Tropical Forest Community Ecology

Symmetric Neutral Theory 147

must sum to unity, we can find the value ofP0,k
from this sum, and all other terms as well.
Actually, the master equation (Equation (9.2))
applies much more generally than to neutral the-
ory alone. It can also describe the dynamics of
non-neutral communities if we let species have
species-specific birth, death, and speciation rates,
and it is completely general in regard to what fac-
tors may control these rates. So, for example, the
birth and death rates could be density dependent,
they could depend on competition or predation,
and so on. But for now, hewing to the philos-
ophy of adding complexity in small, considered
steps, consider a symmetric neutral community
ofSspecies that are all alike on a per capita
demographic basis, such that they all have the
same per capita birth and death rates, that is,
bn,k≡bnanddn,k≡dn(i.e., the species identifier
kdoes not matter). We can introduce speciation
by recognizing a special “birth rate” in this gen-
eral metacommunity solution, that is,b 0 =ν, the
speciation rate. The mean number of species with
nindividuals,〈φn〉, in a community ofSidentical
species is simply proportional toPn:

〈φn〉=SP 0

n∏− 1

i= 0

bi

di+ 1

(9.4)

What does all this have to do with Fisher’s
logseries? It turns out that Equation (9.4) is
Fisher’s logseries if we make birth and death
rates density independent. Herein lies one of
the most profound insights to come from neu-
tral theory: obtaining Fisher’s logseriesnecessarily
impliesdensity independencein population growth
on metacommunity spatial scales. If one’s rela-
tive species abundance data fit Fisher’s logseries
on large biogeographic scales, one can defini-
tively conclude that the population dynamics
of species on large scales behave in a stochas-
tically density-independent manner. This theo-
retical result has potentially paradigm-shifting
implications in ecology for the scale depen-
dence of population regulation, for the structure
and dynamics of communities on large spatial
scales, for conservation biology, and for the evo-
lution of biotas and phylogeography (Hubbell
2008a).

We can easily derive this result. What does

it mean to have density-independent population

growth? It means that the per capita birth and

death rates remain constant as population density

varies. Mathematically, we write that the abso-

lute birth rate of a species of current abundance

nis simplyntimes the birth rate of a species

with abundance 1, that is,bn=nb 1 , the defini-

tion of density independence. Similarly, suppose

that the death rates are density independent,

dn = nd 1. Substituting these expressions into

Equation (9.4), we immediately obtain Fisher’s

logseries:

〈φn〉M=SMP 0

b 0 b 1 ···bn− 1

d 1 d 2 ···dn

=θ

xn

n

(9.5)

where the subscriptMrefers to the metacommu-

nity,x=bn/dn=–b 1 /d 1 =b/d,b 0 =ν, and

θ=SMP 0 ν/b=αof Fisher’s logseries.

The derivation of Equation (9.5) reveals that

the mysterious parameterxof the logseries is,

in fact, biologically interpretable as the ratio of

the average per capita birth rate to the average

per capita death rate in the metacommunity. Note

that when one introduces speciation, parameter

xmust be slightly less than 1 to maintain a

finite metacommunity size. At very large spatial

scales, the total birth and death rates have to

be nearly in mass balance, resulting in a meta-

communityb/dratio only infinitesimally less than

unity. The very slight deficit in birth rates ver-

sus death rates at the metacommunity biodiversity

equilibrium is made u pby the very slow in put of

new species.

NEXT STEP: ADD A BIT MORE

COMPLEXITY – LOCAL

COMMUNITIES AND DISPERSAL

LIMITATION

A few years after Fisher and company’s paper,

Preston (1948) published a critique of Fisher’s

logseries. The logseries predicts that the rarest

abundance category of singletons – species sam-

pled only once – should have the most species;

and indeed, the curve described by Equation (9.1)