Ecology, 83(12), 2002, pp. 3344–3356
q 2002 by the Ecological Society of America

SAMPLING THE SPECIES COMPOSITION OF A LANDSCAPEJOSHUA B. PLOTKIN1,2,3 AND HELENE C. MULLER-LANDAU2,4
2

1
Institute for Advanced Study, Olden Lane, Princeton, New Jersey 08540 USA
Department of Ecology and Evolutionary Biology, Princeton University, Princeton, New Jersey 08544 USA

Abstract. The abundances and spatial distribution of species is central to biogeography
and conservation. Several theories have been offered to explain landscape-scale species
distribution patterns. The verification of biogeographic theories, as well as conservation
decisions, must be based upon empirical data gathered from necessarily restricted censuses.
It is necessary, therefore, to understand the relationship between an underlying landscapescale pattern and the corresponding pattern it produces upon sampling small subregions.
The similarity of species composition between two samples depends not only on the species
composition of the underlying landscape from which the samples are drawn, but also on
the underlying distribution of species abundances, the degree of conspecific spatial clustering, and sample size. In this paper, we investigate how sampling expectations change
depending upon species abundance distributions and upon spatial distributions. We derive
analytical results for the expected species overlap between two sampled regions under awide range of conditions. We compare these results with data from a 50-ha tropical forest
census. These methodologies provide useful tools for assessing beta diversity, for testing
macro-ecological theory, and for designing landscape-scale sampling schemes.
Key words: beta diversity; landscape-scale species distribution; negative binomial; sampling
diversity; similiarity indices; spatial aggregation; tropical forest.

INTRODUCTION
Spatial patterns of species diversity are of great interest to ecologists. This interest is in part practical: tobest conserve biodiversity, we must know where species richness is highest and how species assemblages
change in space (Cody 1986). But diversity patterns
are also of theoretical interest, providing material totest theories of why diversity varies among sites andhow species turnover increases with intersite distance(Condit et al. 2002).
Species diversity in a landscape has long been divided into two parts: alpha diversity and beta diversity(Whittaker 1960). Most studies have focused on alphadiversity, the diversity of species within individualsites. Yet even the most diverse sites hold but a tiny
fraction of the planet’s species. The high diversity of
species overall is due mainly to beta diversity, thechange in species composition between sites. An understanding of the mechanisms and patterns of beta
diversity is critical for understanding the local to landscape scaling of biodiversity in general.
Despite its obvious importance, beta diversity hasreceived relatively little attention in the empirical literature. Those empirical studies that have examined
species turnover across a landscape have used a variety
Manuscript received 27 August 2001; revised 25 April 2002;
accepted 3 May 2002.
3 E-mail: plotkin@ias.edu
4 Present address: National Center for Ecological Analysis
and Synthesis, 735 State Street, Suite 300, Santa Barbara,
California 93101-5504 USA.

of methods (not always comparable), and have focusedon qualitative patterns (e.g., Routledge 1977, Cody
1986, Harrison et al. 1992, Harrison 1997, Potts et al.
2002). Recent contributions (Chave and Leigh, inpress, Leigh et al. in press) make quantitative predictions regarding the spatial turnover of species composition under a neutral model of community dynamics(Hubbell 1995, 1997, 2001). Such work provides theoretical predictions of beta diversity patterns that canbe tested with empirical data sampled from a landscape(Condit et al. 2002).
In order to assess landscape-scale species composition patterns with limited data, we must first understand the sampling distributions of our measured statistics. That is, given some underlying species similarity between two communities, what similarity do we
expect to observe between two small samples fromthose communities? And thus what can we deduceabout the underlying species similarity between two
communities from the observed similarity between two
small samples? Unfortunately, this question cannot ingeneral be answered exactly. Previous studies haveused computer simulations to investigate the robustnessof community similarity measures to sample sizes (e.g.,Morisita 1959, Ricklefs and Lau 1980, Wolda 1981).
These studies show that similarity indices computedfrom small samples almost always underestimate thetrue similarity of the underlying communities fromwhich the samples are drawn. The sampling distribution of similarity indices is so fundamental to biogeography that numerical approximations fitted to com-3344

December 2002

SAMPLING DIVERSITY

puter simulations (Wolda 1981) have even been incorporated into popular textbooks on ecological methodology (Krebs 1999).
Previous studies of similarity indices assume that
individuals are all sampled independently and randomly in space. For many types of communities andsampling practices this assumption is severely violated. For example, studies of contiguous plots intropical tree communities reveal that nearly every
species is spatially aggregated (He et al. 1997, Condit et al. 2000), and that this aggregation significantly
affects large-scale ecological patterns such as species-area curves (Plotkin et al. 2000 b ). Clustering ofconspecifics will further bias measures of species
similarity between sites that are already biased bysmall sample sizes.
In this paper, we investigate how indices of community composition are affected by conspecific clustering within samples, as well as by abundance distributions and by sample size. We incorporate conspecificclustering into the theory of sampling distributions byusing the negative binomial distribution. We derive analytic expressions for the expected value of the mostbasic similarity measure, the proportion of species incommon to two samples, under a large variety of conditions. In order to test our results, we use informationon conspecific clustering and the abundance distribution in a 50-ha tropical forest plot to calculate expectedsimilarities between subplots, and compare these to observed similarities. Finally, we present numerical results for a variety of similarity indices, showing theeffects of conspecific clumping on their samplingmeans and variances. We conclude by discussing theimplications of our results for biogeographic theory
and for landscape-scale sampling design.

3345

FIG. 1. Examples of five continuous abundance distributions represented as probability density functions on a log
scale (top), and as rank–abundance curves (bottom). Parameter values are chosen so that all distributions have the samemean, 1000. The truncated hyperbolic, lognormal, and gamma
distributions have the same variance, 2 3 106. The varianceof the exponential distribution is 106. The variance of thecontinuous logseries distribution is 7.5 3 106. Exact parameter values are as follows: exponential l 5 1/1000 ; gammab 5 1/2 and l 5 1/2000; lognormal m 5 2 log(1000) 2
log(1000Ï3) and s 5 [2 log(1000Ï3 2 2 log(1000)]½; truncated hyperbolic m 5 15.2954 and M 5 5984.7; continuous
logseries x 5 0.999882.

