Theoretical Population Biology 62, 153?168 (2002)
doi:10.1006/tpbi.2002.1597
M
u
e
R
Received August 30, 2001To represent species turnover in tropical rain forest, we use a neutral model where a tree?s fate is not
affected by what species it belongs to, seeds disperse a limited distance from their parents, and speciation
is in equilibrium with random extinction. We calculate the similarity function, the probability F (r) that two
trees separated by a distance r belong to the same species, assuming that the dispersal kernel P (r), the
distribution of seeds about their parents and the prospects of mortality and reproduction, are the same for
all trees regardless of their species. If P (r) is radially symmetric Gaussian with mean-square dispersal
distance r; F ?r? can be expressed in closed form. If P (r) is a radially symmetric Cauchy distribution, then, in
two-dimensional space, F (r) is proportional to 1=r for large r : Analytical results are compared with
individual-based simulations, and the relevance to field observations is discussed. & 2002 Elsevier Science (USA)
Key Words: data diversity; long-distance dispersal; neutral model; tropical forest
1. INTRODUCTION
During the last 20 years, our understanding of the
processes maintaining tree diversity at a given tropical
site has greatly increased, thanks to the establishment of
large permanent sampling plots where all stems > 1
cm dbh (diameter at breast height, which is stem
diameter 1:3 m above the ground) are mapped, marked,
measured and identi?ed (Condit, 1995). The ?rst of
these large plots was the 50-ha plot on Barro Colorado
Island (BCI), Panama, ?rst censused in 1982 (Hubbell
and Foster, 1983, 1986; Condit, 1995; Leigh, 1999).
BCI?s 50-ha plot is now just one of more than 15 large
sample plots in the Neotropics, Africa and South-East
Asia that have been censused and monitored according
to a protocol established by the Smithsonian Tropical
Research Institute?s Center for Tropical Forest Science
(Manokaran et al., 1990; Condit, 1998).
We know much less about the factors which govern b-
diversity (species turnover) as manifested in the diver-
gence of tree species composition between forest plots in
different places, and how differences in soil or climate
affect this divergence. Indeed, the facts of species
turnover among tropical trees are only just now being
elucidated, through the study of regional networks of
small forest plots. Ashton (1964, 1976) and Ashton and
Hall (1992), the pioneers in this question, established a
series of small plots in South-East Asia. Other regional
systems of plots of 1 or 2 ha apiece have since been
established in Amazonian Peru (Terborgh and Andre-A Spatially Explicit Neutral
Tropical Forests
Je?ro?me Chave1
Laboratoire d?Ecologie Terrestre, CNRS UMR 5552, 13, Aven
Department of Ecology and Evolutionary Biology, Princeton Univ
and
Egbert G. Leigh Jr.
Smithsonian Tropical Research Institute, Apartado 2072, Balboa,15
1To whom correspondence should be addressed. E-mail: chave
@cict.fr.odel of b-Diversity in
e du Colonel Roche, F-31029, Toulouse, Cedex 4, France; and
rsity, Princeton, New Jersey 08544-1003
epublic of Panamasen, 1998; Pitman et al., 2001), Ecuador (Pitman et al.,
1999) and central Panama (Pyke et al., 2001). Very little
3 0040-5809/02 $35.00
# 2002 Elsevier Science (USA)
All rights reserved.
Chave and Leigh154has yet been done to order these new facts on b-diversity
(Pitman et al., 2001; Condit et al., 2002). Using an
extension of Hubbell?s (2001) neutral theory of tree
diversity, this paper will show how speciation and
limited seed dispersal can contribute to species turnover.
One function of BCI?s 50-ha plot was to test theories
of why tropical forests have so many tree species.
Prominent among these was Hubbell?s (1979) neutral
theory, which embodied the fundamental insight (Mac
Arthur and Wilson, 1967) that diversity represents a
balance between speciation and extinction. Hubbell
(1979) supposed that extinction is a purely random
process. All trees have the same prospects of mortality
and recruitment, regardless of their species, and
each dead tree is immediately replaced by the young
of a tree chosen at random from among its survivors
in the forest. The independent accidents of which
trees reproduce, and which die ?rst, change the
abundances of different tree species, and drive some to
extinction.
How can such a null model be useful? Hubbell?s
(1979) assumption that a plant?s fate is independent of
what species it or its neighbors belong to is false for
seeds and seedlings. As Janzen (1970) and Connell
(1971) predicted, seeds and seedlings with many
conspeci?c neighbors or close to their parent plants
are more likely to die (Givnish, 1999; Wills and Condit,
1999; Wright, 2002; Harms et al., 2000), probably
because of the disproportionate in?uence of specialized
pests and of pathogens spreading from conspeci?c
neighbors. Although seeds fall close to their parents
(Hubbell et al., 1999), in most tropical tree species, the
proportion of saplings belonging to a given tree?s species
does not increase when one approaches the tree (Condit
et al., 1992; Okuda et al., 1997). Moreover, the rate of
appearance onto 10  10 or 20  20 m subplots of
saplings of a given species over 1 cm dbh per adult of
the same species is lower on subplots with higher
basal area (total cross-sectional area of the trunks) of
adult conspeci?cs (Wills et al., 1997, Wills and Condit,
1999).
On the other hand, these effects are much weaker for
large trees ?510 cm dbh? at larger scales. For most
species, the rate of appearance of saplings per adult on
subplots 50 m or more on a side does not depend
detectably on the density or basal area of adults
conspeci?cs on the subplot (Wills et al., 1997). More-
over, plot-scale analyses of the dynamics of individual
tree species reveals density-dependent population reg-
ulation in only very few species (Hubbell et al., 1990).
This suggests that density dependence may not affect the
dynamics of large trees. Although Hubbell?s model failsfor small saplings on local scales, can it work for larger
trees on larger scales?
Another problem with Hubbell?s (1979) model is that
it assumes a panmictic community totally isolated from
the rest of the forest, leaving no room for metapopula-
tion processes. In a newer version of his model, when a
tree 510 cm dbh dies, its replacement has probability
1  m of being chosen at random at that plot, and
probability m of being chosen from a large, panmictic
pool of N source trees (Hubbell, 1997). In the source
pool, each new individual has probability n of being a
new species. This speciation balances extinction, so the
pool?s distribution of tree species abundance follows a
log series characterized by the parameter a ? Nn
(Watterson, 1974). This makeshift enabled Hubbell
(1997) to ?t the distribution over species of trees 510
cm dbh on plots of 50 ha or less by adjusting the
parameters m and a:
Hubbell?s (1997) model faces two challenges. First, its
picture of speciation is too simplistic (Hubbell, 2001):
species rarely begin as single mutant individuals. None-
theless, most species may well evolve from peripheral
isolates of ancestral species (Mayr, 1954). According to
Hubbell (2001), the results of his model will be little
altered if species begin from small isolated populations
rather than single individuals. Second, Hubbell?s (1997)
model cannot predict species turnover, because a local
community?s species composition depends only on its
relation to the source pool, not on the species present in
neighboring communities. Terborgh et al. (1996) and
Pitman et al. (1999) argued that widely separated plots
in western Amazonia were more similar in species
composition than seemed plausible under Hubbell?s
neutral model. To evaluate their objection, we need a
version of the neutral model for a very large forest where
seeds disperse only a limited distance from their parents,
as is the case for any tree (Hubbell et al., 1999; Harms
et al., 2000). Tree species turnover has traditionally been
ascribed to habitat differences, not to dispersal limita-
tion (Whittaker, 1972), often on good evidence (Ashton,
1964, 1976; Gentry, 1982, 1988). Can a null model that
incorporates the joint effects of speciation and limited
seed dispersal account for how the divergence in tree
species composition between two plots increases with
