy.

Correspondence:

[email protected]

Abstract

Remarkably irregular peaks characterize the dynamics of many plant and animal

populations. As such peaks are often associated with undesirable consequences (e.g. pest

outbreaks, epidemics, forest ®res), the forecast of the forthcoming peak is a problem of

major concern. Here we show, through the analysis of a number of models and of some

of the longest and most celebrated ecological time-series, that the intensity of the

forthcoming peak can often be predicted simply from the previous peaks. When this is

possible, one can also predict the time of occurrence of the forthcoming peak.

Keywords

Nonlinear dynamics, chaos, quasiperiodic behaviour, models, peak-to-peak dynamics,

populations, forecast, strange attractors, fractal dimension, extreme episodes.

Ecology Letters (2001) 4: 610±617

I N T R O D U C T I O N

After transient, any deterministic, dissipative, nonlinear

system settles on an attractor (an equilibrium, a limit cycle, a

torus or a strange attractor) and remains there forever if it is

not perturbed (Strogatz 1994). Insights about the attractor

can be obtained, even in the absence of a mathematical

model, if a single variable y of time t has been recorded for a

long period, while the system was on the attractor (Hastings

et al. 1993). In particular, one can extract from the record all

peaks (local maxima) of the variable y, say yi (i � 0, 1, 2,¼),

and plot them one against the previous one, thus obtaining a

set of points (yi, yi+1), called peak-to-peak plot (PPP). The same

plot is sometimes called next amplitude plot (Olsen & Degn

1985) or next-maximum plot (Scott 1994) or Lorenz plot

(Strogatz 1994). From the same record one can also extract

the times of occurrence of the peaks ti and compute the

return times si+1 � ti+1 ± ti.

If the regime is periodic and there are k peaks per period,

the PPP is composed only of k distinct points. By contrast, if

the regime is quasiperiodic or chaotic, i.e. if the attractor is a

torus or a strange attractor, the points of the PPP are all

distinct and sometimes display ®liform geometries (i.e. the

PPP has fractal dimension close to 1, see Strogatz 1994). More

precisely, when the attractor is a high-dimensional strange

attractor, the PPP is a cloud-like set. Conversely, the points

of the PPP lie on a closed regular curve (the slice of a torus)

when the regime is quasiperiodic and lie roughly on a curve

when the attractor is a low-dimensional strange attractor.

We show in the next section that when the PPP is

®liform, i.e. when the attractor is low-dimensional, the

intensity of the forthcoming peak and its time of occurrence

can be predicted with remarkable accuracy from the

previous peak or from the two previous peaks. The ®rst

possibility has been pointed out by Lorenz himself in his

pioneering paper on chaos (Lorenz 1963) and then

described in more detail by various authors (e.g. Olsen

& Degn 1985, in the context of biology). Peak-to-peak

analysis is a special case of the standard technique for

reconstructing strange attractors (Packard et al. 1980),

popularized in the 1980s among ecologists in a series of

ground-breaking articles (e.g. Schaffer & Kot 1985a, b;

Schaffer 1985; Olsen et al. 1988). Peak-to-peak analysis

requires only very little effort (it can be performed by hand

in a few minutes), especially if it is compared with other

techniques used in ecology (Sugihara & May 1990; Casdagli

1989; Ellner & Turchin 1995; Costantino et al. 1997). As far

as we know, only six ecological models have been studied

until now through peak-to-peak analysis (Funasaki & Kot

1993; Boer et al. 1998; Rinaldi & Solidoro 1998; Blasius &

Stone 2000; Rinaldi et al. 2001a, b).

M A T E R I A L S A N D M E T H O D S

To illustrate why a PPP can be ®liform and to detect the

consequences of this fact in terms of peak predictability, we

consider in this section a paradigmatic example of chaotic

ecosystems, namely the Rosenzweig±MacArthur tritrophic

food chain (Hastings & Powell 1991)

dx1

dt� rx1 1ÿ x1

K

� �ÿ a2x1

b2 � x1

x2 �1�

Ecology Letters, (2001) 4: 610±617

Ó2001 Blackwell Science Ltd/CNRS

dx2

dt� c2

a2x1

b2 � x1

x2 ÿ d2x2 ÿ a3x2

b3 � x2

x3 �2�dx3

dt� c3

a3x2

b3 � x2

x3 ÿ d3x3 �3�

where x1, x2, x3 are the abundances of prey, predator and

top-predator, r and K are intrinsic growth rate and carrying

capacity of the prey and (ai, bi, ci, di) i � 2,3 are maximum

predation rate, half-saturation constant, ef®ciency and death

rate of predator and top-predator, respectively. For suitable

parameter values (see caption of Fig. 1) the model is chaotic

(Hastings & Powell 1991), its strange attractor A looks like a

tea-cup and has fractal dimension slightly greater than 2 (see

Fig. 1A).