SPECIES ABUNDANCE DISTRIBUTIONSIn this section, we review several common modelsof species abundances for which we will later computethe expected similarity between subsamples. The relative abundances of species in a large region will havea significant effect on the observed similarity between
two small samples from the region. If a few common
species dominate, then similarity in species composition between samples will be high; if there are manyrare species, similarity will be low.
The distribution of abundances within ecologicalcommunities is typically described in one of two ways.
The number of species in different abundance classes(often doubling classes) may be plotted as a histogram(Preston 1962). Alternatively, species may be ranked
from most abundant to least abundant, and their abundance, generally on a log scale, plotted against theirrank to produce a rank-abundance curve. These twoalternatives are entirely equivalent and the connection
between them is straightforward (May 1975). Fig. 1
illustrates five continuous probability distributions
commonly used to model abundances of species. The

differences between the distributions are more apparent
when plotted as density functions rather than rankabundance curves.
For the purpose of analytically calculating the expected species overlap between sites, it is convenient
to summarize abundances with a continuous probability
density function f(n). The expression f(n) dn denotes
the probability that a randomly chosen species fromthe community will have an abundance between n and
n 1 dn. The continuous density function is normalized
to total probability one: #`0 f(n) dn 5 1. In reality, thedistribution of species abundances is discrete. Nevertheless, a continuous approximation is useful and commonly employed in the ecological literature (May1975). Furthermore, we will demonstrate that continuous abundance distributions yield almost precisely thesame results as their discrete analogues. Table 1 summarizes the abundance distributions reviewed below.
The table indicates the names of the distributions, thediscrete analogues (where applicable), and the relevant
equations.

JOSHUA B. PLOTKIN AND HELENE C. MULLER-LANDAU

3346

TABLE 1. A summary of the species abundance distributions, with the discrete analogue of each continuous abundance distribution, where applicable, and the relevant equations for the probability density functions.
Continuous distribution

Discrete analogue

Lognormal
Truncated hyperbolic
Exponential
Gamma
Continuous logseries

Eqs.

logseries

Gamma distribution

A popular model of species abundances is the lognormal distribution, for which the histogram of species
numbers is normal when abundances are binned (plotted) on a log scale. The lognormal distribution takes
the form[]1(ln(n) 2 m) 2
exp 2
.
2s 2
nsÏ2p

(1)

This distribution is expected to arise if many independent factors act multiplicatively to determine abundances (May 1975). The lognormal distribution has two
free parameters, m and s. A one-parameter subfamily
called the canonical lognormal is also common in theliterature (Preston 1962, May 1975).

Truncated hyperbolic distribution
Another common abundance distribution is the socalled geometric series distribution, whose rank–abundance curve is linear when the abundance axis is logarithmic (Fig. 1b). The continuous analogue of the geometric series is given by a truncated hyperbolic probability density function:1

n ln M
1m2
f (n) 5 0(3)Note that this distribution is in no way related to thetruncated hyperbolic distribution, which has been referred
to as ‘‘exponential’’ by some authors. Because it has one
parameter, the variance and mean of the exponential distribution cannot be varied independently.

Lognormal distribution

f (n) 5

distribution is the continuous analogue to MacArthur’sdiscrete ‘‘broken stick’’ distribution (Longuet-Higgins
1971, May 1975). The exponential distribution has asingle parameter, l, and takes the form
f(n) 5 le2ln.

1
2
3
4
5–7

geometric series
broken stick

Ecology, Vol. 83, No. 12

for m # n # M

We also investigate the gamma distribution, which
has two parameters. The gamma distribution is very
flexible and it matches empirical abundance distributions well, although it is rarely used in the ecologicalliterature:f (n) 5lb nb21 e2ln.
G(b)When the shape parameter b equals one, the gamma
distribution simplifies to the exponential distribution.

Logseries distribution
The four preceding abundance distributions are allcontinuous. Nevertheless, due its widespread use inecology, we will also include the logseries distribution,
which is discrete. The logseries was introduced by
Fisher et al. (1943) and, much like the truncated hyperbolic, describes the abundances of species that compete for a single resource. The distribution is often
characterized by two parameters, x and a. If we letf(N) denote the number of species with exactly N individuals, then the logseries satisfies
f (N ) 5(2)otherwise.

This distribution has two parameters, m and M, which
represent the minimal and maximal abundances in thecommunity. This distribution arises when the community is dominated by a single resource factor, and
if division of this resource proceeds in a hierarchicalmanner. The most successful species occupies a fraction of the resource, and the next most abundant species
occupies the same fraction of the remaining resource,and so on. Ecologists have often referred to the truncated-hyperbolic as the ‘‘exponential’’ distribution, because of the shape of its rank–abundance curve. Nevertheless, we will reserve the term ‘‘exponential’’ forthe abundance distribution given by Eq. 3.

Exponential distribution
We also use the exponential probability distribution
to describe the abundances of species. The exponential

(4)

ax N
N

(5)

where N is a positive integer and the parameter x lies
between zero and one. In this formulation, the discretedensity function f is not yet normalized; the total number of species present in the assemblage is given by S
5 Sf(N) 5 2a ln(1 2 x). If the value of S is fixed,
then x can be obtained from S, leaving only a as a freeparameter.
For our purposes we do not need to know the total
number of species present in the ambient large region. Hence, we will use the normalized form of thelogseries distribution, which depends upon a single
parameter x :f (N ) 5

xN

1

2

.

(6)

1
N ln
12x

We refer to Eq. 6 as the logseries distribution.
The natural continuous analogue of the logseries distribution—just as the exponential distribution is the

SAMPLING DIVERSITY

December 2002

analogue of MacArthur’s broken stick distribution(Longuet-Higgins 1971)—is given by the following
density function:xnnG(0, 2ln(x))

f (n) 5 
0for n $ 1
(7)
for n , 1.

In this equation, G(x,y) represents the partial gammafunction defined by

E`

G(x, y) 5

t x21 e2t dt.

t5y

We refer to Eq. 7 as the continuous logseries distribution. As we shall see in Expected species overlap
between samples, the continuous analogue is an excellent approximation to the standard, discrete logseries.
SAMPLING INDIVIDUALS FROM A LANDSCAPE
Given a species that occurs in a large region, thenumber of its individuals that occur in a small sampleof the region depends on the total abundance of thatspecies in the larger region, the size of the sample, andany conspecific spatial aggregation of the individuals.
We now formulate analytic expressions that describe
the probability of a species’ occurrence in a small sample, given knowledge of its abundance in a larger ambient region. As is standard in such analyses (May1975), we assume that the density of individuals of all
species combined is uniform in space. This assumption
is appropriate for many ecosystems, including closedcanopy forests (e.g., Hubbell 2001).

