~. ':WI). '992, ~ 219-)05. I t92 b7 1M ~ SocictJ at A-a

COMPLEX DYNAMICS IN ECOLOGICAL TIME SERIES'

PETER TURCHIN AND ANDREW D. TAYLORSOIIIhn" Forf'st Expm'melll Slatioll. 2500 Shmepon Hi,ll_y, PiIleVi/I,., LouisiQIIQ 71 J60 USA

Abstract. Although the possibility of complex dynamical behaviors-limit cycles.quasiperiodic oscillations. and aperiodic chaos-has been recoanized theoretically. mostecologists are skeptical of their imponance in nature. In this ~per we develop a meth-odology for reconstructina endogenous (or deterministic) dynamics from ecolQlical timeseries. Our method consists of fittina a response surface to the yearly population changeas a function of lagged population densities. Usina the version of the model that includestwo lags. we fitted time-series data for 14 insect and 22 venebrate populations. The 14insect populations were classified as: unreaulated (I case). exponentially stable (three cases).damped oscillations (six cases). limit cycles (one case). quasiperiodic oscillations (two cases).and chaos (one case). The vertebrate examples exhibited a similar spectrum of dynamics.although there were no cases of chaos. We tested the results of the response-surface meth-odology by calculatina autocorrelation functions for each time series. Autocorrelation pat-terns were in agreement with our findings of periodic behaviors (damped oscillations. limitcycles. and quasiperiodicity). On the basis of these results. we conclude that the completespectrum of dynamical behaviors. ranaina from exponential stability to chaos. is likely tobe found among natural populations.

Key words: aulocorrelalion funclion; chaos; comp/~x d~t~ministic dynamics; d~/a)'t'd Mnsily d~-Pf'ndenc~; dynamical behaviors of popu/41ions; ins«t populallon dynamICS; limil C)'C/~s; long-t~mpopulation r«ord.s; nonlinear time-snin mode/ling; quasi/H'rlodicilY; time-series analysIS.

. INTRODUC'TJON

The relative importance of density-dependent vs.density-independent facton in determining populationabundances and dynamics is a central issue in ecolOJY.Much of the debate over this question has focused ontwo opposina viewpoints (e..., Nicholson 1954, An-drewanha and Birch 1954). Accordina to the first view-point populations are regulated around a stable pointequilibrium by density-dependent mechanisms. whilethe second one maintains that population chanae islargely driven by density-independent factors. Thereis. however, a third possibility. In addition to stablepoint equilibria. density-dependent processes can pro-duce complex population dynamics-limit cycles,quasiperiodic oscillations, and aperiodic chaos. Whilethe possibility of such dynamics has been recognizedtheoretically since the 1970s (May 1974. 1976), mostecologists have remained skeptical oftbeir importancein nature.

One well-known attempt to determine the frequencyof various kinds of dynamic behavion in insect pop-ulations was made by Hassell et aI. (1976). They con-cluded that most natural populations show monotonicdamping (the most stable kind of equilibrium behav-ior). with only 1 case (out of24) of damped oscillations.I case of a limit cycle. and ~') cases of chaos. D.:o;pitea number of caveal~ listed 0) Hassell 1.'1 ill. (1916). this

result was very inftuential in convincina ecologists thatcomplex dynamics are rarely found in nature (e.g., Ber-ryman and Millstein 1989, Nisbet et aJ. 1989). In thispaper we aflue that the results obtained by Hassell etal. (1976) laracly resulted from their overly simplemethod of analysis. Most imponantly, they used a sin-ale-species model that lacked delayed density depen-dence. Delayed density dependence, however, is ex-pected to arise as a result of biotic interactions inmultispecies communities and as a result of populationstructure (Royama 1981, Murdoch and Reeve 1987:L. R. Ginzbura and D. E. Taneyhill, unpublishedmanuscript), and in fact is found in many insect pop-ulations (Turchin 1990), Using a single-species modelwithout delayed density dependence biases the resultsin favor of stability, since complex dynamics are morelikely in hiaher-dimensionaJ systems, and mistakenlyanalyzing such systems in fewer dimensions will tendto hide this complexity (Guckenheimer et al. 1977,Schaffer and Kot 1985).

One approach to hiaher-dimensional analysis of eco-logical time series has been advocated by Schaffer andco-workers(Schaffer 1985,SchafferandKot 1985.1986,Kot et al. 1988), who used the method of"phase-spacereconstruction" in which unknown densities of inter-acting populatl,'ns are re-prcsented wIth I:!~ged d~"O;I-ties or :he studl.:d population. Sc;;h41lfcr ..:1,.1 Kot (1 \}/\o,examined time series of several natural populationsand concluded that reconstructed dynamics of these;'!opulati('r1~ rtsem~~,'d ch3ns. The major "e3k~'such anal)sl5. ho~t:vcr, is Its reliance on visual .!il.J

I Manu-,cnpt received 18 June 1990; revised 10 ~em~r1990: accel'ted 7 Febru:l~ 1991: final versIon received "March 1991.

290 PETER TURCHIN AND ANDREW D. TAYLOR Ecology, Vol. 73, No.

inal investigators' selection of populations for study isinherently biased. In particular. forest pests exhibitingoutbreaks clearly are over-represented.

Considerations of space prevent us from fully dis-cussing our results for all 36 time series (Table I andTable 2). As a compromise, we show the completespectrum of results for all series in one group-insects.We selected insects for detailed discussion panly be-cause we are most familiar with this group. More im-ponantly, insect data sets tend to be more reliable.since the majority of insect data were collected withthe specific goal of quantifying insect population fluc-tuations, unlike the data extracted from fur returns orbag records. Nevertheless, as will be seen later, manyof the patterns found in mammal and bird data setsare very similar to insect patterns.

therefore inherently subjective) examination of recon-structed attractors (Berryman and Millstein 1989. Ell-ner 1989).

In this paper we build on ideas of both Hassell et al.(1976) and Schaffer and Kot (1986). Our goal is todevelop an objective methodology for extracting de-terministic dynamics from shon and noisy ecologicaltime series. Unlike Hassell et al. (1976) who specifieda panicular equation with which to model data. weused a general and flexible methodology described byBox and Draper (1987), the response-surface meth-odology (RSM). We followed Schaffer and Kot (1985)by using lags to represent the multidimensional dy-namics of the system (e.g., unknown densities of in-teracting species or age structure). We used our meth-odology to reconstruct deterministic dynamics fromlong-term records of population fluctuations of 14 in-sects (with some funher comparisons to 22 mammaland bird species).