Let y(t) � x3(t) and assume that top-predator abundance

is measured for a long period of time. The points in state

space where y(t) is either maximum or minimum are

characterized by dy/dt � 0, and lie on the plane x2 � b3d3/

(c3a3 ± d3) (see Eq. (3) with x3 ¹ 0 and dx3/dt � 0). This

plane, indicated by S in Fig. 1(B), is a particular PoincareÂ

section, namely a manifold transversal to the attractor

(Strogatz 1994). The intersection of A with S provides the

points of the attractor where y is maximum or minimum: of

course, after a maximum (point 0) we have a minimum

(point 1) and then a maximum again (point 2) and so on. In

other words, when the system is on the attractor, Eqs. (1)±(3)

de®ne a transformation, called PoincareÂ map, from any

maximum (point 0) to the next maximum (point 2) on a

subset P of S. Moreover, if the dimension of A is close to 2,

then the dimension of P is close to 1, i.e. P is a ®liform set,

namely a fractal set resembling a thick curve. Thus, in

practice, each contour line y � const. on S intersects P at one

or more distinct points. In the present case (see Fig. 1C),

each contour line (straight line x3 � const.) intersects P only

at one point. This is a fortunate situation: it implies that the

contour line y � yi, where yi is the amplitude of the ith peak,

intersects P only at one point which is then transformed by

the PoincareÂ map into a next maximum point at the

intersection of P with the contour line y � yi+1. Thus, a

unique peak yi+1 is associated to the peak yi, i.e. the ®liform

PPP can be represented by a function

yi�1 � F yi� � �4�as shown in Fig. 1(D). By contrast, when some contour line

y � const. intersects the ®liform set P at h ³ 2 points, the

PPP is still ®liform but cannot be described by Eq. (4),

because the prediction of the next peak yi+1 cannot be

performed exclusively on the basis of the last peak yi.

However, it can be shown (Candaten & Rinaldi 2000) that

independently upon the value of h, each peak yi+1 is uniquely

determined by the two previous peaks, i.e.

yi�1 � F�yi ; yiÿ1� �5�Thus, in conclusion, if the attractor is a torus or a strange

attractor with fractal dimension close to 2, the PPP

is ®liform and successive peaks obey simple recursive

Figure 1 Peak-to-peak dynamics of the

Rosenzweig±MacArthur tritrophic food

chain model (Eqs. 1±3). Parameter values are

r � K � c2 � c3 � 1, a2 � 1.66, b2 � 0.33,

d2 � 0.27, a3 � 0.05, b3 � 0.5, d3 � 0.01.

(A) The teacup strange attractor A in state

space; (B) the PoincareÂ section S where

top-predator abundance is either maximum

or minimum; (C) the subset P where top-

predator abundance is maximum and the

contour lines x3 � const.; (D) the peak-to-

peak plot.

Peak-to-peak dynamics 611


equations. When this happens we say that the system has

peak-to-peak dynamics (PPD). Such PPD are called simple in the

case of Eq. (4) and complex when Eq. (5) holds.