Random spatial distributions
If all individuals are located randomly in space, then
the number of individuals of a species found in a sample
follows a binomial distribution, and is well approximatedby a Poisson distribution. (Equivalently, if a given numberof individuals are sampled from an area by picking theindividuals in random locations, then the number of individuals per species will follow a binomial distribution.)
Consider a small sample whose area constitutes a proportion a , 1 of the large ambient region. According to
the Poisson distribution, the probability c(a,n) that a species will be encountered in the sample, given that it hasabundance n in the larger region, isc(a, n) 5 1 2 exp(2an).

(8)

The Poisson distribution is the simplest model ofsampling individuals from a landscape. In previous
studies of similarity indices, authors have almost always assumed Poisson sampling (e.g., Morisita 1959,
Ricklefs and Lau 1980, Wolda 1981).

Aggregated spatial distributions
The Poisson distribution stipulates that the probability that a sampled individual is of a particular species

3347

is independent of the probability that another individualin the sample is of that species. However, if samples
are taken within a contiguous area, then conspecificaggregation often leads to nonindependence of individuals, elevating the probability of having more or
fewer individuals of a particular species, depending onwhether a clump is encountered or not.
Clustering of conspecifics within samples can bemodelled using the negative binomial distribution inplace of the Poisson. Specifically, if a species has abundance n in the large area, then its probability of occurrence in a sample of proportional size a may be
represented by1c(a, n) 5 1 2 1 1

2k

2

an
k

(9)

where k is a parameter that reflects the degree of overdispersion or clustering (Wright 1991, He and Gaston
2000). Parameter values k . 0 imply spatial aggregation. Notice that as k → 6`, the probability of occurrence approaches 1 2 exp(2an). In other words, as
k → 6`, the spatial distribution approaches the Poisson(random) case. Parameter values k , 0 reflect spatial
patterns that are more regular than random placement.
Extensive analyses in tropical forests reveal thatnearly every species is aggregated, a few are random,
and almost none exhibit spatial regularity (He et al.
1997, Condit et al. 2000, Plotkin et al. 2000 b). We
therefore assume that the clumping parameter k always
exceeds zero.
Both Eqs. 8 and 9 and indicate that the probabilityof a species’ occurrence in a sample of area zero iszero, as we would naturally expect. In addition, we
would expect that a species that occurs in the large
ambient region will also occur in the sample with probability one as the size of the sample approaches thesize of the large region, a → 1. Unfortunately, neither
Eq. 8 nor 9 satisfies this criterion. This difficulty arises
because the Poisson and negative binomial both represent sampling with replacement. After an individualis sampled, it is replaced back into the pool of individuals in the large region (even though, strictly speaking, it should be removed). Although the Poisson andthe negative binomial can be altered to amend thisshortcoming (He and Legendre 2002), the differences
between sampling with replacement and without replacement are negligible when, as will always occur inpractice, the sample size is significantly smaller than
the ambient community from which it is drawn.
EXPECTED SPECIES OVERLAP BETWEEN SAMPLES

General framework
Given a large region for which we know (or assume)an underlying abundance distribution of species, we
now investigate the expected proportion of species incommon between two small samples. The number of
species in common divided by the average number of3348

JOSHUA B. PLOTKIN AND HELENE C. MULLER-LANDAU

Ecology, Vol. 83, No. 12

the Sørensen similarity, x(a,b), in all cases except thelognormal abundance distribution. Using the same
techniques, analytic results are also possible for mostother measures of similarity based on species presence/
absence.
As before, let f(n) denote the distribution of species
abundances in the large ambient region. Let c(a, n)denote the probability of encountering a species in asample that covers a proportion a of the larger region,
given that the species has abundance n in the larger
region. With these definitions, the expected Sørensen
similarity between the two samples is given byE`

x (a, b) 5

1
2

[E

f (n)c(a, n)c(b, n) dn

0

`

0

E

`

f (n)c(a, n) dn 1

0

f (n)c(b, n) dn

]

.

(10)

FIG. 2. The expected Sørenson similarity between subplots of increasing area (a 5 b) on linear axes (top) and loglinear axes (bottom). Areas a and b are measured as proportions of the large ambient ecosystem. We compare the analytic
expressions in Eqs. 11 and 12 (lines) against the mean similarity of 100 simulated samples (dots). Species abundanceswere drawn from an exponential distribution with l 5 0.01.
Individuals were sampled via either the Poisson distribution(dashed line) or the negative binomial distribution with a fixedclumping parameter k (solid lines). The similarity is shownfor various values of the clumping parameter (k 5 0.2, 0.4,
0.8, 1.6, 3.2; starting from bottom curve). The mean similarity
found by numerical simulations of sampling with replacement(from a large region of 30 000 individuals) matches the analytic predictions. Note that spatial aggregation can significantly alter the similarity curve, but that as k → ` the similarity curve approaches the Poisson (random) case.

species in two samples is called the Sørensen index ofsimilarity (Legendre and Legendre 1998). We will also
use the term species overlap. The Sørenson index is
attractive and popular because of its simplicity and its
relation to the species–area curve.
We will consider two sampled regions whose areas
constitute proportions a and b, respectively, of a larger
ambient region. We assume throughout that the possible species pool in the two samples is the same. (Fordiscussion of cases where the species pool varies see
the Discussion and the Appendix.) We examine theexpected species overlap between the sampled regions
under each of four abundance distributions and underboth random (Poisson) and aggregated (negative binomial) spatial arrangements of individuals. Under
these conditions, we derive an analytic expression for

The numerator represents the number of species incommon to the two samples, and the denominator represents the average of the number of species present
in each sample. Notice that the denominator of Eq. 10
is essentially a species–area relation. Although the denominator contains a species–area relation, note thatthe dependence of the Sørenson index on sample area
is not simply equivalent to the scaling of the species
diversity with area.
For any abundance distribution f of the species in
the large region and for any type of sampling of individuals c (e.g., randomly placed individuals or aggregated individuals), Eq. 10 determines the expected
number of species in common to two small subregions
whose areas constitute proportions a and b of the larger
ambient region. Notice that Eq. 10 does not involve
the total number of species contained in the ambient
region. Counter to our intuition, we can compute theexpected similarity between two samples without
knowledge of the total diversity contained in the ambient region. This fact justifies our use of a normalized
abundance distribution f(n) whose total species diversity is left unspecified.
Strictly speaking, Eq. 10 is only approximately correct because it treats the numerator and denominatorof Sørensen’s index as independent quantities. Nevertheless, when the samples are small compared to theambient region (as will always be the case in practice)
and when the ambient ecosystem is diverse enough sothat its abundance distribution is approximately continuous, then the simplifying assumption of independence is justified and Eq. 10 is extremely accurate. (Theaccuracy of Eq. 10, even for samples as large as 20%of the ambient region, is demonstrated below by comparison with simulations.)

