Tropical Forest Community Ecology

20 Jérôme Chave

beta-diversity of tropical trees usin gtheoretical
results from population genetics (Malécot 1948).
Let us consider that a community is saturated
withρindividuals per unit of area, such that
any dyin gindividual is immediately replaced by
a youn gindividual, not necessarily of the same
species. New species may appear in the system as
a result of point-wise speciation or long-distance
dispersal (immigration from outside the commu-
nity). Thus, a dyin gindividual is replaced at rateν
by an individual belonging to a species not yet rep-
resented in the landscape. Recruitment will occur
through seed dispersal from existing individuals to
neighboring sites, some of which may be empty.
The probability that a seed fallsrmetersaway
from its parent is defined asP(r), and is commonly
called adispersal kernelin the theoretical literature
(Kotet al. 1996, Clarket al. 1999b, Chaveet al.
2002). A crucial feature of this model is disper-
sal limitation, that is, seeds are more likely to fall
close to the parent than far from it. The dispersal
kernel may take a broad array of mathematical
forms. The Gaussian dispersal kernel is defined by
P(r)=( 1 /( 2 πσ^2 ))exp(−r^2 /σ^2 ), hence the vari-
anceσ^2 is the only parameter of the dispersal
model. In general the Gaussian dispersal func-
tion provides a poor fit to empirical data (Clark
et al. 1999b, Nathan and Muller-Landau 2000).
The so-called “2Dt” dispersal kernel (Clarket al.
1999b), defined as

P(r)=

p

πu

(

1 +r^2 /u

)p+ 1 (2.5)

provides a better fit (this kernel is parametrized by
pandu, both positive). The 2Dt kernel is similar
to a Gaussian dispersal function if bothuandp
become large, such that 2σ^2 =u/p.Forr^2 larger
thanu, elementary calculations on Equation (2.5)
show that the dispersal function is approximately
equivalent to a power law:P(r)≈pup/πr^2 p+^2 ∼
1 /r^2 p+^2. Hubbell (2003) proposed the use of
another “fat-tailed” class of dispersal kernels,
namely Lévy-stable dispersal kernels (for a tech-
nical definition, see Gnedenko and Kolmogorov
1954). Lévy-stable dispersal kernels have not been
used in the literature because they are difficult to
manipulatebothmathematicallyandnumerically.
They behave as power laws at large values ofr, but
so does the 2Dt dispersal kernel. Results obtained

with the 2Dt dispersal model should therefore

be qualitatively similar to those obtained with

any dispersal kernel with a power-law tail, and

the mathematically more tractable kernel should

always be preferred.

In a spatially structured neutral model, it is

possible to find how the species overlap index in

Equation (2.3) varies with the geographical dis-

tance between pairs of sites. Chave and Leigh

(2002) callF(r)the similarity function, the proba-

bility that two individuals taken from two different

sites belon gto the same species. Since the only

parameters in the Gaussian neutral model are

the speciation/immigration rateν, the dispersal

parameterσ, and the density of individualsρ, the

similarity function can be exactly expressed as a

function ofν,σ, andρ(for an exact expression, see

Chave and Leigh 2002). A simple approximation is

F(r)≈−

2

2 πρσ^2 +ln( 1 /ν)

ln

(

r

√

2 ν

σ

)

(2.6)

The similarity function decreases logarithmically

with increasin gdistance, and this approximation

is valid as lon gas 1≤r/σ<< 1 /

√

2 ν. Assuming

values of 50 m forσand 10−^8 forν(Conditet al.

2002),therangeof validityof thisequationwould

be 50≤r <<7000, in meters. Two remarks

should be made at this point. First, Equation (2.6)

is not valid in the ranger/σ ≥ 1 /

√

2 ν, and it

should be replaced by an exponentially decreas-

in gfunction (Chave and Lei gh 2002). Lar ge-scale

analyses confirm an exponentially decayin gpat-

tern at larger scales (Nekola and White 1999,

Qianet al. 2005). Second, in the case of a 2Dt

kernel, the similarity function is parametrized by

uandp, rather thanσonly. Here again an exact

formula and useful approximations are available

(Chave and Leigh 2002). For instance, it can be

shown that for larger,F(r)∼ 1 /r^2 p.

TESTING THEORIES

Landscape-scale patterns of tree

diversity

The confrontation of niche-based and dispersal-

based theories in community ecology has

turned into an active field of research since