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Fitting Ecological Process Models to Spatial Patterns Using Scalewise Variances and MomentEquations.Author(s): Matteo Detto and Helene C. Muller-LandauSource: The American Naturalist, Vol. 181, No. 4 (April 2013), pp. E68-E82Published by: The University of Chicago Press for The American Society of NaturalistsStable URL: http://www.jstor.org/stable/10.1086/669678 .

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vol. 181, no. 4 the american naturalist april 2013

E-Article

Fitting Ecological Process Models to Spatial Patterns

Using Scalewise Variances and Moment Equations

Matteo Detto* and Helene C. Muller-Landau

Smithsonian Tropical Research Institute, Apartado Postal 0843-03092 Balboa, Ancon, Panama

Submitted June 7, 2012; Accepted October 29, 2012; Electronically published February 19, 2013

Online enhancement: zip file with appendixes, supplemental figure.

abstract: Ecological spatial patterns are structured by a multiplicityof processes acting over a wide range of scales. We propose a newmethod, based on the scalewise variance—that is, the variance as afunction of spatial scale, calculated here with wavelet kernel func-tions—to disentangle the signature of processes that act at differentand similar scales on observed spatial patterns. We derive exact andapproximate analytical solutions for the expected scalewise varianceunder different individual-based, spatially explicit models for sessileorganisms (e.g., plants), using moment equations. We further de-termine the probability distribution of independently observed scale-wise variances for a given expectation, including complete spatialrandomness. Thus, we provide a new analytical test of the null modelof spatial randomness to understand at which scales, if any, thevariance departs significantly from randomness. We also derive thelikelihood function that is needed to estimate parameters of spatialmodels and their uncertainties from observed patterns. The methodsare demonstrated through numerical examples and case studies offour tropical tree species on Barro Colorado Island, Panama. Themethods developed here constitute powerful new tools for investi-gating effects of ecological processes on spatial point patterns andfor statistical inference of process models from spatial patterns.

Keywords: moment equations, spatial logistic model, spectral analysis,negative density dependence, spatial patterns, wavelet variance.

Introduction

Nonrandom spatial patterns are ubiquitous in ecology andprovide important information on processes structuringcommunities, including dispersal, competition, density de-pendence, and habitat associations (Law et al. 2009; Mc-Intire and Fajardo 2009). However, similar spatial distri-butions can emerge from different mechanisms ordifferent combinations of mechanisms. A first step in dis-entangling different processes of pattern formation is todefine the scales at which the pattern occurs and thereby

* Corresponding author; e-mail: [email protected].

Am. Nat. 2013. Vol. 181, pp. E68–E82. � 2013 by The University of Chicago.

0003-0147/2013/18104-53892$15.00. All rights reserved.

DOI: 10.1086/669678

link the pattern to some mechanism operating at thesescales (Kershaw 1963; Levin 2000). Large-scale variability(∼0.1–10 km), for example, may be driven by topographicfeatures or soil heterogeneity operating at the same scales.Small-scale structure (!∼100 m) may be related to localdispersal and interindividual interaction.

Most of the widely used tools for characterizing spatialpatterns have limited ability to separate scales. For example,Ripley’s K, pair correlation densities, and functions relatedto second-order product densities more generally are de-signed to estimate average local densities around the focalindividual as a function of distance. By their cumulativenature, values at any distance reflect the combined effectsof mechanisms acting at different scales (Loosmore andFord 2006). For example, aggregation at short distances isfound in a habitat specialist plant with long dispersal dis-tance (fig. 1A) as well as a generalist with short dispersal(fig. 1B) and, of course, a specialist with short dispersal (fig.1B). This limits our ability to investigate basic ecologicalmechanisms contributing to pattern formation.

To overcome some of these limitations, we need to movefrom a framework where variability is analyzed as a func-tion of space to a new approach where the variance isdecomposed scale by scale. Spectral analysis is such anapproach, and it allows for a clear view of how differentfrequencies contribute to the pattern of interest (Bartlett1964; Platt and Denman 1975). Here, we employ spectralanalysis and focus in particular on the scalewise varianceusing wavelets, a second-order, consistent estimator of thespectral density (Percival 1995). Wavelet transforms havebeen employed in several studies of ecological time series(Cazelles et al. 2008; Detto et al. 2012) but less in spatialecology (e.g., Dale and Mah 1998; Keitt and Urban 2005;Mi et al. 2005), and we are not aware of any applicationsin point process analysis. The mathematical and statisticalproperties of wavelet transforms are well understood fromapplications in other fields (e.g., Kumar and Foufoula-Georgiou 1997). The advantage of this technique is that


mailto:[email protected]


Scalewise Decomposition of Ecological Point Processes E69

Figure 1: Contrasting spatial patterns, together with their estimated pair correlation densities and wavelet variances. A, Poisson process, amodel of global dispersal in a homogenous (left) and inhomogeneous (right) habitat. B, Simulated individual-based model with shortdispersal in a homogenous and inhomogeneous habitat. Note that the wavelet variance, unlike the pair correlation density, separates processesby scale. For example, habitat association at larger scales will elevate pair correlation statistics at all smaller distances, while the waveletvariance is substantially unaltered for scales at which habitat association is not operating.

it makes it possible to isolate the scales of the process ofinterest and analyze them separately, even in the presenceof nonstationarity (i.e., habitat heterogeneity, a case thatis not addressed in this study).

Individual-based, spatially explicit models are anotherkey tool in exploring the determination of spatial patterns(Pacala 1986; Durrett and Levin 1994). Such models cap-ture spatiotemporal dynamics with a plant’s-eye view or,more generally, an individual-based view. Dispersal andspatial interactions among individuals located a specifieddistance apart are represented by kernel functions describ-ing the mechanisms that govern the interaction or thedispersal/movement event. Moment closure can be appliedto describe the dynamics of mean densities and pair cor-relation densities and thereby obtain analytical results(Bolker and Pacala 1997; Law and Dieckmann 2000).These methods have improved our understanding of thestructure of plant communities, showing how endogenousaggregation patterns affect species interactions and coex-istence (Murrell and Law 2002; Bolker et al. 2003). Mo-ment equations can also be developed—and, indeed, aremore easily handled—in the Fourier transformed space(Bolker et al. 2000; Ovaskainen and Cornell 2006b). How-ever, little has been done to apply moment methods toreal data in a statistical inference framework. Even in caseswhen the mechanisms of pattern formation are well un-derstood, the quantification of their effects remains a for-midable challenge (McIntire and Fajardo 2009; Thompsonand Katul 2011).

One of the fundamental problems in pattern analysisconcerns understanding the limits to the information thatcan be extracted from a single random sample, includinghow many processes can be unambiguously discerned. For

example, the same static patterns of the distributions oftropical tree species in the large forest plot (50 ha) of BarroColorado Island have alternatively been used to investigatehabitat associations (Plotkin et al. 2000; Harms et al. 2001),effective dispersal distance (Anand and Langille 2010), andnegative density dependence (Bagchi et al. 2011). Thisraises the question of how all these processes can be an-alyzed simultaneously without interference of one processwith another.