distance between the plots, or must we invariably invoke
the effects of environmental heterogeneity?
One measure of species turnover, or b-diversity
(change in species composition of two plots with
distance between them) is how rapidly increasing r
decreases the probability F ?r? that two randomly chosen
trees a distance r apart are the same species (Condit
et al., 2002). Leigh et al. (in press), have calculated F ?r?;
Model of b-Diversity in Tropical Forests 155using a population genetics model of Nagylaki (1974) to
represent a suitably modi?ed version of Hubbell?s model
where seeds are dispersed about each parent, regardless
of species, according to the same radially symmetric
Gaussian density. Theoreticians have extended this type
of model to various other dispersal kernels (Durrett and
Levin, 1996; Hubbell, 2001; Chave et al., 2002). In this
paper, we provide a general and exact derivation for
F ?r? using any radially symmetric dispersal kernel, and
apply it to the Gaussian and Cauchy dispersal kernels.
Population geneticists such as Mal!ecot (1948), Weiss
and Kimura (1965), Maruyama (1972), Nagylaki (1974,
1976), and Sawyer (1976, 1977a,b) devised very similar
models to solve a problem posed by Wright (1943) and
Kimura (1953): given a multi-allelic genetic locus in a
continuously distributed population where offspring
disperse only a limited distance from their parents, or
a population on an archipelago where only nearby
islands exchange migrants, what is the correlation
between allelic states at this locus in individuals
separated by a distance r? Such models have been used
as null hypotheses to probe causes of spatial patterns in
the genetic composition of selected populations, includ-
ing humans (Kimura and Ohta, 1971; Cavalli-Sforza
et al., 1994; Rousset, 2000). Nagylaki?s (1974) model
readily translates into Hubbell?s null model of tree
diversity if we let an individual in a continuously
distributed population represent a tree in a continuously
distributed, multi-species forest and a haploid indivi-
dual?s allele at the multi-allelic locus in question
represent a tree?s species. Harada et al. (1997) used this
model to calculate how, in a continuous monospeci?c
stand where plants reproduced both clonally and
sexually, the probability that two plants were of
different genotype increases with distance between them.
Here, clones propagate only a limited distance but
sexually produced seeds are dispersed uniformly
through the population.
To develop our model?s predictions, we ?rst derive
F ?r? when seeds are dispersed about each parent in a
radially symmetric Gaussian distribution. This formula
is known (Mal!ecot, 1948) and approximations for it
(Nagylaki, 1976) and asymptotic results (Sawyer, 1977a)
have been obtained, but we believe a detailed derivation
would bene?t readers. Indeed, current derivations are
abbreviated, hard to ?nd, or intended for mathemati-
cians, and colonies are often substituted for individuals.
Next, we let the seed dispersal kernel be proportional to
1=r3 when r is large, which implies that mean-square
dispersal distance is in?nite, thus violating the assump-
tions of Sawyer?s (1977a) limit theorem. This leads us to
new results. Next, we use numerical simulations of ourmodel to test the accuracy of our model?s predictions of
the average value of F ?r?; for given r at equilibrium,
assess the variability of F ?r? about this average, and
learn how initial conditions affect the approach to
equilibrium. Finally, we discuss how to test the model
with data from sample plots of forest.
2. THE MODEL
Consider two plots A and B, with known ?oristics,
with S species in all. Let the proportion of individuals
belonging to species s be pAs on plot A and p
B
s on plot B,
where pAs and p
B
s can be 0. The similarity of species
compositions of these two plots may be measured by the
index of codominance (Leigh et al., 1993), the chance
that two trees chosen at random, one from plot A and
the other from plot B, are the same species, or
F ?A; B? ?
XS
s?1
pAs p
B
s :
In population genetics, this index is the coef?cient of
consanguinity between samples from two parts of a
population, that is the chance that genes from the same
locus, one from sample A and the other from sample B,
are the same allele (Crow and Kimura, 1970). In ecology
1  F ?A; A? measures local, or a-diversity, and is related
to Simpson?s index (Simpson, 1949; Magurran, 1988).
Similarity, indices S?A; B? normalize F ?A; B? to make
S?A; A? ?S?B; B? ? 1: Thus, the Morisita?Horn simi-
larity index MH?A; B? is 2F ?A; B?=?F ?A; A? ? F ?B; B?
(Horn, 1966) while Nei (1987) calls I?A; B? ? F ?A; B?=?????????????????????????????
F ?A; A?F ?B; B?
p
the genetic identity. We focus on
F ?A; B? because, unlike normalized indices, codomi-
nance between two widely separated forest plots A and
B is an unbiassed estimator of the probability that a
randomly chosen tree near A and a randomly chosen
tree near B are the same species (Leigh et al., 1993).
Assuming that F ?A; B? depends only on the distance
between plots A and B, and not on different environ-
mental conditions in these plots, we seek the probability
F ?r? that two trees a distance r apart are the same
species, given that seeds disperse a limited distance from
their parents. Our problem has analogs in spaces of one
or three dimensions, which we also consider. We
accordingly let Fd ?r? represent the probability that two
organisms separated by a Euclidean distance r in d-
dimensional space are the same species. We assume that
trees are distributed homogeneously over a surface, with
a density of r per unit area.
Chave and Leigh156We begin by letting F2?x; y; t? be the probability that a
tree chosen randomly at time t from a neighborhood
dSx ? dx1 dx2 including the site x ? fx1; x2g is conspeci?c
with a tree simultaneously chosen in a neighborhood dSy
including the site y ? fy1; y2g: How is F ?x; y; t ? dt?
related to F ?x; y; t?? To ?nd out, we assume that (i) the
spatial distribution of trees is a continuous Poisson
point process whose mean is r; (ii) any tree alive at time
t has probability dt of dying by time t ? dt independent
of the fate of any other tree, so that the average lifetime
of any tree is one time unit, regardless of its species; (iii)
a proportion P ?x; y? dSy of the seeds of a tree at site x
land in the neighborhood dSy surrounding y; where
P ?x; y? is the ??dispersal kernel?? of each tree?s seeds; and
(iv) each new tree has probability n of being an entirely
new species.
To calculate F2?x; y; t ? dt? from F2?x; y; t?; recall
that with probability 2dt; one of the two trees dies
between time t and t ? dt: Let the dying tree be the one
at x; and let it be replaced immediately by the young of a
nearby tree. Call the probability that this parent is
conspeci?c with the tree at y; Q?x; y; t?: If the
young has a probability n of being an entirely new
species, then
F2?x; y; t ? dt? ? ?1  2dt?F2?x; y; t?
? ?2dt??1  n?Q?x; y; t?: ?2:1?
The continuous time limit of this equation is
@F2?x; y; t?
@t
? 2F2?x; y; t? ? 2?1  n?Q?x; y; t?: ?2:2?
To ?nd Q?x; y; t?; recall that the probability that the
tree at x is replaced by the young of a tree in the
neighborhood dSu of u ? fu1; u2g is P ?u; x? dSu; in which
case this young has probability ?1  n?F2?u; y; t? of being
conspeci?c with the tree at y: If the tree at x is replaced
by a young of a tree in a neighborhood dSy of the tree at
y; its parent has probability 1=?r dSy? of being the tree at
y; and probability 1  1=?r dSy? of being another tree in
the neighborhood, which latter would have probability
F2?y; y; t? of being conspeci?c with the tree at y: Notice
that 1  1=?r dSy? is positive only if dSy > 1=r; that is if
there is at least one individual in the neighborhood dSy:
This condition sets the lattice spacing in the ?ne-grained
model. Q?x; y; t? may be expressed as
Z
u
F2?u; y; t?P ?u; x? dSu ?
1  F2?y; y; t?
r dSy
P ?y;x? dSy
?
Z
u
F2?u; y; t?P ?u; x? dSu
?
1  F2?y; y; t?
r P ?y; x?: ?2:3?Notice that in these equations the condition dSu > 1=r
can be relaxed without leading to impossible results (i.e.,
negative probabilities).
Now suppose that the probability two trees are the
same species depends only on their relative positions, so
that F2?x; y; t? ? F2?x y; t? and let P ?x; y? ? P ?x y?:
Then the master equation is
@F2?x; t?
@t
?  2F2?x; t? ? 2?1  n?

