JTBI=20002158=Binni=Venkatachala=BG

J. theor. Biol. (2000) 207, 81}99
doi:10.1006/jtbi.2000.2158, available online at http://www.idealibrary.com on

Species-area Curves, Spatial Aggregation, and Habitat Specializationin Tropical Forests
JOSHUA B. PLOTKIN*-, MATTHEW D. POTTS?, NANDI LESLIEA, N. MANOKARAN#,JAMES LAFRANKIEB AND PETER S. ASHTON***Institute for Advanced Study and Princeton University, Olden Lane, Princeton, NJ 08540, U.S.A.
?Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, U.S.A.,
AProgram in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08540, U.S.A.
#Forest Research Institute of Malaysia, Kepong, Malaysia, BCenter for Tropical Forest Science, National
Institute of Education, 1025 Singapore and **Department of Organismal and Evolutionary Biology,
Harvard University, Cambridge, MA 02138, U.S.A.
(Received on 26 April 2000, Accepted in revised form on 26 July 2000)
The relationship between species diversity and sampled area is fundamental to ecology.
Traditionally, theories of the species}area relationship have been dominated by randomplacement models. Such models were used to formulate the canonical theory of species}area
curves and species abundances. In this paper, however, armed with a detailed data set froma moist tropical forest, we investigate the validity of random placement and suggest improved
models based upon spatial aggregation. By accounting for intraspeci"c, small-scale aggregation, we develop a cluster model which reproduces empirical species}area curves with high"delity. We "nd that inter-speci"c aggregation patterns, on the other hand, do not a!ect the
species}area curves signi"cantly. We demonstrate that the tendency for a tree species toaggregate, as well as its average clump size, is not signi"cantly correlated with the species'
abundance. In addition, we investigate hierarchical clumping and the extent to which aggregation is driven by topography. We conclude that small-scale phenomena such as dispersal andgap recruitment determine individual tree placement more than adaptation to larger-scale
topography.
 2000 Academic Press

Introduction
Faced with an ecological assemblage, a natural
question is &&How many species are found ina given area?'' This question has puzzled naturalists since the early 19th century (Connor& McCoy, 1979). Aside from an intriguing question, the species}area relationship is a conceptual
cornerstone for almost all theories of community
ecology (McGuinness, 1984; Rosenzweig, 1995).
Referring to their seminal work on biogeography,
RAuthor to whom correspondence should be addressed.
E-mail: plotkin@ias.edu0022}5193/00/210081#20 $35.00/0

MacArthur and Wilson write, &&Theories, like islands, are often reached by stepping stones. The&species}area' curves are such stepping stones''(MacArthur & Wilson, 1967). Beyond their theoretical signi"cance, species}area curves also
form the basis for almost all estimates of extinction due to habitat loss (May, 1995; Pimm
& Raven, 2000). More generally, the species}area
relationship helps ecologists to assess the relative
importance of those factors*competition, dispersion, adaptation to environment, chance,
etc.* which determine species' geographic
ranges.
 2000 Academic Press

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82

J. B. PLOTKIN E¹ A¸.

The most common characterization of the species}area relationship (SAR) posits a power law:
the number of species is proportional to a constant power of sampled area. This relationship,S"cAX, was "rst postulated by Arrhenius (1921).
Nevertheless, this model did not enjoy universal
popularity until the pioneering work of Preston(1962) followed by MacArthur & Wilson (1963).
Macarthur and Wilson considered the power law
in light of a dynamic equilibrium of species exchanges between islands.
In 1975, May endowed the power-law model ofthe SAR with a "rm theoretical underpinning byrelating it to the relative abundances of species(May, 1975). May argued that in a complex community the relative abundances of species should
follow the lognormal distribution: the proportionof species with n individuals is a Gaussian function of log n. Assuming this abundance distribution, May showed that the power-law SAR must
hold over a wide range of spatial scales. In orderto perform this derivation, May assumed that
individuals are drawn independently from the species abundance distribution, yielding a species}individual curve. In order to translate thespecies}individual curve into the species}area
curve, May assumed that individuals are placedin space with a constant density. The entire process is equivalent to drawing individuals from theabundance distribution and placing them inspace (Poisson-) randomly.
Following May's approach, Coleman developed a random-placement null model of theSAR (Coleman, 1981; Coleman et al., 1982). Coleman provided explicit equations for the species}area curve which results from given species
abundances. Speci"cally, given N species with
abundances n , n , n ,2, n , Coleman derived  ,
equations for the species}area curve assumingthat individuals are placed in space randomly
and independently. As Coleman states, the randomness assumption &&presupposes a lack of correlation in the locations of individuals. [As such,]
it can be considered a zeroth-order hypothesis''.
In other words, Coleman's model ignores thepossibility of inter- and intra-speci"c spatial aggregation. Although the random-placement hypothesis is violated in many ecological systems,
few data sets were available in 1981 by which toform or judge more explicit models. (Coleman

did employ avifaunal data from Pymatuning
Lake, Pennsylvania for comparison to his theory.
Nevertheless, the region contained fewer than 40
identi"ed species.)
Since the work of Coleman and May, there
have been relatively few explicit e!orts to extend
the &&zeroth-order hypothesis'' of a random placement SAR. In fact, current theories of biogeography and diversity often still utilize Coleman's
model (e.g. Hubbell, 1997). The prevailing wisdom generally holds that the species}area curveis driven by random sampling from the abundance distribution*although this belief is notheld without exceptions (Hubbell & Foster,1983). The explicit, theoretical e!orts towards
a non-random theory include work by Leitner& Rosenzweig (1997), Buckley (1982), McGuinness (1984), and Gotelli & Graves (1996). Recently, Kunin (1998) and Ney-Ni#e & Mangel(1999) have developed spatial models based upon
the geographic range of each species. In general,these spatial-re"nements upon Coleman's model
have seldom been compared to extensive, empirical data.
With less of an emphasis on a spatial model forthe species}area curve, previous research has
analysed spatial patterning in tropical forests. He
et al. (1987) analyse conspeci"c patterning byusing nearest-neighbor statistics and the Donnelly clumping index. They also investigate theinterplay of aggregation and topography. A
comparison of our results with theirs will prove
interesting; their analysis does not quantify topography as a hierarchical aggregation e!ect. Batista & Maguire (1998) provide a comparative
overview of stochastic point processes and theirapplication to tropical forests. They focus on thee!ect of the forest canopy on its understory. Condit et al. (2000) o!er an extensive, comparative
study of aggregation across tropical forest plots.
In a seminal, qualitative investigation, Hubbell& Foster (1983) provide a largely biological discussion of spatial patterning in the canopy ofa tropical forest, focusing on the maintenance ofdiversity. Most of their analysis relies on visualcharacterization of spatial patterns. Their discussion of the biological and ecological factors
driving aggregation provides a uniquely well-informed interpretation of the more quantitative
results in this paper.