Here, we show how scalewise variances and, specifically,wavelet variances can be derived analytically from spatiallyexplicit, individual-based models. We derive new analyticalresults on wavelet variances expected under models withlocal dispersal and neighborhood competition, using mo-ment methods in homogeneous environments. We furtherderive the asymptotic statistical properties of these waveletvariances and show how these can be used to estimateprocess parameters from observed spatial patterns. A keyresult is the derivation of an approximate likelihood func-tion that is consistent with the process underlying themodel and makes it possible to estimate the most likelyparameters and, importantly, their uncertainty. We dem-onstrate these methods with numerical examples and acase study of four tropical tree species. The methods pre-sented here constitute a powerful new tool for ecologiststo investigate spatial processes and patterns.

Theory

The Scalewise Variance: A Tool for Scale-By-ScaleDecomposition of Spatial Patterns

Consider a spatial point process that generates a randompattern of points that are the locations of individuals in



E70 The American Naturalist

a two-dimensional space of finite area A. A realization ofthis process can be represented as the sum of Dirac d

functions centered at the n point locations andx p (x, y)0 everywhere else: .

np(x) p � d(x � x )iip1

The first spatial moment, the mean density, is simply

1 nN p p(x)dx p . (1)�A A

The second-order spatial moment, the pair correlationdensity, holds information on how individuals are dis-tributed across space:

1C(r) p p(x)[p(x � r) � d(r)]dx, (2)�A

where r is the spatial lag and the d function removes thefocal individual.

Application of a Fourier transform, C(q) p, reduces the pair correlation density to a simple�jqrC(r)e dr∫

product minus a constant (the Fourier transform of a d

function is unity):

1 *˜ ˜ ˜C(q) p p(q)p (q) � N, (3)A

where is the angular frequency, is the˜q p (q , q ) p(q)x y

Fourier transform of p(x) (i.e., , with jn �jqx ip(q) p � eip1

the imaginary unit), and is its complex conjugate.*p (q)The first term on the right-hand side of equation (3) isthe two-dimensional spectral density function of p(x), orsimply the power spectrum, (Bartlett 1964). Knowl-G(q)edge of the function makes it possible to assess theG(q)scalewise variance Vw, that is, the expected variance inpopulation density sampled with a given kernel w(x)(Daley and Vere-Jones 2003, p. 304):

2˜V p G(q)Fw(q)F dq. (4)w �For convenience, we introduce the operator ⎡f(q)⎤ p

, where , so equation (4) can2˜f(q)H(q)dq H(q) p Fw(q)F∫be rewritten as . The functions w are essentiallyV p ⎡G(q)⎤w

smoothing filters whose properties depend on the type ofkernel and associated scaling parameter, s. They enablestudy of features of the process whose detail matches theirscale parameters, that is, broad features for large s and finefeatures for small s. For this purpose, kernels should befunctions with compact support in both the frequency andthe spatial domain, essentially compromising between fre-quency and spatial resolution in order not to lose locali-zation in physical space (in accordance with the Heisen-berg uncertainty principle, which in this context states thatit is impossible to achieve spectral and spatial resolutionsimultaneously). The scale parameter s is associated with

a physical scale to allow ready interpretation; for example,for isotropic problems, it is convenient to express s as afunction of the one-dimensional Fourier period l p

, which represents the distance among peaks of an2p/FqFoscillation.

Wavelets constitute a suitable family of kernel functionssatisfying the above requirements (Kumar and Foufoula-Georgiou 1997). Among many choices of wavelets, theMorlet (Daubechies 1992) is one of the most commonlyused wavelets and has proved effective in spatial ecologicalapplications (Mi et al. 2005). It can assume a simplifiedisotropic form, and its Fourier transform is

2�1 �(sFqF � q )0w(q) p exp , (5)( )c s 2w

where cws is a normalization factor to ensure that the wave-let has unit energy at all scales. The shape parameter q0

regulates the width of the support in the frequency space:for , the scalewise variance Vw(s) is basically identicalq 1 80

to its Fourier counterpart when the spectrum is relativelyfeatureless (e.g., it obeys a power law over a certain intervalof frequencies).

For this wavelet, the ratio between the equivalent Fou-rier period (l), and the wavelet scales (s), Ff exists andcan be derived analytically (Meyers et al. 1993); we findthat it is

l 4pF p p . (6)f 2�s q � 4 � q0 0

Using equations (3), (4), and (6) and normalizing by N(to permit comparisons among populations with differentdensities), we obtain the normalized scalewise variance asa function of the Fourier period:

C(q)2n (l) p � 1 . (7)

N⎡ ⎤This is the key spatial statistic we explore in the remainderof the article, although the theory above is not restrictedto wavelets or this particular type of wavelet. The waveletvariance can be thought of as a smoother and more con-sistent estimate of the Fourier spectrum, which is usuallyplagued with noise and in need of some arbitrary manip-ulation (e.g., tapering, binning, windowing). However, dif-ferences exist; while the Fourier coefficients are influencedby the process on its entire domain, the wavelet transformcoefficients are influenced by local events. This makes thewavelet spectrum a better measure of variance attributedto localized structures. Equations (6) and (7) are new re-sults that enable this work.

For more background on wavelets and wavelet vari-




ances, see Percival (1995), Kumar and Foufoula-Georgiou(1997), and Torrence and Compo (1998).

Statistical Properties of the Scalewise Variance Estimator

For complete spatial random (CSR) distributed popula-tions, the normalized scalewise variance is equal to 1 atall scales, just as for a Poisson process, the variance isequal to the mean density. Eventually, at small scales, thenormalized variance of any observed pattern tends to 1;as the probability to encounter two individuals in the samearea becomes negligible, the variance approaches the mean,and locally the process approaches a Poisson process(Daley and Vere-Jones 2003).

The null hypothesis of complete spatial randomness canbe tested using the properties of the Poisson distribution.For a CSR process, the number of individuals containedin a box of area is a Poisson variable with mean equal2�to . Given k independent realizations X1, X2, ..., Xk,

2N�drawn from a Poisson distribution with mean , the sumXof k normalized square deviations is approximately equalin distribution to a x2 with degrees of freedom:k � 1

2k (X � X)i w 2p x .� k�1Xip1

This relationship directly provides the confidence intervalsas a function of block size for box counting methods,which corresponds to kernel-like indicator functions:

�1, FxF, FyF ≤

2w(x) p .�{0, FxF, FyF 1

2

The degrees of freedom are simply the numbers of boxesminus 1: .2k p A/�

Our work shows that the same approach can also beused to estimate the confidence intervals of the waveletvariance for a CSR process. In this case, the determinationof the degrees of freedom is not so straightforward becausethe wavelet is not perfectly localized in physical space (incontrast with indicator functions). The degrees of freedomdepend on the number of independent points averagedby the wavelet filter, which varies with the analyzed scale.We would expect that in the same manner as2k ∝ A/lthe box counting method. Large sample theory ensuresthat, under fairly general conditions (Serroukh et al. 2000),the estimator is asymptotically normally distributed2n (l)with mean and finite large sample variance. For a2n (l)Gaussian process, the large sample variance is equal to