Analytical results
We now integrate Eq. 10 under the assumptions ofvarious abundance and spatial distributions. We startSAMPLING DIVERSITY

December 2002

1 2 F 12k, 2k; 2k 1 1; aM2Mkk
J 5 k ln 1 2 2 1 2 F 1k, k; k 1 1;
m
am
am2
k
k
1 1 2 F 1k, k; k 1 1;
.
(14)
aM
aM2

with the situation in which two samples are drawn fromthe same ambient region whose species abundance distribution is exponential (Eq. 3). Assuming that the spatial distribution of individuals is random (or, equivalently, that individuals are chosen for sampling at random), then integrating Eq. 10 with Eq. 8 yields the
expected Sørensen similarity between two samples
x (a, b) 5

2ab (a 1 b 1 2l).
(a 1 b 1 l)(2ab 1 al 1 bl)(11)

Next we address expected sample similarity when
the individuals are aggregated in space and samples arecontiguous. For a species with abundance n and conspecific clustering parameter k, we use c(a, n) givenby the negative binomial formulation (Eq. 9). We startby assuming that all species are clustered in the sameway, i.e., that they all have the same value of the parameter k, and that this parameter is invariant acrossspatial scales. (Below we address variation in the cluster parameter k.) In this case, Eq. 10 is integrable provided that the two sampled regions have the same area
(i.e., a 5 b), yielding

3349

2

1 k
2 aM

2k

k

k

k

In these equations, F denotes the generalized hypergeometric function (Gradshtein and Rhyzhik 2000).
For gamma-distributed abundances (Eq. 4) and Poisson
sampling, the expected Sørensen similarity is given by
x (a, b)52[(a 1 l)2b 1 (b 1 l)2b 2 (a 1 b 1 l)2b 2 l2b ].
(a 1 l)2b 1 (b 1 l)2b 2 2l2b
(15)

In the case of gamma-distributed abundances andconspecific spatial aggregation (Eq. 9), the expectedsimilarity between two subplots of proportional area aequals x(a, a) 5 X/Y where

X 5 1 1 {[ G(k)G(2k 2 b)F1 (b, 1 1 b 2 2k, H )x (a, a)2 2G(2k)G(k 2 b)F1 (b, 1 1 b 2 k, H )]5 {(k 2 1)G(k)[2H kG(1 2 k, H )4 [G(k)G(2k)H 2b ]}2 H 2kG(1 2 2k, H ) 2 exp(2H )]}1 {[ H G(b 2 2k)F1 (2k, 1 2 b 1 2k, H )k

4 {[ G(k 1 1) 2 G(k)]3 [H kG(1 2 k, H ) 2 exp(2H )]}where H 5 kl/a. Eq. 12 matches simulated data exactly,
and shows that similarity is lower when individuals areaggregated as opposed to random (Fig. 2).
When the underlying abundance distribution is truncated hyperbolic (Eq. 2) and the spatial distribution is
Poisson (Eq. 8), then the expected Sørensen similarity
between two subplots is given by
x (a, b) 5

5 ln 1Mm2

2 G(am) 1 G(aM ) 2 G(bm)

1 G(bM ) 1 G[(a 1 b)m] 2 G[(a 1 b)M ]

[12

4 ln

1

6

M
1
1
1
2 G(am) 1 G(aM ) 2 G(bm)
m
2
2
2

]

1
G(bM ) .
2(13)

Under negative binomial sampling, we find a closedform solution for equal-sized samples: x(a, a) 5 I/J
where

12 1 2 1
2kk
1 21 2 F 1k, k; k 1 1;
aM
aM2
1 k
k
1 1 2 F 12k, 2k; 2k 1 1;
2 am
am2
k

I 5 k lnMkk
22
F k, k; k 1 1;
m
am
am
k

2k

2 2G(b 2 k)F1 (k, 1 2 b 1 k, H )](12)

4 [G(b)H 2k ]}Y512
2H bG(k 2 b)F1 (b, 1 1 b 2 k, H )G(k)H kG(b 2 k)F1 (k, 1 2 b 1 k, H )
G(b)

(16)

where H 5 kl/a and F1 denotes the Kummer confluent
hypergeometric function. Even through the analytic expressions above are complicated, they match simulations exactly (not shown) and they also accurately
match empirical data (see Comparison with EmpiricalData). As usual, aggregated spatial distributions show
distinctly lower similarity than random (Poisson) spatial distributions.
Unlike all of the other abundance distributions we
consider, the standard logseries distribution is discrete.
In this case, the equivalent discrete version of Eq. 10
takes the formO f (N )c(a, N )c(b, N )x (a, b) 5
. (17)1O f (N )c(a, N ) 1 O f (N )c(b, N )]
2[
`

N51

`

`

N51

n51

In general, it is much more difficult to reduce Eq. 17
into closed form than it is integrate the quantities in
Eq. 10. Nevertheless, for the logseries distribution (Eq.
5 or 6) when individuals are distributed randomly in3350

JOSHUA B. PLOTKIN AND HELENE C. MULLER-LANDAU

Ecology, Vol. 83, No. 12

space (Eq. 8), then we have the following closed form
solution for the expected species overlap:
x (a, b) 5 {2[ln(1 2 xe2a ) 1 ln(1 2 xe2b )2 ln(1 2 x) 2 ln(1 2 xe2a2b )]}4 [ln(1 2 xe2a ) 1 ln(1 2 xe2b )2 2 ln(1 2 x)].

(18)

If we use the continuous analogue of the logseries (Eq.
7), then under Poisson sampling we obtainx (a, b) 5 {2[ G(0, 2ln(x)) 1 G(0, a 1 b 2 ln(x))
2 G(0, a 2 ln(x)) 2 G(0, b 2 ln(x))]}4 [2G(0, 2ln(x)) 2 G(0, a 2 ln(x))
2 G(0, b 2 ln(x))].