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83

SPATIAL AGGREGATION IN TROPICAL FORESTSThis paper aims to develop a spatially explicit,
theoretical framework for the species}area curvein a tropical forest. By doing so, we will assess therelative importance of small-scale (e.g. dispersal,
gap-recruitment) vs. large-scale (e.g. topographic)
phenomena as determinants of tree-placement. In
the spirit of Coleman and May, we will assume
throughout that the abundance of each species isknown. Given these abundances, we will "rstinvestigate the impact of local aggregation on thespecies}area curve. Then, we will investigate heirarchial clumping and larger-scale e!ects of environment. We develop our models of the SAR
hand in hand with comparisons to extensive data.
Tropical Forest Data
There could hardly be an ecological system
more well-suited for species}area and spatial-aggregation research than tropical forests. Unlike
avifauna, trees have the obvious advantage ofa sedentary life history*which makes them relatively easy to locate and identify. Moreover,tropical forests boast an astounding diversity oftree species (up to 1000 within 50 h), providing
excellent resolution for our analyses. Finally, due
the impressive e!orts of the Smithsonian Center
for Tropical Forest Sciences, we have currently
identi"ed the location and species for a sum-totalof over two million individual trees.
In this paper, we investigate spatial aggregation and the SAR in a 50-h, fully censused treeplot from the Pasoh forest on peninsular
Malaysia. Compared with several other 50-ha
censuses, Pasoh has the advantage of a fairly
homogenous environment and a relatively richspecies diversity. These qualities make Pasoh an
excellent choice as the focus of our study. In
addition, throughout the paper we also refer totwo other 50-ha plots as veri"cation of the generality of our methods. Each of the 50-ha Forest
Dynamics Plots is part of a long-term research
program coordinated by the Center for Tropical
Forest Science. The plots are located in the following forests: Pasoh Forest Reserve, Peninsular
Malaysia, 1996 census; Huai Kha Khaeng Wildlife Sanctuary (HKK), Thailand, 1995 census;
Lambir Hills National Park, Sarawak, Malaysia,
1999 census. In every plot, each woody stem
'1 cm diameter has been identi"ed to species,

TABLE 1¹ree density and diversity at each of three 50-h
plots of tropical forest.
Plot nameLocation

Stems

SpeciesPasoh
LambirHKK

Malaysia
Malaysia
Thailand

320 902325 335
96 072

817
1171
251

We develop and test our spatial model at Pasoh, and then we verify
the results at Lambir and HKK. The table indicates the total
number of woody stems '1 cm in diameter and the total number of
species found in each plot. For a complete list of references, consult
the CTFS web site at http://www.si.edu/ctfs.

measured for girth, and spatially mapped to atleast 1 m. The number of such stems, and thenumber of species among them, varies greatly
from plot to plot (Table 1).
We will include all free-standing stems '1 cm
diameter throughout our analyses. In general,aggregation patterns bene"t from a separate
large-tree/small-tree treatment*especially forcomparison with the theories of Janzen (1970)
and Connell (1971). Nevertheless, for the purposeof investigating the species}area curve, it is best
to include every individual in the data set. Almost
all of our results remain true if we analyse, instead, stems '5 cm in diameter. When summarizing results, we will often classify species as
either rare or abundant. Following Hubbell& Foster (1986), we de"ne a species as rare if it
has one or fewer stems per hectare, on average.
The Random-placement Model
We begin with a brief review of Coleman's&&zeroth-order'', random-placement theory of species}area curves. Consider a region of total area
A within which individuals of various speciesare located. Assume that there are N species andthat the i-th species is represented by n indiG
viduals. Consider any sub-region of area A(A .

Under the assumption of independent, random
placement of individuals, the probability thata given member of the i-th species does not residein a sub-region of size A is simply (1!A/A ).

Similarly, the probability that all members ofspecies i lie outside of A is given by (1!A/A )LG .
JTBI=20002158=Binni=VVC

84

J. B. PLOTKIN E¹ A¸.

Thus, the probability that at least one member ofthe ith species resides in the sub-region A is1!(1!A/A )LG . This in turn yields an expres
sion for the mean number of species in A, denoted
S(A), and the variance, denoted p(A) (Coleman,
1981):,
S(A)"N!
(1!A/A )LG ,

G

(1)

,
,
p(A)" (1!A/A )LG ! (1!A/A )LG .


G
G
(2)
It is often assumed that, given the abundancesof the species, eqns (1) and (2) provide an adequate model of the SAR in tropical forests (e.g.
Hubbell, 1997).
Fortunately, it is easy to test the adequacy ofeqns (1) and (2) for tropical forests. Early tests by
Hubbell at Barro Colorado Island (BCI) indicate
a failure of the random model for 186 species in
the canopy '20 cm in diameter (Hubbell& Foster, 1983). Our independent test of therandom-model will include all trees '1 cm indiameter. In the interest of precision we mustde"ne the species}area curve carefully. Following
Harte et al. (1999) and Plotkin et al. (2000), wede"ne the species}area curve for a rectangularregion of total area A . Let A "A /2G denote theGarea of a rectangular patch obtained after i bisections of A (bisections chosen perpendicular tothe longer dimension). There are 2G disjoint
patches of area A which naturally partition A .
G
Loosely speaking, the empirical species}area
function S(A) is de"ned as the average number of
species found in an area of size A. By longstanding convention, we evaluate S(A) by averaging over disjoint patches of area A (Connor& McCoy, 1979). In this manner, we let S(A )Gdenote the average number of species found ina patch of area A . In other words, we de"neG
1 j"2G(C species in the j-th patch
S(A )"G
2G H
of area A ).
G

(3)

FIG. 1. The random placement model signi"cantly
overestimates diversity for areas in the 0.04- to 45-ha range.
The "gure shows graphs of the actual SAR measured at
Pasoh ($1 S.D.) compared to the SAR predicted by the
random-placement model ($1 S.D.). The inset repeats thegraphs on log-linear axes; the solid lines demark the $1
S.D.-con"dence interval predicted by Coleman's model. As isclear from the graphs, the observed SAR is far outside of therandom-placement prediction. The same result holds at
HKK and Lambir (not shown).

In particular, S(A ) denotes the total numberof species found in the entire plot. We de"ne M as
the total number of individuals in the plot, i.e
M" , n .
G
The G
SAR predicted by random placement
signi"cantly overestimates diversity at Pasoh(Fig. 1). In fact, for all three forests the measured
SAR is well outside of the two-standard-deviation con"dence interval of the random model.
This con"dence interval is given by 2p from Eq.
(2). The discrepancy between the random modeland the empirical data is even larger than thediscrepancy originally measured for large trees
alone at BCI (Hubbell & Foster, 1983). A cursory
glance at Fig. 1 may suggest that Coleman's
model is inaccurate for mid-sized areas, but very
accurate near 0 and 50-ha. This is an artifact. By
construction, Coleman's model must agree withthe actual data at 0 and 50 ha.
The extent to which random placement fails to
capture the SAR is, in fact, quite severe. Coleman's model signi"cantly overestimates diversity
for all areas within the range 0.04}45 ha (see Fig. 1,
inset). For example, at Pasoh the measured valueof S(A )"S(3.1 ha) is more than nine standarddeviations (S.D.) less than the predicted value.
On average over areas ranging from 1.5 to 25 ha,

