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# Predicting species diversity in tropical forests

Communicated by John W. Terborgh, Duke University, Durham, NC (received for review April 15, 2000)

### This article has a Correction. Please see:

## Abstract

A fundamental question in ecology is how many species occur within a given area. Despite the complexity and diversity of different ecosystems, there exists a surprisingly simple, approximate answer: the number of species is proportional to the size of the area raised to some exponent. The exponent often turns out to be roughly 1/4. This power law can be derived from assumptions about the relative abundances of species or from notions of self-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 the power law is a reasonable, zeroth-order approximation of our data, 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 other available method.

A primary motivation for modern ecological research is the effort 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 to species extinction? How many tree species must remain in an exploited forest if primate species are to survive in it? What is the best possible design of a natural reserve that maximizes the number or genetic diversity of surviving species? All of these questions 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: the number of species, *S*, found in a census area,
*A*, is given by
1
where *c* and *z* are constants
(12). Empirical observations suggest that *z* is about 1/4
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
assuming that species' abundances follow a lognormal distribution. The
canonical lognormal distribution implies that *S* ≃
*cN ^{1/4}*, 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*= 1/4.

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
independent of *A*, then the spatial distribution of
species is self-similar. Let *A*_{i} =
*A*_{0}/2^{i} denote
the area of a rectangular patch obtained after *i* bisections
of the total sampled area *A*_{0}. Denote
by *S _{i}* the average number of
species found in a patch

*A*. If the ratio

_{i}*a*=

_{i}*S*/

_{i}*S*

_{i−1}does not depend on

*i*then the assemblage is self-similar (18). Self-similarity is equivalent to

*S*=

_{i}*cA*

_{i}

^{z}with

*z*= −

*log*

_{2}

*a*. Unlike the canonical lognormal, self-similarity does not provide an

*a priori*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 tropical forests cover only 7% of the Earth's land surface, they contain more than half of the world's species (6). Tropical forests are well known 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 the following 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, and the number of species among them, varies greatly from plot to plot (Fig. 1).

Fig. 2*a* shows the species-area
relationship for the five tropical forests 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 power law 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. 2*b* shows the
dependence of the parameter *a _{i}* =

*S*/

_{i}*S*

_{i−1}on area. The ratio

*a*describes the average fraction of species that persist upon the

_{i}*i*th bisection; therefore we call

*a*the spatial persistence parameter. Self-similarity would require that this parameter be independent of area (18). As the figure shows, however,

_{i}*a*is not constant for any range of areas between 1 m

_{i}^{2}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 *a _{i}* and the area,

*A*, in all five forests (Fig. 2

*b*). 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 a continuous extension of

*a*. In the Appendix, we use the persistence values of our data to derive a canonical, two-parameter model of

_{i}*a*(

*A*). Once this function has been derived, we obtain the SAR by solving the differential equation 2 Using the diversity measured in a small area as the initial condition 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 solution
of the form *S* =
*cA*^{z}exp[P(A)], where
*P*(*A*) is an infinite polynomial in *A*.
We can truncate after the first *n* terms to obtain an
approximate solution. Truncating after the first term leads to the
expression
3
Here *c*, *z*, and *k* are constants
determined by *a*(*A*). This approximate 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 of simplicity. If we let
*n* → ∞, then we recover the full solution to Eq.
2; if we let *n* → 0, then we reduce to the power
law. Hence the power law is a zeroth-order, special case of our general
model for the SAR.

Eq. 2 accurately predicts diversity given only a small
amount of data. Because the persistence curves are similar across the
five plots (Fig. 2*b*), 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 predict the 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
compared with 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 over other previous
methods. It is 1- to 7-fold more precise than Fisher's alpha (23), and
5- to 10-fold more precise than the power law. 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 of persistence 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 location-mapped
trees in tropical forests. We find that the SAR shows consistent
deviations from the power law on all spatial scales that were studied,
ranging from 1 m^{2} 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.) These results might have some bearing on the longstanding
controversy surrounding SARs. Previous research has focused on
why the SAR has different slopes in different ecosystems (11), but in
our extensive data the SAR does not possess a constant slope
whatsoever.

Instead of self-similarity we propose a model of the SAR, based on the spatial persistence function, which generalizes the power 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 of diversity that regenerates, as estimated from a small census (24). Hence, an accurate method to extrapolate 50-ha diversity from a 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 landscape-scale 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 the observed persistence patterns offers an important challenge to ecology. In the meantime, our theory provides a valuable tool for conservation planning and a practical method for estimating diversity in the field.

## Acknowledgments

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 and editorial input. J.B.P. is supported by a National Science Foundation graduate fellowship.