(19)

In Fig. 3, below, we see that the continuous solution(Eq. 19) agrees almost perfectly with the exact discrete
solution (Eq. 18), and that both solutions agree withnumerical simulations. This result validates our use ofcontinuous abundance distributions instead of their discrete analogues (e.g., the exponential distribution instead of the broken stick distribution, the truncatedhyperbolic instead of the geometric, etc.).

Comparison with previous literature
Previous studies have investigated species overlap
between samples. Wolda (1981) analyzed the situationin which abundances are distributed according to thelogseries distribution, and individuals are sampled randomly. (Wolda also investigated many other similarity
indices.) By fitting numerical simulations, Wolda obtained an approximate equation relating expected species overlap with sample sizes. Wolda gives approximations for four different parameter values of the logseries distribution. In Fig. 3, we compare Wolda’s approximate equations with our exact results and withsimulations (which match the exact results).
As seen in Fig. 3, Wolda’s approximations are fairly
accurate within the range of sample sizes and for theparticular values of the logseries abundance distribution that he considered. Unfortunately, his numericalresults cannot be generalized to other situations. Eqs.
11 through 19, on the other hand, provide exact similarity predictions for a large range of abundance distributions, for arbitrary abundance-distribution parameters, and for both random and aggregated spatial distributions.

Variation in aggregation across species and acrossspatial scales
So far, we have treated spatial aggregation via a negative binomial sampling function whose clumping parameter, k, does not change with spatial scale nor depend upon species. If the clumping parameter k depends
upon area, as it will in many ecosystems, then we cansimply replace the parameter k in Eqs. 11–19 with the

FIG. 3. The expected Sørensen similarity between subplots of increasing sizes on linear axes (top) and log-linear
axes (bottom). The dots show the simulation results, thedashed line shows our analytic prediction, and the solid line
shows the approximation of Wolda (1981). Samples weredrawn randomly from a community that matches one of theexact cases simulated by Wolda (1981): a community of
100 000 individuals in which the abundance distribution of S
5 580 species is logseries with a 5 81.54. Wolda (1981)
states that in this case the Sørensen similarity should vary asx(A, B) 5 1.137 2 3.375A20.347 2 281 3 1027 B, where A is
the number of individuals in the smaller sample and B is thenumber in the larger sample. We fix the size of the larger
sample, B 5 1000, and plot similarity as a function of thesmaller sample’s size, A. Numerical results (dots) represent
the mean of 100 simulated samples. Our analytic formulae(Eq. 18, based up on the discrete logseries, and Eq. 19, based
upon the continuous logseries) are indistinguishable fromeach other, and they match the numerical simulations exactly.

dependency on area, k(a) and k(b). Thus we can easily
accommodate for changes in the intensity of aggregation with spatial scale.
Studies have found that within a given species, koften varies with the mean number of individuals sampled. Empirical estimation of k at multiple spatial scales
from data on Barro Colorado Island shows that k increases approximately linearly with the area sampled.
In this case, there is less of a difference between thenegative binomial and the Poisson at large areas, andmore of a difference at small areas.
Species may also differ from one another in theirtendency to aggregate. Extensive analyses of tropicalforests have demonstrated that species exhibit a rangeof clumping intensities (Condit et al. 2000, Plotkin etal. 2000b). Moreover, the range of clumping parameters

SAMPLING DIVERSITY

December 2002

cannot be collapsed to a single mean value without
losing information about the accumulation of species
diversity with area (Plotkin et al. 2000a, Plotkin andLevin 2001). Tropical forest analyses do not reveal any
significant correlation between a species’ tendency toaggregate and its abundance (Plotkin et al. 2000 b). This
does not in itself imply that a species’ clumping parameter is uncorrelated with its abundance, because of
possible sampling biases in estimating k. Nevertheless,
it is a relatively safe assumption that k and abundance
are only weakly correlated.
We can incorporate interspecific variation in clumping into our analysis of similarity indices. If the distribution of clumping parameters is denoted by g(k),independent of abundance, then the expected similarity
between subplots a and b is given by simplyE`

x (a, b) 5

g (k)x(a, b, k) dk.

(20)

3351

FIG. 4. A histogram of species abundances in the 50-haplot on Barro Colorado Island, with the best-fit gamma distribution (b 5 0.245, l 5 0.000321) . The first histogram bar
represents the number of species with n individuals, 1 # n, 2. The second bar represents those species with 2 # n ,4, etc.

k50

For most distributions of k values, Eq. 20 is not easily
integrated into a closed form solution. Nevertheless,this expression can certainly be evaluated numericallyfor any given distribution of clumping parameters. As
we will see in the next section, however, we can often
safely ignore interspecific variation in k and still predict
intersite similarity with accuracy.
COMPARISON WITH EMPIRICAL DATA
The methodologies developed above are intended for
applications to landscape-scale patterns of species turnover. Nevertheless, we can evaluate the accuracy of ouranalytic results by using data from the 50-ha tropicalforest census on Barro Colorado Island, Panama (Hubbell
et al. 1995). In this case, we consider the BCI census as
the ‘‘large ambient region,’’ and we sample multiple subplots within the region, computing their observed species
similarities. Within the 50-ha plot on BCI every freestanding woody stem .1 cm in diameter has been spatially mapped to 1 m accuracy and identified to species.
The 50-ha plot contains 229 070 such individuals comprising 300 species (1995 census year).
In order to apply the techniques developed here to
BCI, we must first choose a probability density function
to model the abundances of species within the BCI
census. We find that the abundances are best modelled
by using a gamma distribution. In our case, we requirethat the mean abundance of the best-fit distributionagree with the empirically observed mean abundance(763.57 individuals), so that the total number of individuals represented by 300 species agrees with theobserved total number of individuals. Therefore, we
require that b/l 5 763.57. Among this one-parametersubfamily of gamma distributions, we choose the parameters (b 5 0.2452, l 5 0.0003211) that minimize
the sum of the squared differences from the observed
abundances, when binned (plotted) on a logarithmic(base two) scale (Fig. 4).