JTBI*20002158*Binni*VVC
SPATIAL AGGREGATION IN TROPICAL FORESTS

FIG. 2. The observed SAR of Pasoh compared to the
SAR predicted by Coleman's random-placement model, on
a log-linear graph. Unlike 50-ha curves in Fig. 1, the curves
displayed here are generated from a single subplot of area
A +0.78 ha. At this small-scale, the random-placementprediction is much closer to the observed SAR than at the
50-ha scale.

the random-placement model overestimates diversity 7.2 S.D.s at Pasoh, 4.0 S.D.s at HKK, and17.2 S.D.s at Lambir. As these results indicate,beyond a doubt the random-placement model
fails to account for the observed SAR. The actual
data must therefore follow some non-random,
spatial patterning.
Figure 1 also reveals another signature of spatial aggregation. For each i, the diversity found inpatches of area A shows larger variance in reality
Gthan in Coleman's model. This large variancesuggests spatial aggregation: some patches havelarge diversity, while others are dominated bya few, clumped species. Observed variance ismuch larger than Coleman's model in all threeplots.
The departure from random placement is apparent to us because of the large scale at which
the plots have been censused. We test this bytemporarily restricting our attention to a single
subplot of small area. In particular, we choose
a single subplot from Pasoh of area A +0.78 ha,measure the abundances within the subplot, andagain compare Coleman's model to the observed
SAR. When restricting our attention to this subplot, the random-placement model provides anadequate characterization of the SAR (Fig. 2). At
small spatial scales, the entire subplot is smaller
than the correlation length of spatial clumps;
hence random placement describes the spatial

85

pattern fairly well. This trend*that the randomplacement model can be rejected for large plots,
but not for small plots*suggests that random
placement will provide an increasingly poor
model for plots even larger than 50 ha.
Taken together, the results in Figs 1 and
2 underscore the need for a decidedly non-random model of tropical forests, especially for large
areas. Throughout the sequel, we aim (i) to analyse the spatial aggregation in the forest plots, (ii)to develop a model which characterizes this aggregation, producing the observed species}area
curves, and (iii) to investigate the relative strengthof topographic e!ects on aggregation.
A Preliminary Measure of Aggregation: proportion of Neighbors which are Conspeci5c
We begin our analysis of the aggregation patterns with a somewhat non-standard approach.
A more traditional analysis*which eventually
yields a spatial model*will be delayed until thenext section. Nevertheless, the statistic which weinvestigate in this section will later provide important information for determining the parameters of our spatial model.
Most univariate measures of aggregation essentially ask the question &&How far apart are two
conspeci"c trees?'' Nevertheless, in this preliminary section we will ask the inverse question:&&Given two trees a distance d apart, how often
are they conspeci"c?'' This question yields aninteresting measure of aggregation. We will calculate this aggregation statistic for each plot andcompare it to the value predicted under a random-placement model. Speci"cally, for each distance d we de"ne P as the proportion of trees inB
the plot, distance d apart, which are the same
species. We calculate this proportion by considering each pair of trees in turn. Notice that P is
B
inherently inter-speci"c: it requires knowledge oftree locations for all species.
In a random-placement model, P clearly doesB
not depend on the distance d. P is determined byB
the relative abundances of the species alone. If
individuals are located randomly and independently, then P is simply the chance that two
Brandomly-chosen individuals are of the same species. This probability does not depend upon distance, and it is easily expressed as a quantity

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86

J. B. PLOTKIN E¹ A¸.

tances greater than 250 m. On the other hand,clumping seems to be more widespread at HKK
and Lambir. These preliminary observations will
prove useful in later sections.

5 The Ripley-K Measure of Aggregation

FIG. 3. Graph of the aggregation statistic P !Pfor
B
P?LBMK
each of the forests Pasoh, HKK, and Lambir. All three
forests possess an elevated proportion of conspeci"cs at
distances d(200 m. The x-intercept of the graph
P !Pyields an upper bound on mean dispersal disB
P?LBMK
tance and gap-size. HKK demonstrates the most aggregation at large distances (d(385 m).

closely related to the Simpson index:, n (n !1)
G G
P
"
.
P?LBMK
M(M!1)
G

Unlike the statistic P !P
, which requires
B
P?LBMK
information about the location of all individuals
in the plot, we henceforth focus on strictly intraspeci"c aggregation measures. In particular, we
will employ the well-understood, second-momentmeasure called Ripley's K (Ripley, 1976). As withthe statistic P !P
, we will compare theB
P?LBMK
observed values of K to those predicted under theassumption of random placement. Eventually, we
aim to parameterize our spatial model using theinformation gleaned from Ripley's K*which isthe primary reason why we choose to measure K.
In the meantime, we will use Ripley's K to assess
whether each species is signi"cantly clumped or
not.

(4)

As indicated by Monte-Carlo simulations of ranis small
dom placement, the S.D. of P
P?LBMK((0.005) in all three plots. A measured value of
P greater than P
indicates a higher proporB
P?LBMK
tion of conspeci"cs distance d apart relative tothe random model. Trees generally propagate
locally, and hence we expect that P 'P
for
B
P?LBMK
small distances, but P )P
for large distanB
P?LBMK
ces. This trend is veri"ed in all three forests bygraphing P !P(Fig. 3). (Note that the
B
P?LBMK
value of P
is di!erent in each forest because
P?LBMK
the relative abundances di!er.)
Figure 3 illustrates the aggregation metric
P !Pin the three forest plots. For example,
B
P?LBMK
Fig. 3 reveals that on average, when standing at
a given tree in Pasoh, one "nds an elevated proportion of conspeci"c trees at distancesd(250 m away, but thereafter one "nds no more
conspeci"cs than expected at random, given therelative abundances of the species in Pasoh. Thisobservation provides an upper-bound on thosefactors which determine correlation length forsmall-scale cluster formation*e.g. dispersal distance, gap size, etc. In particular, at Pasoh we do
not expect signi"cant clumping to occur at dis-COMPUTING RIPLEY'S K

If the individuals of a given species are placed
randomly in the plot*i.e. via a Poisson-processwith intensity j*then the expected number ofstems within a circle of radius d is simply jnd.
Ripley's K quanti"es the departure from the randomized situation. Clustering increases K, whileregularity decreases it. Speci"cally, given a pointprocess on the plane, Ripley's K-function is de"ned as
K(d)"j\ $ (number of extra events within adistance d from an arbitrary event).

