Predicting species diversity in tropical forests
Joshua B. Plotkina,b, Matthew D. Pottsc, Douglas W. Yud, Sarayudh Bunyavejchewine, Richard Conditf, Robin Fosterg,
Stephen Hubbellh, James LaFrankiei, N. Manokaranj, Lee Hua Sengk, Raman Sukumarl, Martin A. Nowaka,and Peter S. Ashtonm
aInstitute for Advanced Study and Princeton University, Princeton, NJ 08540; cDivision of Engineering and Applied Sciences, and mDepartment of Organismal

and Evolutionary Biology, Harvard University, Cambridge, MA 02138; dCentre for Population Biology, Imperial College, Ascot, Berkshire SL5 7PY, United
Kingdom; eSilvicultural Research Division, Royal Forest Department, Chatuchak, Bangkok 10900, Thailand; fCenter for Tropical Forest Science, Smithsonian
Tropical Research Institute, Balboa, Republic of Panama Box 2072; gBotany Department, The Chicago Field Museum, Chicago, IL 60605; hBotany Department,
University of Georgia, Athens, GA 30602; iCenter for Tropical Forest Science, National Institute of Education, 1025 Singapore; jForest Research Institute of
Malaysia, Kepong, Malaysia 52109; kSarawak Forestry Department, Kuching, Sarawak, Malaysia SW 93750; and lCentre for Ecological Science, Indian
Institute of Science, Bangalore, India 560 012
Communicated by John W. Terborgh, Duke University, Durham, NC, July 26, 2000 (received for review April 15, 2000)

A fundamental question in ecology is how many species occur
within a given area. Despite the complexity and diversity ofdifferent ecosystems, there exists a surprisingly simple, approximate answer: the number of species is proportional to the size ofthe area raised to some exponent. The exponent often turns out tobe roughly 1兾4. This power law can be derived from assumptions
about the relative abundances of species or from notions ofself-similarity. Here we analyze the largest existing data set of
location-mapped species: over one million, individually identified
trees from five tropical forests on three continents. Although thepower law is a reasonable, zeroth-order approximation of ourdata, we find consistent deviations from it on all spatial scales.
Furthermore, tropical forests are not self-similar at areas <50
hectares. We develop an extended model of the species-area
relationship, which enables us to predict large-scale species diversity from small-scale data samples more accurately than any otheravailable method.

A

primary motivation for modern ecological research is theeffort to save as many species as possible from the sixth
great mass extinction that currently threatens them (1, 2). How
does habitat loss and destruction of tropical forests relate tospecies extinction? How many tree species must remain in an
exploited forest if primate species are to survive in it? What isthe best possible design of a natural reserve that maximizes thenumber or genetic diversity of surviving species? All of thesequestions underscore the necessity to understand the relationship between species diversity and sampled area (3–8)—a longstanding and controversial subject in ecology (9–11).
The earliest model of the species–area relationship (SAR) was
introduced by Arrhenius in 1921 and posits a power law: thenumber of species, S, found in a census area, A, is given byS ⯝ cAz,[1]

where c and z are constants (12). Empirical observations
suggest that z is about 1兾4 for many ecosystems (13). The
power law is a cornerstone for theories of biogeography(14 –16). In 1975 May (17) derived the power law by assumingthat species’ abundances follow a lognormal distribution. The
canonical lognormal distribution implies that S ⯝ cN1/4, where
N is the total number of individuals and c is a constant.
Assuming that N is proportional to the area A, we immediately
obtain Eq. 1 with z ⫽ 1兾4.
More recently, Harte et al. (18) have shown that the power law
is equivalent to self-similarity. If the fraction of species in an area
A that are also found in one-half of that area is independentof A, then the spatial distribution of species is self-similar. Let
Ai ⫽ A0兾2i denote the area of a rectangular patch obtained afteri bisections of the total sampled area A0. Denote by Si the average
10850 –10854 兩 PNAS 兩 September 26, 2000 兩 vol. 97 兩 no. 20

