The Moran effect predicts that a plot ofpopulation correlation versus environmen-tal correlation will yield a straight line ofunit slope if the two synchronized systemsare linear. By measuring re in the mannerintended by Moran2 (Fig. 1b, legend), wefind that the nonlinear threshold model1

provides results that scale very closely to thelinear relationship (Fig. 1d), so the Moraneffect survives. When rp and re are plottedover the entire range of parameters used inref. 1 (Fig. 1a and b, respectively), the twographs are almost identical and reörp, aclear demonstration of the Moran effect.

Why then did Grenfell et al. fail to repro-duce the Moran effect? In their approach,the environmental correlation r, as set bytheir model’s control parameters, is oftenunrepresentative, as shown by the followingargument. The environmental shocks actu-ally experienced by the sheep populations

have, by construction, correlation r onlywhen the populations are both above orbelow a fixed threshold level; but when onepopulation is above threshold and the otherbelow, a situation that can sometimesconstitute half of any simulation run, therespective environmental shocks at the twoislands are, again by construction, uncorre-lated. These long periods of decorrelationcan cause the true environmental correla-tion, re, calculated over the entire simula-tion run, to be markedly less than the valuer defined by the model parameters. It is dueto this misestimation of correlation (andnot because of the presence of a nonlinearthreshold1,3,4) that Grenfell et al.1 require anunusually high environmental correlationto yield the observed sheep synchrony.

Grenfell et al. have demonstrated thepotential of using nonlinear statisticalmodels for ecological investigations.

846 NATURE | VOL 406 | 24 AUGUST 2000 | www.nature.com

Ecology

Nonlinearity and theMoran effect

The study of synchronization phenom-ena in ecology is important because ithelps to explain interactions between

population dynamics and extrinsic environ-mental variation1–5. Grenfell et al.1 haveexamined synchronized fluctuations in thesizes of two populations of feral sheepwhich, although situated on close but iso-lated islands, were nevertheless stronglycorrelated (observed value of the popula-tion correlation, rp, 0.685). Using a nonlin-ear threshold model, they argue that thislevel of population correlation could onlybe explained if environmental stochasticitywas correlated between the islands, with theenvironmental correlation, re, higher than0.9 “on average” (Fig. 1a). This unusuallyhigh environmental correlation is fargreater than would be predicted by theMoran effect2, which states that the popula-tion correlation will equal the environmen-tal correlation in a linear system. Grenfell etal.1 imply that a simple nonlinearity in pop-ulation growth can mask or even destroythe Moran effect1,3,4. Here we show thatthese surprising results are an artefact of thetechniques used to measure noise correla-tions and synchronization.

In their simulations1, Grenfell et al.based all correlation calculations on a timeseries of n4800 model samples, which isfar larger than the n418 pairs of observedsheep-population data. To obtain a correctcomparison of the model with theirobserved results, we repeated their calcula-tions, but examined only n418 pairs ofsimulated data samples for each model run;these smaller sample sizes should tend toincrease statistical variability in the analysis.

Our study of an ensemble of simulationruns revealed that only a very weak envi-ronmental correlation re (calculated as forFig. 1b) is required for the model popula-tions frequently to achieve a correlation rp

greater than or equal to the observed corre-lation r40.685. As Fig. 1d reveals, theobserved correlation (or higher) can befound by chance alone, appearing in aboutone in 20 simulations for re40.3, and thenmuch more frequently as re increases.

The main conclusion of Grenfell et al.,that very strong environmental correlation(re¤0.9) is responsible for the observedsynchronized sheep-population fluctua-tions, therefore needs further considerationif a far smaller environmental correlation(re40.3) can equally well explain the samedynamics. By failing to take into accountthe lengths of the ecological time series,their analysis may have misinterpreted thecauses of synchrony in these feral sheeppopulations.

brief communications

a b

c d

r p r e

r p

00.5

1.0

0

0.5

1.00

0.5

1.0

rbra 0

0.51.0

0

0.5

1.00

0.5

1.0

rbra

0.2

0.4

0.6

0.8

1.0

0 0.2 0.4 0.6 0.8 1.00

0.2

0.4

0.6

0.8

1.0

r

Figure 1 Simulation results using the nonlinear threshold model with density-dependent switching between two independent environ-

mental noise components, as described by Grenfell et al.1. a, Correlation between island sheep populations, rp, as a function of the

model’s noise parameters ra and rb (Fig. 3a of ref. 1). The model1 is set up so that ra and rb represent the correlation of the environmental

noise affecting the two sheep populations when they are both either above or below threshold, respectively (see c for colour calibration).