(Percival 1995), where2A (l)/As

2 2A (l) p G (q)H (q)dq. (8)s � s

Using an equivalent degrees of freedom argument, it fol-lows that is distributed as a x2 with h degrees2 2ˆhn (l)/n (l)of freedom given by

4An (l)h(l) p . (9)

A (l)s

The central limit theorem ensures that the Fourier trans-form of a Poisson process, , is asymp-

n �jqx ip(q) p � eip1

totically (complex) Gaussian because it is a sum of in-dependent, identically distributed random variables (for

). Solving the integrals (4) and (8) for the Morletn 1 5wavelet with the conditions and , we2G(q) { 1 n (l) { 1find

A2 2�h(l) p 2c F , (10)w f 2l

which is directly proportional to , as expected from2A/lthe box counting analogy. Monte Carlo simulations agreevery well with our resulting predicted confidence intervals(fig. 2). According to equation (10), the large sample prop-erties of the normalized wavelet variance for a CSR processare independent of the mean density (provided that thereare at least five individuals in the sampled area), a resultconfirmed by simulations. This makes the wavelet variancea pivotal discriminant statistic for testing the null hy-pothesis of nonrandomness for populations with differentnumbers of individuals.

It is important to note that the wavelet variance esti-mates at different scales are not completely independent,because the functions overlap for small spectral in-H (q)s

tervals. This can create complications when the leastsquares regressions or likelihood functions are computed.One approach would be to choose the interval betweenscales sufficiently large to minimize this problem, but thishas the drawback of losing resolution and important fea-tures of the wavelet variance. Alternatively, under the as-sumption that for two scales li and lj, wavelet variancescomputed from independent realizations are asymptoti-cally normally distributed with mean and , the joint2 2n ni j

probability of observed wavelet variances can be approx-imated by a multivariate normal distribution

(fig. S1, available online). We thus obtain the2MVN(n , S)(new) result that for a Gaussian process, the covariancematrix can be defined, similar to the large sample varianceabove, as , whereS p 2A /Aij ij

2A p G (q)H (q)H (q)dq. (11)ij � i j




2 4 8 16 3210

100

101

wav

elet

var

ianc

e

Fourier scale λ (m)

A

.95 1 1.05.95

1

1.05 λ = 2

pred

icte

d

.9 1 1.1.9

1

1.1 = 4

.8 1 1.2.8

1

1.2 = 8

observed quantiles

pred

icte

d

.8 1 1.2 1.4

.8

1

1.2

1.4 = 16

observed quantiles

C

D E

B

λ

λλ

-1

Figure 2: A, Wavelet variance estimates from 1,000 simulated point patterns of 500 individuals over 1 ha displaying complete spatialrandomness (gray lines), compared with the expected value (black line) and predicted 95% confidence intervals (red lines). B–E, Predictedand observed quantiles for different scales. Note that the width of the confidence interval increases with increasing scale (red lines in A),because the number of degrees of freedom declines with increasing scale. The quantiles predicted using the x2 approximation closely matchthe observed.

When is not known a priori, an estimate can be2 ˆG (q) Aij

used instead (Percival 1995).

Model Fitting

The statistical properties of the observed scalewise varianceenable parameter estimation using maximum likelihoodtechniques. Let be the observed wavelet variance. Then,2n

given a parametric model defining the expected waveletvariance as a function of a parameter vector b, and2n (b)assuming that the estimated wavelet variance deviations

are jointly normal distributed with disper-2 2n (l ) � n (l )i i

sion matrix S, we find that the likelihood function L isdirectly proportional to

12 2 2 T �1 2 2ˆ ˆ ˆL(n Fb) ∝ exp � (n � n (b)) S (n � n (b)) . (12)[ ]2

If the model is known analytically or by numerical2n (b)integration, maximum likelihood estimation can be doneusing a generalized nonlinear least squares algorithm (app.B, available online). More generally, it can be done usingstochastic methods, that is, by simulating point patternsfor specified parameters (for a review of such stochasticmethods, see Hartig et al. 2011).

The dispersion matrix S is known for Gaussian pro-cesses from and equation (11); however, it isS p 2A /Aij ij

unknown for the general case. A first estimate of the pa-rameters can be made using the Gaussian assumption. Ina second step, simulations should give the correct S. Thisprocess can be repeated iteratively until model parametersand their uncertainties converge. We found that one it-

eration was usually enough. The likelihood function (eq.[12]) is a new result that constitutes the foundation of thefitting method.

Models

A General Spatial Logistic Model for Population Dynamicsand Its Expected Scalewise Variance

Consider a spatial-temporal process p(x, t) that representsthe dynamics of a single population of identical sessileindividuals, such as plants, on a two-dimensional spatialdomain. Individuals die at rate m and reproduce at ratef, and propagules (e.g., seeds) are dispersed from the parentaccording to dispersal kernel D(x). The establishmentprobability of a propagule depends on the positions of allthe conspecific individuals in its neighborhood, with theinfluences of neighbors described by interaction kernelK(x). At each time t, the process is regarded as a realizationof a point pattern process and described by the functionp(x, t). The probability of finding a new individual atlocation x is given by the convolution f D(x �∫

. The density-dependent probability of estab-′ ′ ′x )p(x , t)dxlishment of new individuals due to competition withneighbors (including parents) is expressed by the convo-lution .′′ ′′ ′′1 � a K(x � x )p(x , t)dx∫

Assuming isotropy of the dispersal and competition ker-nels and a homogeneous environment, the temporal dy-namics of the first and second moments (Bolker et al.2000; Dieckmann and Law 2000) follow




dNp (f � m � fa(D * K)(0))N

dt

′ ′ ′� af (D * K)(r )C(r )dr , (13a)�1 dC(r)

p f ND(r) � fD * C(r)2 dt

′ ′ ′′ ′′ ′ ′ ′′� af D(r )K(r � r )T(r , r � r)dr dr�� mC(r), (13b)

where * represents a two-dimensional convolution integraland is the third moment (for details, see app.′′ ′T(r , r � r)A, available online).

We obtain the exact analytical solution for the expectedscalewise variances of this model under the simplified case,in which establishment is purely spatial independent(model I), and the approximate solution for the full model(model II).