number of species found in a patch Ai. If the ratio ai ⫽ Si兾Si⫺1
does not depend on i then the assemblage is self-similar (18).
Self-similarity is equivalent to Si ⫽ cAiz with z ⫽ ⫺log2a. Unlike
the canonical lognormal, self-similarity does not provide an apriori estimate of the exponent z.
Tropical Forest Data
To test the basic principles of SARs—with an aim toward
generalizing the power law—we have analyzed five 50-hectare(ha) plots of tropical forests across the globe. Although tropicalforests cover only 7% of the Earth’s land surface, they containmore than half of the world’s species (6). Tropical forests are wellknown as the most genetically diverse, terrestrial communities
on Earth (19). Moreover, animal diversity in tropical forests
depends crucially on the diversity of plants (20).
Each of the 50-ha plots that we analyze is part of a long-term
research program coordinated by the Smithsonian’s Center
for Tropical Forest Science. The plots are located in thefollowing forests: Huai Kha Khaeng (HKK) Wildlife Sanctuary, Thailand (first census); Lambir Hills National Park,
Sarawak, Malaysia (third census); Pasoh Forest Reserve,
Peninsular Malaysia (third census); Barro Colorado Island(BCI), Panama (first census); and Mudumalai Wildlife Sanctuary, India (second census). For each of our plots, every
free-standing, woody stem over 1 cm in diameter has been
identified to species. We include all such stems in our analyses.
(Our qualitative results are unchanged if we include, instead,stems over 5 cm in diameter.) The number of such stems, andthe number of species among them, varies greatly from plot toplot (Fig. 1).
Fig. 2a shows the species-area relationship for the five tropicalforests compared with the best-fit power law. The average slope
for the forests is z⯝0.25. As suggested by May, the power law
tends to overestimate the slope at large areas and underestimate
the slope at small areas (17). But the extent to which the powerlaw fails is often poorly recognized in the ecological literature (3,18, 21). Previous research has uncovered the power law’s failure
for small areas, but has downplayed its deviations for areas larger
than 2 ha (22). Fig. 2b shows the dependence of the parameterai ⫽ Si兾Si⫺1 on area. The ratio ai describes the average fractionof species that persist upon the ith bisection; therefore we call ai
the spatial persistence parameter. Self-similarity would requirethat this parameter be independent of area (18). As the figure
Abbreviations: SAR, species–area curve; ha, hectare; BCI, Barro Colorado Island; HKK, Huai
Kha Khaeng.
bTo whom reprint requests should be addressed at: Institute For Advanced Study, Olden

Lane, Princeton, NJ 08540. E-mail: plotkin@ias.edu.
The publication costs of this article were defrayed in part by page charge payment. This
article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C.
§1734 solely to indicate this fact.

shows, however, ai is not constant for any range of areas between
1 m2 and 50 ha. Hence, tropical forests are conclusively not
self-similar at these scales. The empirical form of the spatial
persistence curve, and its departure from self-similarity, may
result in part from aggregation of conspecifics—a possibility that
we explore in detail elsewhere (29).
A Differential Equation Approach
Instead of self-similarity we find a consistent functional relationship between ai and the area, A, in all five forests (Fig. 2b).
This observation is striking in light of the forests’ disparate
geographic locations, climates, and overall species diversities.
We now introduce the spatial persistence function, a(A), as acontinuous extension of ai. In the appendix, we use the persistence values of our data to derive a canonical, two-parametermodel of a(A). Once this function has been derived, we obtain
the SAR by solving the differential equation⫺log2 关a共A兲兴 ⫽

A dS
䡠
.
S dA

[2]

Using the diversity measured in a small area as the initialcondition in Eq. 2, we may predict the diversity of a much larger
area, if we know a(A).
In the appendix, we derive Eq. 2 and find a general solutionof the form S ⫽ cAzexp[P(A)], where P(A) is an infinite
polynomial in A. We can truncate after the first n terms to obtainan approximate solution. Truncating after the first term leads tothe expression
S ⯝ cAze⫺kA.

