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Taylor was able to interpret the curves he obtained from processing speciation data in
a number of ways using the two parameters. If the slope of the curve and the intercept
are both equal to one,
a
1, then the variance and mean are equal to one another.
This equality holds only for a Poisson distribution, which, when it occurred, allowed
him to interpret the number of new species as being randomly distributed over the field,
with the number of species in any one plot being completely independent of the number
of species in any other plot. If, however, the slope of the curve was less than unity,
b
=
b
=
1,
the number of new species appearing in the plots was interpreted as being quite regular,
like the trees in an orchard. Finally, if the slope of the variance versus average was
greater than one,
b
<
1, the number of new species was interpreted as being clustered
in space, like disjoint herds of sheep grazing in a meadow.
Of particular interest to us here is the mechanism postulated nearly two decades later
to account for the experimentally observed allometric relation [
79
]:
>
We would argue that all spatial dispositions can legitimately be regarded as resulting from the
balance between two fundamental antithetical sets of behavior always present between individ-
uals. These are, repulsion behavior, which results from the selection pressure for individuals to
maximize their resources and hence to separate, and attraction behavior, which results from the
selection pressure to make the maximum use of available resources and hence to congregate
wherever these resources are currently most abundant.
It is the conflict between the attraction and repulsion, emigration and immigration,
which produces the interdependence of the spatial variance and the average population
density. The kind of clustering observed in the spatial distribution of species number,
when the slope of the power curve is greater than one, is consistent with an asymptotic
inverse power-law distribution of the underlying data set. Furthermore, the clustering
or clumping of events is due to the fractal nature of the underlying dynamics. Recall
that Willis, some forty years before Taylor, established an empirical inverse power-law
form of the number of species belonging to a given genus. Willis used an argument
associating the number of species with the size of the area they inhabit. It was not until
the decade of the 1990s that it became clear to more than a handful of experts that the
relationship between an underlying fractal process and its space-filling character obeys
ascalinglaw[
44
,
45
]. It is this scaling law that is reflected in the allometric relation
between the variance and average.
Note that Taylor and Woiwod [
80
] were able to extend Taylor's original argument
from the stability of the population density in space, independent of time, to the
stability of the population density in time, independent of space. Consequently, the
stability of the time series, as measured by the variance, can be expressed as a power of
the time-series average. With this generalization in hand we can apply Taylor's ideas to
time series.
2.4.1
Fractal heartbeats
Mechanisms producing the observed variability in the time interval from one heartbeat
to the next apparently arise from a number of sources. The sinus node (the heart's