(5)

Given a particular map of n events s , s ,2, s Lwithin a region of area A (in our case, the
locations of stems of a "xed species), the canonical edge-corrected estimator of K is given by
Ripley (1976):LL
w(s , s )\II(#s !s #)d)/n.
KK (d)"jK \G H
GHG H H$G(6)

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SPATIAL AGGREGATION IN TROPICAL FORESTS

In eqn (6), jK "n/A estimates the intensity, II is
the indicator function of an event, and the weightw(s , s ) is the proportion of the circumference of
G H
the circle centered at s , passing through s , whichG
H
lies in A . Given j, if the underlying point-pro
cess is stationary and isotropic, then KK is anunbiased estimator of K (Cressie, 1991). jK is a biased estimate of j, but only slightly so. Assuming
random placement, K(d)"nd.
We have computed Ripley's K using eqn (6) for
every species in Pasoh, HKK, and Lambir. Morespeci"cally, we calculate KK (d) for 20 values ofd equally spaced between 0 and d . The quantiK?Vty dsigni"es the largest distance at which weK?Vmeasure clumping in the plot. We use Fig. 3 asa guideline for choosing d
in each plot: we use
K?Vd "250 m at Pasoh, d "385 m at HKK
K?VK?Vand Lambir. As an example, at Pasoh we havecalculated KK (d) for d"12.5,25,37.5,2,250 m. In
order to test spatial randomness, we compile theinformation about KK (d) into a single Cramer-vonMises-type statistic ((KK (h)!h(n) dh.

k"

(7)

In practice, we evaluate this statistic using
a Riemann sum from 0 to d . In order to test
K?V
random placement, for each species we compare
the measured k-statistic to the maximum k-statistic generated from 19 Monte-Carlo simulationsof a Poisson process (i.e. of random placement). If
the measured k is larger than the maximum
simulated k, denoted k , then we conclude at
K?V
the 5% con"dence level that the species is not
distributed at random (Diggle, 1983).
In principle, this procedure allows us to classify
each species as either clumped or not clumped.
Of course, we only consider species with *2stems. &&Not-clumped'' is the null hypothesis; failure to reject the hypothesis should not compel us
to accept it. With this warning in mind, we classify a species as not clumped meaning, simply,
that it was not possible to reject this hypothesison the basis of the k-statistic. When a species israre (n )50), then the power of the k-statistic*Gor any other statistic*is dramatically diminished. We will take special care when interpreting
the results of our statistics on rare species.

87

RESULTS OF RIPLEY-K ANALYSIS

Of the 798 species at Pasoh with at least twostems, 661 of them (83%) are classi"ed as aggregated via k. Of these aggregated species, 513 ofthem (78%) have more than 50 stems. On theother hand, out of the 137 non-aggregated species, most of them (85%) have 50 stems or less. In
other words, fewer of the rare species are classi"ed as clumped, relative to the abundant species.
This trend, which He et al. (1987) noticed as well,is sometimes overemphasized in the literature.
The trend certainly results as an artifact of thestatistics on small sample sizes. In general, whenn (50 at least one Monte-Carlo simulation reG
ceives a very large k-statistic, making it di$cult
to reject the null-hypothesis of random-placement. In other words, when a species is rare, forstatistical reasons alone we often cannot justify
labeling it as clumped.
Ignoring the rare species for the moment, of the
534 species with '50 stems, 513 of them (96%)
are aggregated according to k. Species at HKK
and Lambir demonstrate a comparable tendency
to aggregate. In short, almost all species which
are not rare are clumped. These results contrast
sharply with analyses of a temperate forest(Szwagrzyk & Ptak, 1981; Bodziarczyk et al.,
1999; Bodziarczyk & Szwagrzyk, 1996), but agreewith recent, comprehensive, inter-plot analyses oftropical forests by Condit et al. (2000).
We may also use the k-value to rank species by
their tendency to aggregate (cf. Condit et al.,2000). Both a larger number of clumps anda tighter average clump-size increase the k-statistic. Hence, the k-statistic provides a "rst-cut, agglomerate measure of overall aggregation. To be
precise, the value of k itself means little without
comparison to the distribution of k-statistics generated by many simulations of random placement. Given an observed k-value, k, if
M@QCPTCB
a very large number of random simulations werefeasible then the quantile in which klies
M@QCPTCB
would yield a good index of aggregation. As
a surrogate to this computationally intensive index, we use 19 simulations and the value ofk!kin order to rank the species by aggregaK?Vtion. For each species, the value k!kindiK?Vcates the extent to which the species is more (orless) aggregated than the most aggregated-looking random-placement simulation.

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88

J. B. PLOTKIN E¹ A¸.

Figure 4 illustrates species at Pasoh whichrange from the most-clumped towards the leastclumped according to their k!kranking.
K?VThe rankings seem to agree with an intuitive
assessment of aggregation. For example, the species with the largest k!k , Pentace strychK?V
noidea, is visibly more aggregated than any otherspecies. As Fig. 4 indicates, this ranking allows us
to stratify all the species by their aggregation
tendencies.
Poisson Cluster Model
We have seen that the k-statistic, compiled
from the entire Ripley K(d)-curve, yields a convenient, single measure of aggregation. Nevertheless, the K(d)-curve contains far more information than the value of k alone. In particular, we
desire a more speci"c characterization of aggregation than a single index. We desire a modelof the aggregation patterns. In this spirit, wechoose to characterize local aggregation patternsby (i) a measure of the number of clumps and (ii)a measure of the mean clump size. Furthermore,for each species, we wish to distill these twoparameters, denoted by o and p, from the observed K(d)-curve.
SPECIFICATION OF THE CLUSTER MODEL

We model the spatial pattern of each species byan independent, Poisson cluster process. This
point-process is well understood theoretically(Cressie, 1991; Neyman & Scott, 1958), and it has
the advantage of simplicity. Despite its simplicity,
as we shall see, this model of aggregation captures enough information about spatial-patterning to reproduce the observed species}area curvewith great accuracy. Most important, we choose
the Poisson cluster process because its probabilistic properties (in particular, its expected Ripley
K-curve) are well-understood in the literature;
this fact will allow us to estimate parameters
without much di$culty.
We use the following axiomatic de"nition of
the Poisson cluster process.
1. &&Parents'' form a Poisson process in theplane with intensity o.
2. Each &&parent'' produces a random numberof &&o!spring'', drawn independently froma "xed distribution.

3. The positions of the &&o!spring'' relative totheir parent are drawn independently froma "xed bivariate probability density function h.
4. The "nal pattern consists only of the &&o!spring'' events.
More speci"cally, we stipulate that the o!spring of a parent follows a radially symmetric
Gaussian distribution with distribution functionh(x, y)"(2np)\ exp

!(x#y)
.
2p

(8)