Given the best-fit gamma abundance distribution, Eq.
16 predicts the expected similarity between two randomly chosen, equal-sized subplots, under the assumption of conspecific spatial clustering. In order toapply these equations, we must first measure the negative binomial parameter, k, at various quadrat areas(Krebs 1999). Several empirical studies have investigated the relationship between clustering k and quadrat
area. In our data, the scaling of k is, in general, best
described by a power law of the formk(A) 5 cAz 1 d(21)

where A denotes quadrat area in square meters and c,d, and z are constants.
The equations derived above (Eqs. 12, 14, and 16)
assume that every species is described by the sameclumping parameter k(A), which may vary with area
A. In reality, however, there is considerable interspecific variation in clumping parameters. Using the observed abundances found in multiple quadrat draws
from BCI, we have estimated the best-fit parameter k(A)for each species and for each quadrat area A via amaximum likelihood procedure (Krebs 1999). Alternatively, we can use the same maximum likelihood
method to determine a best fit parameter k(A) that holds
across all species simultaneously. Fig. 5 shows the bestfit cluster parameters for two representative species at
BCI, as well as the best-fit cluster parameter for all
species overall. The scaling of the overall cluster parameter k(A) is accurately described by Eq. 21 with d
5 0.8604, c 5 0.002923, and z 5 0.5450 (r2 . 0.99).
The area is measured in square meters.
Fig. 6 shows the observed subplot Sørensen similarities at BCI as well as the predicted similarity usinga gamma abundance distribution and a power-law relationship between clustering and area. We sample
square subplots of equal areas a 5 b constituting, at
most, 12.5% of the entire 50-ha region.

3352

JOSHUA B. PLOTKIN AND HELENE C. MULLER-LANDAU

Ecology, Vol. 83, No. 12

mulation assumes sampling with replacement, whereas
there is no replacement when sampling in reality.
We can test the relative importance of these threesources of error by using numerical simulations. For
instance, we can use the true observed abundances toquery the extent of error attributable to the gamma distribution assumption. Such simulations (not shown) indicate that all three potential sources of error do, indeed,
contribute small amounts to the observed discrepancy
between model and data. Despite these minor sourcesof error, Fig. 6 demonstrates a remarkably good fit.
FIG. 5. The observed relationship between spatial scale
and the clustering parameter k at BCI. All BCI species are
more clustered (lower k) at small scales than at large areas.
There is, however, considerable interspecific variation in thescaling of k with area. For example, the x’s in the figure show
k values of Calophyllum longifolium (n 5 1000), whereas theopen circles show Xylopia macrantha (n 5 1133). The dotsshow the clumping parameter at each scale that is the best fit
to all species simultaneously; the solid line shows the bestfit power-law model of k(A) 5 0.8604 1 0.002923 3 A0.545(r2 . 0.99).

As seen in Fig. 6, the negative binomial model
matches the observed similarity curve in the BCI 50ha plot fairly well. The predicted similarity is far moreaccurate when we account for aggregation than when
we assume a Poisson model (Fig. 6). The Poisson model consistently overestimates similarity, but falls within
two standard errors of the data as the subplot area becomes .5% of 50 ha. The Poisson approximation cannot be rejected at larger areas because the k values
increase with area (Fig. 5). In all cases, however, thenegative binomial model predicts mean similarity moreaccurately than the Poisson model.
Although it accurately describes the empirical data,
even the negative binomial model seems to overestimate similarity slightly. Such errors generally arise because we are testing the theory at a relatively small(50-ha) scale instead of a true landscape scale. There
are three distinct potential sources of error between our
analytic theory and the observed BCI similarity data.
First, we have modelled the species abundances at BCI
by using the gamma distribution (Fig. 4). Although the
gamma distribution fits the BCI data better than any
other well-known distribution, any discrepancies willcause errors in the predicted similarity between samples. Second, we have assumed that every species at
BCI has the same clumping parameter k(A), and that
this parameters scales as a power-law in area. But inreality we know that there is a range of clustering parameters across species. Third, we have assumed thatthe spatial distribution of a species can be modelled by
the negative binomial sampling distribution. Although
the negative binomial is a reasonable and popular model of spatial aggregation, it need not match the observed
sampling distribution c(a, n) exactly. In particular, as
we have discussed before, the negative binomial for-

NUMERICAL RESULTS AND OTHERSIMILARITY INDICES
Exact results such as the ones presented above can beeasily derived only for the expected value (the samplingmean) of presence/absence similarity indices, such as
Sørenson’s index. Analogous results are more difficult toobtain for quantitative similarity indices that incorporate
information on species’ abundances. Sampling variances
are also difficult to calculate analytically. Thus, further
exploration of the sampling distributions of similarity indices requires numerical simulations.
Several studies have used simulations to examine avariety of similarity indices and their dependence uponsample size, as well as on species diversity, always
under the assumption of random sampling (Morisita
1959, Ricklefs and Lau 1980, Wolda 1981, Smith andZaret 1982, Mueller and Altenberg 1985, Smith 1985).
In this section, we examine the effects of spatial ag-

F IG . 6. Observed mean Sørenson similarity between
equal-sized subplots of the 50-ha plot on Barro Colorado
Island (dots 6 2 SE ), compared with the similarity predictedby two analytic models. The dashed line shows similarity
predicted according to Poisson sampling (Eq. 15), and thesolid line shows negative binomial sampling (Eq. 16). Predictions are based upon the fitted gamma abundance distribution ( b 5 0.245, l 5 0.000321) and the scale-dependent
clumping parameter k( A), as in Fig. 5. Observed similarity
was calculated, for each subplot size, as the average overmany (at least 64) randomly paired, disjoint subplots. The
negative binomial model accurately predicts the observed
similarity, whereas the Poisson model overestimates similarity.

SAMPLING DIVERSITY

December 2002

3353

gregation on sampling means and variances of a presence/absence and a quantitative similarity index.
As before, we will use Sørensen’s index as a typicalpresence/absence index. We use the Renkonen index(Renkonen 1938) to illustrate the general behavior ofquantitative similarity indices. Whereas the Sørensen’s
index is essentially the number of species in common
to two samples divided by the average number of species in each sample, the Renkonen index may bethought of as essentially the number of individuals incommon divided by the average number of individuals
in each sample. The Renkonen index is defined as