[3]Here c, z, and k are constants determined by a(A). Thisapproximate solution is less accurate than the complete solution
to Eq. 2, and it is only valid for a limited range of areas.
Nevertheless, the approximation has the obvious advantage ofsimplicity. If we let n 3 ⬁, then we recover the full solution to
Plotkin et al.

Eq. 2; if we let n 3 0, then we reduce to the power law. Hencethe power law is a zeroth-order, special case of our general modelfor the SAR.
Eq. 2 accurately predicts diversity given only a small amountof data. Because the persistence curves are similar across the fiveplots (Fig. 2b), we may use the canonical form of a(A) fit at one
plot to predict diversity for another plot. For example, using
BCI’s persistence curve to determine a(A) and using the diversity in a single ha of Pasoh as the initial condition, we can predictthe 50-ha diversity of Pasoh within 3% on average (Fig. 3).
Conversely, Pasoh’s persistence function predicts BCI’s total
diversity with 4% average error, and Lambir’s diversity with 9%
error, from a single ha of data.
Fig. 3 illustrates the extrapolative ability of Eq. 2 as comparedwith the classical models of the SAR. The precision of our
method—namely, the ability to predict 50-ha diversity within 5%
at Pasoh and BCI and 10% at Lambir—is an improvement overother previous methods. It is 1- to 7-fold more precise thanFisher’s alpha (23), and 5- to 10-fold more precise than the powerlaw. On the one hand, the increased precision of our method is
not surprising; we have used two parameters to describe a(A), as
opposed to the classical models, which generally require one
parameter. On the other hand, given the interplot similarity ofpersistence curves, in practice we need only measure one parameter—the diversity of a single ha—to extrapolate diversity
via Eq. 2.
Implications and Conclusions
We have analyzed the largest existing data set of locationmapped trees in tropical forests. We find that the SAR shows
consistent deviations from the power law on all spatial scales thatwere studied, ranging from 1 m2 to 50 ha. Hence, self-similarity
does not hold over this range of areas. (There is the possibility
that tropical forests are self-similar over scales greater than 50
ha, but in the absence of further data this remains speculative.)
PNAS 兩 September 26, 2000 兩 vol. 97 兩 no. 20 兩 10851

ECOLOGY

Fig. 1. The locations of five tropical forest plots across the globe. Each census encompasses 50 ha of forest within which every woody stem greater than 1 cm
in diameter has been identified to species, measured for girth, and spatially mapped to ⬍1-m accuracy. The name, country, number of trees, and number of
species is indicated for each plot. The forests vary widely in species diversity and environment. Pasoh and Lambir (Malaysia) are evergreen, dipterocarp rainforests;
BCI (Panama) is a lowland, moist forest, with a 4-month dry season; HKK (Thailand) and Mudumalai (India) are the only forests that are regularly subject to fires.
For a complete list of references, consult the Center for Tropical Forest Science web site at http:兾兾www.stri.org.

Appendix
In this appendix we provide the details behind the derivation of
Eq. 2, its solution, and its application to extrapolating diversity.
Derivation of Eq. 2. Given the definition of the spatial persistence
parameter, ai ⫽ Si兾Si⫺1, we start by deriving its relationship to
the slope of the SAR. All logarithms are henceforth taken basetwo:⫺log共a i 兲 ⫽ ⫺ log冉 冊Si
Si ⫺ 1⫽log共S i ⫺ 1 兲⫺ log共S i 兲
log2⫽

log共S i ⫺ 1 兲⫺ log共S i 兲.
log共A i ⫺ 1 兲⫺ log共A i 兲[4]The last equality follows because Ai⫺1兾Ai ⫽ 2. The final quantity
in Eq. 4 measures the slope of a small chord on the log-log SAR
to the right of log(Ai). We conclude that⫺ log共a i 兲 ⯝Fig. 2. Graphs of the SAR and the spatial persistence parameter for each of
five tropical forests. (a) Log-log graph of the observed species-area data. Eachplot encompasses a total area A0 ⫽ 50 ha. We measure the mean species
diversity, Si, found in disjoint patches obtained by repeated bisections of A0.
The log-log species-area data are concave down for all five plots. The SAR isapproximated loosely by the power law, z ⫽ 0.25, whose slope is indicated by
the trapezoid (red). (b) The persistence parameter, ai ⫽ Si兾Si⫺1, provides asensitive tool for analyzing SARs and testing self-similarity. Self-similarity
would predict constant a ⯝ 2⫺0.25 ⯝ 0.84, shown in red. All five persistence
curves are seen to depart from the power-law model over the entire range ofareas.

