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

Spring 1962
CANONICAL DISTRIBUTION
211
FIG.25. A family of generalized parabolas, and the
locus of points of maximum curIrature.
point of sharpest curvature. In this sense there
is a theoretical "break" in the curve, though it
is not a discontinuity of any sort.
The curvature of any curve at any point is
given by
and if we differentiate this once more and set
and solve for x, we can define the point of maximum
curvature in terms of the abscissa, or by
solving for y we can define it in terms of the
ordinate. which is usually (in our problem) more
satisfactory.
In this case, we find that the point is given by
On Fig. 25 I have plotted the locus of the maximum
curvature. It is probably the break-point
that Cain was seeking. In practice, however, it
is not easy to determine such a point by graphical
methods. It is much easier, even with a curve
perfectly free from experimental or statistical
errors, to find the point by equation (32), and for
that purpose we have first to find the value of n.
Clearly, if the curve really is a parabola, this is
most easily done by taking logarithms of both
ordinate and abscissa, whereon the equation becomes
11 log y = log a + log x (33)
which is a linear relation between log (Species)
and log (Area) : i.e. the curve ought to be a
straight line. This log-log plotting I have elsewhere
(' Preston 1960) called an Arrhenius plot-
' I am indebted to Dr. R. E. mould of Preston Laboratories,
Inc., no\v Xtnerican Glass Research, Itlc., for this
result,
FIG.26. The Cain & Castro Species-Area curve on a
log-log (Arrhenius) basis.
ting. Using it on the data of G.A. Black and S. A.
Cain as reported in Cain and Castro (1959)) for
savanna vegetation in Para, Brazil, we get Figure
26, which, except for the first 2 points, is a pretty
fair straight line. The slope is given by k = l/n
= 0.14, or n = 7 approximately, and the curve,
over 225 orders of magnitude, is a pretty good
generalized parabola.
The Gleasofi plotting
Plant ecologists seem to have made less use of
the Arrhenius "log-log" plotting than of the
Gleason "semi-log" plotting. It is necessary, in
order to space the points fairly uniformly across
the graphs, to take the logarithm of the area, but
it is not essential to take logarithms of the number
of species. Furthermore, because species increase
so slowly with increasing area, the logarithm of
the number of individuals is a linear function of
the number of species over rather wide intervals.
Thus if the Arrhenius plot gives a straight line, so
will the Gleason plot, over an interval of an order
of magnitude or more in the size of quadrat or
time of observation.
In Figure 27 we illustrate this point by means
of Thomas' data on the 10-years of breeding bird
counts on the 65 acres of Neotoma in south-central
Ohio. The same data are plotted by the Arrhenius
and by the Gleason methods. In each case
the abscissa is a logarithmic scale of years-ofobservation.
The ordinate for the lower curve is
FIG.27. Gleason and Arrhenius curves compared over
a range of one order of magnitude (3.3 doublings or
octaves).