Finally, it is worth noticing that when PPD are simple the

peak yi identi®es a unique point on the PoincareÂ section

(point 0 in Fig. 1B) so that Eqs. (1)±(3) can be integrated

from that point until after si+1 units of time, the next peak

yi+1 (point 2 in Fig. 1B) is reached. This means that the

return time si+1 is a function of the previous peak, i.e.

si+1 � T(yi), so that the time of occurrence of the

forthcoming peak can also be predicted from the previous

peak through the formula ti+1 � ti + T(yi). Of course, the

function T(yi) must be substituted by a function T(yi, yi±1) in

the case of complex PPD. Moreover, it can be shown

(Candaten & Rinaldi 2000) that the existence of PPD

implies that the return times satisfy a ®rst-order recursive

equation of the form

si�1 � H �si� �6�or, in the most complex cases, a second-order equation

si�1 � H �si ; siÿ1� �7�The fact that the return times associated with a quasiperi-

odic or low-dimensional chaotic regime obey the laws of an

autonomous system has been noticed for the ®rst time in

physics through very simple experiments on dripping taps

(Shaw 1984). But the same phenomenon is sometimes

present also in biology, for example, in the study on the

in¯uence of light stimuli on the locomotor behaviour of

Halobacterium (see Fig. 1g in Schimz & Hildebrand 1992).

The forecast of the time of occurrence of the next peak is of

great importance in ecology, where in many cases the

amplitude of the peak is hardly measurable, whereas its time

of occurrence is perfectly known (see Rinaldi et al. 2001a for

the forecast of insect pest-outbreaks and ®res in Mediter-

ranean forests).

Results

In order to show that PPD are frequent in ecology, we ®rst

present in Fig. 2 a selected sample of PPPs derived from

various models.

Figure 2(A) refers to the plant-herbivore-carnivore model

recently used (Blasius et al. 1999; Blasius & Stone 2000) to

explain the phase synchronization of hare and lynx popu-

lations in Canada. The model is essentially a Rosenzweing±

MacArthur model (Eqs. 1±3) with very high prey carrying

capacity and top-predator predation rate. Moreover, a small

positive term is added at the right-hand side of Eq. (3) to

accommodate for the alternative food sources available to

the lynx. The time-series obtained through simulation shows

that the return times are almost constant, whereas the peak

amplitudes vary remarkably. Although the changes in

abundance look erratic, they can be perfectly explained with

Eq. (4), as pointed out by the PPP. In other words, the model

suggests that each peak of the lynx population can be

predicted years ahead from the previous peak.

Panel B of Fig. 2 comes from a prey±predator model

(Eqs. (1) and (2) with x3 � 0) with periodically varying food

supply. It has long been known that seasonal perturbations

in prey±predator assemblies can give rise to quasiperiodic

and chaotic behaviours. A general discussion of this topic

can be found in Rinaldi et al. (1993), and a recent application

to the population dynamics of Fennoscandian voles is

available in Turchin & Ellner (2000). In the case examined

in Fig. 2(B) the prey time-series is chaotic, with return time

roughly double than the period of the food supply.

However, as the PPP is ®liform, the randomness of the

peaks can be fairly well explained through Eq. (4).

Figure 2(C) has been obtained from the discrete-time

three-stage (Larvae-Pupae-Adults) insect population model

discussed in Costantino et al. (1997)

Lt�1 � bAt exp ÿcel Lt ÿ ceaAt� � �8�Pt�1 � 1ÿ ll� �Lt �9�

At�1 � 1ÿ la� �At � Pt exp ÿcpaAt

ÿ � �10�As the number of points of the PPP (obtained through

simulation of the model) is practically unlimited, all the

details of the PPP can be detected. Thus, it is possible to

distinguish between chaoticity (open regular curves) and

quasiperiodicity (closed regular curve) without performing

complicated tests or computing Lyapunov exponents

(Costantino et al. 1995, 1997). (Obviously, the same distinc-

tion is not possible in the case of a limited number of

recorded peaks). For the parameter values indicated in the

caption, the regime is quasiperiodic because the points of the

PPP lie on a closed regular curve, so that the PPD are

complex and the next peak can be predicted with a model of

type (5).

Figure 2(D) has been obtained from the very simple

insect model with delayed recruitment rate discussed in

Gurney et al. (1980)

dx�t�dt� ÿax�t� � bx�t ÿ s�exp�ÿx�t ÿ s�� 11�

where x is biomass, a and b are mortality and maximum

recruitment rate and s is the time-delay. Periods of very low

abundance regularly alternate with periods of high abun-

dance characterized by bursts of high-frequency oscillations.

