Spatial Variation in Tree Species Composition 13compare only equal-sized subsamples, a method
called “rarefaction” (Hurlbert 1971), or to assume
an underlyin gspecies abundance distribution. For
instance, if the species abundance distribution fol-
lows Fisher’s logseries, then an unbiased index of
alpha-diversity is Fisher’sα. This assumption has
been tested in several tropical tree communities
(Conditet al. 1996), but it would be interestin gto
test it further in other forests.
Many biological questions relate to species
turnover, or changes in species composition from
one community to another, rather than just local
diversity as defined above. In such cases, one can
define a relationship between alpha- and gamma-
diversities, coined beta-diversity by Whittaker
(1972). Beta-diversity is useful for studyin geco-
logicalprocessessuchashabitatspecializationand
dispersal limitation, but also large-scale patterns
of abundance, rarity, and endemism.Two extreme
cases may occur: alpha-diversity may be much
smaller than gamma-diversity, when most species
are spatially clumped; in this case, beta-diversity is
large. Conversely, alpha-diversity may be on the
same order as gamma-diversity, in which case
most species would be represented in any local
samplin gof the re gion, and beta-diversity would
be low. More precisely, Whittaker (1972) defined
beta-diversity as the ratio of gamma-diversity over
alpha-diversity:βW=Dγ
Dα(2.1)
whereDαandDγare the expected species diver-
sities at local and regional scales, respectively. A
statement equivalent to Equation (2.1) is that
gamma-diversity is equal to the product of alpha-
and beta-diversity, that is, there exists a multiplica-
tive partition of gamma-diversity into a strict local
contribution and a spatial turnover contribution.
Lande (1996) pointed out that an additive parti-
tion of diversity into alpha- and beta-diversity is
more natural than Whittaker’s multiplicative par-
tition. He defined beta-diversity as the difference
of gamma-diversity minus alpha-diversity:βL=Dγ−Dα (2.2)This additive partitionin gscheme simply results
from the exact definition of the diversity indices.
Lande (1996) chose to define local diversity as
the probability of two random chosen individ-
uals to belon gto different species, a quantity
also known as Simpson diversity. Gamma diver-
sity may be defined usin gexactly the same sta-
tistical interpretation but at the regional scale.
Because probabilities are additive, the scheme of
Equation (2.2) make more sense than that of
Equation (2.1). For further details on this addi-
tive diversity partitionin gscheme, the reader is
referred to Lande (1996), Cristet al. (2003), and
Jost (2006).
Wilson and Shmida (1984) reported six dif-
ferent measures of beta-diversity based on pres-
ence/absence data. More recently, Koleff et al.
(2003) reported a literature search of 60 papers
quantifyin gbeta-diversity, in which they found no
fewer than 24 different beta-diversity measures
based on presence/absence data. While all of these
measures are increasin gfunctions of the num-
ber of shared species, as intuitively expected for
a measure of species overlap, their mathematical
behavior differs broadly, and this contributes to
obscuration of the discussion on patterns of beta-
diversity. Wilson and Shmida (1984) and Koleff
et al. (2003) proposed a terminology for these
indices (which I here follow), together with a use-
ful interpretation in terms of two samplin gunits
with overlappin gspecies lists. Ifaandbare the
number of species restricted to samples 1 and 2
respectively, andcthe number of shared species
(Krebs 1999, Koleffet al. 2003; Figure 2.1), then
the total number of species isDγ =a+b+c,
and the local diversity can be defined as the aver-
age number of species in the two samples:Dα=
(a+c)/ 2 +(b+c)/2. Thus it is easy to see that
βL=(a+b)/2. Many other overlap measures
have been used in the literature, but I shall men-
tion just two. The Sørensen indexβSørensenis the
number of shared speciescdivided by the average
number of species in the two samples:βSørensen=
2 c/(a+b+ 2 c). The Jaccard indexβJaccardis
the number of shared species divided by the total
number of species:βJaccard=c/(a+b+c). Evi-
dently this generalizes to more than two sites;
beta-diversity indices are then defined for any pair
of sites,βi,j. The diagonal terms of this diversity
matrix compare any plot with itself, so that, for
instance,βWi,i=1 andβLi,i=0.