These results might have some bearing on the longstanding
controversy surrounding SARs. Previous research has focusedon why the SAR has different slopes in different ecosystems (11),but in our extensive data the SAR does not possess a constantslope whatsoever.
Instead of self-similarity we propose a model of the SAR,
based on the spatial persistence function, which generalizes thepower law. This framework allows us to predict 50-ha diversity
from small-scale samples with greater accuracy than ever before.
Candidate logging protocols often are assayed at the 50- to
100-ha scale, and they are evaluated on the proportion ofdiversity that regenerates, as estimated from a small census (24).
Hence, an accurate method to extrapolate 50-ha diversity froma small census will greatly benefit in the formulation of protocols
for sustainable forestry and for biodiversity surveys (25). Furthermore, our methods may be extended to estimate landscapescale diversity (see Appendix). These advances may induce
ecologists to focus on the persistence curve itself as a unifying
concept. The search for a biological mechanism that explains theobserved persistence patterns offers an important challenge toecology. In the meantime, our theory provides a valuable tool for
conservation planning and a practical method for estimating
diversity in the field.
10852 兩 www.pnas.org

冏d log S,d log A log共 A i兲

[5]where i is now a continuous variable. To be more precise, thechord to the right of log(Ai) is an estimate of the derivative atlog(A(i⫹(i⫺1))/2). Note that Eq. 5 clearly illustrates the equivalence between self-similarity (constant a) and the power law(constant dlogS兾dlogA).
Eq. 2 follows easily from Eq. 5, using the fact that dlogS兾
dlogA ⫽ A兾S 䡠 dS兾dA. Note that Eq. 2 is a strict generalizationof self-similarity: if a(A) is constant, then Eq. 2 reduces to thepower law. If a(A) ⫽ exp(⫺1兾logA), then Eq. 2 reduces toS(A)⬀logA. Hence Eq. 2 also generalizes the logarithmic law
suggested by Gleason (26).
Diversity Extrapolation. Eq. 2 together with the interplot similarity of the persistence curves provide a method for extrapolating diversity over many spatial scales. For example, to
extrapolate diversity from a subsample of Pasoh, we use BCI
data to fit a(A), and then we solve Eq. 2 according to the small,initial condition measured at Pasoh. In effect, this processtranslates BCI’s log-log SAR so that it coincides with Pasoh’s
initial condition; nevertheless, the universality between theforests is described more simply in terms of the persistence
parameter, ai.
We choose to model the persistence values ai with thesimple, two-parameter family of curves 1⁄4⌽[(␣ ⫺ i)兾␤] ⫹ 3⁄4.
(Other choices, such as a cubic model, are also possible andproduce accurate predictions at these scales and beyond. See
below.) Here ⌽(x) is the ‘‘error function’’ given by the cumulative distribution of the Gaussian: ⌽(x) ⫽ (2兾公␲)䡠兰x0e⫺t2dt.
The parameter ␣ moves the inf lection point of the persistence
curve horizontally, and ␤ determines the slope at the inf lection
point. Hence ␤ measures the maximal ‘‘acceleration’’ ofdiversity with area, and ␣ measures the spatial scale at which
acceleration is maximized. The best-fit at BCI is given by ␣ ⫽8.56, ␤ ⫽ 8.08. For Pasoh, ␣ ⫽ 7.73, ␤ ⫽ 7.41; for Mudumalai,␣ ⫽ 7.06 and ␤ ⫽ 7.76.
We may express all solutions to Eq. 2 in the form S ⫽cAzexp[P(A)], where P(A) is a polynomial in A of arbitrary
degree n, with no constant term. Once we specify ␣ and ␤, we
expand ⫺log(a(A)) in a Taylor series of order n around the pointA ⫽ 25 ha. The resulting separable equation can always be solvedin closed form, yielding P. For example, using Pasoh’s ␣ and ␤
to determine a(A), the approximate solution of order n ⫽1 is
Plotkin et al.