Since the methodology proposed here is new. we donot know how well it succeeds at reconstructing com-plex dynamics from data. This is especially true fordetecting chaos. However, methodolOlies for detectingperiodic behaviors (e.g., limit cycles) are well under-stood. Accordingly, we begin by using one of thesemethodologies, which is based on estimating the au-tocorrelation function (ACF) for each data set. We useACF patterns to characterize presence or absence ofperiodic behaviors in natural populations. and thencompare ACF results to conclusions reached with theresponse-surface methodolOJY. Our loaic is that ifRSMis not capable of extracting limit cycles from data, thenthere is little hope that we can use it to detect chaos.If, on the other hand, we can accurately reconstructone kind of non equilibrium behavior ,limit cycles, thenconfidence in our ability to reconstruct another kind,chaos, is correspondingly enhanced:

Investigating time series with autocorrelationfunctions

As the first step in our analysis of the populationtime series, we used the qualitative diagnostic tech-niques based on estimating the autocorrelation func-tion (ACF: Box and Jenkins 1974: for discussions ofACF in ecological context see Fineny [1980], Nisbetand Gurney [1982». Prior to the analysis the values ofpopulation density at each year, N" were log-trans-formed, L, - loa Nt. The au~ocorrelation function isestimated by calculating the correlation coefficient be-tween pairs of values L,_. and L, separated by lag T(1' = I, 2, . . . ). These correlation coefficients are thenplotted as a function of lag T.

The shape of the estimated ACF provides insightsregarding two aspects of population dynamics: station-arity and periodicity. A process is stationary if its dy-namical propenies do not change during the period ofthe study. Stationary processes fluctuate around con-stant mean levels, with constant variances. As will beseen later, our ability to reconstruct the endogenousdynamicsofa system depends considerably on whetherthey are stationary, or not. ACFs of stationary pro-cesses are characterized by an exponential decay tozero, either monotonic or oscillatory (Box and Jenkins1974).

Other ACF patterns indicate various forms of non-stationarity. A possible cause of non stationary dynam-ics is density independence, perhaps arising becausedensity regulation only occurs at extreme levels-.,ftoars" and "ceilinp"-that were not encountered bythe population during the study. Such a populationundergoes a "random walk," in w~;,.h !h;; ..-,,;;'..:..:.,"&radually wanders away from its initial density. Thereis, then. no true mean around which fluctuations occur.Alternatively. environmcnlal ..:iI..nges ~;"I.'urr:~c ~'~ .

time scale comparable to the length of the obsen'edtime series could produce a gradual trend in the mean.In either of these situations, the ACF will decay slowcrthan e~ponentially, and will become incrcasingly neg-ative at long lags (Fig. I A).

METHo~

The data set

We collected and analyzed every ~trial animalpopulation time series we could obtain, subject to thefollowing criteria: (I) Data were annual and continu-ous; if a time series had missing data, only the longestunintenupted period was used. (2) Time series had tocontain at least 18 yr of continuous census data, sothat no less than half the total degrees offreedom wouldalways be available for the error tenn in our responsesurface model (see Reconstructing endogenous d)'na,n-ics . . . . below). (3) The data were for a single locality(spatial scale having been determined by the originalauthor). Where several time series were available forthe same species, we selected the longest one. to avoidoverrepresentation of much-studied species.

We exercised no selectivity beyond applyina thesecriteria. Nonetheless, our data set cannot be regardedas representative of natural populations. since the orig-

COMPLEX DYNAMICS IN ECOLOGICAL DATA 291February 199

- I. Summary of insect time series.- .-

Species

Ph.l-lloprrrha hart/cola (prden chafer)Choristont'UFa .fumiferana (spruce budworm)Dendrolimus pini (pine spinner moth)H.\"loicus pinastri (pine hawk-moth)Dendroaonus .frontalis (southern pine beetle)Panolis }lammeD (pine beauty moth)L \.mantr/a monacha (nun moth)Fiupalus piniarius (pine looper)H.\phantria CUrIeG (fall webworm)I.espula spp. (wasps)Drepanosiphum platanoides (sycamore aphid)L.\.,nantria dispar <lYPSy moth)Zeiraphera diniaM (larch budmoth)Ph)'l/aphis fag; (beech aphid)

TABLE--

Time period

1947-197'1945-19721881-19401881-193019S8-19871881-19401900-19411881-19401937-19'81921-19461969-19871954-19791949-19861969-1987

Reference-, ..

Milne 1984Royama 1981Schwerdtfeaer 1941Schwerdtfeaer 1941Turchin et aI. 1991Schwerdtfeaer 1941Bejer 1988Schwerdtfeaer 1941Morris 1964Southwood 1967Dixon 1990MODt8Qmery and Wallner 1987BalteDSweiler and Fischlin 1987Dixon 1990

Using lags to represent multidimensionaldynamics of the s.vstem

Numerical chances of a population typically affectand are in turn affected by the population abundancesof resources. natural enemies. and competitors. Thus.in order to understand and predict how a populationchanaes with time. we need information about theabundances of interacting species. However. usuallydata are available only for a single population. and wenever know abundances of all populations in the com-munity. This difficulty can be overcome by consideringthe population change from the previous year t - I to

the current year t as a function of not only previousyear's density, N,-I, but also of densities N'_2' N,-". . . . The mathematical justification for this method-ology is provided by a theorem proved by Takens(1981). and the method has been successfully used inmany physical and chemical applications (e.g.. Argoulet aI. 1987), where it is called "attractor reconstruction

TA8U 2 Summary or venebrate time series

Species Time period.1821-19341819-19301834-192.51834-192.51834-192.51914-19.511914-19.511914-19.511841-19031884-19081933-19.551932-19541932-19541932-19.541932-19.541152-184911~2-1'1111862-1'/:I~191.5-191219.51-1913

IQ'4-ICJ521911-1"41

Reference

Moran 1953Elton 1942Elton 1942Eton 1942Elton 1942Keith 1963Keith 1963Keith 1963Leigh 1968Naumovl971Naumov 1971~butin 1960L'lbutin 1 Q60L'lbutin 1960~butln 1960Jones 1914jon...,. I'll J\1!Jdl...lon 1, '..Kclth 1963Loeryand

Nichols 1985L1A."k IQ(~liII:.. I ~)4

Nonstationarity can also be caused by externallydriven periodic changes in the mean. The resultingdynamics have been called "phase-remembering qua-si-cycles" (Nisbet and Gurney 1982), since the exOl-enous forcinJ factor maintains the reaularity of theoscillation despite random perturbations in abun-dances. The ACF of such a system miJht look like theone in FiJ. I C: it does not decay to zero, but ratheroscillates around zero with constant amplitude. Theperiod of oscillation of the ACF is determined by theperiodicity of the external forcinJ factor. In ecolOJY themost important such periodic factor is seasonality. 8yusina only data sets that reported population densitieson a yearly basis, however, we have avoided the com-plications of seasonality.