Model I: Dispersal Only

If establishment is purely spatial independent (e.g.,), then the spatial pattern is due entirely to�1K(x) p A

dispersal. The equation for the first moment reduces tothe classic logistic model

dN2p (f � m)N � af N (14a)

dt

and the second moment to

1 dC(r)p

2 dt

(f ND(r) � fD * C(r))(1 � aN) � mC(r). (14b)

It is convenient to rewrite equation (14b) in the Fourierdomain, because the convolution integral simplifies to aproduct among transformed variables and the solution isdirectly comparable to the scalewise variance (by meansof eq. [7]):

ˆ1 dC(q)p

2 dt

˜ ˜˜ ˜(f ND(q) � fD(q)C(q))(1 � aN) � mC(q). (15)

The stable steady state solution, which requires N p, is(f � m)/af

˜ND(q)C(q) p . (16)˜1 � D(q)

This can be expressed in terms of the normalized scalewisevariance as

12n (l) p , (17)˜1 � D(q, c )⎡ D ⎤

where cD is the dispersal kernel parameter. The exact sameresult would be obtained if establishment is purely densityindependent ( ).a p 0

Equation (17) can be used to estimate the dispersalparameter from the observed scalewise variance for anygiven dispersal kernel. This is completely straightforwardif the Fourier transform of the dispersal kernel (i.e., itscharacteristic function) is known analytically (table 1) andcan be done numerically in other cases. The wavelet var-iance provides a clear signature of the effects of differentdispersal parameters (fig. 3A) and also differs among dif-ferent dispersal kernels (fig. 3B).

One important property of this dispersal-only model isthat for large scales ( ), the logarithmic slope of thel k cD

variance is independent of the type of kernel and asymp-totically approaches 2. This suggests defining an equivalentGaussian factor for each kernel such that for large scalesall the curves collapse to the solution obtained with theGaussian kernel. The Gaussian factor for a particular ker-nel is obtained analytically from the limit forD(q, c )D

of the ratioq r 02 2�c q /21 � e

2G p lim . (18)f ˜1 � D(q, c)qr0

When the curves obtained with different kernels are plot-ted as a function of the Fourier scale normalized by theequivalent Gaussian dispersal distance, their wavelet var-iances differ only in a small region around the normalizedscales 1–10 (fig. 3B).

This model is simulated starting from an initial randompattern of n individuals on a torus of area A (or larger tominimize edge effects). A new individual is generated ata random distance, chosen according to the dispersal ker-nel distribution D(r, cD), from a randomly chosen parent.Then, a randomly chosen individual is removed from thepoint pattern. Simulation of births and deaths continuesuntil the second-order spatial moment is approximatelyconstant.

Simulations show that the analytical model correctlycaptures the mean spatial pattern (fig. 4C). Unlike thePoisson process, for patterns generated by this cluster pro-cess, the confidence intervals around the wavelet varianceestimates—and thus around —depend on the numbercD

of individuals n (fig. 3D). As , the assumption ofn r �normality is increasingly well satisfied, the dispersion ma-trix S for the likelihood function approaches the analyticalexpression derived above (eq. [11]), and the confidence




Table 1: Various one-parameter dispersal kernels, their Fourier transforms, and their Gaussianfactors, Gf

Name Dispersal kernel D(r, c)Fourier transform of kernel

D(q, c)Gaussian factor

Gf

Gaussian2 22 �1 �r /2c(2pc ) e

2 2�c q /2e 1Exponential 2 �1 �FrF/c(2pc ) e 2 2 �3/2(1 � c q ) �3/3Laplace 2 �1(2pc ) B (FrF/c)0

2 2 �1(1 � c q ) �2/2Uniform 2 �1(pc ) r ≤ c

0 r 1 c2J (cq)/cq1 2

Epanechnikov 2 �1 2 22(pc ) (1 � r /c ) r ≤ c0 r 1 c

�28J (cq)(cq)2�6

Biweight 2 �1 2 2 23(pc ) (1 � r /c ) r ≤ c0 r 1 c

2 2Hypergeometric0F1 (4, � 1/4c q ) �2 2

Note: B0 is the modified Bessel function of the second kind, Jn is the Bessel function of the first kind, and Hyper-

geometric0F1 is the confluent hypergeometric function (Abramowitz and Stegun 1965). Gf is a multiplicative factor of

the dispersal parameter such that the normalized wavelet variance (eq. [17]) is asymptotically equal to that for a

Gaussian kernel for large scales.

intervals approach those obtained analytically (fig. 3D).Maximum likelihood fits to simulated data show that thereis a small bias in parameter estimates (1%–4%), especiallyfor small data sets (table 2). The standard deviations ofthe estimates, , decline with increasing n. ObservedjcD

are lower than expected values for small n and veryjcD

accurate for (table 2; app. A).n ≥ 200

Model II: Dispersal and ConspecificDensity-Dependent Establishment

When spatial density-dependent effects are present, equa-tions (13) do not have a closed analytical solution; anapproximate solution can be obtained through momentclosure (app. A; Bolker et al. 2000; Murrell et al. 2004):

1 dC(r) ≈ f ND(r) � fD * C(r) � mC(r)2 dt

� af N D * C(r) � ND(r) � K * C(r) � NK(r) (19)[′ ′ ′ 2� D * K(r )C(r )dr � ND * K(0) � 2N .� ]

The steady state solution for the scalewise variance is

2n (l) ≈II

1(20)˜ ˜1 � (f/m)(1 � aN)D(q, c ) � (f/m)aNK(q, c )⎡ D K ⎤

(app. A). This model now involves all the parameters andthe mean density, N.

It is useful to further simplify this model for the pur-poses of examining its qualitative behavior. If we assumelong-range dispersal and competition, then N ≈ (f �

, and the scalewise variance becomesm)/af

12n (l) ≈ . (21)II ˜ ˜1 � D(q, c ) � [(f � m)/m]K(q, c )⎡ D K ⎤

The qualitative behavior of the wavelet variance under thismodel is a function of two nondimensional derived pa-rameters: a nonspatial parameter , whichP p (f � m)/m1

reflects the importance of negative density dependence(NDD), and a spatial parameter , which reflectsP p c /c2 K D

the relative scale of competition to dispersal.The influence of the density dependence parameter P1

on the wavelet variance clearly shows how NDD reducesaggregation relative to what would be expected from dis-persal alone (eq. [21]; fig. 4A). When competition anddispersal scales are equal and density-dependent effectsequal density-independent ones ( ), the distributionP p 11

is indistinguishable from a CSR process (fig. 4A, greenline). When density-dependent mortality is greater thandensity-independent mortality ( ), spatial distribu-P 1 11

tions are disaggregated, because the probability of findingnew individuals increases with distance from parents fordistances smaller than cD, exactly the pattern posited byJanzen (1970) in his classic article on the influences ofnatural enemies (fig. 4A, red line).

The influence of parameter P2 on the wavelet varianceshows how the relative scales of dispersal and competitiondetermine the shape of the wavelet variance function (fig.