In particular, the mean squared distance from an
o!spring to its parent is 2p and the mean distance is p(n/2. As desired, o measures the density of clumps and p measures clump size. We
choose to describe o in units of clumps-persquare-meter and p in units of meters.
In practice, once the parameters (o ,p ) have
G G
been estimated for species i, we simulate the cluster process by placing Wo 5;10(m)#1/2
XG&&parents'' in the plot according to a uniform
distribution. Next, we assign each of n stems toGa randomly chosen parent, and position the stem
according to eqn (8), relative to the parent. In
particular, the expected number of stems perclump is given by n /(o 5;10). We impose
G G
toroidal boundary conditions in the event thata stem is placed outside of the 50-ha plot. Finally,
we erase the &&parents''*which were only used inorder to position the individual clumps. For eachspecies i, the cluster process is completely determined by the parameters (n , o , p ).
G G G
By choosing this model, we certainly imply
that clusters arise from local propagation. In thissense, the Poisson cluster process models aggregation caused by local seed dispersal or gap
recruitment. The observed clump size for eachspecies is surely determined by a number of biotic
and abiotic factors other than seed dispersal andgap recruitment. Nevertheless, dispersal and recruitment are prominent among these factors.
We have used the words &&parent'' and &&o!spring''
as an analogy to these processes. We emphasize,
however, that our notation does not imply, however, that every observed cluster arises froma single &&parent'' tree. Nor do we believe that ourJTBI*20002158*Binni*VVC
SPATIAL AGGREGATION IN TROPICAL FORESTS

stochastic process represents the actual mechanics which caused the formation of clusters in theforest. Instead, we are using the cluster process
because it o!ers a convenient, phenomenologicalmodel to describe the pattern which results fromthe myriad of mechanistic forces which determine
local aggregation patterns. More global patternsof aggregation (e.g. those guided by altitude
speci"city) and hierarchical aggregation will beconsidered later.
ESTIMATING CLUSTER PARAMETERS

Having speci"ed the cluster model, all thatremains is to estimate the parameters (o,p) for
each species. We estimate the best-"t parameters
by using the measured KK (d) curve. To this end, werely on the well-established fact that a Poissoncluster process (o,p) results in the following K(d)curve (Cressie, 1991):

K(d).!."nd#o\ 1!exp

 
!d
4p

. (9)

Given the empirical values KK (0), 2, KK (d ), we
K?Vchoose (o, p) such that K(d).!. most closely "ts
the observed values. Choice of the upper limitdhas a signi"cant e!ect on the resulting paraK?Vmeter "t. For instance, using d (385 m at theK?VHKK plot does not provide near as good as "t asd "385 m. In this sense, the statistic
K?V
P !P
illustrated in Fig. 3 provides crucial
B
P?LBMK
information for each plot: it indicates the proper
range of distances over which to parameterize
our spatial model.
Diggle (1983) suggests "tting (o, p) by minimizing the integral
BK?V (K(h)A!(K(h).!.)A)dh,(10)

for some tuning constant c+1/2. (We found thatc"1/4 was most e!ective.) In order to minimize
eqn (10) we must specify an initial parameter
guess for (o, p). This choice has a dramatic e!ecton the eventual parameter "t. Let (dM , KK (dM )) be themaximum point on the observed KK (d) curve (ina tie, choose the largest such dM ). Following
Diggle, we specify our initial parameter guess as(o, p)"(1/KK (dM ), dM /4). We have estimated para-

89

meters by "tting eqn (9) to the observed values
via minimization [eqn (10)] and via a more traditional curve-"tting technique (minimizing thes merit function of the sum squared residuals).
Both methods provide comparable parameters.
Having computed the parameters (o , p ) for
G Geach species, we may compare the resulting Poisson cluster model to the actual data. Visually, thecluster model of each species is strikingly similar
to the actual data (Fig. 4). Of course, the simplistic, radially symmetric model does not reproduce
the "ne details of all tree placement, but our
method has the ability to separate a species intoa best-"t number of clusters and best-"t cluster
size. Careful inspection of Fig. 4 reveals that theparameter estimation, although largely accurate,
is not perfect. For example, the method appears
to overestimate slightly the number of Mallotus
leucodermis clusters, and underestimate the average Phaeanthus ophthalmicus cluster size. The
method also fails to capture the sharpness of thecluster margins in some cases*e.g. Pentace
strychnoidea and Cleistanthus sumatranus. To
a "rst approximation, however, the model matches our complex, visual intuition of aggregation(Fig. 4).
A rigorous goodness-of-"t between model anddata, for each species at each plot, is certainly
possible. Nevertheless, we refrain from this statistical exercise. Given the topic of our investigation, the important criterion by which to judge
the cluster model should be its ability to reproduce the observed species}area curve. This criterion provides a practical, ecologically motivated metric. Moreover, whether or not themodel produces the correct SAR, we will deduce
information about the extent and manner inwhich individual-level aggregation e!ects species-level patterning.
Before examining the SAR, we pause brie#y to
inspect the distribution of clump sizes estimatedby our best-"t parameters. The distribution ofp values is right-skewed normal at all three plots.
Given p, recall that p(n/2 yields the mean distance from an individual to the center of theclump. This corresponds to mean patch size, andis determined primarily by dispersal distance andgap-sizes. Figure 5 shows a histogram of mean
clump radius (p(n/2) for those species at Pasohwhich are classi"ed as clumped. From Fig. 5 we

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90

J. B. PLOTKIN E¹ A¸.

FIG. 4. Examples from Pasoh of the observed spatial pattern of a species (upper rectangle of each pair) compared with thepattern simulated by the Poisson cluster model (lower rectangle). Each rectangle encompasses 50 ha. The species are ranked
according to their values of k!k ; their ranked position is denoted in parentheses. This aggregation statistic appears toK?Vagree with an intuitive measure of clumping. Despite minor discrepancies, the Poisson cluster model provides a visuallyacceptable reproduction of the spatial pattern for each species.

see that 250 m is a decent upper-bound on mean
patch size at Pasoh*which agrees with ourearlier estimate in Section 4 (cf. Fig. 3).

THE SAR GENERATED BY THE CLUSTER MODEL

For Pasoh we have simulated the entire 50-haplot by overlaying each independently simulated

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91

FIG. 5. A frequency chart of mean clump radius (estimated by p(n/2) for aggregated species at Pasoh. Noticethat 250 m provides a decent upper bound on dispersion
distance, as seen independently in Fig. 3.

species. The resulting model SAR is almost indistinguishable from the actual SAR. The model
very slightly overestimates diversity for areas(0.2 ha, but accurately characterizes the SAR
from 0.2 to 50 ha. For this large range of areas,
the observed SAR falls within the con"dence intervals for the model (which are extremely tight,
smaller than the rectangular dots used Fig. 6). We
have repeated the entire method ("tting parameters and simulating the point process) at HKK
and Lambir; in all three plots we "nd an excellent
agreement between model and data (Fig. 6). In
short, we may conclude that the cluster model
accurately reproduces enough information about
spatial patterning to generate the correct species}area curve, given the abundances of species.
In this sense, the Poisson cluster model completes
the &&zeroth-order analysis'' pioneered by Coleman.
Given the simplicity of the cluster model, the"delity with which it reproduces the SAR in each
forest is somewhat surprising. We have accounted for clustering only on a relatively local
scale. We have not accounted for larger-scale,environmentally driven patterns. We have notused information about tree-diameter. Most importantly, we have ignored all inter-speci"c spatial patterns. The Poisson cluster model, the Ripley-K measurements, and the (o , p ) parameter
G G
estimation are all intrinsically univariate. Despite
these simpli"cations, the resulting SAR agrees

FIG. 6. The observed species}area curve at Pasoh, Lambir, and HKK along with the predictions via a randomplacement model and the Poisson cluster model. In contrast
to the random-placement model which overestimates diversity, the cluster model reproduces the SAR accurately*especially for areas '0.2 ha. In each graph, the inset displays
the same information on log-linear axes: Random placement
model, } } } ; Cluster model, ** ; Observed data, ) ) ) )with the empirical SAR. In other words, whatever
non-random, inter-speci"c patterns or environmental patterns exist, they do not in#uence thespecies}area curve. For each species, the SAR
depends only on the location of conspeci"c trees;JTBI=20002158=Binni=VVC

92

J. B. PLOTKIN E¹ A¸.

the placement of other species is relatively unimportant. In some sense, this result suggests thatinterspeci"c competition may have only a limited
or indirect e!ect on the species}area curve at thisscale. Our results also suggest that, despite its
prominence in ecology at large, the species}area
curve may be a somewhat insensitive indicator ofcommunity structure.