The PPP obtained by using all local peaks is a cloud-like set

(not shown), whereas the PPP obtained by retaining only

the highest peak (superpeak) of each burst (marked with *)

has a ®liform, although complex, geometry. This means that

the dynamics of the superpeaks are described by Eq. (5), i.e.

each superpeak can be predicted from the two previous

612 S. Rinaldi, M. Candaten and R. Casagrandi


superpeaks, a rather astonishing result. It is worth noticing,

however, that the structure of the PPP can be fully identi®ed

only if the population is observed for a very long time. For

example, if the available record would be that of Fig. 2(D)

(nine superpeaks), one would obtain eight points in the

space (yi, yi+1) (points marked with · in the right column of

Fig. 2D), which do not reveal all the details of the PPP.

Actually, one would most likely conjecture that these eight

points lie on a closed regular curve and hence conclude that

the regime is quasiperiodic. Of course, other records of the

same length, obtained with the same model, would evidence

other parts of the PPP and therefore suggest chaoticity.

Figure 2 shows only a small sample of ecological models

displaying PPD. In fact, we have also identi®ed PPD in the

following 15 models:

· Ricker's model (Ricker 1954);

· The discrete-time logistic model (May 1976);

· Three age-structured population models (Guckenheimer

et al. 1977; Caswell & Weeks 1986; Higgins et al. 1997);

· A model for host±parasitoid interaction (Lauwerier

& Metz 1986);

· Three rodent-mustelid models (Hanski et al. 1993;

Turchin & Hanski 1997; Turchin & Ellner 2000);

· A model of scramble competition (FerrieÁre & Gatto

1995);

· A plant±herbivore±omnivore model (McCann & Hastings

1997);

· A prey±predator evolutionary model (Abrams & Matsuda

1997);

· A model of tritrophic food chains with allochtonous

resources (McCann et al. 1998);

· An algal competition model (Gragnani et al. 1999);

· A model for snowshoe hare demography (King

& Schaffer 2001).

A ®le on these models and their corresponding PPD is

available on request from Renato Casagrandi.

By contrast, the identi®cation of PPD from ®eld and

laboratory data poses some problems. There are three reasons

Figure 2 Peak-to-peak dynamics in four

ecological models. On the left of each panel

there is a short segment of a very long

simulated time-series used to construct the

PPP shown on the right, where the points ·correspond to all pairs of subsequent peaks

of the short segment. (A) Carnivore popu-

lation of the tritrophic food chain model

used in (Blasius et al. 1999) (see Eqs. (1)±(3)

with an extra-term equal to 0.06 in Eq. (3)).

Parameter values are r � d2 � c2 � c3 � 1,

K � a3 � b3 � 8, a2 � 4, b2 � 2, d3 � 10.

(B) Prey population of a prey±predator

model (Eqs. (1), (2) with x3 � 0) with peri-

odically fed prey. Parameter values are

r � d2 � 2p, a2 � 4p, b2 � 0.3, c2 � 1, and

K � K0(1 + esin(2pt)) with K0 � 1.44,

e � 0.65. (C) Larvae abundance of a three-

stage insect model (see Eqs. 8±10) with

cpa � 0.075. All other parameter values are as

in Costantino et al. (1997), namely b � 6.598,

cel � 0.01209, cea � 0.01155, lL � 0.2055,

la � 0.96. (D) Adult individuals of an insect

model with delayed recruitment (see Eq. 11)

(the peaks marked with * are the superpe-

aks). Parameter values are s � 1, a � 8.21,

b � 821.



for this. First, even the longest population time-series are too

short for obtaining reliable estimates of PPD and other

dynamic invariants when a sound supporting model is not

available. Second, false peaks due to measurement and

process noise have a severe impact on the PPP. In fact, a false

peak f between two successive true peaks a and b implies the

loss of point (a, b) in the PPP and the addition of two false

points (a, f ) and ( f, b). In principle, data ®ltering should be

used to eliminate false peaks and ®nd reliable estimates of real

peaks. In practice, exclusion of false peaks through common

sense (as done in Fig. 3) can also be effective. Finally, the data

used to construct a PPP should be collected when the system

is on the attractor. Thus, the initial peaks of any laboratory

time-series should be excluded from the analysis because they

usually correspond to a transient towards the attractor. The

elimination of transient peaks may dramatically reduce the

length of an already too short time-series. For this reason, in

our analysis we have systematically sacri®ced only the ®rst

peak of each laboratory time-series.