10-1 100 101 102 103

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LaplaceGaussian

lUniform

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100

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BA

Fourier scale c DG f

c = 0.1D

c = 10D

c = 1D

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simulationsaverage95% CImodel

N = 25

N = 50

N = 200

N = 500

N = 1000

χ2

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ianc

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C D

Figure 3: Wavelet variances for model I, in which spatial patterns are driven entirely by dispersal. A, Analytical solutions for a two-dimensional Laplace dispersal kernel under different dispersal parameters. B, Analytical solutions for different dispersal kernels as a functionof the normalized Fourier period, where Gf is the equivalent Gaussian factor (table 1). C, Observed wavelet variances for 1,000 simulationson a square arena of ha with Gaussian dispersal kernel, with parameter values m and (gray lines), with their averageA p 1 c p 3 N p 200D

(black line) and 95% confidence intervals (CIs; black dashed lines under red line), compared with the analytical model (red line). D, 95%CIs of 1,000 simulations with m and ha, relative to the theoretical 95% CIs for Gaussian processes (dashed red lines).c p 3 A p 1D

4B). If competition occurs over shorter distances than dis-persal ( ), individuals can be disaggregated at smallP ! 12

scales, while at larger scales they start to form patternsbecause of dispersal limitation (fig. 4B, blue line). If com-petition occurs at longer scales than dispersal ( ),P 1 12

the spatial pattern for is insensitive to NDD andl ! cK

the solution resembles that for dispersal only, while atlarger scales, clustering is reduced far below that expectedunder dispersal alone (fig. 4B, red line). The latter case

results in the formation of a pattern with a characteristicscale (corresponding to the peak of the spectrum), just asoccurs in reaction-diffusion systems when a slow diffusingactivator (here dispersal) combines with a fast diffusinginhibitor (here competition; Turing 1952).

These qualitative insights are useful, but unfortunately,the underlying equation for the scalewise variance is justan approximation that is valid only for a limited range ofparameter values. Numerical simulations have shown that




Π1 < 1

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Figure 4: Wavelet variance for model II, in which both dispersal and density-dependent establishment affect spatial patterns. A, B, Analyticalsolutions under varying values of the nondimensional parameters and , compared with the equivalent model IP p (f � m)/m P p c /c1 2 K D

(black dashed lines). C, Observed wavelet variance for 1,000 simulated processes, with , , m, ha (grayN p 100 a p 5 c p c p 3 A p 1D K

lines), their average (black solid line), and their 95% confidence intervals (black dashed line), relative to the solution for model I with(red line). D, Log likelihood as a function of the model parameters, when the true value of the dispersal distance ( ) is known.c p 3 c p 3D D

Dashed lines indicate the true parameter values: , .a p 5 c p 3K

the power-1 closure used to obtain the approximate so-lution in equations (19) and (20) may fail for moderateoverdispersion (Murrell et al. 2004) and high density (Ra-ghib et al. 2011). Moreover, the simplified model (eq. [21])explored above is limited to long-range dispersal or com-petition (relatively large cD or cK) or moderate NDD (smallP1). Thus, we next consider the exact behavior and im-plications for parameter estimation using numericalexamples.

This process can be simulated through a small variation

on the previous algorithm. Offspring are still generatedfrom a randomly chosen parent and displaced accordingto the dispersal kernel, but they establish successfully (andthus are retained) only with probability 1 � a K(x �∫

. If the offspring fails to establish, another par-′ ′ ′x )p(x )dxent is randomly chosen and another offspring randomlydisplaced.

Simulations confirm the qualitative insights from theanalytical approximation hold also for shorter-range dis-persal and competition (e.g., fig. 4C). They also demon-




Table 2: Effect of sample size (specifically, numberof individuals [n]) on parameter estimates ofmodel I

n ÊcD jcDÊjcD

25 3.12 .67 .7550 3.09 .59 .64200 3.09 .36 .39500 3.06 .29 .301,000 3.03 .24 .25

Note: and are the average and standard deviationÊc jD cD

of the parameter estimates from 1,000 simulations (A pha, Gaussian kernel m). is the average esti-ˆ1 c p 3 EjD cD

mated standard deviation using , where J is2 T �1 �1j p (J S J)b

the Jacobian and , the covariance2 2ˆ ˆS p Cov [n (l ), n (l )]ij i j

matrix of the wavelet variances (for details, see app. B, avail-

able online).

strate the feasibility of estimating parameters of model IIfrom wavelet variance data using maximum likelihood,since the likelihood is maximized around the true param-eter values (fig. 4D). However, there is a relatively largeregion of parameter space with very similar likelihoods,even when one of three parameters is assumed known.Estimation of all three parameters simultaneously leads toeven larger regions of equifinality and thus wider confi-dence intervals.

Case Study

We analyzed spatial patterns of four tropical tree speciesin the 50-ha forest dynamics plot on Barro Colorado Is-land, Panama (Leigh et al. 1996). All freestanding woodystems 11 cm in diameter were identified to species andmapped to 0.1 m, with censuses in 1982, 1985, and every5 years through 2010 (Hubbell and Foster 1986; Condit1998). Barro Colorado data are deposited in the NationalCenter for Ecological Analysis and Synthesis datarepository (http://knb.ecoinformatics.org/knb/metacat/nceas.298.6/nceas).

Methods

For each species, a raster was created by counting thenumber of individuals in each -m grid cell. The Mor-1 # 1let wavelet variance ( ) and the abundance (first andq p 8o

second moment) were calculated for each census for 39scales log-evenly spaced between 2 and 100 m. We averagedthe seven individual census normalized wavelet variancesand analyzed this ensemble average as a single realization.(If the spatial patterns in different censuses were statisti-cally independent, the ensemble average would follow ax2 distribution with degrees of freedom equal to the sumof the degrees of freedom of individual realizations. Turn-

over rates average 10% per census interval, so the patternsin different censuses are far from independent.)

We first tested whether spatial patterns deviated signif-icantly from random and, if so, at what scales, by com-paring observed wavelet variances with the expectationunder CSR.

Model I (eq. [17]) and the simplified model II (eq. [21])were then fitted for each species, using a Laplace kernelfor dispersal and competition (Van den Bosch 1988; Bolkerand Pacala 1997; Ovaskainen and Cornell 2006a). Themodels, , were fitted using generalized nonlinear2n (l, b)least squares (details in app. B). The fitting scheme differedslightly from the numerical examples because the disper-sion matrix S was not known a priori. Thus, we obtaineda first estimate of the parameters using . WeS p 2A /Aij ij

then simulated 1,000 processes using these estimated pa-rameters, used the simulations to compute the exact matrixS, and reestimated the model parameters and their un-certainties. The differences between the first and secondestimates were small because the likelihood functions arequalitatively very similar (but the parameter uncertaintiesare underestimated using the first matrix S, especially forsmall N); thus, no further iterations were necessary. Infitting model II, we set the initial value of as equalc p cD K

to the best-fit dispersal parameter from model I, and(thus, also ).P p 0 a p 01

We compared the models in terms of their modifiedAkaike information criterion (AIC; Burnham and Ander-son 2002):

2k(k � 1)AIC p 2k � 2 ln (L) � ,

J � k � 1

where k is the number of parameters and J is the numberof observed scales. Further details are given in appendixC, available online.