Z
u
P ?u?F2?x u; t? dSu

?
1  F2?0; t?
r P ?x?

: ?2:4?
At equilibrium
F2?x1; x2?
? ?1  n?
Z
u1;u2
P ?u1; u2?F2?x1  u1; x2  u2? du1 du2

? 1  F2?0; 0?r P ?x1; x2?

: ?2:5?
Mal!ecot (1948, Eq. (1), p. 57; 1969, Eq. (3.3.1), p. 68)
?rst derived an equation like (2.5) for the correlation in
allelic state of genes at the same diallelic locus sampled
from two individuals at a distance r apart in a diploid
population (so that he replaces our r by 2d) where
successive generations are distinct (so that he replaces
our n by 1  ?1  k?2 ? 2k).
3. MALE?COT?S INTEGRAL EQUATION
Now we derive a formal solution of Eq. (2.5) similar
to that of Mal!ecot (1948). First, we take the Fourier
transform of Eq. (2.5). Let p?k; m? be the Fourier
transform of P ?x; y? and f2?k; m? that of F2?x; y?: Then
f2?k; m? ? ?1  n?
 f2?k; m?p?k; m? ?
1  F2?0; 0?
r p?k; m?
 
:
?3:1?
Since j?1  n?p?k; m?j51; we may set
f2?k; m? ?
1  F2?0; 0?
r
?1  n?p?k; m?
1  ?1  n?p?k; m?
? 1  F2?0?r
X1
j?1
?1  n?jp?k; m?j: ?3:2?
Assume next that P ?x; y? and F2?x; y? are functions only
of r ?
???????????????
x2 ? y2
p
: Then their Fourier transforms p and
Model of b-Diversity in Tropical Forests 157f2 are functions only of q ?
????????????????
k2 ? m2
p
; and Eq. (3.2)
becomes
f2?q? ?
1  F2?0?
r
X1
j?1
?1  n?jp?q?j: ?3:3?
The two-dimensional Fourier transform of an isotropic
function Z?r? and its inverse are
z?q? ? 2p
Z 1
r?0
Z?r?J0?rq?r dr;
Z?r? ?
1
2p
Z 1
q?0
z?q?J0?rq?q dq; ?3:4?
where J0?x? is the Bessel function of the ?rst kind and
order 0. Taking the inverse Fourier transform of (3.3),
we ?nd
F2?r? ?
1  F2?0?
r
1
2p
Z 1
q?0
X1
j?1
?1  n?jp?q?jJ0?rq?q dq:
?3:5?
Here again, the series is absolutely convergent for all
n > 0: Therefore, we can use Fubini?s theorem to
interchange the sum and integral as follows:
F2?r? ?
1  F2?0?
r
1
2p
X1
j?1
?1  n?j
Z 1
q?0
p?q?jJ0?rq?q dq:
?3:6?
So far, we have assumed nothing about the dispersal
kernel P ?r? except that it is a probability density (which
implies that it has a Fourier transform).
Equation (3.6) has a much simpler form if the
dispersal kernel P ?r? has a ?nite variance 2s2; and if
r  s: To be more speci?c, let us de?ne s2 by
s2 ?
Z
x;y
x2P ?x; y? dx dy ?
Z
x;y
y2P ?x; y? dx dy:
?3:7?
When r  s we may neglect the term multiplying 1 
F2?0; 0? in Eq. (2.5) and obtain
F2?x; y? ? ?1  n?
Z
P ?w; z?F2?x  w; y  z? dw dz: ?3:8?
If the dispersal kernel P ?r? decreases with r faster than a
negative exponential (as is true for a Gaussian dispersal
kernel) we may expand F2?x  w; y  z? in Taylor series
around x; y and neglect terms higher than second order.We, thereby, obtain
F2?x  w; y  z? ? F2?x; y?  w
@F2?x; y?
@x
? z @F2?x; y?
@y
 
?
1
2
w2
@2F2?x; y?
@x2
? 2wz
@2F2?x; y?
@x@y

? z2 @
2F2?x; y?
@y2

: ?3:9?
Since P ?x; y? is isotropic,
R
x;y xyP ?x; y? dx dy ? 0: Sub-
stituting Eq. (3.9) into Eq. (3.8), we obtain
F2?x; y? ?
?1  n?s2
2n
@2F2?x; y?
@x2
? @
2F2?x; y?
@y2
 
? ?1  n?s
2
2n DF2?x; y?; ?3:10?
where D is the Laplacian operator.
If we rewrite Eq. (3.10) in polar coordinates, and
assume n  1; we obtain the Bessel equation
d2F2?r?
dr2
?
1
r
dF2?r?
dr
? ?2n=s2?F2: ?3:11?
The only positive solution such that limr!1 F2?r? ? 0 is
F2?r? ? cK0
r
?????
2n
p
s
 !
: ?3:12?
K0?x? is the modi?ed Bessel function of zeroth order and
the second kind, and c is a constant. As we shall show
almost immediately (Eq. (4.7)), Eq. (3.12) is accurate for
r  s; showing that no harm came from ignoring terms
in the Taylor expansion higher than second order. On
the other hand, K0?r? diverges logarithmically as r
approaches 0: Eq. (3.12) cannot be valid for small r: For
large r; the function K0?r? is further approximated by
K0?r? ?
?????
p
2r
r
exp?r?
which is the form suggested in Harada et al. (1997) for
the same model, but in a different biological context.
4. GAUSSIAN DISPERSAL KERNELS
In this section, we solve Eq. (3.6) when the dispersal
kernel is a radially symmetric Gaussian density, show
how large r must be for F2?r? to approach the solution of
Eq. (3.12), and evaluate the coef?cient c in that
equation.
Chave and Leigh1584.1. Two Dimensions
If the Fourier transform or characteristic function
(Feller, 1971) of the probability density P ?r? is p?q?;
p?q? j is the Fourier transform of the probability density
of a sum of j independent random variables, each with
probability density P ?r?: The Fourier transform of the
radially symmetric Gaussian density P ?r? ? ?1=2ps2?
exp??r2=2s2?; is p?q? ? exp??s2q2=2?: Therefore,
the probability density whose Fourier transform is
exp?js2q2=2? must be ?1=2pjs2? exp?r2=?2js2?:
If we use the above expressions in Eq. (3.6), then
F2?r? ?
1
2ps2
1  F2?0?
r