According to ref. 1, rp equals the observed value of ro40.658 only if the environmental correlations ra and rb are larger than “0.9 on aver-

age”. b, True environmental correlation, re, as a function of the model parameters ra and rb. To calculate re, we stored all environmental

shocks received by each model sheep population and then simply determined their correlation: hence re is viewed as the correlation co-

efficient of shocks directly experienced by the two model sheep populations. The surface for re is almost identical to that of rp in a,

demonstrating a Moran effect. c, Colour-coding of correlations depicted in a and b. d, Analysis of systems in which model environmental

correlation parameters r4ra4rb (that is, systems along the diagonal of parameter space in a and b). Red curve, cut through surface in a

above the line r4ra4rb. The correlation between sheep populations rp is plotted against the model’s environmental correlation parameter

r; the deviation from the reference diagonal (black) seems to indicate the absence of a Moran effect, as argued in ref. 1. Blue curve, rp vs

re, where re (horizontal axis) is the true environmental correlation (see b). Note that this line is very close to the reference diagonal (black),

where rp4re, as predicted by the Moran effect. Magenta line, effects of small sample size (n418) due to statistical scattering in an

ensemble of 1,000 simulation runs: for each value of environmental correlation re (horizontal axis), 5% of the simulations have rp values

larger than the value plotted. Green horizontal line, the measured correlation between the two observed sheep populations is ro40.685.

The point of intersection of the green and magenta lines yields the minimum value of the ‘true environmental correlation’, re, for which we

can expect to find the observed sheep correlation by chance alone (that is, in at least 1 in 20 runs, or 5%, when re40.3). Further details

are available from the authors.

© 2000 Macmillan Magazines Ltd

Refining these techniques should increaseour understanding of the relation betweenenvironmental fluctuations and populationdynamics, in the spirit of Moran.Bernd Blasius, Lewi StoneThe Porter Super-Center for Ecological andEnvironmental Studies,Department of Zoology, Tel Aviv University,Ramat Aviv, Tel Aviv 69978, Israele-mail: lewi@lanina.tau.ac.il

1. Grenfell, B. T. et al. Nature 394, 674–677 (1998).

2. Moran, P. A. P. Aust. J. Zool. 1, 291–298 (1953).

3. Stenseth, N. C. & Chan, K.-S. Nature 394, 620–621 (1998).

4. Hudson, P. J. & Cattadori, I. M. Trends Ecol. Evol. 14, 1–2

(1999).

5. Blasius, B. et al. Nature 399, 354–359 (1999).

Grenfell et al. reply — The Moran effectrefers to systems of population dynamicsthat are linear: under these circumstances,the long-term correlation between popula-tion densities will be the same as the corre-lation between the random environmentalperturbations. The Soay sheep exhibit sig-nificant nonlinearity in their density depen-dence (Fig. 2a of ref. 1). At low populations,numbers tend to increase exponentially,with mean growth rate r40.24, whereas athigh densities (above a threshold of 1,172animals), the population tends to decline,with mean r410.29. Thus, when popula-tions on two adjacent islands are bothabove their thresholds, both will tend todecline, and when both are below theirthresholds, both will tend to increase.

This immediately introduces a positivecorrelation between population dynamicson adjacent islands in the absence of anyenvironmental forcing. When one popula-tion is above the threshold and one isbelow, the expectation is of no short-termcorrelation because the trends will tend tocancel out. Adding noise to the system hastwo effects. Correlated noise tends to pushthe dynamics into synchrony, because bothpopulations tend to crash to low densitiesduring the same years (for example, thosewith the most extreme winter weather). Thenonlinearity means, however, that the samestochastic event could drive one populationbelow the threshold but leave another pop-ulation above it (for example, if initial pop-ulation densities are sufficiently different).In this case, the two populations wouldexperience different regimes of densitydependence during the same year and syn-chrony would be reduced. Intuitively, then,in the presence of nonlinear density depen-dence, environmental forcing has to bestronger if it is to drive the populations intosynchrony and keep them there.

Blasius and Stone have pointed out twoproblems with our analysis. First, they showthat we had the two populations experienc-ing different — hence uncorrelated — noiseduring periods when the two populationswere on opposite sides of the threshold.Correcting this mistake reduces the level

of noise correlation (rn) required to producethe observed level of population correlation(rp) (rp40.685) from rn¤0.9 to around0.8 for large samples. This means that theextra-Moran effect is reduced, but notabolished.