Results

All four species showed highly significant spatial structure,with wavelet variances significantly 11 for spatial scalesk10 m (fig. 5). Model I reproduced the basic features ofthe wavelet variance for most species. However, the slopeof the observed wavelet variance was generally !2, thevalue expected under model I; thus, model II was a betterfit for all four species (fig. 5). This implies that NDDreduces clustering created by dispersal. Dispersal distanceestimates under model II were invariably less than thosefor model I, and the difference increased as P1 increased.

For Desmopsis panamensis, an abundant understory spe-cies, the wavelet variance was significantly higher than 1at scales 110 m but increased less rapidly than predictedby model I. Furthermore, there was disaggregation at small


http://knb.ecoinformatics.org/knb/metacat/nceas.298.6/nceas

http://knb.ecoinformatics.org/knb/metacat/nceas.298.6/nceas



0

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dataModel IModel IIci 95% CSR

4 8 16

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1

N = 230 (#ha )

c = 8.03 (0.42)

c = 3.26 (1.80)c = 1.24 (1.11)Π = 0.34 (0.58)

ΔAIC = -98

D(I)

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1

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-1

N = 56 (#ha )

c = 6.05 (0.42)

c = 3.37 (1.80)c = 1.42 (1.19)Π = 0.23 (0.48)

ΔAIC = -5

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1

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-1

N = 37 (#ha )

c = 7.13 (0.73)

c = 5.6 (2.30)c = 0.78 (0.88)Π = 0.07 (0.26)

ΔAIC = -3

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N = 24 (#ha )

c = 15.38 (2.12)

c = 12.02 (3.4)c = 2.44 (1.5)Π = 0.08 (0.29)

ΔAIC = -3

D(I)

K(II)

1

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D(II)

-1

Figure 5: Observed spatial patterns (left) of four tropical tree species in the 50-ha plot on Barro Colorado Island, Panama, along with theirobserved (right; dots) and fitted (right; solid lines) wavelet variances. The focal species were chosen to represent the range of patternsobserved among the 100 most abundant species. The observed wavelet variances are compared with 95% confidence intervals for a completespatial randomness process (red dashed lines). Fitting statistics are shown on the right; best-fit parameter values are shown with theirstandard deviations; all distances are in meters. In all cases, model II provides a significantly better fit than model I (negative DAICII-I); thedifference is greatest by far in the case of Desmopsis panamensis.

scales, which was correctly captured by model II but notmodel I (fig. 5). This is the classic case that fairly strong( ) conspecific NDD acts at smaller scales thanP p 0.341

dispersal ( ).c ! cK D

In contrast, Guatteria dumetorum, a less abundant can-opy species, appeared less aggregated than D. panamensis.Small scales were entirely bounded by the 95% confidence

intervals for a CSR. Both models estimated longer dispersalfor Guatteria than for Desmopsis. Though NDD was sign-ificant (model II does better), it was much weaker thanin Desmopsis (fourfold lower P1).

Coussarea curvigemmia and Protium tenuifolium, bothsmall trees of moderate abundance, showed the highestvalues of the wavelet variance, especially at larger scales.




Coussarea was estimated to have only weak NDD, whileProtium had strong NDD whose effect was particularlyobvious at large scales.

Discussion

Ecologists have long examined spatial patterns for insightsinto ecological processes (e.g., Kershaw 1963). A first focushas been to evaluate whether spatial patterns are signifi-cantly nonrandom. The test developed here has two keyadvantages over previous approaches. First, it is analytical;it does not required Monte Carlo simulations. Second, itidentifies at which scales the process departs significantlyfrom CSR. In contrast, randomization tests based on Rip-ley’s K (e.g., Plotkin et al. 2000) or pair density correlations(e.g., Condit et al. 2000) cannot distinguish whether, forexample, departures from CSR at small distances are dueto causes acting at small scales or at larger scales (Loosmoreand Ford 2006).

Once spatial patterns are established to be nonrandom,the next challenge is to extract information about the pro-cesses underlying this nonrandomness. Here, we providenew tools for calculating the likelihood of particular pat-terns under alternative models and alternative parametervalues and, thereby, a method to fit models and chooseamong them on the basis of static patterns. This representsa major advance over previous approaches because it pro-vides a clear analytical link to models formulated explicitlyin terms of ecological processes. Some previous studieshave related patterns to classic cluster algorithms (Waa-gepetersen and Guan 2009; Wiegand et al. 2009), but suchlinkages provide little insight into, for example, how dis-persal and density dependence contribute to pattern for-mation. Linking to individual-based models derived usingbasic ecological principles (e.g., reproduction, mortality,dispersal, competition) allows easier interpretation (e.g.,Anand and Langille 2010). Further, our methods enablequantification of uncertainties in resulting parameter es-timates, critical information for comparing and integratingresults of multiple studies. An important caveat, as men-tioned above, is that the approximation used in the mo-ment equation is not uniquely defined and the error maybe dependent on the model parameter (Murrell et al.2004), a clear drawback for inferential analysis that shouldbe more carefully addressed in future work.

A major advantage of linking ecological patterns withindividual-based, spatially explicit models is that all modelassumptions are clearly stated in terms of ecological pro-cesses, and these can easily be kept in mind when resultsare interpreted. Taking the above case study results as anexample, we noted that dispersal distance estimates undermodel II are invariably less than those for model I, andthe difference increases as P1 increases. This makes sense,

since the dispersal estimates from model I are best thoughtof as estimates of effective dispersal distance—that is, thedispersal distance of successfully establishing individuals—which becomes ever larger than actual dispersal distanceas conspecific NDD becomes stronger. Even for model II,it is important to note that estimated dispersal distancesare not expected to correspond exactly to real dispersaldistances, because the fitted model assumes all individualsare reproductive. Given that in reality only the larger in-dividuals are reproductive, real dispersal distances mustnecessarily be longer than those estimated under thismodel. Similarly, estimates of NDD obtained from fittingmodel II may be biased because in the real world com-petition for resources is size dependent, while the fittedmodel lacks size structure (for an age-structured model,see Murrell 2009). Interpretation must also consider thesuite of alternative models examined and unexamined.Models that incorporate NDD in survival, growth, or fe-cundity are likely to produce similar patterns to model II,which has NDD establishment.

The models analyzed here do not include habitat het-erogeneity, which affects spatial structure in many pop-ulations. Where habitat variability operates scales muchlarger than the scales of dispersal and competition, thespatial patterns at these smaller scales would not be af-fected, and to a first-order approximation, habitat effectscan be neglected in estimating dispersal and competitionparameters. However, in other cases, habitat heterogeneitymay act on scales comparable to dispersal and competition,in which case the spatial pattern will depend on all threeprocesses (habitat association, dispersal, and competition)and it will be difficult to disentangle their influences. TheBarro Colorado Island plot can be divided into five broadhabitat types: slope, low plateau, high plateau, streamside,and swamp (Harms et al. 2001). Of our focal species,Desmopsis panamensis is positively associated with highplateau, Guatteria dumetorum is positively associated withslope and negatively with high plateau, and Protium tenu-ifolium is negatively associated with the swamp, accordingto conservative habitat randomization tests (Harms et al.2001). Given the large spatial scale of these habitats, theyare expected to affect wavelet variances at scales of ∼100m and greater. In contrast, dispersal and competition op-erate mostly on smaller scales in our system. Thus, it isnot surprising that we are able to obtain a good fit tosmall-scale spatial structure in these species with a modelthat ignores habitat heterogeneity.