X1
j?1
?1  n?j
j
exp  r
2
2js2
 
: ?4:1?
Equation (4.1) corresponds to Mal!ecot?s (1948, p. 62)
solution for a Gaussian dispersal kernel, his 2k  k2
corresponds to our n; and his 2d to our r: To ?nd F2?0?;
we set r ? 0 in Eq. (4.1) to obtain
F2?0? ?
1
2ps2
1  F2?0?
r
X1
j?1
?1  n? j
j
: ?4:2?
Since ln?1=n? ?
P1
j?1 ?1  n?
j=j; we ?nd
F2?0? ?
ln?1=n?
2rps2 ? ln?1=n?: ?4:3?
Community ecologists know 1  F2?0? as Simpson?s
diversity index. For monodominant communities, 1 
F2?0? is nearly 0. Substituting Eq. (4.3) into Eq. (4.1), we
?nd
F2?r? ?
1
2rps2 ? ln?1=n?
X1
j?1
?1  n?j
j
exp  r
2
2js2
 
: ?4:4?
When does this solution approach the approximations
of Nagylaki (1974) and Sawyer (1977a)? If we set C ?
1=?2ps2 ? ln?1=n?; then Nagylaki?s approximation for
Eq. (4.4) is (Leigh et al., in press)
F Nagylaki2 ?r? ? 2C K0
r
?????
2n
p
s
 !
 K0
r
???
2
p
s
 ! !
; ?4:5?
where his 2n corresponds to our n (we assume over-
lapping, he assumes distinct, generations) and his 2r to
our r (we consider haploid, he, diploids).
In Eq. (4.4), let n5106; so that we can replace
?1  n?j by exp?jn?; and let r  s: The sum in Eq. (4.4)
is a Riemann seriesX1
j?1
?1  n?j
j
exp 
r2
2js2
 
?
Z 1
y?0
exp ny  r
2
2ys2
 
dy
y
: ?4:6?
This integral is 2K0?r
?????
2n
p
=s? (Gradshteyn and Ryzhik,
2000, (3.471.9)). Therefore,
F2?r? ?
1
2rps2 ? ln?1=n? 2K0
r
?????
2n
p
s
 !
: ?4:7?
This approximation is equivalent to that of (3.12), only
we now know the coef?cient c of (3.12) is equal to 2C: If
we now remember that when r  s; K0?r
?????
2n
p
=s? 
K0?r
???
2
p
=s?; this result agrees with Eq. (4.5). Numerical
evaluation of (4.4) shows that Eq. (4.7) is accurate to
within 1% for r51:149s if n ? 106:
Sawyer (1977a, Eq. (3.2)) proved that Eq. (3.12) is
valid for r  s for any dispersal kernel with ?nite
variance, if we set
c ? 2
2rps2 ? ln?1=n? ? C0
with C0 a constant that depends only on the shape of the
dispersal kernel P ?r?:
Now let r  s and set exp?r2=2js2? ? 1  r2=2js2
in (4.4) to obtain
F2?r? ?
1
2rps2 ? ln?1=n?
X1
j?1
?1  n? j
j
1 
r2
2js2
 
? F2?0? 
1
2rps2 ? ln?1=n?
r2p2
12s2: ?4:8?
Here we have used the fact that when n50:01;P1
j?1 ?1  n?
j=j2 ? p2=6: Nagylaki?s (1974) equation
agrees rather poorly with (4.8) for small positive r:
Numerical evaluation of (4.4) shows that Eq. (4.8) is
accurate to within 1% for r40:827s if n ? 103; and for
r41:00s if n ? 106:
4.2. Multi-Dimensional Gaussian Dispersal
Kernels
We may generalize Eq. (4.4) to Euclidean spaces of
any dimension d: The radially symmetric Gaussian
dispersal kernel is
P ?x1; . . . ; xd? ?
1
?
?????
2p
p
s?d
exp 
Pd
i?1 x
2
i
2s2
 !
: ?4:9?
Since the Fourier transform p?q? of the radially
symmetric Gaussian density P ?r? is exp?s2q2? for any
speciation is not needed to maintain diversity. The
similarity in one, two and three dimensions of space are
compared in Fig. 1.
A limit theorem of Sawyer (1977a) completes this
picture. It establishes that, for dispersal kernels with
?nite moments:
(1) In one dimension, for any dispersal kernel such
that
R
r5P ?r? dr51; F1?r? is proportional to Eq. (4.12),
with a correction term of magnitude O?n?1 ? 1=r??:
(2) In two dimensions, for any dispersal kernel such
that
R
r3P ?r? dr251; F2?r? is proportional to Eq. (4.7) to
leading order, with a correction term of magnitude O?n?:
(3) In three dimensions, for any dispersal kernel such
that
R
r2P ?r? dr351; F3?r? is proportional to Eq. (4.13)
to leading order, with a correction term of magnitude
O?n1=2?:
Here, we say that a function f ?r? is proportional
to a function g?r? iff limr!1?f ?r?=g?r?? is a con-
stant independent of r: Therefore, for any
dispersal kernel P ?r? that decreases fast enough for
large r; Eqs. (4.12), (4.7), and (4.14), accurately describe
the behavior of Fd?r?; d43 for large r: For slightly
weaker hypotheses of this limit theorem, see Sawyer
(1977a).
Model of b-Diversity in Tropical Forests 159dimension d we ?nd, as in Eq. (4.1)
Fd?r? ?
1  Fd?0?
r
X1
j?1
?1  n? j
?s
???????
2jp
p
?d
exp  r
2
2js2
 