Fig. 3. The actual SAR at Pasoh (black) compared with the SAR predicted by our model and three classical models. Assuming that individuals scale linearly witharea, MacArthur’s ‘‘broken stick’’ distribution of relative abundances, the canonical lognormal distribution, and Fisher’s log series each yields a one-parametermodel of the SAR (green, red, and blue, respectively). The first two of these models were parameterized by using 25 ha of Pasoh data; 1 ha was used toparameterize the log series. The log series provides a fairly accurate model, but it overestimates 50-ha diversity by 21%. The canonical log normal accounts forsteeper slopes at small areas and gentler slopes at large areas, and hence it is more accurate than the broken stick (17). Our persistence method (yellow) requires
two parameters, fit by using any other forest, and an initial condition obtained from 1 ha of Pasoh data. The persistence method extrapolates 50-ha diversity
with 3% average error. The figure indicates a 1-SD confidence interval around the extrapolation.

given by S(A) ⫽ S(1ha)䡠AzekA, where z ⫽ 0.125 and k ⫽⫺5.66䡠10⫺4. For Mudumalai, z ⫽ 0.161 and k ⫽ ⫺5.41䡠10⫺4. Thisapproximation is only valid for A ⱕ 50 ha, but its accuracy
compares well with the complete numerical solution: it predicts
50-ha diversity with average error 4% at Pasoh, 9% at BCI, and
16% at Lambir.
In practice, a numerical solution of Eq. 2 yields the mostaccurate SAR. We used the Fehlberg-Runge-Kutta method togenerate the prediction shown in Fig. 3. The initial condition
S(1ha) for Fig. 3 was determined by the diversity of a single,
random, 1-ha subplot of Pasoh. The confidence interval wasconstructed from 1,000, independently sampled, 1-ha initialconditions.
We have divided our five plots into two categories: those that
suffer regular disturbance and those that do not. The two
tropical forests subject to regular fires, HKK and Mudumalai,
generally should not be modeled via the persistence curve froma more stable, moist tropical forest. The values of a1 to a6 are
generally smaller at HKK and Mudumalai than the other three
forests. This reflects the fact that HKK and Mudumalai aresubject to more disturbances, causing greater patchiness. The
persistence curve at Mudumalai can predict HKK and conversely
within 17%, given 1 ha of data. Compared with BCI, Pasoh, andLambir, the accuracy of Eq. 2 has been decreased by the
disturbances at Mudumalai and HKK. Nevertheless, 17% error
is still preferable to the 28% error-rate or worse given by Fisher’s

be useful to assess logging protocols, ⌽ provides a simple,
two-parameter model of the persistence curve. For larger
areas, however, a cubic model (which works as well as ⌽ at 50ha) is often more effective. For example, we can use a cubic
persistence curve calibrated at Pasoh to extrapolate the diversity of the entire BCI, which occupies 1,500 ha, from a 1-hasample. Using Eq. 2, the predicted diversity for all of BCI is436 ⫾ 32 species (1 SD). This estimate compares favorably with
Croat’s f loral count of 450 tree and shrub species on the island(27). For even larger areas, the persistence curve should be
parameterized by using multiple, small censuses spread acrossthe landscape (as in ref. 28), although such techniques require
further development.

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Plotkin et al.

alpha or a power law on these disturbed forests. In practice, whenestimating diversity in a new forest, the ecologist should first
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We sincerely thank Alice Chen, Jay Rosenheim, Stuart Davies, Egbert
Leigh, Bill Bossert, Burt Singer, Steve Pacala, Jerome Chave, and Simon
Levin for many helpful discussions. We are especially grateful to Helene
Muller-Landau, Bob Holt, and Robert May for their conceptual andeditorial input. J.B.P. is supported by a National Science Foundation
graduate fellowship.

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