O min(n (i), n (i))
z(a, b) 5
1
O n (i) 1 12 O n (i)
2
S

a

i51
S

i51

b

S

a

i51

(22)

b

where na(i) and nb(i) are the abundances of species i insample a and sample b, respectively, and S is the total
number of species. The value of z(a, b) always lies
between 0 and 1; z(a, b) equals 1 only if the two samples a and b contain the same set of species with exactly
the same abundances. The Renkonen index has also
been referred to as the Steinhaus index (Motyka 1947),
the ‘‘Sørensen index with cover,’’ the ‘‘percent similarity,’’ and the ‘‘coefficient of community.’’ The onecomplement of the Renkonen index is often called the
Bray-Curtis coefficient (Bray and Curtis 1957) or
Odum’s percentage difference (Odum 1950).
Conspecific aggregation decreases expected similarity and increases sample variance in both Sørensen andRenkonen similarity. This behavior is apparent from
Fig. 7, which shows simulation results using parametervalues consistent with the BCI 50-ha plot. Note that
the Renkonen index is much more sensitive to conspecific aggregation, in both its mean and variance.
This sensitivity is a general phenomenon of quantitative similarity indices that are responsive to the numberof individuals in each sample, rather than simply theirpresence or absence. The degree to which quantitativeand presence/absence indices differ depends on the diversity of the community: for less skewed abundance
distributions and more species-rich communities, quantitative indices behave more like presence/absence indices. Quantitative similarity indices are also more sensitive to variation in clumping among species within
the community (not shown).
While most presence/absence similarity indicesshow the same general response to sample size as the
Sørensen, and most quantitative similarity measures
behave like the Renkonen, there are a few notable exceptions. The probability that an individual randomly
drawn from area a is the same species as an individual
randomly drawn from area b is unaffected by sample
size; this index, called Morisita’s index (Morisita
1959), has been used in recent analytic work (Chaveand Leigh, in press, Leigh et al., in press) and empirical

FIG. 7. The sampling means 6 2 SD of the Sørensen (top)and the Renkonen (bottom) similarity indices under Poisson(solid) and negative binomial (dashed) sampling. We simulated 1000 replicate communities, each containing 300 species whose abundances were drawn independently from a
gamma distribution with l 5 0.245 and b 5 0.000321 (asfound on BCI). Negative binomial sampling assumed that k
was constant for all species and varied with sample size a as
in Fig. 5. Replicate similarity indices for a given sample size
and type were distributed normally; hence the 62 SD interval
shown encompasses 95% of the variation.

tests thereof (Condit et al. 2002). Under Poisson sampling, the sampling mean of Morisita’s index is unaffected by sample size, while the variance decreases
steadily with increased sample size. Negative binomial
sampling decreases the value of Morisita’s index to alevel that is dependent on k alone and independent ofarea; thus, under negative binomial sampling, Morisita’s index will increase with area only if k increases
with area. The other quantitative similarity indices examined, Horn’s information theory index (Horn 1966),
Euclidean distance (Ricklefs and Lau 1980), and Morisita’s index as adjusted by Horn, show lower similarity
under negative binomial sampling than under Poisson,
and within each sampling scheme, exhibit decreasing
similarity at smaller sample sizes (not shown).
DISCUSSION
Knowledge of the sampling distributions of similarity indices is a prerequisite for testing biogeographic
theories of landscape-scale community variation, especially because such theories must be tested from

3354

JOSHUA B. PLOTKIN AND HELENE C. MULLER-LANDAU

small samples (e.g. Condit et al. 2000, Potts et al.
2002). In this paper, we have developed analytic techniques that synthesize information about the underlying
species abundance distribution and conspecific spatial
aggregation to yield exact mathematical predictions for
species overlap between sampled subregions. Such results can be used to detect and analyze variation inunderlying species composition, i.e., species turnover,from small censuses spread across a landscape.
We have demonstrated that local clustering of conspecifics significantly reduces the similarity between
sampled subregions. Clustering also increases the variance of sampled similarity indices. Nevertheless, we
can incorporate the effect of aggregation into analytic
formulae for the expected species overlap between
samples. These results are important in light of thespatial aggregation that is nearly ubiquitous in tropicalforests and other ecosystems. In fact, without accounting for spatial aggregation, we cannot accurately explain the observed species overlap between subplots of
the 50-ha tropical forest census at BCI.
We have limited our analysis to the simple, null casewhen two samples are drawn from the same underlying
region. In other words, we have assumed that a species’
abundance in the two sample locations is perfectly correlated, which is a reasonable assumption when thesamples are close together. If, instead, the samples arefrom two very different, distant regions, and species
abundances in one region are not correlated with theirabundances in the other, then the species overlap between the samples is a function of the joint abundance
distribution in the two regions; the integrals over abundances in Eq. 10 would be replaced by double integrals
over the abundances in the two regions (see the Appendix). Solutions for two special cases of joint distributions are given in the Appendix. The appropriate
form of such joint abundance distributions for intermediate cases in which the distributions in the tworegions are partially correlated has not yet been addressed and is an important topic for further research.
Our results, especially those regarding spatial aggregation, have implications for the sampling designof beta diversity studies. If the degree of spatial contagion at the scale of measurement is known, then it
can be incorporated into expected similarity as we haveshown. Alternatively, sampling efforts can be designed
to minimize the influence of local clustering by either(1) arranging long narrow plots (cf. Gentry 1988), or(2) spreading the same total sampled area over a greater
region. The disadvantage of such irregular sampling
schemes is that they compromise our ability to characterize any remaining effects of aggregation. Furthermore, it would be difficult to compare similarity
measures calculated between samples taken in different
ways; sampling should above all be done consistently.
When the objective is to assess similarity between different regions, the fail-safe way to account for contagion and sample size is to include replicate plots in