: ?4:10?
As in the two-dimensional case, we set ?1  n?j
? exp?nj?; replace the sum by an integral and use
formula (3.471.9) in Gradshteyn and Ryzhik (2000) to
obtain for r > s
Fd?r? ?
1  Fd ?0?
r
2
?2ps2?d=2
r2
2ns2
 2d=4
 Kd=21
r
?????
2n
p
s
 !
: ?4:11?
For biological problems, only the versions of this
formula in one, two, or three dimensions are useful.
Equation (4.11) is particularly simple for d ? 1 or 3,
because K1=2?x? ? K1=2?x? ?
?????????????
?p=2x?
p
exp?x?: When
d ? 1
F1?r? ?
1
1 ? rs
?????
2n
p exp  r
?????
2n
p
s
 !
: ?4:12?
F1?r? declines exponentially with distance (Mal!ecot,
1948; Kimura and Weiss, 1964; Nagylaki, 1974). Unlike
the two-dimensional case, approximation (4.12) for F1?r?
does not diverge as r approaches 0. Indeed, numerical
evaluation shows that if n  1; the value of F1?0?
estimated from Eq. (4.12) approximates the exact
solution quite closely.
In three dimensions, we ?nd
F3?r? ?
1  F3?0?
2prs2r exp 
r
?????
2n
p
s
 !
: ?4:13?
We must compute F3?0? from (4.10), because (4.13) has
no clear limit as r approaches 0. With an error not
exceeding 1% for n4104; we may set
X1
j?1
?1  n?j
j3=2
?
X1
j?1
1
j3=2
? z?3=2? ? 2:612;
where z?s? ?
P1
j?1 1=j
s is Riemann?s zeta function. Then
F3?r? ?
?????
2p
p
r?2ps2?3=2 ? z?32?
s
r
exp  r
?????
2n
p
s
 !
: ?4:14?
As dimensionality d increases from 1 to 3, Fd ?0? becomes
progressively less sensitive to the speciation rate n:
F1?0? ? 1=?1 ? rs
?????
2n
p
 depends on
??
n
p
; F2?0? ?
1=?1 ? 2prs2=ln?1=n? depends only logarithmically on
n; and Fd?0?; d53 does not depend on n at all, provided
n is small enough. Moreover, as the speciation rate n
approaches 0, Fd ?r?  1=rd2 for large r; as it should,according to a powerful theorem of Bramson and
Griffeath (1980). The formula for F3?0? shows that in
the three-dimensional case with limited dispersal,
0 1000 2000 3000 4000 5000
Distance
10?8
10?7
10?6
10?5
10?4
10?3
10?2
10?1
Si
m
ila
rit
y 
fu
nc
tio
n
d=1
d=2
d=3
FIG. 1. Theoretical predictions for the similarity function in one,
two, and three dimensions of space with the Gaussian dispersal kernel
(full lines) and with the Cauchy dispersal kernel (dashed line) in two
dimensions of space. For the Gaussian dispersal kernel, s ? 8 and
n ? 106 in all three cases, and for the Cauchy kernel, c ? 1: For a
better comparison, the function corresponding to the Cauchy kernel
has been multiplied with 3.
Chave and Leigh1605. POWER-LAW DISPERSAL KERNELS
If the third moment of the dispersal kernel is not
?nite, Sawyer?s limit theorem does not apply. Dispersal
kernels with in?nite variance were long considered
irrelevant for ecological problems, because ?nite sam-
ples have ?nite variances (for a recent discussion, see
Clark et al., 2001). In some studies, however, the next
item sampled might greatly increase the whole sample?s
variance, no matter how many were sampled before, in
which case the sample may be best ?t by a distribution
with in?nite variance. Biologists have long realized that
exponentially bounded dispersal kernels cannot account
for the speed with which trees reoccupied regions
vacated by retreating glaciers at the end of the
Pleistocene as measured by the pollen record (Skellam,
1951; Davis, 1963; Clark et al., 1998). Clark (1998),
using the formalism developed by Kot et al. (1996),
showed that ??fat-tailed?? dispersal kernels, where some
seeds disperse quite long distances, can explain the speed
with which trees spread in the Pleistocene. One
suggested dispersal kernel is (Clark et al., 1999)
P ?r? ?
p
pc2?1 ? r2=c2?p?1
: ?5:1?
For computational convenience, we have replaced the u
in Clark et al. (1999), by c2: The kth moment is ?nite if
and only if k52p: Thus, Sawyer?s theorem is valid if
and only if p > 32: Clark et al. (1999) ?tted dispersal
kernels of three types, including (5.1) and the radially
symmetric Gaussian, to the distributions of dispersal
distances 5100 m of seeds of 24 tree species using a
maximum likelihood method. In 14 of these 24 tree
species, seed dispersal was best described by dispersal
kernel (5.1). For 11 of the 14 species, the value of p
which best ?t the data was 51; for nine, it was 512; the
smallest value tried.
What form does F ?r? take when the dispersal kernel is
the ??fat-tailed?? (5.1)? In two dimensions, the Fourier
transform of (5.1) is (Gradshteyn and Ryzhik, 2000,
formula (6.565.4))
p?q? ? 2p
Z 1
0
pc2p
p?c2 ? r2?p?1
J0?rq?r dr
? 2p cq
2
 
p Kp?cq?
G?p ? 1?; ?5:2?
where, as above, Kp is the modi?ed Bessel function of
the second kind. Here, the median dispersal distance isc
???
3
p
; and F2?r? is
F2?r? ?
1  F2?0?
r
1
2p
X1
j?1
?1  n? j