Despite these dif®culties, we have constructed the PPPs

of some of the longest available ecological time-series and

the results (described in the following) are encouraging: not

only do the points of the PPPs lie on regular curves, but

such curves are shaped as those predicted from the models.

The results are summarized in Fig. 3, which presents four

well-known population time-series. The PPPs obtained

from these series using all their peaks are not shown,

because they do not display sharp geometries. Instead, the

second column of Fig. 3 displays the PPPs obtained by

eliminating the false peaks (marked with h) and the transient

peaks (marked with s). In each one of these ®gures a dotted

curve (obtained without performing any sort of formal

calibration) has been drawn to help the reader in focusing

on a plausible alignment of the points of the PPP. In other

words, the dotted curves are the result of the most crude

and model-free analysis of the data: plot in a plane all pairs

of subsequent peaks and decide (subjectively!) if the points

of the PPP lie or not on one or more curves.

Figure 3 Four population time-series (®rst

column) and corresponding PPPs (second

column). Peaks marked with s and h are

considered as transient and false peaks,

respectively, and are not used for deriving

the PPP. See text for the interpretation of

the dotted lines. (A) Lynx fur returns (in

thousands) in the MacKenzie River Region

(Elton & Nicholson 1942); (B) Paramecium

aurelia (in hundreds) in a controlled experi-

ment (Veilleux 1979); (C) larvae of ¯our

beetle Tribolium castaneum (in hundreds)

(see Fig. 2 with cpa � 0.25 in Costantino

et al. 1997); (D) Nicholson's long experiment

on blow¯y Lucilia cuprina (in thousands)

(Nicholson 1957). The peaks marked

with * are the superpeaks.



Panel A of Fig. 3 shows the pattern of the abundance of

the Canadian lynx in the MacKenzie River Region over

almost one century (Elton & Nicholson 1942). The data are

estimates of fur returns obtained from the archives of the

Hudson's Bay Company. Similar data, available for the

period 1913±40, have not been considered because they

come from different archives and most likely are in a ®xed

but unknown ratio with the data of Fig. 3(A). Schaffer (1984)

was the ®rst to suggest that the randomness of the lynx time-

series could actually be low-dimensional chaos. After that,

various methods have been applied to these data with

somewhat controversial results. By contrast, the eight points

of our PPP align pretty well on a regular curve, thus

suggesting the existence of a low-dimensional strange

attractor. Moreover, the shape of the dotted curve in

Fig. 3(A) recalls those of Figs 1(D) and 2(A), which have

been obtained from food chain models. This is not surprising

because the most convincing interpretations of the 10-year

lynx cycle are indeed based on plant±herbivore±carnivore

models (Gamarra & SoleÂ 2000; King & Schaffer 2001).

Figure 3(B) shows Veilleux's data (Veilleux 1979) on the

evolution of a protozoan population (Paramecium aurelia) in

the presence of its predator population (Didinium nasutum).

During the experiment the protozoa were fed periodically

every 2 days and the return times were roughly double. If

the two peaks marked with h are considered as false peaks

and, thus, eliminated and the ®rst peak is sacri®ced (as

suggested by Veilleux himself) the seven points of the PPP

nicely lie on a curve shaped as that of Fig. 2(B), obtained

from a prey±predator model with periodic food supply.

Panel C of Fig. 3 reports an 80-week time-series of larvae

of the ¯our beetle Tribolium castaneum (®fth time-series of

Fig. 2 in Costantino et al. 1997). In this case the PPP is

composed of 12 points roughly distributed on a closed

curve. This suggests that the regime is quasiperiodic or

weakly chaotic, in good agreement with the estimate of an

almost vanishing Lyapunov exponent given in Costantino

et al. (1997).

Finally, Fig. 3(D) reports the outcome of Nicholson's

long experiment on the sheep blow¯y Lucilia cuprina

(Nicholson 1957). Ups and downs of abundance vary

remarkably in amplitude and multiple local peaks, attributed

to time-delays in the reproductive cycle (Gurney et al. 1980),

often characterize periods of high abundance. If we eliminate

the ®rst (transient) superpeak (marked with s) and we retain

the other superpeaks (marked with *) we obtain eight points

in the space (yi, yi+1), which suggest that the attractor is a low-

dimensional strange attractor. It is worth noticing that these

eight points are not enough to fully evidence the geometry of

the complex PPP of the supporting model (see Fig. 2D). A

similar result holds for Nicholson's short experiment

(Nicholson 1954). All this supports the idea that a simple

recursive relationship regulates the superpeaks and compares

quite favourably with the outcome of some statistical

methods, which suggest that these time-series are almost

chaotic (Ellner & Turchin 1995).