In addition to externally driven nonstationary pe-riodicity, stationary periodicity may arise from the en-doJenous dynamics of the system. Population fluctu-ations with an end~ous periodic component ("phase-foraettinJ quasi-cycles," Nisbet and Gurney 1982) winbe produced when the deterministic dynamics aredamped oscillations (around a stable point equilibri-um), a limit cycle, or "weak" chaos (Poole 1977). TheACF of these systems is characterized by an oscillatorydecay to zero (FiJ. 1 D). In contrast. a nonperiodic sta-tionary system, resultina from exponential stability (ofa point equilibrium), will have a monotonically de-cayinJ ACF (Fig. 18).

As a diapostic tool the estimated ACF is much moreuseful than "eyeballinJ" the observed time series. 8yaveraging over, and thus smoothing, the noisy timeseries, ACF reveals the periodic pattern in the data ifit is present. The average period of oscillations is readi-ly determined by observing at which lags ACF achievesits maxima. The speed with which ACF maxima ap-proach zero reveals the strenJth of the periodic com-ponent, that is. how long the process "rememb..'rs" itshistory. Finally, a quick. Jlthough crude. test of thehypothesis that there is a periodic component in p0p-ulation fluctuations can be performed by determlninJwhether ACF at the !:lg equal to one period is grcall'rthan the QS% conl1dcnce limit.

LynxFoxes (two species)Colored foxManenAn:tic foxMinkMuskratCoyoteSnowshoe hareVaryina hareSquirrelBelyak hareLynxFoxWolfParchment beaverWolverineRahbllRulfcd GrouseBlack-capped

ChickadeeHeronCircal I I

Ecology, Vol. 73~ No.PETER TURCHIN AND ANDREW D. TAYLOR292

B)(A}

..'CC""c-'

LAG

Lq)

-I I -I '

FIG. I. Theoretical shapes of autocorrelalion functions for (A) a process with nonstationary mean and no periodicity; (B)a stationary process with exponential return to equilibrium; (C) a process driven by an exogenous periodic force. or phase-remembering quasi-cycle; (D) a stationary process with endogenously generated periodicity. or phase-forgetting quasi-cycle.

is, the maximum lag time beyond which a past valueof population density has no direct effect on the currentpopulation change (autocorrelations can persist muchlonger than p. since past values of N affect intermediatevalues, which in turn affect present).

in time delay coordinates" (Schaffer 1985, Ellner 1989).Representing the unknown densities of interacting spe-cies with delayed density dependence is also a vener-able tradition in population ecology (H utchinson 1948,Moran 1953, Benyman 1978, Royama 1981). Essen-tially, one replaces the "true" multivariate system de-scribing deterministic population change

N,' = GI(N:-" Hl-I' . .., N~-I)N; = G2(N:-1, Hl-" . . ., N~_'>

N," = GIr(Nl-a, Hi-., . . . , N:-1)

(where N,' is the density of species i at time t. and G'is a function describing the change in the density ofspecies i with respect to the densities of interactingspecies) with a single equation for one species thatinvolves lags

N,I=P(Nl-1,Nl-z,...,Nl-,,). (1)

It is important to note that Nt can refer not only topopulations of interacting species, but also to abun-dances of different cohorts of the same species, if thepopulation has age, physiological, or spatial structure.

The above argument leads us to the following generalmodel:

N,=F(N"I,N'-2,...,N,_p,E,), (2)

where we have added the exogenous component E, tothe equation for population change. ("Endogenous"refers to dynamical fcedbacks affecting the system, in-cluding those that involve a time lag, e.g., natural en-emies. "Exogenous" refers to density-independent fac-tors that are not a part of the feedback loop.) We willmodel the exogenous component as a random, nor-mally distributed variable with mean zero, and vari-ance (71. The quantity p is the order of the process. that

Reconstructing endogenous d.vnamics withresponse-surface methodology

Our major goal in this paper is to develop a meth-odology that would objectively determine the type ofthe dynamic behavior that characterizes the endoge-nous component of population change. Several ap-proaches have been suggested. all based on the methodof reconstructing the attractor in time-delayed coor-dinates described in the preceding section. One is toestimate the dimensionality of the reconstructed at-tractor (for the explanation of this approach see Ellner[1989]). This approach appears to work for perfectlyaccurate data even with relatively shon time-series (50points). although dealing with noisy data sets remainsproblematic (Ellner 1989). Another approach relies onthe direct estimation of Lyapunov exponents from ex-perimental time series (Eckmann and Ruelle 1985. Wolfet al. 1985; for an explanation of Lyapunov e~ponentssee Abraham and Shaw [1983». This method requiresenormous amounts of data: a minimum of severalthousand data points is needed to characterize a low-dimensional attractor (Vastano and Kostelich 1986).The method of Sugihara and May (1990). which usesnonlinear forccasting to detect chaos in noisy time se-ries. also requires substantial amounts of data (500-1000 points in their applications).

Making as few assumptions as possible about thenature of the process that has produced the observedtime series is a powerful feature of the above m~thods.but It is also their weakness. Such non parametrIc. mod-

February 1992 COMPLEX DYNAMICS IN ECOLOGICAL DATA 293

'\,~",p~

.~~~:..:.:..:::~:!::~..:;;;.~ ~

(0) with

XFIG. 2. Results of ftttina a function Y - aX""

polynomials of second (-) and third order (- --)

el-independent approaches typically require plentifuldata points. In ecology, where the length of time seriesrarely exceeds 20-30 yr, one is forced to use a para-metric approach, which is much more frupl with datapoints.

The approach that we followed in this paper consistsof approximating the function Fin Eq. 2. This functiondescribes the behavior of trajectories on the recon-structed attractor, and thus knowins its properties givesus a complete description of the system dynamics. Forexample, the dynamic behavior of F could be formallycharacterized by calculatins its dominant eisenvalueand dominant Lyapunov exponent. Alternatively, onemay determine the type of dynamics simply by iter-ating Eq. 2 on the computer, and observing the re-sultins dynamics. We have followed the latter coursein this paper.

A potential problem associated with usina a para-metric approach, however, is that one may happen tochoose an inappropriate model with which to approx-imate F. This possibility can be minimized by usinsthe seneral method of response-surface flttins de-scribed by Box and Draper(1987). Briefty, this methodis similar to regular regression in that it employs poly-nomials for approximating the shape of F. However,both the response (dependent) variable and the pre-dictor (inde~dent) variables are transformed usingthe Box-Cox power transformation (Box and Cox 1964),with the transformation parameter (the exponent) be-ing also estimated from the data. In the followins par-agraphs we describe the logic and details of the ap-proach with which we have extracted endosenousdynamics from ecological time series.