Z 1
q?0
2p
cq
2
 
p Kp?cq?
G?p ? 1?
 j
J0?rq?q dq:
When p ? 12; Eq. (5.1) is the two-dimensional Cauchy
density c=?2p?c2 ? r2?3=2: This density, like the corre-
sponding Cauchy density in any dimension has Fourier
transform p?q? ? exp?cq? (Feller, 1971). The d-dimen-
sional Cauchy density is Cdc=?c2 ? r2??d?1?=2: here, C1 ?
1=p; C2 ? 1=2p; and C3 ? 1=p2: We now show that
If the dispersal kernel is the Cauchy distribution with
Fourier transform p?q? ? exp?cq?; and if d > 1 the
similarity function Fd ?r? is independent of r if r  c; while
F2?r?  r1d if r  c:
The similarity function in d-dimensional space corre-
sponding to the Cauchy kernel is
Fd?r? ? Cd
1  Fd ?0?
r
X1
j?1
?1  n? j jc
?r2 ? j2c2??d?1?=2
: ?5:3?
If r is so small that r2 ? c2j2 ? c2j2 for those j
contributing the overwhelming majority of the sum in
(5.3), then
Fd ?r? ? Cd
1  Fd ?0?
r
X1
j?1
?1  n?j
jdcd
: ?5:4?
For these values of r; Fd ?r? depends very slightly on r
and hardly at all on speciation rate n: For r  c;
Fd?r?  r1d (see Appendix A). When d ? 2;
F2?0? ?
p
12c2r? p ?5:5?
and F2?r?  1=r for large r: For arbitrary p in (5.1), the
technique used in Appendix A to bound F2?r? for large r
no longer works, but we conjecture that if p51;
F2?r?  1=r2p for large r: In the two-dimensional case,
the similarity function corresponding to the Cauchy
kernel is compared with the similarity function corre-
sponding to the Gaussian kernel in Fig. 1.
In the one-dimensional case
F1?r? ?
c
prc ? ln?1=n?
X1
j?1
?1  n?jcj
c2j2 ? r2
; ?5:6?
F1?r? tends to a non-zero value for large r: Very crudely,
for d52; the behavior of Fd ?r? with a Cauchy dispersal
kernel tends to resemble the behavior of Fd?1?r? with a
Gaussian dispersal kernel. In the Cauchy case, F2?0? is
independent of the speciation rate n and F2?r?  1=r for
large r as is true for F3?r? in the Gaussian case. In the
Cauchy case, F1?0? depends on ln?1=n? in the same way
Model of b-Diversity in Tropical Forests 161F2?0? does in the Gaussian case. On the other hand, F1?r?
in the Cauchy case behaves very differently from F2?r? in
the Gaussian case.
6. VOTER MODEL AND THE NO
SPECIATION LIMIT
Now let us consider the limit as the speciation rate
approaches zero. When s51; the diffusion approxima-
tion (3.10) applies in which DF ?r? ? 0 when n ? 0: Here,
the only stable solution is F ?r? ? C: Equation (2.5)
shows that C ? 1: This implies that a single species must
eventually take over the whole forest. Sawyer (1977b)
describes the process by which ever larger monospeci?c
patches grow when there is no speciation. If, in a forest
without speciation containing N trees, every tree were a
different species at time t ? 0; then, after a large number t
of tree generations, the total number of species remaining
would be  N ln?t?=t: Moreover, for large t and large r
F2?r?  1 
ln?gr2=s2?
ln?t?
; ?6:1?
where g ? 0:57722 is Euler?s constant (Cox and
Griffeath, 1986, Eqs. (3.2) and (3.3)). Thus, if s51;
F2?r? always converges to 1, but very slowly.
When s51; the change in F ?r? over time may be
simulated using an analogy with the ??voter model??
(Holley and Liggett, 1975; Cox and Griffeath, 1986;
Dornic et al., 2001). In this model, a ??voter?? is located
at each site of a square lattice. At the beginning, let each
voter prefer a different candidate, and suppose that, at
each timestep each voter randomly chooses one of his
four nearest neighbors and adopts his preference. This
model also represents a square lattice of trees where each
dead tree is replaced by the young of one of his four
neighbors (Bramson et al., 1998).
This model is in turn formally equivalent to a model
of coalescing random walks (Holley and Liggett, 1975;
Liggett, 1985; Chave et al., 2002), an equivalence that
provided the basis for the coalescence theory in genetics
(Kingman, 1982). At time t; the voter at site x has a
preference passed on through exactly one source or
antecedent, for every time t  t5t: Let Ax?t? be the
position of this antecedent at time t  t; so that
Ax?0? ? x: This is a simple random walk, where the
distance jjAx?t?  Ax?t? 1?jj2 travelled at each timestep
is 1. As t increases, t  t decreases: these random walks
travel backwards in time. If, at time t ? 0; each voter
prefers a different candidate, then the probability
F2?x; y; t? that at time t the voters at x and y prefer thesame candidate is the probability that their preferences
were passed on from the same antecedent, which is the
probability that, for some t Ax?t? ? Ay?t?: At the
smallest such t; the random walks of these sites
??coalesce??: that is to say, for all larger t they are one
walk because for these t; the antecedent sources of both
voter?s preferences are the same. If we let the voter
model represent a forest where at time t ? 0 each tree is
a different species, as mentioned above, then F ?x; y; t? is
the probability that at time t the trees at x and y are the
same species.
If preferences are passed on faithfully (no speciation),
the voter model predicts that, for systems in one and two
dimensions, one preference will eventually prevail, while
in three and more dimensions, different preferences
coexist inde?nitely (Holley and Liggett, 1975) for
exactly the same reason that different random walks
almost surely meet, given time enough, in one and two
dimensions, but not in three or more (Feller, 1971).
If at time 0, each of N voters in a square lattice has a
different preference, the mean time t needed for one
preference to spread to all N is 2N ?ln?N ??2=p (Cox,
1989). If, at time 0, a proportion ps of the N voters
prefer candidate s; then the time t required for one
preference to spread to all N is (Cox, 1989):
t ? 2HSp N ln?N ?; HS ? 
XS
s?1
ps ln?ps?: ?6:2?
Here, HS is the ??entropy?? of the initial state.
If s is in?nite, then if n > 0 the solution of Eq. (5.3) is
the unique stable solution of Eq. (2.5). If n ? 0;
however, the solution of Eq. (5.3) still exists, but it is
unstable. The only stable solution is F2?r? ? 1; yet,
unlike the case where s is ?nite, diversity may persist
inde?nitely.
7. NUMERICAL EXPERIMENTS
The formulas derived in Sections 2?5 applied to
systems of in?nite extent. Real systems, however, are
?nite. Cox?s theorem (Eq. (6.2)) shows that when n ? 0;
the time required for one species to take over the forest
depends on the forest?s size and initial species composi-
tion. If n > 0; the ?niteness of a forest may affect the
similarity function F ?r? in various ways. Here, we use
numerical simulations to address these questions (see
also Harada et al., 1997).
To simulate Hubbell?s neutral model with dispersal
limitation, we consider, with Durrett and Levin (1996), a
predictions with the similarity function F ?r? averaged
over 50 replicate runs using the nearest-neighbor
dispersal kernel. Since s ? 1; we expect Eq. (4.7) to ?t
the data for all r large enough that the discrepancy
between a Gaussian dispersal kernel and the nearest-
neighbor kernel ceases to affect F ?r?: According to (4.7),
if r ? 1; F ?r? ? cK0?r=x? where the coef?cient c is
2=?2ps2 ? ln?1=n? ? C0? and the correlation length x is???????????????
s2=?2n?
p
: The correspondence between the predicted
values of x and c; and those ?tted from our simulated
F ?r? (Fig. 2) are in agreement with theoretical predic-
tion (Table 1). Replicate runs vary remarkably (Fig. 2,
right panel), and the standard deviation is greatest for
small speciation rates n (results not shown). For n ?
3:81  106; the maximal relative standard deviation
around the similarity is  0:03:
We also test our model?s predictions against simula-
tions for the nearest we can come to a Gaussian
dispersal kernel on a square lattice (Fig. 3), assuming
Chave and Leigh162forest of N ? L2 trees on a square lattice where, at each
elementary timestep:
(1) a tree chosen at random, without regard to its
species, dies;
(2) the dead tree is immediately replaced by a
young of one of its neighbors;
(3) with probability 2pP ?r?r dr; the distance from
the newly dead tree to its replacement?s seed?
parent is between r and r ? dr; where P ?r?; the
dispersal kernel, is the same for trees of all
species, and
(4) with probability n; the replacement is a mutant
of an entirely new species.
A Monte Carlo (MC) timestep is de?ned to be one
tree generation, or N ? L2 elementary timesteps. During
one MC timestep, each site experiences an average of
one tree-death apiece. Young trees grow up only in the
sites of their dead elders, thereby avoiding the tendency
characteristic of continuous models for individuals to
pile up at particular points, as remarked upon by
Felsenstein (1975).
We run simulations on a 1024  1024 square lattice
(N sites), with free boundary conditions. A square lattice
of this size is large enough to avoid strong boundary
effects, as veri?ed by a few simulations on a square
lattice of size 2048  2048: We vary both the speciation
rate n and the shape of the dispersal kernel P ?r?: For
each choice of n and P ?r?; we run 50?200 replicates to
assess the sample-to-sample variance. We evaluate the
similarity function F ?r? for each replicate as follows.
When equilibrium is attained (see below), we draw
100N ? 108 pairs of individuals at random from the
lattice and ask whether they are conspeci?c. The maximum
distance between members of a pair is 1448 cell units.
We actually simulate the random walk version of the
voter model. The system starts at pseudo-time t ? 0
with one random walk per site. Each random walk
moves once on average during a MC timestep. Random
walks always merge upon encounter. A speciation event
represents the annihilation of a random walk (Bramson
et al., 1998; Chave et al., 2002): by de?nition, the ?rst
representative of a new species has no conspeci?c
ancestor. All the walks that had previously coalesced
with the annihilated walk de?ne a set of sites (the initial
positions of these random walks) to which we af?x a
species label. We run simulations until all random walks
have either coalesced, or suffered annihilation. At this
time, every site in the lattice has a species label, and we
call such a con?guration an equilibrium distribution of
species. In the random walk algorithm, one timestep
corresponds to two timesteps of the direct version. Thus,the effective parameter s2 of the dispersal kernel we use
in the analytical formulas of the similarity function
should be multiplied by 2. Indeed, the 2-step dispersal
kernel is the self-convolution of the 1-step dispersal
kernel, therefore, its variance of the 2-step dispersal
kernel is just twice the variance of the 1-step dispersal
kernel. Attaining equilibrium takes longer, the lower the
speciation rate n: When n ? 3:81  106 average time to
equilibrium is 106 MC time steps, about 2=n genera-
tions. Our simulations take no more than a few minutes
per replica on a powerful PC.
Let dead trees be replaced only by young of one of
their four nearest neighbors. Let the distance between
lattice points be 1, so r ? 1: We compare the model?s
FIG. 2. Similarity function F ?r? in the neutral model with nearest-