## 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,
*a _{i}* =

*S*/

_{i}*S*

_{i−1}, we start by deriving its relationship to the slope of the SAR. All logarithms are henceforth taken base two: 4 The last equality follows because

*A*

_{i−1}/

*A*= 2. The final quantity in Eq. 4 measures the slope of a small chord on the log-log SAR to the right of log(

_{i}*A*). We conclude that 5 where

_{i}*i*is now a continuous variable. To be more precise, the chord to the right of log(

*A*

_{i}) is an estimate of the derivative at log(

*A*

_{(i+(i}

_{−}

_{1))/2}). Note that Eq. 5 clearly illustrates the equivalence between self-similarity (constant

*a*) and the power law (constant dlog

*S*/dlog

*A*).

Eq. 2 follows easily from Eq. 5, using the fact
that dlog*S*/dlog*A* = *A/S* ⋅
d*S*/d*A*. Note that Eq. 2 is a strict
generalization of self-similarity: if *a*(*A*) is
constant, then Eq. 2 reduces to the power law. If
*a*(*A*) = exp(−1/log*A*), then Eq.
2 reduces to *S*(*A*)∝log*A*. 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 process translates BCI's
log-log SAR so that it coincides with Pasoh's initial condition;
nevertheless, the universality between the forests is described more
simply in terms of the persistence parameter,
*a*_{i}.

We choose to model the persistence values
*a*_{i} with the simple, two-parameter
family of curves ¼Φ[(α −
*i*)/β] + ¾. (Other choices, such
as a cubic model, are also possible and produce accurate predictions at
these scales and beyond. See below.) Here Φ(*x*) is the
“error function” given by the cumulative distribution of the
Gaussian: Φ(*x*) =
(2/)⋅∫_{0}^{x}*e*^{−t2}d*t*.
The parameter α moves the inflection point of the persistence curve
horizontally, and β determines the slope at the inflection point.
Hence β measures the maximal “acceleration” of diversity with
area, and α measures the spatial scale at which acceleration is
maximized. The best-fit at BCI is given by α = 8.56, β =
8.08. For Pasoh, α = 7.73, β = 7.41; for Mudumalai,
α = 7.06 and β = 7.76.

We may express all solutions to Eq. 2 in the form
*S* =
*cA*^{z}exp[*P*(*A*)],
where *P*(*A*) is a polynomial in *A* of
arbitrary degree *n*, with no constant term. Once we specify
α and β, we expand −log(*a*(*A*)) in a Taylor
series of order *n* around the point *A* = 25
ha. The resulting separable equation can always be solved in closed
form, yielding *P*. For example, using Pasoh's α and β to
determine *a*(*A*), the approximate solution of order
*n* =1 is given by *S*(*A*) =
*S*(1ha)⋅*A*^{z}*e*^{kA},
where *z* = 0.125 and *k* =
−5.66⋅10^{−4}. For Mudumalai,
*z* = 0.161 and *k* =
−5.41⋅10^{−4}. This approximation is only
valid for *A* ≤ 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 most
accurate SAR. We used the Fehlberg-Runge-Kutta method to generate 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 was constructed from 1,000,
independently sampled, 1-ha initial conditions.

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 from a more stable, moist tropical
forest. The values of *a*_{1} to
*a*_{6} are generally smaller at HKK and
Mudumalai than the other three forests. This reflects the fact that HKK
and Mudumalai are subject 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,
and Lambir, 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 alpha or
a power law on these disturbed forests. In practice, when estimating
diversity in a new forest, the ecologist should first determine the
frequency of disturbances (*e.g.*, fires, hurricanes, or
roaming elephants) and choose a model forest, where α and β are
known, accordingly.

### Extrapolation Beyond 50 ha.

For 50-ha predictions, such as would be useful to assess logging protocols, Φ provides a simple, two-parameter model of the persistence curve. For larger areas, however, a cubic model (which works as well as Φ at 50 ha) 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-ha sample. Using Eq. 2, the predicted diversity for all of BCI is 436 ± 32 species (1 SD). This estimate compares favorably with Croat's floral 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 across the landscape (as in ref. 28), although such techniques require further development.

## Footnotes

↵b To whom reprint requests should be addressed at: Institute For Advanced Study, Olden Lane, Princeton, NJ 08540. E-mail: plotkin{at}ias.edu.

## Abbreviations

- SAR,
- species–area curve;
- ha,
- hectare;
- BCI,
- Barro Colorado Island;
- HKK,
- Huai Kha Khaeng

- Received April 15, 2000.
- Accepted July 26, 2000.

- Copyright © 2000, The National Academy of Sciences

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