The flnt step is to decide on the number of lags pto include in the model, that is, the "embeddins di-mension" (Schaffer 1985). Ideally, since the correct pis unknown, one should start with a low-dimensionalmodel and then increase the dimension until the resultdoes not depend on funher increase in dimensionality.In practice, due to data limitations (primarily the lensthof a time series) only a few lags can be examined. Intheir attempt to extract deterministic dynamics fromdata, Hassell et aI. (1976) used a model with only onela& (only direct density dependence):

N, - AN,-.(l + aN'-I)-d. (3)

We took the next step and used a model with two lags(in other words, we added delayed density depen-dence). Thus, the aeneral model (Eq. 2) becomes

N, - F(N, I' N, ~. I,). (4)

Biolosical considerations indicate that F can be rep-resented 3S a product of V, I and the per-capIta re-placement ratej(JV,-I, ,V, ,. t,). In geno:ral,J'wili havea simpler form, and can be approximated with a poly-nomial of one order lower than F. For example, Iff isa monolonll:ally decr"';j~lns function of," I and .,.it can be approximated with a first-order polynomial

y

I

- .'.

(together with appropriate transformations of the pre-dictor variables), while the function F will have a max-imum and will need to be approximated with a qua-dratic polynomial. These considerations lead us to thefollowing model:

N, - N,_J"{N,_" ,V,-~, E,). (.5)

We are now in position to estimate / by fitting a re-sponse surface to the observed replacement rateN,IN'-I as a function of N'_I and N,_~. Hiihly nonlineardependence of the replacement rate on laaaed popu-lation densities in several data sets and in some the-oretical models (P. TUKhin and A. D. Taylor, personalobservation), necessitated using polynomials of at leastsecond order (see Box and Draper 1987). However,polynomials by themselves are notoriously bad at ap-proximating both the function and its derivative, es-pecially for I~-like functions that are characterized byrapidly changing derivatives. Consider, for example,data plotted in Fig. 2. Fittina a quadratic polynomialto the nonlinear function represented by points, we findthat at high values of the predictor variable, the fittedfunction has a positive slope, while the actual functionhas a negative slope, Correct estimation of the slopeof/is crucial to the success of accurately reconstructinaendoaenous dynamics. since whether an equilibriumis stable or not, and whether the attractor is periodicor chaotic will depend on the derivatives of.(. Usinghigher-desree polynomials does not help, even thoughthey provide a progressively better approximation tof. since higher-dqree polynomials "oscillate" aroundthe true function (e.g" the cubic polynomial in Fig. .2).In addition. such an approach is very wasteful of de-gr~ of freedom. A better approach, proposed by Bo~and Cox (1964), is to power-transform either predictor.or rl.'sponse \;Irlablcs. IJr rn)111 the I(:-~arllhm:, .It-ur:llly cmbedded in the power translormatlon lam II)',unce lettina' - 0 is equivalent to a loa transformation(Box and Cox 1964; see also SobI and Rohlf 1981:4.2~'::rll

Whllc transformana predictor vanablcs alrcct~ uni~

PETER TURCHIN AND ANDREW D. TAYLOR Ecology. Vol. 73. No.294

ACF

3':5

.2~OJ0~

i:~0.-4

];~,

~o 30cy~

~o 50 6()a:o ,~ ~O1:0

YEARACF

44

3

,.tJI0~

,~2

t'~:

2

!'~

10 20' ,020 3&~~YEARYEAR

ACF

6

5

4

~ 3t7I0 2

~,-i:010 y

11

1.0 20 30 40 50 10 20 30 .10

YEAR YEAR

FIO.3. Log-transformed (base 10) time series and autocorrelation functions (ACF) for insect populations: IA) Phl'lhJPf'rthahtlftictJla. IB)Chonslo"eura (uml(l'ra"a. CC) Ill-ltJICUS pinaslrl. CD) Pant/Iii Ilam,nra. IE) 1.I'mantria dlipllr, If) Zt'lruphrra