C O N C L U S I O N

Our analysis has revealed that PPD are almost the rule in

standard ecological models (indeed, we are now aware of 25

different ecological models with PPD). Moreover, the crude

analysis of four of the longest ®eld and laboratory

population time-series suggests that PPD are quite plausible,

and the qualitative comparison with some supporting

models reinforces this conclusion. These results are in

agreement with the ®ndings of many authors about the role

of density dependence mechanisms in the dynamics of

ecosystems.

Once PPDs have been identi®ed, the population can be

described by a very simple reduced-order model involving

only the peaks of the variable of concern (see Eqs. (4) and

(5)) or the corresponding return times (see Eqs. (6) and (7)).

These models can be of paramount importance in applica-

tions, because high population peaks are often associated

with undesirable consequences, which can be attenuated if

predicted in good time. In this respect, it is worth noticing

that if one has a reliable mathematical model of the standard

form dx/dt � f(x), or equivalently x(t + 1) � f(x(t)), the

forecast of the forthcoming peak requires the knowledge of

the state x of the system at a given initial time. But this is

never the case in real situations. For example, one may know

the lynx population in a given year, but not the hare

population and/or the density of grass and plants. By

contrast, PPD models are in the form (4)±(7) and can

therefore be used even if the available information is

restricted to the peaks of the variable of interest.

In closing this paper it is fair to make a few comments on

the identi®cation of PPD from real, i.e. noisy, data. The

technical literature on this matter clearly indicates that,

without the support of a model, hundreds of peaks are

needed for estimating the invariants (Eckmann et al. 1986;

Farmer & Sidorowich 1987; Casdagli 1991; Kennel

& Isabelle 1992; Mitschke & Damming 1993; Aguirre

& Billings 1995), a condition which is far from being

satis®ed by ecological time-series. Hence, it is mandatory to

make a combined use of data and models if the target is the

identi®cation of the PPD of a given population. A very

quick, but also very naõÈve way of proceeding has been

indicated in this paper, where the supporting models have

been used only for a qualitative consistency check. Better

estimates of PPD should be obtained by ®xing at realistic

values all parameters of the supporting model, except a

couple of strategic parameters which could then be varied to

®nd the best ®t of the simulated PPP with the data. This

calibration procedure is certainly justi®ed if the aim is to



build up an operational forecasting technique based only on

information on past peaks. For improving the estimates of

the PPD one should also consider the possibility of

pre®ltering the data, as there is here a very speci®c feature,

namely the fact that the structure of the PPP is sensitive to

high frequency noise (see discussion on false vs. true peaks).

General and effective pre®ltering rules that might counter-

act this peculiar weakness are hard to discover, but the

challenge is worth trying. By contrast, in studying speci®c

populations it might be possible to ®nd ways to circumvent

speci®c obstacles. This is, for example, what has been done

in Rinaldi & Solidoro (1998), where the impact on PPD

identi®ability of the plankton sampling period and of

measurement noise and weather variability has been

evaluated, together with the ef®ciency of various pre®ltering

algorithms.

A C K N O W L E D G E M E N T S

This study has been ®nancially supported by Consiglio

Nazionale delle Ricerche, Project ST/74 `Mathematical

Methods and Models for the Study of Biological Phenomena'.

The authors are grateful to Alessandra Gragnani and Carlo

Piccardi for their help and advice. Three anonymous

reviewers and Mercedes Pascual are also acknowledged for

their constructive criticisms.

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B I O S K E T C H

Sergio Rinaldi is interested in theoretical population dynamics.

At the moment he is involved in studies on tritrophic food

chains, plankton dynamics and forest ecosystems.

Editor, M Pascual

Manuscript received 7 May 2001

First decision made 18 May 2001

Second decision made 8 August 2001

Manuscript accepted 30 August 2001



Evidence of peak-to-peak dynamics in ecology

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