Abundance and Aggregation
The methods developed thus far are su$cient
to rank species by aggregation (Fig. 4), to reproduce intra-speci"c patterns in all three plots(Fig. 4), and to characterize accurately the species}area curve (Fig. 6). Thus, we may have somecon"dence in the practical and biological relevance of our model and its estimated parameters. We now investigate which biological features are correlated with the parameters of ourmodel. We will address the relationship between
a species' abundance and its tendency to aggregate.
Figure 7 shows a graph of species abundancen versus the k-statistic for Pasoh (compare withG
Fig. 3 in He et al., 1987). As mentioned before, thestatistic k alone, without reference to its quantile
position in Monte-Carlo trials, may be a poor
choice of aggregation index due to autocorrelation problems. Alternatively, we could use
k!kover 19 simulations. These provisos
K?Vaside, we will investigate the relationship betweenn and k.
G

FIG. 7. Graph of abundance vs. the clumping index k for
Pasoh. We only graph those species with abundance greaterthan 50 stems. Spearman rank correlation reveals a statistically signi"cant, but very small, negative correlation between n and k for these species.
G

We "nd that there is a statistically signi"cant,but extremely slight negative correlation between
abundance and aggregation tendency at Pasoh,
indicated by k (cf. Fig. 7). This result agrees withthe parallel result of He et al. (1987) based upon
the Donnelly index of aggregation. Similar results hold at HKK and Mudumalai. Our correlation statistics are based only on those species
with n '50. We cannot place much con"denceGin the clumping index for less abundant species
where k is likely prone to autocorrelation errors.
This is unfortunate because roughly one-third of
the species in Pasoh have fewer than 50 stems.
Although the standard Pearson correlation coef"cient of (n , k ) is insigni"cant (r"!0.0884), theG G
Pearson correlation coe$cient on the log}log
transform is statistically signi"cant (r"!0.2688). The latter statistic should be trusted
more; each variable follows a nearly lognormal
distribution. As further evidence, the Spearman
rank coe$cient of correlation between n and k isGalso statistically signi"cant (r"!0.2644) We
have tested signi"cance using the fact that"r"((n!2)/(1!r) follows a Student's t distribution.
We emphasize, however, that statistical signi"cance here by no means implies biological orpractical importance. On the contrary, thenegative correlation between abundance and aggregation is extremely slight. The statistical signi"cance arises only out of the large number ofdata points. In fact, the correlation coe$cientreveals that only r, or less than 7% of thevariation in k may be predicted from the abundance n .
GIf, however, we include all species with n '5,G
we "nd a somewhat stronger negative correlation
between abundance and k (Pearson r"!0.4792), suggesting that rarer species are moreaggregated. This trend is very likely a statisticalartifact, and we cannot be con"dent that it has
much biological signi"cance. Even when trees areplaced at random, the k statistic becomes large
when n is small. In other words, the k statisticG
su!ers from autocorrelation problems when n isGsmall (Fig. 8). When considering all species withn '5, the alternative statistic k!kyields an
G
K?Vinsigni"cant Spearman coe$cient (r"!0.0677),
suggesting that aggregation is not correlated with
abundance after all. The experiment in Fig. 9

JTBI*20002158*Binni*VVC
SPATIAL AGGREGATION IN TROPICAL FORESTS

FIG. 8. Graph of abundance versus the maximum (upper) ) ) ) )), mean () ) ) )), and median (lower ) ) ) ) ) )) k-statistic obtained over 19 Monte-Carlo trials of spatial randomness.
Even when trees are located randomly, k automatically
becomes large for small n . Hence, the k statistic alone isGa poor choice for comparing aggregation with abundance.

FIG. 9. The results of an experiment which tests the biological impact of abundance on aggregation. We graph the
SAR generated by the Poisson cluster model of Pasoh (as inFig. 6), as well as the SAR generated by the cluster model
whose aggregation parameters (o ,p ) have each been asG G
signed to a randomly chosen abundance n . The resultingH
species}area curves are almost identical, and they both agreewith the actual SAR at Pasoh. In other words, randomly
shu%ing the abundances of the species does not e!ect theSAR. The same phenomenon also occurs at HKK and
Lambir (not shown). Hence, abundance and aggregation areconclusively uncorrelated insofar as the species}area curveis concerned. Before shu%ing (*), after shu%ing (} } }).

should help to resolve this statistical quandary ina practical way.
Aside from overall aggregation (k), we may
investigate the more speci"c issues of abundancecorrelation with clump density (o) or with clump
size (p). In fact, given autocorrelation di$culties

93

with the k-statistic, a comparison between n andGo may be more useful. Among the 534 species at
G
Pasoh classi"ed as clumped (k'k ) with
K?Vn '50, there is no statistically signi"cant corG
relation between abundance and o (Spearman
rank correlation r"!0.0057). This gives further
evidence that the negative correlation between
abundance and aggregation tendency (k) is probably spurious.
We also "nd a slight negative correlation between abundance and p (Spearman rank correlation r"!0.1152)*suggesting, if anything, thatrare species are less tightly clumped than common species.
Given our results about o, p, and k!k , we
K?Vbelieve that the observed correlation between
k and abundance re#ects a statistical artifact
more than a biological reality. This claim may
con#ict somewhat with analyses of Hubbell(1979) or recent, extensive analyses of Condit etal. (2000). The biological importance of the correlation may be tested*insofar as the species}area curve is concerned*in a revealing,practical manner. Recall that three parameters,
n , o , and p , determine the distribution pattern
G G
Gof a species in the Poisson cluster process. As anexperiment, we may assign the previously measured parameters (o , p ) to the abundance of
G G
a randomly chosen species, n . In other words, weHrandomly shu%e the abundances of the species,and re-run the cluster simulation. This has the
e!ect of removing whatever correlation may haveexisted between abundance and cluster parameters.
For all three plots, upon randomly shu%ing abundances the resulting SAR of the cluster model isalmost identical to the SAR with non-shu%ed
abundances (Fig. 9). This somewhat surprising result re#ects the fact that abundance was not strongly correlated with aggregation to start with.
Although we originally "t parameters (o , p )G G
to each species i, our model produces the correct
SAR even if we randomly draw each species'
parameters from the distribution of the "ttedparameters (Fig. 9). This result should allow us topredict an SAR fairly well without detailed
knowledge of best-"t, species-by-species parameters. For example, if we use the abundances of
Pasoh and draw aggregation parameters randomly from the distribution of parameters "t for
Lambir, we nevertheless obtain a fairly accurate