Ecology, Vol. 83, No. 12

each habitat or region being compared. In this way,
similarity between plots in the same region can be contrasted with similarity of plots from different regions.
We have primarily focused on the Sørenson index ofsimilarity. This simple index measures the proportionof species that overlap between two samples. Although
we have seen that the Sørenson index is not as sensitive
as other, quantitative indices, Sørenson has several distinct advantages. Foremost, the Sørenson index only
requires knowledge of species presence/absence in a
census, which is often the extent of information collected in a landscape survey (Gaston 1994). Moreover,the species overlap between samples is closely related
to the species–area curve in the underlying region.
Novel theories that unify species overlap and species–
area curves (Harte et al. 1999) are being developed.
Despite the vast literature on biogeography, there are
relatively few unified theories of community assembly,
based on biological mechanisms such as speciation anddispersal, that predict the change in species composition across a landscape (Hubbell 2001). The most recent contributions provide theoretical predictions forone similarity index (Chave and Leigh, in press, Leighet al., in press), but predictions for species overlap
require further research. Such biogeographic theories
must eventually be tested against empirical data, bearing in mind the effects of sample size on measured
statistics. Similarly, the design and placement of refugia for the preservation of landscape-scale biodiversity must be guided by a sound understanding of thesampling distributions for species turnover.
ACKNOWLEDGMENTS
Foremost, we thank Rick Condit for posing the problem ofSorenson similarity to us, and for his conceptual input. We
also thank Fangliang He and Jonathan Dushoff for stimulating
discussions, and John Harte, Jess Green, and an anonymous
reviewer for helpful comments. We are indebted to the principal investigators behind the 50-ha Forest Dynamics Plot on
Barro Colorado Island: Robin Foster, Steve Hubbell and Rick
Condit. The Forest Dynamics Plot of Barro Colorado Island
has been made possible through the generous support of the
U.S. National Science Foundation, The John D. and Catherine
T. MacArthur Foundation, and the Smithsonian Tropical Research Institute, and through the collaborative efforts of many
scientists over the past two decades. The BCI Forest Dynamics Plot is part of the Smithsonian’s Center for Tropical Forest
Science, a global network of demographic tree plots. Finally,
we gratefully acknowledge support to J. B. Plotkin in theform of a National Science Foundation graduate fellowship,
and from the Teresa and H. John Heinz III Foundation and
the Burroughs Wellcome Fund, and to H. C. Muller-Landau
from the Smithsonian Tropical Research Institue, the Mellon
Foundation, and the Princeton Environmental Insititute.
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APPENDIXEXPECTED SPECIES OVERLAP WHEN ABUNDANCES ARE NOT PERFECTLY CORRELATED
In general, the expected species overlap between two samples is a function of the joint distribution, f(m, n) of species
abundances in the regions from which they are sampled,

where m is the abundance in the region from which proportiona is sampled, and n is the abundance in the region from which
proportion b is sampled:JOSHUA B. PLOTKIN AND HELENE C. MULLER-LANDAU

3356

[E E
5 [E EEE`

x (a, b) 5`

f (m, n)c(a, m)c(b, n) dm dn

0

4

0

1
2

`

[12 1 2 1

]

`

1k

0

`

1

0

f (m, n)c(b, n) dn dm .

0

[E

`

0

E

]

0

512 [E (n) (a, n) dn
f

0

1

E

]6

(A.2)

This situation is somewhat unrealistic, since it suggests thatexactly the same species are present in the two communities,
but that a species’s abundance in one location is completely
uncorrelated with its abundance in the other. Nevertheless,an examination of the extreme case of completely uncorrelated abundances may be useful in indicating the limitationsof similarity indices at detecting amounts of correlation.
When the two samples each comprise the same proportional
area of their respective ambient regions, a 5 b, then Eq. A.2
reduces to x(a, a) 5 #`0 f(n) c(a, n)dn. In other words, in thiscase the expected species overlap between the two samples
is simply the expected proportion of all the species in one
ambient region that occur in a sample of size a. So, in thisspecial case, the similarity between two subregions scales
linearly with the species–area curve, regardless of the abundance or spatial distribution.
Below we present closed-form integrations of Eq. A.2 fora range of abundance and spatial distributions, parallel to our
results in Eqs. 11–19.
Truncated hyperbolic abundance distribution
For Poisson sampling and uncorrelated abundance distributions, Eq. A.2 reduces to the following form:

5[ 1 2]Mx (a, b) 5 2 ln
1 G(0, am) 2 G(0, aM )m

[12
5 1 2[ 1 2
2 ln

2ab
.
2ab 1 bl 1 al

(A.5)

Under negative binomial sampling, we findx (a, b)

a
b

21
523
1

kl
kl 

a 2 e kl/a klE k,b 2 e kl/b klE k,
a
b 


1 2

1 2

(A.6)
where E(x, y) 5 #`1 e2yt t2x dt is the exponential integral.

]6]6

x (a, b)522(l2b 2 (a 1 l)2b 2 (b 1 l)2b 1 (a 1 b 1 l)2b).
(22l2b 1 (a 1 l)2b 1 (b 1 l)2b)(A.7)
Under negative binomial sampling, we find that x(a,a) 5 N/
D whereN 5 {G(b) 1 a22k (kl) k
3 [(kl) k F (2k, 1 2 b 1 2k, H )G(b 2 2k)
2 2a k F (k, 1 2 b 1 k, H )G(b 2 k)]}/ [G(b)]22a2b (kl)b F (b, 1 1 b 2 k, H )G(k 2 b)G(k)1

a2b (kl)b F (b, 1 1 b 2 2k, H )G(2k 2 b)
G(2k)

D512
2

G(b)21 a2k (kl) k F (k, 1 2 b 1 k, H )G(b 2 k)
G(b)a2b (kl)b F (b, 1 1 b 2 k, H )G(k 2 b)G(k)Logseries abundance distributionM
1 G(0, am) 2 G(0, aM )m

1 G(0, bm) 2 G(0, bM ) .

Under Poisson sampling, with uncorrelated abundances wefind that

where H 5 kl/a.

M
3 ln
1 G(0, bm) 2 G(0, bM )mMmx (a, b) 5

Gamma abundance distribution

f (m)c(b, m) dm .

0

4 ln

When the abundances are uncorrelated and spatial distribution is random, we have

c

`

M

f (m)c(b, m) dm

`

4

k

Exponential abundance distribution`

f (n)c(a, n) dn

k

(A.1)

When the samples are taken from the same region such thatm 5 n for all species, then this equation reduces to Eq. 10. It
follows also that if samples are taken from regions that sharea proportion p of their species, with those species having thesame abundances in both regions and other species not shared,
then the expected species overlap between the samples is simply p 3 x(a, b), where x is calculated as in Eqs. 11–19.
If the species pool is identical at the two samples, and theabundance distributions have the same exact shape in bothregions, but the abundances of individual species in the tworegions are completely uncorrelated, then the expected similarity is given by
x (a, b) 5

] [ ]1aM2 F 1k, k; k 1 1; aM2 4 ln 1m2 .
k

(A.4)

]6

`

2

k

M
k
k
2k
F k, k; k 1 1;mamam

x (a, b) 5 ln

f (m, n)c(a, m) dm dn0

Ecology, Vol. 83, No. 12

Under Poisson sampling, with uncorrelated abundances wefind thatx (a, b) 5 {2[ln(1 2 xe2a ) 2 ln(1 2 x)](A.3)

For negative binomial sampling and uncorrelated abundanceswith constant k:

3 [ln(1 2 x) 2 ln(1 2 xe2b )]}4 {ln(1 2 x)[ln(1 2 xe2a ) 1 ln(1 2 xe2b )2 2 ln(1 2 x)]}.

(A.8)