146 Stephen P. Hubbell
we expect? This is clearly a non-equilibrium com-
munity in the taxonomic sense because any given
species experiences a finite lifespan with a “birth”
and a “death.” But there is nevertheless a non-
trivial stochastic steady-state distribution of rel-
ative species abundance in the metacommunity
among the slowly turning over species. Neutral
theory proves that this distribution is Fisher’s
logseries (Hubbell 2001, Volkovet al.2003).
Fisher’s logseries emerged from one of the two
most celebrated papers ever written on relative
species abundance, one by Fisheret al. (1943),
and the other by Preston (1948), which sparked
a theoretical controversy – about which more
will be said in a moment – a controversy that
persists to the present day (McGill 2003, Volkov
et al.2003, 2007, Dornelaset al.2006). Fisher
found an excellent fit with the logseries to rela-
tive abundance data on Lepidoptera from Britain
and Malaysia. Under the logseries, the expected
number of species withnindividuals〈φn〉 is
given by
〈φn〉=α
xn
n(9.1)
where parameter xis a positive number less
than (but very close to) unity, andαis a diver-
sity parameter known as Fisher’sα. One of the
remarkable properties of Fisher’sαis that it is
relatively stable in the face of increasing sam-
ple size. This stability makes Fisher’sαone of
the preferred measures of species diversity, but
why is it so stable? Until the development of
neutral theory, Fisher’s logseries was simply a
phenomenological statistical distribution fit to rel-
ative abundance data. There was no clear biologi-
cal explanation for either Fisher’sαor parameter
xin the logseries that could be derived from
population biology.
One of the most remarkable results of neu-
tral theory is proof that the celebrated diversity
parameter, Fisher’sα(θin neutral theory), is pro-
portional to the product of the speciation rate
and the size of the metacommunity. This offers
an explanation of the stability ofα: these are two
very stable numbers, one the average per capita
speciation rate in the entire metacommunity, a
very small number, and the other the size of the
metacommunity – the sum of the population sizes
of all species in the metacommunity–averylarge
number (Hubbell 2001). Fisher’sα(orθ)isa
biodiversity number that crops up all over neu-
tral theory, so in a real sense it is a fundamental
number in the theory. But what is the biological
meaning of parameterx?
To explain parameterx, we need to introduce
the so-called master equation of neutral theory,
which describes the stochastic population dynam-
ics of species in the metacommunity (Volkovet al.
2003).Letbn,kanddn,kbetheprobabilitiesof birth
anddeathof anarbitraryspecieskatabundancen.
Letpn,k(t)be the probability that specieskis at
abundancenat timet. Then, the rate of change of
this probability is given bydpn,k(t)
dt=pn+1,k(t)dn+ 1 +pn−1,k(t)bn− 1−pn,k(t)(bn,k+dn,k) (9.2)This equation is not hard to understand. The first
term on the right represents the transition from
abundancen+1ton, due to a death. The sec-
ond term is the transition from abundancen− 1
tondue to a birth. The last two terms are losses
topn,k(t)because they are transitionsawayfrom
abundancento eithern+1orn−1 through a
birth or death, respectively. When one first sees
Equation (9.2), it appears to be little more than
a book-keeping exercise, but it is actually much
more. Note that it is a recursive function of abun-
dance, so we can use it to find an equilibrium
solution for species of arbitrary abundancen.
If we set derivatives at all abundances equal to
zero, then each abundance transition is in equi-
librium. LetPn,kdenote this equilibrium. Then
Pn,k =Pn−1,k·[bn−1,k/dn,k], and more gener-
ally, this corresponds to an equilibrium solution
for the metacommunity:Pn,k=P0,kn∏− 1i= 0bi,k
di+1,k(9.3)
Note that the probability of being at abundance
nis a function of the product of the birth rate
to death rate ratios over all abundances u pto
n−1. Because thePn,k’s are probabilities and