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J. B. PLOTKIN E¹ A¸.

model of the Pasoh species}area curve. In otherwords, only the distribution of aggregation parameters matters, and the distributions at Lambir
and Pasoh are fairly similar.
Environment and Aggregation: Heirarchial
Clumping
Thus far we have modeled clumping on a local
scale. The cluster model loosely mimics local
aggregation patterns. Although aggregation at
this scale alone has been demonstrated to determine the SAR (insofar as 50-ha data sets may
verify), larger-scale aggregation driven by habitat
certainly occurs as well. In this section, we investigate the extent and importance of environmentally driven aggregation. We are particularlyinterested in the possibility of hierarchical aggregation in which the local patches are themselves clustered following a more global pattern. We
will employ our Poisson cluster model in order tofactor out local clumping, and thereby properly
examine larger-scale patterns.
Topography, soil di!erentiation, water stress,
etc. are all examples of environmental factors

which we suspect in#uence spatial patterns at
scales larger than mean dispersal distance andgap size (cf. Hubbell & Foster, 1983). In addition,
there is certainly an undersurge of abiotic in#uences on all spatial scales. We demonstrate ananalysis of heirarchial clumping driven by topography at the Pasoh forest. As with local aggregation, we follow a simple, "rst-cut approach.
In particular, we focus on the main topographic
gradient found at Pasoh: the single hill within theplot.
In particular, we divide the Pasoh 50-ha plot
into two habitats called simply &&on the hill'' and&&o! the hill''. Our methods easily generalize tomultiple environments*such as in a valley, on
the slope, on a plateau, on humult soils, in oldgrowth forest, etc. Subdividing the plot into 800
squares each 25;25 m large, we de"ne eachsquare as either on the hill or o! the hill. Thissubjective process relies only on the topographic
contour lines (Fig. 10). According to our (subjective) de"nition, the &&hill'' at Pasoh accounts for
37% of plot's total area; 36% of all stems in theplot are on the hill.

FIG. 10. A schematic diagram depicting our subjective de"nition of the two environments at Pasoh: &&on the hill'' and &&o!
the hill''. The example species Barringtonia macrostachya in (a) is used as a general guide by which we select a cut-o!
topographic contour (b). Altitudes above the cut-o! are de"ned as on the hill. (c) illustrates the 25;25 m boxes into which we
subdivide the plot. Boxes with a circle are de"ned to be on the hill, and the other boxes o!. As de"ned, the hill occupies about
37% of the 50-ha plot. Barringtonia macrostachya has 88% of its stems o! the hill.

JTBI*20002158*Binni*VVC
95

SPATIAL AGGREGATION IN TROPICAL FORESTSRANDOM-PLACEMENT TESTS OF TOPOGRAPHICCORRELATION

Given a speci"cation of the two habitats,a straightforward analysis of aggregation driven
by topography would proceed via a s test (cf.
Basnet, 1992). For each species, the s test determines if an extraordinary number of trees are on(or o!) the hill, assuming that tree locations are
independent of one another. As we have demonstrated already, the independence assumption iscertainly violated by most species. Therefore, wewill design a slightly more sophisticated method
which accounts for local aggregation. Our
method o!ers an alternative to the elegant,torus-randomization method of Harms (1997);
Harms randomizes the locations of the habitats,
while we randomize the locations of the trees. In
preparation for our more sophisticated test, we
begin by re-formulating the s test in a MonteCarlo setting. For a given arrangement of trees(of a particular species i) in the plot, we de"ne1
C"proportion of trees on the hill"nG(number of trees on the hill).
(11)
In order to test if the observed arrangement ofspecies i is signi"cantly hill-correlated (positively
or negatively), we perform 1000 Monte-Carlosimulations. For each simulation, we placen trees of species i in the plot randomly, and weGmeasure C. If the true, observed value of C falls ineither the top 2.5% or bottom 2.5% tail of thesimulations, we can reject the null hypothesis ofno correlation at the 5% con"dence level.
The C test with random-placement MonteCarlo simulations is absolutely equivalent to thestandard s test. As veri"cation, 1000 simulations
reveal that the two methods di!er on (1% of
the species. The s test indicates that roughly
two-thirds of all species at Pasoh are hill-correlated (Table 2). Figure 11 illustrates the six mosthill-correlated species (positively or negatively) as
ranked by their s value.
Figure 11 illustrates that the topography at
Pasoh can have a strong e!ect on spatial patterning. Nevertheless, one of the species shown in
Fig. 11, Pentace strychnoidea, despite its extremely high s value, would likely not meet our intuitive notion of topographic speci"city. Pentace

TABLE 2
¹he proportion of all 817 species which are classi,ed as hill-correlated (leftmost column)AllspeciesSpeciesSpecieswith n '50 with n '50,GG '3n
AJSKNQ

C!testrandom-model 511 (62.5%) 423 (79.2%) 361 (79.2%)
C!testcluster-model 241 (29.5%) 212 (39.7%) 183 (40.1%)
Results are given for the C test with random-placement MonteCarlo simulations (equivalent to a s test) and for the C test withcluster-placement. Both tests were performed at the 5% signi"cancelevel. Species by species, the latter test indicates whether or not theclusters of trees are themselves clustered on (or o!) the hill. The
cluster test reveals that, by taking account of local aggregation,
topography in#uences the geographic range of nearly half as many
species as suggested by a naive s-analysis. The same result holds
among the 534 species with '50 stems (middle column) or amongthe 456 species with '50 stems and '3 clumps (rightmost
column).

strychnoidea is an extremely clumped species (infact, it has the highest aggregation index in Fig.
4), and it is unclear whether its association withthe hill is driven by environment per se. Perhaps,
instead, this species is found only on the hill
because its single, tight, dispersal-driven clump
happens by chance to lie on the hill. Thes test*or, equivalently, the C test with random-placement simulations*cannot distinguishbetween these two possibilities. Many species at
Pasoh follow patterns similar to P. strychnoidea.
We desire a new test to disentangle aggregation
on di!erent scales.
A CLUSTER-BASED TEST OF TOPOGRAPHICCORRELATION

Fortunately, we can adjust our C test so as totake advantage of our knowledge of local aggregation. Instead of using random-placementfor our Monte-Carlo simulations, we may use the
Poisson cluster process with the best-"t parameters for each species. In essence, the C test withcluster-based Monte-Carlo simulations will determine whether or not an extraordinary numberof clumps of each species are on (or o!) the hill.
For each species, the improved C test provides
a viewpoint which is more coarse then the indi-

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96

J. B. PLOTKIN E¹ A¸.

FIG. 11. Spatial maps of the six most topographically driven species at Pasoh, as indicated by their s scores. Three of thespecies are positively correlated with the hill, and three negatively. Even though the s test ranks Pentace strychnoidea asstrongly correlated with the hill, it seems plausible that the single, tight, clump of this species lies on the hill by chance, as
opposed to genuine topographic speci"city. This example highlights the need for a more sophisticated test than thes analysis.