295COMPLEX DYNAMICS IN ECOLOGICAL DATAFebruary 1992

ACFACF

42

3

~tJI0

~ ].~0~

2

1

"40 so30

YEARJo ~~.o201.0

y~ACF

~~

~~~0~

~8'M

1;

,~

30~o'1..201.0

YEARYEAR

ACF

tX)

],

24.

~

~..

010~

1.~-

010.-4

2

,:..I1

')GI~~Xo40 ~o)01.0 ~OYEAR YEAR

diftlana, (GI Drt'p6nl'(If'hllm platan()/dt's. (H) PII,'iIaphl.f liJ'(l (I) L,',nunf';a m()nulJ: a , (J) Bupalu.f plnlarlus. (K.) H\'ph.J"trl~

lUftt'cl. Ill; cfpulll ~pp.

296 PETER TURCHIN AND ANDREW Do TAYLOR EcoIOlY. Vol. 73. No. I

5

4

, 3~

~ 20

~

This model has a total of eight parameters (six param-eters defining the quadratic surface and two transfor-mation exponents). The best transformations (8-val-ues) of the predictor variables for any specific systemare unknown, and need to be estimated from the data.The transformations were estimated by fitting the mod-el (Eq. 6) by least squares for all combinations of 8.and 82 equal to {-I, -0.5,0, . ..,2.5,31 (using 10&-transform when 8, = 0) and selecting the 8.values thatresulted in the smallest residual sum of squares (Boxand Draper 1987). The farther the estimated 8, is fromI, the more nonlinear is the transformation.

The type of RSM-extracted dynamics was deter-mined by iteration of the model (Eq. 6) on the com-puter. The initial values HI and H2 were set equal tothe mean population density of the observed series.This procedure decreased the likelihood of beina mis-led by multiple attractors, if any were present. Thesimulated trajectory was plotted as an Nt vs. Nt-I phaseplot. If the trajectory approached a single point, thesystem was classified as stable. If the trajectory settledonto several points, the dynamics were classified as alimit cycle. In many cases the trajectory would notsettle onto a finite number of points, but instead allthe points in the phase space would be Iyina on anellipse (after discarding transients). Such dynamical be-havior, called "quasiperiodic" in mathematical liter.ature, results when the period of the oscillation is ir-rational, so that the solution never repeats itself exactly(Schaffer and Kot 1985). This kind of behavior is com.monly found in discrete models of order> I, such asthe model (Eq. 6). From the ecologist's point of view,the distinction between limit cycles and quasiperiodicdynamics is not very important, so we will treat themtogether as a sinaie category. FinaUy, a "Stranae" at.

;1.

'1:0 20 30

YEAR

40 so 60

4

~

~2

1.0 20 30 40 50 60

YEAR

FIG. 4. The time series of Drndrol;mus pin; and its au-tocorrelation function (ACF). (A) actual data; (B) quadrati-cally dettended data.

ACF

1."

,~

10 20 30Yt::AH

FIG. 5. Population ftuctuationl in Df'nd'I,,',onllS ",'",uilsand Its autocorrclatlon function (A('F). Dala a~ plolll-d onthe anlhmetlc (not loa-lranslorrnl-dl sc:alc.

the functional shape of f. transformation of the re-sponse variable also affects the error structure. Popu-lation data are non-negative. often ript-skewed. andmore variable when the mean is larae. Taking log-transforms of the response variable. a standard pro-cedure in population ecology (Moran 1953. Fineny1980. Royama 1981. Pollard et al. 1987). tends toalleviate all these problems at the same time (Ruppen1989). AccordinalY. we loa-transformed the replace-ment rate N,I N, ,. obtaining the rate or populationchange. " . 10g("',I.\', .). Defining X - I'~ I and Y 0=I\~ :. the above argument leads to the rollowing modelfor extracting deterministic dynamics rrom data:

" ~ an + a,X + alY + ailX'

+- a~:~ + a,~.rY + f,. (6)

February 1992 COMPLEX DYNAMICS IN ECOLOGICAL DATA 297

J~3. Summary of reconstructed dynamics.

Species Autocorrelation function Response-surface model resultPhy/lopertha horticola No regulation

Choristoneura fumiferanaDt"ndro/imus pini

Exponentially stableExponentially stablet

H}'/oicus pinastri Damped oscillations

Dendroctonus froma/is Damped oscillations;

Pano/is flammea Exponentially stable

Lymantria monacha Damped oscillations

Bupa/us piniarius Damped oscillations

Hyphantria cuni'a Damped oscillations

Vt'spula spp. Damped oscillations

Drepanosiphum platanoides Limit cycle (2 yr)

Lyman/ria dispar Quasiperiodicity (- 7 yr)

Zeiraphera di"iana Quasi periodici ty (~ 8 yr)

Ph,vl/aphis fag; Chaos

Non-stationaryNon-periodicNon-stationary or a very lonl cycle

Non-stationaryNon-periodic.Non-stationaryNon-periodic.Non-stationaryNon-periodicStationaryNon-periodicStationaryNon-periodicStationarySugestive of periodicity

StationarySugestive of periodicity

StationarySuuestive of periodicity

StationaryPeriodic (2 yr)

StationaryPeriodic (8.5 yr)

StationaryPeriodic (9 yr)StationarySuuestive of periodicity. Autocorrelation function of the detrended series suaeSt5 periodicity.

t Damped oscillations extracted from the detrended series.* Diverling oscillations and chaos extracted from the first and second half of the series, respectively.

'tractor, indicating chaotic dynamics, can look muchlike an ellipse that has been stretched and then folded.Another possibility is for a strange attractor to be sep-arated into several discontinuous pieces (see Schaffer[1987] for the explanation of many routes to chaos,and examples of phase graphs for various kinds ofattractors),

When iterating the model (Eq, 6) using an estimatedresponse surface with noise, or a chaotic response sur-face without noise, the trajectory occasionally jumps

TABLE 4. Estimated response-surface parameten, as defined by Eq. 6.

Species t; ',~ '~ Go Q1 a" a" -ICI

,-I1~

;~~I.'0.00.'0.03.03.00.'1.0

-0.5,0.5

2..1

0:1-1#

l.oj2.0I.'3.0

- 1.0,

1..0.-1.0

0.0-1.0

0.50.030

- 2.6370.0280.1630.3390.~911.306

-1.3700.6550.408~.2412.7222.894

-4.1741.131!

0.399-0.282

0.034-0.217-1.157~ 2.660-0.262

0.003-().(>61-10.251- 3.665-0.208

4.34q.. "- '-

6.280-0.007

-0.66.5

-0.326

0.193-0.010

0.062-1.522

J632-0.646

0.292-8.170-1.7QO-0.452

-0.003

-0.081

0.0000.029

.-0.2110.7000.0540.000U.WI3.9860.6460.005

-1.~800.564

-2.817

0.000

0.0890.005

-0.125-0.000-0.002

0.115G.I:"/)0.303

-0:0271,1'3

-0.124

u,OOl

Phyllopertha hortico/aChoristoneura fumifi'ranaDt'ndro/imus pinniJIy/oicus plnastriDt'ndroctonus frontalisPano/isllammi'aLr,nantria monachaBupa/us PlnluriusI/.rphantria ,uneaVi'spula spp.Dri'panosiphum p/atanoldnLymanlria dlSparZetrap"rra dinzanaPhrllaphI5/1l'"

outside the range of observed N, values. This causes adifficulty, because the shape of the response surfacewhere it is not constrained by data points may be Quitestrange, e.g., the surface could blow up to infinity. Inorder to prevent such occurrences, the values of thefunction/(N,_., N,_:>, at the boundary of the box in theN,_, - N'_2 phase space defined by the maximum and

the minimum of the observed series, were extrapolatedfor areas outside the box. In other words, when thesimulated trajectory left the minimum-maximum box,

.661

.002

.024

.009

.386

.013

.0371J()51!4t1.JSS.108.297.s'"7

298 PETER TURCHIN AND ANDREW D. TAYLOR Ecology, Yolo 73, No.

1..51~.)

c~ t1N~

\

t;0.15

~~.

The Dendroctonus frontalis population exhibited adifferent kind of nonstationarity, in which the meanstayed more or less constant, but the amplitude of theoscillation increased with time, with both the peaksbecoming higher and the troughs lower (Fig. 5). Onepossible explanation of this pattern is increased insta-bility of the Dendroctonus population as a result of aseveral-fold enrichment of this beetle's food base overthe last 30 yr (Turchin et al. 1991).