Their second point is that, with realisti-cally short time series (such as our 18points), variability in the inter-populationcorrelation coefficient generates a relativelyhigh expectation of observing correlationshigher than Moran, leading to inflatedtype-I errors. We have carried out furthercalculations with the corrected model thatagree qualitatively with this. However, evenshort model simulations show the imprintof nonlinearity in their aggregate correla-tion structure — a strong downward bias inpopulation correlation for a given level ofnoise correlation, compared to various lin-ear null models (Fig. 1; for more details, seeref. 2). Thus, the impact of nonlinearity onpopulation correlation is apparent in thecollective behaviour of short simulations, aswell as in individual realizations of themodel’s long-term dynamics.

There are several important directionsfor studies on the interactions betweennoise and determinism in populationdynamics. Most important is an increase inthe realism of the underlying model. Theinclusion of age- and sex-structure effects isessential, because we know that animals ofdifferent ages and sexes experience marked-ly different patterns of mortality3. A furtherimprovement would incorporate thresholddensity as a random variable rather than aconstant (it is intraspecific competition forfood that underlies the density dependence,and food supply determines the sheep den-sity at which competition kicks in). Thiswould allow for island-to-island differencesin the response of food availability to envi-ronmental noise, so that islands with thesame population densities could experience

brief communications

NATURE | VOL 406 | 24 AUGUST 2000 | www.nature.com 847

different density-dependence regimes in thesame year.

The ability to detect extra-Moran corre-lations depends critically on the balancebetween noise and density dependence (B.Blasius and L. Stone, personal communica-tion), so any model refinement thatexplains more variability in terms of popu-lation processes will increase our powers ofevaluation. Technically, developments innonlinear time-series analysis need toencompass estimation of age4 and spatialheterogeneities, as well as the dissection ofprocess noise from measurement error. Thisis particularly important for the relativelyshort time series found in ecology, whereeven linear time-series models can producea complex range of correlation behaviours2.

A thorough understanding of spatialdynamics can only come about once theinteraction between correlated noise andnonlinear density dependence is under-stood through long-term ecological studies,combined with models.B. T. Grenfell*, B. F. Finkenstädt†, K. Wilson‡, T. N. Coulson§, M.J. Crawley||*Zoology Department, University of Cambridge,Downing Street, Cambridge CB2 3EJ, UK†Department of Statistics, University of Warwick,Coventry CV4 7AL, UK‡Institute of Biological Sciences, University ofStirling, Stirling FK9 4LA, UK§Institute of Zoology, Zoological Society of London,Regent’s Park, London NW1 4RY, UK||Imperial College, Silwood Park, Ascot SL5 7PY, UK

1. Grenfell, B. T. et al. Nature 394, 674–677 (1998).

2. www.zoo.cam.ac.uk/zoostaff/matt/GROUP/Download/

3. Grenfell, B. T., Price, O. F., Albon, S. D. & Clutton-Brock, T. H.

Nature 355, 823–826 (1992).

4. Leirs, H. et al. Nature 389, 176–180 (1997).

0 0.5 1.0–0.5

0

0.5

1.0

Correlation (noise), rnC

orre

latio

n (p

op),

r pFigure 1 Scatter plot (blue) of inter-island population correlation

(rp) against true noise correlation (rn) for 3,000 simulations (each

of 18 time points) of the SETAR model, defined as in Table 1 of

ref. 1, with the correction in noise realization proposed by Blasius

and Stone. The black line indicates where population correlation

equals the noise correlation (the expectation of the Moran effect).

See ref. 2 for more details.

Physiology

An actively controlledheart valve

Vertebrate hearts typically have cardiacvalves that are thin and leaf-like andwhich work passively, allowing blood to

move forward during systole and preventingit from flowing back during diastole. Croco-dilian hearts have nodules of connectivetissue, resembling opposing knuckles, orcog-teeth1–3, in the subpulmonary conus justproximal to the pulmonary valves. Here weshow that these cog-teeth act in the estuar-ine crocodile Crocodylus porosus (Fig. 1) as avalve that regulates the flow of bloodbetween the lungs and the systemic circula-tion in response to a b-adrenergic mecha-nism. To our knowledge, this is the firstreport of an actively controlled intra-cardiacvalve in a vertebrate.

Among the vertebrates, only the mam-mals, birds and crocodilians have anatomi-

© 2000 Macmillan Magazines Ltd