vidual level*and the level of coarseness is controlled by the cluster parameters (o , p ). (ForG G
those species for which we estimated very manyclumps or very large clump radius, the cluster-C
test still operates on an individual level.) For
signi"cantly clumped species, the C test with cluster-based simulations will detect the extent towhich the clumps are themselves aggregated on
the hill, i.e. a test of hierarchical clumping (Table 2).
The cluster-based C test reveals that about
one-half of the species classi"ed as on or o! thehill by the s test are not, in fact, signi"cantly
hill-correlated. In other words, by taking accountof their local-clumping properties, only half as

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97

FIG. 12. Three examples of species classi"ed as hill-correlated by the s test, but uncorrelated by the cluster-based C test.
These species do not show an extraordinary number of clumps either on or o! the hill (although they do have an
extraordinary number of individuals on or o! the hill). In other words, insofar as may be inferred from our 50-ha data set, thecluster-based C test indicates that small-scale dispersion/recruitment constricts the geographic ranges of these three species,
but topography does not.

many species are actually guided by topography
per se as compared to the naive estimate. This
would suggest that adaptation to topography
limits the placement of individuals less severely
than dispersal, gap recruitment, and other local
factors. For instance, at the 5% level, we haveseen that 83% of species show dispersion-scale
aggregation, while only 30% show true topographic aggregation.
We may have con"dence that the cluster-model
C test performs its stated task well. Figure 12
illustrates three example species for which thenaive s analysis concludes hill-correlation, but
for which the modi"ed C test indicates no correlation. In these examples, by broadening our
focus to the cluster level (instead of the individuallevel), we do not "nd a disproportionate amountof hill-correlation. In particular, as desired, thecluster-based C test indicates that Pentace strychnoidea is not signi"cantly correlated with the hill.
There are no species which the s test "nds uncorrelated but which the C test indicates arecorrelated; the cluster-based test is uniformallymore conservative. Finally, among those species
which the cluster-based C test classi"es as hillcorrelated, there is no signi"cant disparity between the number positively (116) and negatively(125) correlated. The Pasoh species have apparently adapted to "ll both ecological niches
equally well.
We conclude this section with some generalobservations and provisos about the clusterbased C test. Although we have used it to test
topography, the C test can be used to disentangle
local clumping from*and therefore query thestrength of*any environmental factor which operates on scales larger than 50}100 m. Nevertheless, one should be aware that the C test is inherently conservative. As a null hypothesis we
assume that habitat has no e!ect on the spatial
arrangement; hence, if a species fails the C test, wecannot soundly conclude that habitat has noe!ect, but rather that our particular test did notdiscover an e!ect. In this sense, the results reported in Table 2 are conservative lower bounds.
For example, when properly interpreted, Table
2 indicates that at least 30% of the species in
Pasoh are driven by topography, but possibly
more. Furthermore, the Poisson cluster model
used in the C test assumes that the abundance ofeach species is known. These abundances*and,in particular, the total species richness of theplot*are certainly determined to a large extent
by the diversity of habitats in the plot. Hence, theJTBI=20002158=Binni=VVC

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results of the C test should not be interpreted as
evidence against the importance of environmental
determinants overall, but rather as evidence
against the speci"c, relative strength of topography versus local factors as determinants of spatial
patterning within our particular 50-ha plot.
Discussion and Conclusions
We have modeled local clumping via a Poissoncluster process. Fitting the proper parameters of
this model to each of three tropical forests has
yielded signi"cant results on a few fronts.
First, the cluster model characterizes the species}area curve with extremely high "delity. The
model can be viewed as a completion of Coleman's, zeroth-order random-placement model.
This result highlights those biological factors
which are su$cient*and which are unnecessary*to determine the species}area curve. The factthat the cluster parameters can be shu%ed between the species (or transplanted from one forest
to another) without a!ecting the SAR suggeststhat the species}area curve may be a somewhat
insensitive indicator of community structure.
Second, the best-"t parameters of each species
allow us to address possible correlates between
biological factors (such as abundance) and clump
density or clump size. In the future, one may andshould use these parameters to look for spatial
patterns correlated with genus, functionalgroups, dispersal syndrome, and other biologicalfactors.
Finally, the cluster model*which characterizes clumping at the dispersion distance/gap size
scale*may be used to investigate the heirarchical e!ects of habitat on aggregation. This method
applies to those environmental factors which operate on scales larger then dispersion. In particular, we have used the cluster model to ask if,
within each species, the clusters of trees are themselves aggregated following a topographic gradient. Once parameterized the cluster model allows
us to factor out small-scale clumping, and toinvestigate accurately aggregation driven bylarge-scale topography. We conclude quantitatively that, within the 50-ha plot at Pasoh, topography determines the geographic range ofa species less often than small-scale factors suchas dispersion and gap recruitment.

Despite the merits of the Poisson clustermodel, we must mention some of its severe drawbacks. Foremost, the model is phenomenological.
It does not provide a dynamic understanding of
the processes which form the spatial patterns.
Instead, the model is simply a static, retroactive
characterization of spatial patterns. Indeed, estimating the parameters of the model required
detailed knowledge of all tree locations. Thus, thecluster model does not provide a predictive theory. Instead, the cluster model is useful as a tool
for assessing the e!ects and causes of aggregation, for comparing aggregation parametersacross plots, and for investigating biological correlates to aggregation.
We conclude by recalling one of the early investigations into spatial aggregation in a large
tropical forest plot. In 1983, before the "rst census at BCI was even complete, Hubbell andFoster categorized the signi"cant spatial patterns
into three disjoint groups (direct quote, Hubbell& Foster, 1983):(i) Species which appear to be randomly or
near-randomly distributed over the plot.
(ii) Species which are clumped and whosepatches follow easily recognized topographic features of the plot.
(iii) Species which are clumped but whosepatches are spatially uncorrelated with
topography.
At the time, a limited number of species in the
BCI canopy were identi"ed according to thesecategories, relying mainly upon intuition and visual inspection. We believe that the three categories above, even though they were identi"ed in
the BCI canopy, remain the most descriptive,
qualitative characterization of the major spatial
patterns in tropical forests. The rich theory of
stochastic point processes, however, now allows
us to approach these categories more quantitatively. The Poisson cluster model and the C
test provide a rigorous, systematic method for
categorizing each species according to Hubbell's
original delineations.
The authors sincerely thank Helene Muller-Landau, Simon Levin, Bert Leigh, Bill Bossert, Ran Nathan, Liz Losos,
Richard Condit, and Burt Singer for their helpful suggestions. The authors are especially indebted to Lee Hua SengJTBI*20002158*Binni*VVC
SPATIAL AGGREGATION IN TROPICAL FORESTS

for the use of Lambir data, and to Sarayudh Bunyavejchewin for the HKK data. J.B.P is supported by a fellowshipfrom the National Science Foundation.

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