Tropical Forest Community Ecology

Symmetric Neutral Theory 149

and common species become commoner. This is
because, under low immigration rates, when rare
species go locally extinct, they take longer to re-
immigrate, so that the steady-state number of rare
species locally is lower than would be expected
if local communities were a random sample of
the metacommunity, that is, if dispersal were
unlimited.
Herein lies a potential mechanistic explanation
for both Fisher’s and Preston’s hypotheses: the first
applies to macroecological scales – or for pooled,
multiple scattered samples from a landscape that
overcomes dispersal limitation – and the second to
local scales. Not everyone agrees with this inter-
pretation (e.g., McGill 2003), but we will return
to this question when we confront the theory with
data on tropical tree species abundances.

NOW, ADD YET MORE

COMPLEXITY: SYMMETRIC

DENSITY- AND FREQUENCY

DEPENDENCE

Density- and frequency dependence are often
regarded as the classical signatures of niche-
assembled communities because they imply the
regulation of species abundances by their real-
ized carrying capacities set either by the limiting
resources available to the species in its real-
ized niche, or by top-down predator control, as
by Janzen–Connell effects (Janzen 1970, Connell
1971). However, if these effects are symmetric,
meaning that every species experiences the same
per capita density dependence when it is of equiv-
alent abundance, then neutral theory can be
generalized to accommodate density dependence
(Volkovet al.2005). There are any number of
ways to put density dependence into the master
equation, but the simplest is to make the ratios of
per capita birth and death rates in Equation (9.4)
be functions of abundancen. So, define a density-
dependent birth: death ratio at abundancenas
ˆr=(bn/dn+ 1 )f(n). We can choose any appro-
priate functionf(n)that has the property that
f(n)>1 whennis small andf(n)→1 when
nis large. Note that asn→∞,ˆr→x, wherex
is one of the two parameters of Fisher’s logseries
(recall,x=bn/dn+ 1 =b/d). This means that

whennis small,ˆrwill be greater than unity,

and births outnumber deaths (rare-species advan-

tage). In Fisher’s logseries,b/ddoes not change

with density (i.e., density-independent growth).

In the spirit of choosing simple over complex,

we setf(n) = n/(n+c), which requires only

one additional free parameter, and the same for

all species (symmetry). This function in the mas-

ter equation results in per capita death rates

that are independent ofn, but per capita birth

rates that increase asngets smaller. This for-

mulation results in a very simple and elegant

modification of Fisher’s logseries distribution for

the mean number of species〈φn〉having abun-

dancen:

〈φn〉=θ

xn

n+c

(9.7)

wherecis the density dependence parameter,

with units of individuals, that measures the

mean strength of density dependence in the

community. In neutral theory, we call this dis-

tribution the hyperlogseries because it has sta-

tistical properties similar to the hypergeometric.

How parameter c influences theb/d ratio is

shown in Figure 9.1. Whencis small, only very

small populations enjoy a birth rate larger than

the death rate. However, ascbecomes bigger,

larger and larger populations enjoy a frequency-

dependent advantage. Note how the function for

b/dcrosses the line of population replacement

(b/d = 1) at higher and higher values of n

ascincreases (Figure 9.1). In the symmetric

case described by Equation (9.7), since all species

enjoy the same advantage when rare, this func-

tion describes both intraspecific density depen-

dence and interspecific frequency dependence.

Note that these stochastic population dynam-

ics never lead to population equilibrium. Even

though species on average are near replacement

at largen, they do not exhibit central tenden-

cies to a dynamical attractor at a fixed carrying

capacity. The fate of every species in the the-

ory is extinction, although time to extinction

increases approximately at a rate proportional

ton·ln(n)with increasing abundance (Hubbell

2001).This fact will become important in the later

discussion.