The models here also ignore heterospecific interactionsas determinants of population spatial structure, despite thefact that heterospecific neighbors also have strong influ-ences on the growth and survival of individuals (Uriarteet al. 2004; Comita et al. 2010). A multispecies model thatincludes heterospecific as well as conspecific interactions




has been developed by Law and Dieckmann (2000). Wecan use this model to simultaneously fit the wavelet var-iance of two individual species and their covariance. How-ever, in species-rich communities such as Barro Colorado,with ∼300 tree species in 50 ha, a dilution effect yieldsapproximate species independence (Wiegand et al. 2012).As a result, there is little improvement, if any, in fittingthe multivariate model compared with fitting each speciesindependently. Clearly, there are other communities inwhich heterospecific interactions are relatively more im-portant in shaping spatial patterns and in which a mul-tispecies model would thus provide a better fit and yieldmore information about the processes driving dynamicsin the system.

Our methods have certain limitations, which are generalto any analyses of spatial patterns. First, multiple processescan produce similar spatial patterns, providing limits towhat any analysis of static patterns alone can yield. Forexample, strong NDD can counterbalance dispersal limi-tation effects, producing a pattern that is similar or inextreme cases indistinguishable from a random pattern(long dispersal). In general, this problem arises when thescales of different ecological processes overlap, making ithard to quantify, for these scales, how much variance isproduced (or reduced) by one process or another. Second,analyses of processes with larger scales require data withadequate spatial extension. Thus, for example, estimatesof dispersal distance that are long relative to the scale ofthe data ( ) are less accurate because asymptotic1/2c ∼ AD

trends at larger scales are not well resolved or are absentin the scalewise variances.

Recommendations and Future Directions

Here, we developed new tools for investigating the effectsof ecological processes on spatial point patterns and de-riving information on ecological processes from these pat-terns. Future theoretical work should investigate how hab-itat heterogeneity is expected to influence scalewisevariances (Ovaskainen and Cornell 2006a) and evaluatethe potential for linking species spatial patterns to habitat.Exploration of expected scalewise variances for additionaldispersal and interaction kernels, anisotropic effects, dif-ferent types of NDD (and potentially positive density de-pendence), and more complex models incorporating ageand size structure (Murrell 2009) and/or multiple species(Law and Dieckmann 2000) could further shed light onthe impacts of different ecological processes on spatial pat-terns. These methods can also be extended beyond sessileorganisms by considering adding a movement kernel tothe model, following Law and Dieckman (2000).

Maximum insight into underlying ecological processeswill be gained by judiciously combining the spatial analyses

pioneered here with other sources of information. Fittingeven moderately complex models to spatial data alone re-sults in wide confidence intervals (e.g., fig. 4) because thescales of many ecological processes often overlap (e.g.,short dispersal and NDD), with the consequence that sim-ilar patterns are produced with different combinations ofthe parameters. The likelihood function for the normalizedwavelet variances can be combined with prior informationor analyses of other data sets to narrow confidence inter-vals. For example, a study of seed dispersal might informprior distributions on the dispersal parameter. Data frommultiple censuses may provide further information aboutthe spatial-temporal dynamics (Ovaskainen and Cornell2006b), enabling, for example, independent estimates offecundity and mortality. Wide application of the methodsdeveloped here could shed light on the prevalence andstrength of NDD, the relative scales of dispersal and com-petition, and the distribution of effective dispersal dis-tances within and among populations and communities.

Acknowledgments

We gratefully acknowledge the financial support of theSmithsonian Institution Global Earth Observatories, and wethank R. Condit and S. Hubbell for providing the BarroColorado Island tree data and J. Calabrese and R. Chrislomfor useful comments. The Barro Colorado Island forest dy-namics research project was made possible by National Sci-ence Foundation grants to S. P. Hubbell (DEB-0640386,DEB-0425651, DEB-0346488, DEB-0129874, DEB-00753102, DEB-9909347, DEB-9615226, DEB-9615226,DEB-9405933, DEB-9221033, DEB-9100058, DEB-8906869,DEB-8605042, DEB-8206992, and DEB-7922197), supportfrom the Center for Tropical Forest Science, the SmithsonianTropical Research Institute, the John D. and Catherine T.MacArthur Foundation, the Mellon Foundation, and theSmall World Institute Fund.

Literature Cited

Abramowitz, M., and I. A. Stegun. 1965. Handbook of mathematicalfunctions. Dover, New York.

Anand, M., and A. Langille. 2010. A model-based method for esti-mating effective dispersal distance in tropical plant populations.Theoretical Population Biology 77:219–226.

Bagchi, R., P. Henrys, P. Brown, and D. Burslem. 2011. Spatial pat-terns reveal negative density dependence and habitat associationsin tropical trees. Ecology 92:1723–1729.

Bartlett, M. 1964. The spectral analysis of two-dimensional pointprocesses. Biometrika 51:299–311.

Bolker, B., and S. Pacala. 1997. Using moment equations to under-stand stochastically driven spatial pattern formation in ecologicalsystems. Theoretical Population Biology 52:179–197.




Bolker, B. M., S. W. Pacala, and S. A. Levin. 2000. Moment methodsfor ecological processes in continuous space. Pages 388–411 in U.Dieckmann, R. Law, and J. A. J. Metz, eds. The geometry of eco-logical interactions. Cambridge University Press, Cambridge.

Bolker, B. M., S. W. Pacala, and C. Neuhauser. 2003. Spatial dynamicsin model plant communities: what do we really know? AmericanNaturalist 162:135–148.

Burnham, K. P., and D. R. Anderson. 2002. Model selection andmultimodel inference: a practical information-theoretic approach.2nd ed. Springer, New York.

Cazelles, B., M. Chavez, and D. Berteaux. 2008. Wavelet analysis ofecological time series. Oecologia (Berlin) 156:287–304.

Comita, L. S., H. C. Muller-Landau, S. Aguilar, and S. P. Hubbell.2010. Asymmetric density dependence shapes species abundancein a tropical tree community. Science 329:330–332.

Condit, R. 1998. Tropical forest census plots. Springer, Berlin.Condit, R., P. S. Ashton, P. Baker, S. Bunyavejchewin, S. Gunatilleke,

N. Gunatilleke, S. P. Hubbell, et al. 2000. Spatial patterns in thedistribution of tropical tree species. Science 288:1414–1418.

Dale, M., and M. Mah. 1998. The use of wavelets for spatial patternanalysis in ecology. Journal of Vegetation Science 9:805–814.