Several insect populations appeared to have periodicdynamics: significant periodicity was found in the ACFsof three populations (Fig. 3E, F, and G), and the ACFwas suggestive of an oscillation in an additional fourcases (Fig. 3H,J, K, and L). In each of the three periodiccases the ACF was of the phase-forgetting kind, thatis, the peaks in ACF decayed at higher lags. This sug-gests that oscillations in these populations are drivennot by an exogenous periodic force, but by endogenousdynamics.

f'0.1.0~':;'; 'i~

~, G~.~j1\ !\ r'\. r\Iv v vv

'O'X

p;~1_:1

0..-0;"

:,;:]

'(ct'

A r A~'\ Reconstructed endogenous d.vnamics

Applying response-surface methodology (RSM) tothe insect time series indicated the following spectrumof endogenous dynamics: no regulation (one case); sta-

v ,\I

5 J.O J.5 20 ::5 30

YEAR

FIG. 6. Hyphantria cunea: observed time series (A), andtrajectories predicted by response-surface model (RSM) with-out noise (8), and with noise (q (t, is normally distribuledwith mean zero and standard deviation IT - 0.2).

6

5~4~

3-~

A/\ l"\ f'\}1

~the computer program evaluated the function /(N,_"N,-J at the point on the box boundary nearest to thepoint (N,-.. N,-J. 4

8}3 1\ 1"\: ~

jRESULTS

AUlOCO"ela!ionjunction (ACF) patterns ~Several insect data sets exhibit ACFs suggestive of g

nonstationarity. The ACF of Phyllopertha horticola, .-I

the garden chafer, does not decay to zero, but insteadbecomes progressively more negative as the lag in-creases (Fig. 3A). Two other populations appear tooscillate around a nonstationary mean (Figs. 4 and 3C).The level around which the Dendrolimus pini popu-lation is fluctuating appears to ha ve first decreased, andthen increased, while the population of Hyloicus pi-nas!ri exhibited a downward trend. We estimated thelong-term trend for Dendro/imus by fitting a Quadraticpolynomial to Nt as a function of time. Subtracting theestimated trend from the time series ("detrending").we obtain a series that appears to fluctuate around aconstant level. and whose ACF is of the periodic phase-forgetting kind (although not Quite significantly peri-odic at 95% level) (Fig. 4). This example shows thatthe ability of the ACF to detect periodic behaviors issensitive to whether the underlying process is station-ary or not (see also Box and Jenkins 1974).

2

\1,; ...

c .;

.J

(C

At

J

10 20 30 40

'{EAR

FIG. 7. /./'/ruph('ra dIIllUIIU: obsenl'd tim... ~...n...s (AI. and

trajectorIes predicttd by r...sponse-surtace model (RSMI ~'ith-out noise (8). and with noise (C) (f, is normally distnbuttdwith mean L.:ru oind stan~rd deviation" - 0.2). Note thatthe dcterministic trajectory in (8) does not e\actly repeat t ."~. t

t'v~ry oscillation. This is an ~x3mpl~ of quasI pen odic ~hav-lor. In which thl." period of oscillation IS an irrational number.

COMPLEX DYNAMICS IN ECOLOGICAL DATAFebruary 1 99 2

~01

'""'

f\

"~,O ,,-~! ~

aO' '3

that RSM fitted the vapries of the data rather thanthe actual relationship, producinl a meaningless result.

Equilibrium dynamics: exponemial stability. -Of thethree cases classified by RSM as exponentially stable,one (Dendrolimus pini) had an ACF that exhibited ev-idence of non stationarity. When the Dendrolimus datawere made stationary with quadratic detrending, RSMsugested that this population may be in the oscillatorydamping regime, which agrees with the periodicity ex-hibited by the ACF of the detrended series (Fig. 4).This result demonstrates the sensitivity ofRSM resultsto nonstationarity.

The case of Choristoneurafumiferana presents a puz-zle. Althou&h it was suaested that this population un-dergoes periodic outbreaks (Royama 1984), regressingr, on !ailed population densities did not detect anysigns of density-dependent reaulation. The shape of theACF is consistent with either of the two hypotheses:that the budworm population cycles with a very longperiod. or that it is nonstationary (for example, thepopulation could be trackina a long-term oscillatorytrend in its food base). It is clear that data on morethan a sinaie outbreak will be needed before we areable to reach any conclusions about this insect's dy-namics.

The final case for which RSM indicates exponentialstability is Panolisflammea. This finding is consistentYEAR

FIG. 8. Lyman/ria dispar: observed time series (A), andtrajectories ~ictcd by the response-surface model (RSM)without noise (8), and with noise (C) (f, is nonn"ly distributedwith mean zero and standard deviation G' = 0.2).

2~

A

B

'i:~0~

ble. exponential damping (three cases); stable. oscil-latory damping (six cases); limit cycle (one case); quasi-periodicity (two cases); and chaos (one case) (Table 3).The parameten defining estimated response surfacesare listed in Table 4 for each insect species. In thefollowing subsections we will consider each of thesecategories in turn. paying particular attention to wheth-er the RSM results are consistent with those from theACFs. and to the effects of non stationarity.

No regulation.-lteration of the model estimated byRSM for the garden chafer. Phyl/opertha hort;cola, ex-hibited unstable behavior: at first the population grewat a very slow rate. and then it suddenly crashed (pop-ulation density decreasinl by about five orders of mag-nitude). Regressions of r, on lalled population densi-ties Nt ~ I and N'_2 (Turchin 1990) did not indicate anydensity regulation in this population. sugesting thatthe garden chafer population may undergo a density-independent "random walk." This conclusion is sup-poned by the nonstationary shape of the ACF (Fia.3A). and is In agreement with the previous analysis ofMilne (1984). With a v~ry !;lrgc numhcr ofdala pointsthat were aenerated by a density-independent popu-lation process. RSM wouki fit a level plane to Ihe scat-terplot of r. as a function of N. I and N..~, Since wehad to deal with a limited amount of dat.1. II .1pJ)l.'.1rs

L-2~

~YEAR

FIG. 9. D'I'panosiphuM p/alanoldes: observed time series(AI, and trajectories predicted by the ~sponse.surfar:e modelfRSMI u'~houl nol~ ,III. .lnd ~ilh nol~ () (t, I' ~ '.-',;:i.distnbutcd with ml:.ln Icro and sundard dl:vrallun " ., _:

PETER TURCHIN AND ANDREW D. TAYLOR300 Ecology, Vol. 73, No.

with the shape of the ACF, which rapidly decays tozero and does not show any signs of periodicity there-after. Thus, our result suggests that density fluctuationsof almost three orders of magnitude observed in thispopulation were produced by density-independent fac-tors. Nevertheless, the population is regulated around

i:010

an equilibrium, as indicated by the RSM result of ex-ponential stability and significant regressions of r, onboth N,_, (F,ss = 7.31, P < .01) and N'-2 (F,ss = 7.00,P < .05).

Equilibrium dynamics: damped oscillations. - Twoof the six cases classified as damped oscillations werenonstationary. One, the southern pine beetle (Den-droctonusfrontalis), may have been misclassified, sincewe do not know how to detrend a series with the kindof non stationarity exhibited by the southern pine beetle(constant mean but increasing amplitude of oscilla-tions), The second nonstationary case, H.vloicus pi-nastri showed a trend in the mean. Removing the trenddid not alter the RSM result, but did produce strongerevidence for periodicity in the ACF (ACF was signif-icantly negative at the half-period. but not significantlypositive at the full period).