neighbor dispersal, in log-linear axes. The function F ?r? is drawn up to
L=2 ? 512 space units. Panel A: similarity function averaged over 50
runs for three different speciation rates as in Table 1
(n ? y=N ; y ? 4; 8; 16; from right to left, N ? 10242). Dashed lines
correspond to the function cK0?x=x? plotted for the values of c and x
corresponding to the above values of y: This solution agrees fairly well
for r > s; but not for r5s; as discussed in the text. Panel B: similarity
functions for each of the 50 runs and for y ? 16 (dots), and running
average.
Fd c
Model of b-Diversity in Tropical Forests 163TABLE I
Comparisons between Predicted Values of n and c and Those Found by
m ? h=N and Dispersal Parameters r ?N ?1024 1024)
n s2 Observed S Observe
3:81 106 1 121 0.147
7:63 106 1 222 0.150
1:53 105 1 394 0.162
1:53 105 2 264 0.121
1:53 105 4 221 0.0953
1:53 105 8 198 0.0636
1:53 105 16 184 0.0390
3:05 105 4 405 0.0970that the probability of a young replacing its seed parent
is 0. We vary y between 4 and 128, and s2 between 2 and
16, with 200 replicates for each case. Predicted values of
x underestimate the observed value only when x is larger
than the system size (Table 1). Otherwise, our simula-
tion?s results agree with theoretical predictions.
8. DISCUSSION
We have calculated and simulated the probability F ?r?
that two trees separated by a distance r are the same
6:10 105 8 670 0.0644
1:22 104 16 1165 0.0385
Note. The mean number of species in the system, S; is also provided. Th
kernel (Fig. 2). The last seven lines correspond to runs with the Gaussian
that C0 ? 0:
1 10 100 1000
Distance
0
0.1
0.2
0.3
0.4
0.5
0.6
Si
m
ila
rit
y
1 10 100 1000
Distance
0
0.1
0.2
0.3
0.4
0.5
0.6
FIG. 3. Similarity function in the neutral model with Gaussian
dispersal, and n ? y=N ?N ? 10242?; averaged over 200 replicates in
each case. The function F ?r? is drawn up to L=2 ? 512 space units. Left
panel: simulations with y ? 16 and s2 ? 2; 4; 8; and 16 (from top to
bottom). Right panel: simulations such that the predicted value for the
correlation length x is held constant at x ? 256; both y and s2 are
varied: fy;s2g ? f16; 1g; f32; 2g; f64; 4g; and f128; 8g (from top to
bottom). Observed x (the cross-over point between linear and convex
shape in the panel) is roughly constant (see also Table 1).species, assuming competitive equivalence, limited seed
dispersal, and balance between speciation and extinc-
tion. We have learned how F ?r? declines with increase of
distance between the trees compared, and what factors
affect the speed of decline. The behavior of F ?r? depends
on the dimensionality of the system. For a one-
dimensional system, and presumably, for a long, very
narrow river-side gallery forest, F ?r? declines more
slowly with r than in a forest extending in two
dimensions. Moreover, if seeds are distributed about
their parent tree according to a ??fat-tailed?? Cauchy
distribution, in which a few seeds travel very great
distances, rather than a Gaussian distribution, it is as if
itting cK0?r=n? to F?r? from Simulations with Different Speciation Rates
Predicted c Observed x Predicted x
0.128 341 362
0.134 215 256
0.141 142 181
0.115 239 256
0.0845 345 362
0.0552 585 512
0.0326 1295 724
0.0871 220 256
0.0574 215 256
0.0337 209 256
e ?rst three lines correspond to runs with the nearest-neighbor dispersal
dispersal kernel (Fig. 3). To compute predicted values of c; we assumedthe forest extended in an extra dimension. In a forest
with Cauchy seed dispersal, F2?r?  1=r for large r;
whereas in a similar forest with Gaussian seed dispersal
F2?r?  1=
??
r
p
exp?r=x?; where x is the correlation
length. Spatially explicit, individual-based simulations
show great variation among runs in F ?r?; suggesting that
testing this theory experimentally will require vast
amounts of data (Rousset, 2000).
8.1. The Similarity Function and b-Diversity
Condit et al. (2002) and Leigh et al. (in press)
measured b-diversity by the slope of F ?r? when plotted
against ln?r?; as a crude measure of species turnover, the
rate at which species known from an area of forest
disappear, and other species appear, as the species
composition of progressively more distant areas is
compared with that of the initial one. Unfortunately,
F ?r? is a crude measure of species turnover, or b-
Chave and Leigh164diversity, for the same reason F ?0? is a crude measure of
a-diversity, the diversity of species coexisting in one
region. Indeed, both depend disproportionately on the
more common species, whereas turnover may be more
rapid among rarer species (Pitman et al., 2001). For
example, in 1985, one randomly chosen hectare in BCI?s
50-ha plot had 93 species among its 456 trees 510
cm dbh; and F ?0? ? 0:02562: If the hectare?s 29 species
with one tree apiece were absent, there would be 64
species among 427 trees and its F ?0? would be 0.02548,
only a 5.4% decrease, but the species number would
drop by 31%.
The neutral theory has been compared with data from
networks of plots in western Amazonia and central
Panama (Condit et al., 2002). In Amazonian Ecuador,
setting s ? 54:8 m; n ? 3:6  1011 enabled the F ?r?
given by Eq. (4.4) to ?t observations rather well for 0:1
5r5100 km: When these Ecuadorian plots were
compared with plots 1200 km distant in Amazonian
Ecuador, however, F ?r? was far higher than predicted:
some common species were so widespread that they
must have been spread by natural selection (Condit
et al., 2002). In Panama, where spatial heterogeneity in
climate and soil is much greater than in the upland
forest of western Amazonia, setting s ? 40:2 m; n ?
4:8  108; enabled Eq. (4.4) to ?t the trend of the data
for r between 0.2 and 50 km: The smaller s required to
?t the Panama data, and the greater scatter of these data
about the trend, suggest that the in?uence of environ-
mental heterogeneity on species turnover is greater in
Panama (Condit et al., 2002). The neutral theory of F ?r?
suggested that speciation and limited dispersal in?uence
species turnover and served Condit and his coworkers as
a null hypothesis for distinguishing different in?uences
on species turnover. Few other analyses of the decline in
similarity of species composition between two plots with
increased distance between them have yet been pub-
lished, although Nekola and White (1999) provide one
such analysis for boreal forest.
8.2. On the Relevance of the Neutral Theory
Despite its success in summarizing various arrays of
empirical data (Hubbell, 2001; Condit et al., 2002), the
neutral model does not provide a perspicuous under-
standing of the causes of a-diversity in tropical trees.
Indeed, this model?s extraordinary capacity to match
data is a spectacular warning of the dangers of inferring
process from pattern.
The neutral model has no way of explaining the
contrast between tropical and temperate-zone tree
diversity (Leigh, 1999), or the difference in speciescomposition of ?oodplain and upland rainforest in
Amazonia (Terborgh et al., 1996; Pitman et al., 1999). It
ignores the trade-off between the ability to grow
fast in large clearings and the ability to survive in deep
shade (Brokaw, 1987), whereby the formation of light
gaps allows pioneer tree species to persist in mature
forest (Skellam, 1951; Denslow, 1987; Schnitzer and
Carson, 2001).
Even for forest in homogeneous habitat, the assump-
tion of the neutral model are implausible. The neutral
model assumes a plant?s fate is independent of its
species, or those of its neighbors. Yet most seeds,
seedlings and saplings are more likely to die when close
to conspeci?cs, whether young or adult (Wills and
Condit, 1999; Harms et al., 2000). The neutral model
assigns all species the same fecundity and the same
mortality rate. However, Zhang and Lin (1997) and Yu
et al. (1998) argued that differences in these parameters
irrepairably falsi?ed the neutral model. BCI?s 50-ha plot
has nine species with over 500 trees 510 cm dbh: In ?ve
of these species, the number of trees changed far more
between 1982 and 2000 than was consistent with the
neutral theory (Condit et al., 1995, 1996; R. Condit,
pers. comm.).
On the other hand, null models can be extremely
valuable for developing new theoretical approaches.
Hubbell (1979, 1997) focussed salutary attention on the
circumstance that a-diversity represents a balance
between speciation or immigration and extinction. Can
a neutral theory perform a similar service for b-diversity,
by providing new mathematical approaches, or by
viewing b-diversity as the joint effect of local speciation
and limited dispersal?
Incorporating limited seed dispersal into ecological
theory is an urgent challenge. Ribbens et al. (1994)
evaluated the effect of differences among tree species in
seed dispersal on a model of forest dynamics; Clark et al.
(1999) ?t mathematical formulae to observed dispersal
kernels and showed how the dispersal kernel of a tree
species might in?uence its capacity to spread. Several
authors have already explored the implications of
limited seed dispersal for the neutral model (Durrett
and Levin, 1996; Hubbell, 2001; Chave et al., 2002). The
Gaussian dispersal kernel many theorists employ,
however, is often inconsistent with observed seed
shadows (Clark et al., 1999). Moreover, different tree
species disperse their seeds very different distances,
according to very different kernels (Willson, 1993; Clark
et al., 1999; Muller-Landau, 2001), even though our
neutral model assigns the same dispersal kernel to all
trees. Finally, theorists can only solve the neutral model
of b-diversity if F ?r? expresses an equilibrium between
Model of b-Diversity in Tropical Forests 165speciation and extinction, an equilibrium that takes long
to attain, of the order of 2=n tree generations, in the case
of nearest-neighbor dispersal (Section 7, see also
Maruyama, 1972). It is increasingly evident that for
many tree species, ??fat-tailed?? dispersal kernels ?t
observed distributions of seeds about their parents
better than Gaussian kernels do. Fat-tailed dispersal
kernels may also hasten the approach to equilibrium,
transforming neutral models into more practical null
hypotheses.
More generally, the incorporation of Cauchy dispersal
kernels may improve the usefulness of the models of
isolation by distance now being used by many population
geneticists. Relating the gene ?ow between different locales
to the distance between these locales is a great challenge.
Le Corre et al. (1997) have explored the in?uence of long-
distance gene ?ow on the genetic structure of a tree
population. Our analytical model could prove a useful null
hypothesis for such investigations.
APPENDIX: TAIL OF THE SIMILAR-
ITY FUNCTION FOR THE CAUCHY
DISPERSAL CURVE
In this appendix, we investigate the tail of Fd (r  c)
in the case of a power-law dispersal curve of the Cauchy
type (Eq. (5.3)). The approximation r2 ? j2c2 ? j2c2 is
only valid if j > r=c: One can write
Fd?r? ?
Cd ?1  Fd ?0??
rcd
Xr=c
j?1
j?1  n? j
??r=c?2 ? j2??d?1?=2
 
