148 Stephen P. Hubbell
is almost hyperbolic in shape (i.e.,〈φn〉≈θ/n
whenxis very close to unity). Preston con-
cluded that the shape of Fisher’s logseries was
a sampling artifact. He argued that if sample
sizes were increased, and if relative species abun-
dances were log-transformed, then they would be
nearly normally distributed with the most species
occurring at intermediate log abundance classes.
This meant that the log-transformed distribution
would display a bell shape with an “interior”
mode, not a mode in the abundance class of
singleton species. There were now two compet-
ing statistical hypotheses for the distribution of
relative species abundance: Fisher’s logseries and
Preston’s lognormal. Over the half century since
this debate was enjoined, Preston’s lognormal
has probably been fit to relative species abun-
dance data more often than Fisher’s logseries, but
Preston’s hypothesis could not explain the stabil-
ity of Fisher’sα, and Fisher’s hypothesis could
not explain distributions with an interior mode.
Can neutral theory reconcile these two disparate
explanations?
So far, the theory we have developed above
only gives rise to Fisher’s logseries for the rel-
ative species abundance distribution at steady-
state between speciation and extinction in the
metacommunity. This distribution is appropriate
on macroecological scales of space and time,
but what patterns of relative species abundance
do we expect on small spatial and temporal
scales in local communities? In the metacommu-
nity model, we ignored dispersal and the spatial
substructure of the metacommunity, assuming
thatbirthsanddeaths,andspecies,wererandomly
distributed over the metacommunity landscape.
What happens if we put substructure and dis-
persal into the theory, a bit of added complexity?
Suppose we subdivide the metacommunity of
sizeJMindividuals into local communities of size
Jindividuals, and allow dispersal between the
local communities with probability ratemper
birth in a given local community. We can now
describe the patterns of relative abundance in
local communities as semi-isolated samples of the
metacommunity. What do we get?
It turns out that under dispersal limitation,
when species cannot move with impunity any-
where in the metacommunity in an infinitesimal
time step, the distribution of relative species
abundance in local communities will not be
Fisher’s logseries. The new insight from the
slightly more complex version of neutral the-
ory is that the shape of the local distribution of
relative species abundance depends on the disper-
sal probability,m.Ifmis large and near unity,
then the distribution approaches Fisher’s logseries
(i.e., no dispersal limitation), with no interior
mode at intermediate abundances. However, ifm
is small, such that local communities are very iso-
lated and do not often receive immigrants from
the surrounding metacommunity, then the dis-
tribution becomes more lognormal-like, with an
interior mode.
Although the distributions of local relative
speciesabundancearenotlognormals,theycanbe
closely approximated by (and confused with) log-
normals when the parameters are within certain
ranges. Volkovet al.(2003) derive the analyti-
cal expression for relative species abundance in
a local community undergoing immigration from
a much larger metacommunity, analogous to the
classical island–mainland problem in the theory
of islandbiogeography.Again,let〈φn〉bethemean
number of species withnindividuals. Then〈φn〉=θJ!
n!(J−n)!(γ)
(J+γ)×∫γ0(n+y)
( 1 +y)(J−n+γ−y)
(γ−y)×exp(
−yθ
γ)
dy (9.6)where(z) =∫∞
0 tz− (^1) e−tdtwhich is equal to
(z− 1 )!for integerz, andγ=m(J− 1 )/( 1 −m).
Composite parameterγ is another fundamen-
tal number in neutral theory: it is the scale-
independent fundamental dispersal parameter
that takes out the effect of local community sizeJ.
Equation (9.6) can be solved numerically quite
accurately. As the immigration ratemdecreases,
the distribution of relative species abundance in
the local community given by Equation (9.6)
becomes progressively more skewed, confirming
the simulation-based results in my book (Hubbell
2001). Thus, as islands or local communities
become more isolated, rare species become rarer,