Daley, D., and D. Vere-Jones. 2003. An introduction to the theoryof point processes. I. Elementary theory and methods. 2nd ed.Springer, New York.

Daubechies, I. 1992. Ten lectures on wavelets. Society for Industrialand Applied Mathematics, Philadelphia.

Detto, M., A. Molini, G. Katul, P. Stoy, S. Palmroth, and D. Baldocchi.2012. Causality and persistence in ecological systems: a nonpara-metric spectral Granger causality approach. American Naturalist179:524–535.

Dieckmann, U., and R. Law. 2000. Relaxation projections and themethod of moments. Pages 412–455 in U. Dieckmann, R. Law,and J. A. J. Metz, eds. The geometry of ecological interactions.Cambridge University Press, Cambridge.

Durrett, R., and S. Levin. 1994. The importance of being discrete(and spatial). Theoretical Population Biology 46:363–394.

Harms, K. E., R. Condit, S. P. Hubbell, and R. B. Foster. 2001. Habitatassociations of trees and shrubs in a 50-ha Neotropical forest plot.Journal of Ecology 89:947–959.

Hartig, F., J. M. Calabrese, B. Reineking, T. Wiegand, and A. Huth.2011. Statistical inference for stochastic simulation models: theoryand application. Ecology Letters 14:816–827.

Hubbell, S. P., and R. B. Foster. 1986. Biology, chance, and historyand the structure of tropical rain forest tree communities. Pages314–329 in J. Diamond and T. J. Case, eds. Community ecology.Harper & Row, New York.

Janzen, D. H. 1970. Herbivores and the number of tree species intropical forests. American Naturalist 104:501–528.

Keitt, T. H., and D. L. Urban. 2005. Scale-specific inference usingwavelets. Ecology 86:2497–2504.

Kershaw, K. A. 1963. Pattern in vegetation and its causality. Ecology44:377–388.

Kumar, P., and E. Foufoula-Georgiou. 1997. Wavelet analysis forgeophysical applications. Reviews of Geophysics 35:385–412.

Law, R., and U. Dieckmann. 2000. A dynamical system for neigh-borhoods in plant communities. Ecology 81:2137–2148.

Law, R., J. Illian, D. F. R. P. Burslem, G. Gratzer, C. V. S. Gunatilleke,and I. A. U. N. Gunatilleke. 2009. Ecological information fromspatial patterns of plants: insights from point process theory. Jour-nal of Ecology 97:616–628.

Leigh, E. G., Jr., A. S. Rand, and D. M. Windsor. 1996. The ecologyof a tropical forest: seasonal rhythms and long-term changes.Smithsonian Institution, Washington, DC.

Levin, S. A. 2000. The problem of pattern and scale in ecology.Ecology 73:1943–1967.

Loosmore, N. B., and E. D. Ford. 2006. Statistical inference usingthe G or K point pattern spatial statistics. Ecology 87:1925–1931.

McIntire, E. J. B., and A. Fajardo. 2009. Beyond description: theactive and effective way to infer processes from spatial patterns.Ecology 90:46–56.

Meyers, S. D., B. Kelly, and J. O’Brien. 1993. An introduction towavelet analysis in oceanography and meteorology: with appli-cation to the dispersion of Yanai waves. Monthly Weather Review121:2858–2866.

Mi, X., H. Ren, Z. Ouyang, W. Wei, and K. Ma. 2005. The use ofthe Mexican Hat and the Morlet wavelets for detection of eco-logical patterns. Plant Ecology 179:1–19.

Murrell, D. J. 2009. On the emergent spatial structure of size-struc-tured populations: when does self-thinning lead to a reduction inclustering? Journal of Ecology 97:256–266.

Murrell, D. J., U. Dieckmann, and R. Law. 2004. On moment closuresfor population dynamics in continuous space. Journal of Theo-retical Biology 229:421–432.

Murrell, D. J., and R. Law. 2002. Heteromyopia and the spatial co-existence of similar competitors. Ecology Letters 6:48–59.

Ovaskainen, O., and S. J. Cornell. 2006a. Asymptotically exact anal-ysis of stochastic metapopulation dynamics with explicit spatialstructure. Theoretical Population Biology 69:13–33.

———. 2006b. Space and stochasticity in population dynamics. Pro-ceedings of the National Academy of Sciences of the USA 103:12781–12786.

Pacala, S. 1986. Neighborhood models of plant population dynamics.2. Multi-species models of annuals. Theoretical Population Biology29:262–292.

Percival, D. P. 1995. On estimation of the wavelet variance. Bio-metrika 82:619–631.

Platt, T., and K. L. Denman. 1975. Spectral analysis in ecology. AnnualReview of Ecology and Systematics 6:189–210.

Plotkin, J. B., M. D. Potts, N. Leslie, N. Manokaran, J. Lafrankie,and P. S. Ashton. 2000. Species-area curves, spatial aggregation,and habitat specialization in tropical forests. Journal of TheoreticalBiology 207:81–99.

Raghib, M., N. A. Hill, and U. Dieckmann. 2011. A multiscale max-imum entropy moment closure for locally regulated space-timepoint process models of population dynamics. Journal of Math-ematical Biology 62:605–653.

Serroukh, A., A. T. Walden, and D. B. Percival. 2000. Statistical prop-erties and uses of the wavelet variance estimator for the scaleanalysis of time series. Journal of the American Statistical Asso-ciation 95:184–196.

Thompson, S. E., and G. G. Katul. 2011. Inferring ecosystem param-eters from observation of vegetation patterns. Geophysical Re-search Letters 38:2–5.

Torrence, C., and G. P. Compo. 1998. A practical guide to waveletanalysis. Bulletin of the American Meteorological Society 79:61–78.

Turing, A. 1952. The chemical basis of morphogenesis. PhilosophicalTransactions of the Royal Society B: Biological Sciences 327:37–72.

Uriarte, M., R. Condit, C. D. Canham, and S. Hubbell. 2004. A




spatially explicit model of sapling growth in a tropical forest: doesthe identity of neighbours matter? Journal of Ecology 92:348–360.

Van den Bosch, F., J. Zadoks, and J. Metz. 1988. Focus expansion inplant disease. II. Realistic parameter-sparse models. Phytopathol-ogy 78:59–64.

Waagepetersen, R., and Y. Guan. 2009. Two-step estimation for in-homogeneous spatial point processes. Journal of the Royal Statis-tical Society B 71:685–702.

Wiegand, T., A. Huth, S. Getzin, X. Wang, Z. Hao, C. V. S. Gun-

atilleke, and I. A. U. N. Gunatilleke. 2012. Testing the independentspecies’ arrangement assertion made by theories of stochastic ge-ometry of biodiversity. Proceedings of the Royal Society B: Bio-logical Sciences 279:3312–3320.

Wiegand, T., I. Martınez, and A. Huth. 2009. Recruitment in tropicaltree species: revealing complex spatial patterns. American Natu-ralist 174:E106–E40.

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