The damped-oscillation dynamics reconstructed byRSM for the stationary cases ran the complete spec-trum from rapid to slow convergence to the equilib-rium. The slowest convergence to equilibrium wasfound in the fall webworm population (Fig. 68). whichis one of the populations with ACF suggestively. butnot significantly periodic. It is known that populationscharacterized by oscillations slowly converging to anequilibrium will behave like noisy limit cycles in astochastic environment (e.g., Poole 1977). Thus, add-ing a modest amount of stochastic variation to thedeterministic dynamics extracted by RSM producessustained pseudoperiodic oscillations (Fig. 6C).

Cofnplc.\: d.vnafnics: limit c.vclt's and quasiperiodici-/.v. - The three insect time series that Wl're C!3SS;:;':u byRSM as limit cycles or quasiperiodic dynamics ".erealso the ones for which the ACF had a significantlyperiodic ~I.)mponent (Fig. 3E. F, and G ::1 T;ob:..- 3).Moreover, the period of extracted oscillations was veryclose to the observed period: 8 vs. ') yr for larch bud-moth (Fig. 7), 7 vs. 8.5 yr for gypsy moth (Fig. 8). and

10 20

YEAR

FIG. II. Ph vI/aphis jagi: observed time series (A). andtrajectories predicted by the response-surface model (RSM)without noise (B), and with noise (C) (f, is normally distributedwith mean zero and standard deviation t1 = 0.2).

:.: ~~~::::c=:::=:,,:= ::::£>

FIG. 10. The stranae attractor extracted from the Phyllaphis time series. This pph was produced by iterating Eq. 6 2200times on the computer. The first 200 points were discarded, and the last 2000 points were plotted in the N, vs. N'-I vs. ;V,-,phase space. To aid in visualizing the attractor. a projection of the attractor onto the N'-I vs. N,_, plane is also shown.

COMPLEX DYNAMICS IN ECOLOGICAL DATA JOIFebruary 1992

ACF ACF

2

~tJI0: ].

"20 .0 60

YEAR

eO 3;00 1;aq

ACf' ACF

1.~..~)

3

~ z:i:tit0

'i:010

.-41. 1.

10 20 30 10 20 30 40 50 60 70 80 90100

YEAR YEAR

FIO. 12. Four mammal time series for which the autocorrelation function (ACf) was sianificantly periodic. (A) lynx. (8)belyak hare. (C) colored fox. (0) arctic fox.

of RSM-estimated parameters? We addressed thisquestion by performing a sensitivity analysis on thedata set. We excluded each data point in turn, esti-mated the response surface for the reduced data set,and determined its qualitative behavior. Our resultsindicate that the prediction of chaos in this case wasnot due to a freak combination of "just right" datavalues, since chaos was extracted in 9 out of 17 reduceddata sets, with the rest divided between limit cycles (2cases, with periods of 8 and 5), stability (3 cases), anddiverging oscillations leading to extinction (3 cases).

RSM-predicted dynamics (Fig. 118 and C) \\erecharacterized by penods of exponentl:!! growth I"f -'-4 yr (lines of constant slope on the loa scale) followedby crashes, as well as by periods of rapid oscillations.Some features of the observed trajectory were similarto RSM J:-namics. Observed time senes had t"o r~-

2 vs. 2 yr for sycamore aphid (Fig. 9). The relativeamplitude of the oscillation in the larch budmoth andthe sycamore aphid was also matched by the RSMtrajectories (Figs. 7 and 9), although RSM underesti-mated the amplitude of gypsy moth oscillations. Sucha close correspondence between patterns observed inactual time series and the time series generated byresponse surfaces is a strong indication that RSM is atthe very least capable of correctly reconstructing pe-riodic complex dynamics.

Comple.t" dynamics: chaos.-Finally. in one case.Ph.vl/aphisfaKi. RSM-extracted dynamics were of thecnaollc kind. The "strange" nature of the attractor ex-tracted from this time senes is apparent when it isplotted in the N, - Nt I - N,.! phase space (Fig. 10).It is not dear, howev~r. how robust this result is. Doesthe prediction of chaos d~pend on a delicate balance

PETER TURCHIN AND ANDREW D. TAYLOR Ecology, Vol. 73, No.302

2 Vertebrate data sets(A)

The time series of vertebrate populations exhibiteda similar spectrum of ACF patterns. In particular. ex-amination of ACFs suggested that there were four cy-clic mammal populations (Fig. 12). Vertebrate popu-lations also exhibited many of the same dynamic

I behaviors that we found among insects: 3 cases exhib-

ited unstable oscillations leading to extinction. 6 caseswere classified as exponential damping. 11 cases asdamped oscillations, and 2 as quasiperiodic dynamics.There were no cases of chaos. Of the four mammalpopulations that had significantly periodic ACF. twowere found to have quasiperiodic RSM dynamics (lynxand belyak hare). RSM-reconstructed dynamics for thecolored fox and the arctic fox were damped oscillations.The damped oscillations regime is more plausible thana four-point cycle for these populations because theACF peaks were of rather small magnitude: ACF atthe first peak. 4 yr (ACF[4» was <0.4 (compare thiswith the lynx ACF(IO] = 0.6. or the larch budmoth

, ACF[9] = 0.7). Such a sharp drop-off in ACF reflectsa much noisier-looking time series of the two foxes.compared to either lynx or belyak hare. and thereforeis more consistent with RSM-indicated oscillatory

) damping. than with limit cycles. The period of dampedoscillations predicted by RSM was 4 yr (Fig. 13). thesame as the pattern in the ACF. This result once againdemonstrates the ability of RSM to accurately mimicthe patterns observed in actual time series.

AA/\!\;1,/~v " vv v v

2(B)

~'~~\ ~r,\ r: '\!

~

'i:010

v~ v

!(C)

"1\1'\ 1\~ I\:N vyv-Jv \J v"

1.0 20 30 4C

YEAR

FIG. 13. Colored fox: observed time series (A) (only the

middle 40 yr a= shown), and trajectories predicted by theresponse-surface model (RSM) without noise (8), and with

noise (C) (E, is normally distributed with mean zero and stan-dard deviation.,. - 0.2). DISCUSSION

Our results are very different from those of Hassellet al. (1976), who concluded that all but 2 of their 24insect populations had exponentially stable point equi-libria. By contrast, our response-surface methodology(RSM) found exponential stability in only 3 of our 14insect populations. The remaining populations wereclassified as unregulated (one case), damped oscilla-tions (six cases), limit cycles (one case), quasiperiodicoscillations (two cases), and chaos (one case). The ver-tebrate examples exhibited a similar spectrum of dy-namics, although there were no cases of chaos. We donot wish to claim that all of these classifications (es-pecially the two most extreme ones, Phy//aphis andPhy//opertha) are correct. This fairly small number ofexamples does, however, include convincing cases ofperiodic dynamics (damped oscillations, limit cycles,and quasiperiodicity) and one case with parameter val-ues at least approaching those producing chaos. Weconclude, then. Ihal thl: complete spectrum of dynam-ical behaviors, ranging Irom exponenlial stability 10chaos, is likely to be found among nalural populatIons.

The contrast between our findings and those of Has-sell et al. (1976) resulted from three importanl differ-ences in methodology: (I) fitting actual time-series dalainstead of the two-step method of Hassell et al.. (2)using a model with a much more Ilexible functional

