The Canonical Distribution of Commonness and Rarity: Part I F. W ...

Spring 1962
CANONICAL
"representation" of the species, be no at the modal
octave. Then at the Rth octave from the mode it
is n,ZR individuals per species, and the octave
holds yn,2R individuals, or :
Y = (noyo)[2Re-aeRS]= (noyo)[eRIn e-a2~2]
- noyo (IL&) \-&2(RR~2) 2
2a 2a2 (3)
This is a lognormal curve with the mode dis-
In 2
placed by an amount - octaves : it has the same
2a2
dispersion constant as the Species Curve, and it
has the modal height Yo= noyoe(s)2.
(See Figure 1.)
Equations like (3) become somewhat simpler
in appearance if ive use "natural orders of magnitude"
in place of "octaves," i.e. if we use intervals
in which the frequency or abundance of a
species increases in the ratio 2.718 instead of 2.0,
and if we use F, the standard deviation in orders
of magnitude, instead of the coefficient a.
This method of working is convenient if we
have available adequate tables of natural logarithms,
for then the observed frequencies are easily
DISTRIBUTION 187
classified into their natural orders of magnitude.
I think, however, that it will be more convenient
if we continue, as we have begun, by using "octaves,"
which do not depend on the availabilitv
of such tables, or on thk alternative method df
cotlverting ordinary logarithms to natural ones.
The eflects of a finite number of species
Referring to the Species Curve in Figure 1, we
note that in theory this curve extends infinitely
far both to left and to right, but the long "tails"
are exceedingly close to the R axis as asymptote.
The area under the cunre, or the integral of the
curve, represents the number of species we have
accumulated, as we go from minus infinity to any
given point. This area is at first so small that
not until we are within about 9 octaves of the
mode (for this particular case, where we have
a total of 178 species) have we accumulated
enough area to correspond to a single species.
This is the beginning of the real, finite, distribution.
As we continue to the right we accumulate
species rapidly; then we pass the mode and
accumulate them increasingly slowly. Finally
we reach a point some 9 octaves to the right of
the mode where the remaining area is scarcely
enough to hold one more species. In practice,
/+-
lntermodal Distance
2)ILaz .-I
1
) SCALE OF OCTAVES
FIG.1. The canonical distribution ior an ensemble of 178 species. For this number oi species, the coefficient
a = 0.200 and the standard deviation o is 3.53 octaves for both "Species" and "Individuals" curves. The modal
height of the species curve is y, = X, species. The ordinate scale for the individuals curve is arbitrary: see text
for explanation. The intermodal distance is 8.68 octaves. The real part of each curve is drawn solid : the first
(rarest) and last (comtnonest) species ought to lie at about this distance (8.68 octaves or 2.45 a) from the
mode of the species curve, or perhaps a little, but only a little, more.