?
X1
j?r=c
j?1  n?j
??r=c?2 ? j2??d?1?=2
1
A: ?A1?
In the ?rst sum, the term ?1  n?j is dominant only when
r  c=n: We do not consider these extremely long
distances. We can, therefore, write
?r=c??r=c ? 1?
2
1
2?d?1?=2?r=c?d?1
5
Xr=c
j?1
j?1  n? j
??r=c?2 ? j2??d?1?=2
5?r=c??r=c ? 1?
2
1
?r=c? d?1
: ?A2?
Since r  c
cd1
2?d?3?=2rd1
5
Xr=c
j?1
j?1  n? j
??r=c?2 ? j2??d?1?=2
5 c
d1
2r d1
: ?A3?For the second sum, we also have
X1
j?r=c
?1  n? j
2?d?1?=2jd
5
X1
j?r=c
j?1  n? j
??r=c?2 ? j2??d?1?=2
5
X1
j?r=c
?1  n? j
jd
:
?A4?
These series are uniformly convergent for d > 1: There-
fore, we can approximate them as a sum from r=c to a
very large number n; such that n5r=c=n: This proves
that the term ?1  n?j can also be neglected from these
series. Finally, we replace the summation term by its
corresponding Riemann integral
Z 1
u?r=c
1
2?d?1?=2ud
du
5
X1
j?r=c
j?1  n? j
??r=c?2 ? j2??d?1?=2
5
Z 1
u?r=c
1
ud
du:
Consequently, if d > 1
1
2?d?1?=2
d ? 1
2?d  1?
cd1
rd1
5
X1
j?1
cj?1  n?j
??r=c?2 ? j2??d?1?=2
5 d ? 1
2?d  1?
cd1
r d1
or
Cd ?1  Fd?0??
rcr d1
d ? 1
2?d?3?=2?d  1?
5Fd?r?
5Cd ?1  Fd ?0??rcr d1
d ? 1
2?d  1?
:
?A5?
For a simpler proof of the same result, the sum in
Eq. (5.3) can be approximated by an integral:
Fd ?r? ? Cd
1  Fd ?0?
r
Z 1
u?0
?1  n?u jc
?r2 ? u2c2??d?1?=2
?Cd
1  Fd ?0?
r
Z 1
u?0
uc
?r2 ? u2c2??d?1?=2
du
or
Fd ?r? ? Cd
1  Fd ?0?
cr
1
?d  1?r d1
: ?A6?
Chave and Leigh166In the one-dimensional case, the ?rst sum (A3) is
1
4
5
Xr=c
j?1
j?1  n?j
??r=c?2 ? j2?
51
2
while, for the second sum (A5), we have
1
2
X1
j?r=c
?1  n?j
j
5
X1
j?r=c
j?1  n?j
??r=c?2 ? j2?
5
X1
j?r=c
?1  n?j
j
which is a decreasing function of r; bounded from above
by ln?1=n?: We deduce that
C1?1  F1?0??
2rc
1
2
? ln
1
n
  
5F1?r?
5C1?1  F1?0??rc
1
2
? ln 1n
  
:
ACKNOWLEDGMENTS
We warmly thank Jim Clark, Rick Condit, Steve Hubbell, Liz
Losos, Helene Muller-Landau, Fran
-
cois Rousset, and Doug Yu for
discussions, insights, and advice that greatly improved the quality of
this paper. We also thank associate editor Yoh Iwasa and three
anonymous reviewers for their useful suggestions on the submitted
manuscript.
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