riods of almost exponential growth (3 and 4 yr), withthe first period followed by a crash (what happenedafter the second period is unknown), and there was aperiod of rapid oscillations during the middle portionof the time series (Fig. 11 A). On the other hand, theactual trajectory did not exhibit a rigid regulatory "ceil-ing" that was a characteristic feature of simulations.Another point of similarity between the observed andextracted dynamics was that ACFs of both exhibitedweak periodicities, with a period of 7 yr in the dataand 5 yr in the RSM model.

At this time we know little about the ability of RSMto extract deterministic chaos from data. In addition,the data are sparse. Therefore we cannot make anydefinite statements about whether endogenous dynam-ics of the Ph.vl/aphis population are chaotic or not.However. the fact that RSM did extract chaotic dy-namics in at least one case indicates that the regionwithin the parameter space where the model (EQ. 6) ischaotic overlaps with the region enclosing parameterestimates for actual insect populations. In other words.one does not need to postulate biologically unrealisticvalues of parameters to obtain chaotic dynamics withinthe framework of the model (Eq. 6).

COMPLEX DYNAMICS IN ECOLOGICAL DATA 303February 1992

.A

4

a'"O-o:~

~q~~

24

4

L . :Q .'-.':.;

.,.]. 2 3 . ;.- 6In N,..

",B

FK1. 14. Response function estimated for lan:h budmoth: (A) r, is. function of N,N'_I and N'_2.

only, and (B) r, is a function of both

form, and (3) accounting. albeit indirectly, for the mul-tidimensional nature of population dynamics that couldbe due to interactions with other populations withinthe community, or to population structure. We believethat the last of these differences is the most critical.Indeed. 6ttina data with a nrst-order model (only termsinvolving Nt-I [the previous year's density]), which inaU other respects was identical to model (Eq. 6), pro-duced results very similar to those of Hassell et al.: IIcases of exponential stability, 2 cases of damped os-cillations, and one limit cycle (the sycamore aphid). Itis revealing that this truncated model, as well as theanalysis of Hassell et al. (1976), classified the larchbudmoth population in the Engadine Valley in Swit-zerland (see Baltensweiler and Fischlin 1987) as ex-ponentially stable, although this population is arguablythe most convincing c\amplc of a quasiperIodic at-tractor in our data set. ThIs misclasslhcatlon happenedbecause in this population there is little elfcct of Nt I

on 't (the rate of population change), and a larle effectof ,V, : (compare FIg. 14A 10 I ~B). Wh.:n we rcdu,-c

~

the dimensionality of the model by ianoring jV,-~, weturn a clean, strongly nonlinear response surface inthree dimensions into a cloud oflarlelY uninformativepoints in two dimensions. Fitting the model to thesepoints then yields a gentle slope (Fig. 14A), indicatinamild direct density dependence and thus stability.

The flexibility of our RSM model, provided by in-clusion of both the Box-Cox transformation and qua-dratic terms. also was essential for correct classifica-tion. For example. the quadratic term (although notthe second lag) was necessary for an accurate recon-struction of the sycamore aphid dynamics. Fitting amodel with either one or two lags but no quadraticterms leads to a classification of dampc.-d oscillations.in contrast to the conclusion of a two-point limit CYl:leubtained by a quadratic RSM (with either one or bothlags). Simul:lIIOnS of the RSM model ".I!~ no qu..dratlctc:rms produced an ACF that decayed to zero muchfaster than the ACF of either the data or the full RSMmodel. In addition. the full modd came much closerto rl:producing thc: pert'ect alternation of inl:rl.';1~..', ..;1";

PETER TURCHIN AND ANDREW D. TAYLOR EcoIOlY. Vol. 73. No.

Gurney 1982:55). Our results suggest otherwise. Weargue. therefore. that natural populations cannot beranked within a one-dimensional spectrum goina fromno regulation to tight regulation around a point equi-librium. Instead. a two-dimensional scheme needs tobe emplqyed. with one axis indicating the relativeS1ren&th of the exogenous (density-independent) com-ponent. and the other axis indicating the type of en-dogenous dynamics.

decreases seen in the observed series. Thus. the qua-dratic term was essential for reaching the correct con-clusion in this case.

The preceding examples suggest that leaving out im-portant factors, such as delayed density dependence orstrong nonlinearities, may lead to incorrectly classi-fying a population as more stable than it actually is.In other words, use of overly simple models for re-constructing endogenous dynamics from data may bebiased in favor of finding stability. This may well applyto our own analysis, since regressions of r, on laggedpopulation densities indicate that lags of order higherthan two are not infrequent (P. Turchin and A. DoTaylor, unpublished analysis). Analysis of cases withhigher dimensional response surfaces might well resultin additional findings of complex dynamics, thoughthe feasibility of such expanded analysis will be limitedby the relatively short length of a typical ecologicaltime series.

The methodology used in this paper is by no meansperfect. For instance, it cannot effectively handle sys.tems with multiple equilibria. By applying a standardmodel to each case, we also risk. misclassifying someinstances by miDI an inappropriate model. As we not-ed above, inclusion of additional lags may be appro-priate in a number of cases (subject to data constraints).However, our model may be more complex than need-ed for some systems; whether such overfittina in anyway biases the results is unknown but under investi-gation.

Another limitation is that our approach currentlylacks any means for determinina "confidence inter-vals" around our dynamical predictions. Confidencelimits can be obtained for each parameter estimate ofthe model (Eq. 6), but they tell us nothing about howvariation in parameter estimates will affect our con-clusion about the type of extracted dynamics. Anotherpotential problem is the estimation bias that ariseswhen models such as Eq. 6 are fitted to data withsubstantial observation errors (Walters and Ludwig1987).

In closing we note that much controversy surround-ing the issue of population regulation stems from theone-dimensionai viewpoint, held by many, that at-tempts to place all populations within the spectrumranging from tight control around a stable point equi-librium (regulation) to little or no dynamical feedbackin population density (no reaulation). Bias against com-plex endogenous dynamics is so strong that most dis-cussions or criticisms of population reaulation do notmention (e.I., Wolda 1989)-or even dismiss out-right-the possibility that populations may undergocyclic or chaotic l1uctuations. The following quotationsshow that this view is shared both by experimentalists:"the rarity with which populations fluctuate cyclicallyin nature. . ." (Hairston 1989:6). and theoreticians:"deterministic stability is the rule rather than the ex-ception, at least with insect populations" (Nisbet and

ACKNOWLEDGMENTS

We thank J. Allen. A. Berryman. L. GinzburJ. J. Hayes. P.Kareiva. and W. Morris for readina and commentina on thcmanuscript. A. Berryman. A. Dixon. and W. Morris werevery generous in sharin, data sets.

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