purely mathematical fractals (Schmid, 2000 ). Scale-dependent analysis of this type
has been applied to describe the species-sizedistribution in ecological communities
by assuming proportionality between the physical scale and body size following:
SðL>lÞ/lD ( 8 : 5 )
whereSis the cumulative size distribution of species at scalel larger than a
characteristic size, andDis the fragmentation dimension (Schmidet al., 2002).
Thus, in a homogeneous system, the probability (p) of a measure varies withl as
p(l)/lD.
However, not all cells of scalel contain equal measures or mass and, con-
sequently, averaging these subsets eliminates information that may elucidate
the processes that generate community patterns. Species size-frequency distri-
butions can be considered mathematically as measures that themselves may
display statistical self-similarity. This self-similar pattern could arise through
stochastic resource partitioning among different sized species together with
an inherent resource space, which itself displays fractal characteristics.
Consequently, finer resolutions of body-size patterns would reveal that each
subinterval of size has different scaling exponents, which should not be con-
fused with the concept of ‘multiscaling or mixed fractals’ in which the slope of a
log-log plot changes abruptly at some particular scale (Schmid, 2000 ). If different
subintervals of body-size and their corresponding measures are considered, a
heterogeneous distribution of fractal exponents for such intervals would be
expected. These heterogeneous features are present in many benthic distribu-
tions (Schmid, 2000 ), displaying a multifractal behaviour, and the probability
within eachith subinterval,piscales as:
piðlÞ/l^ i ( 8 : 6 )
where (^) iis the Lipschitz-Ho ̈ lder exponent (see Eq. (8.12)), characterizing scaling in
theith subinterval (Feder, 1988 ). This multifractal type of a distribution can be
analyzed directly when many data are available, and involves a spectrum of
scaling indices, where the mass around different points is characterized by differ-
ent single fractal dimensions. Various formalisms have been developed to
describe the statistical properties of these measures in terms of their generalized
dimensions Dq or the singularity spectrum f( ) (Mandelbrot, 1974 , 1989;
Hentschel & Procaccia, 1983 ;Halseyet al., 1986). A set of different lattices with
cells or subintervals of equal length is required to implement the scaling analysis
of a general size distributionon an interval¼[a,aþL]. For the size spectra in
each stream, a measureis considered, and the interval determined by the
extreme values of all body and particle sizes. A common choice is to consider
dyadic scaling with successive partitions ofof sizeLl ¼L 2 f,whereLis the
length of,l is the size scale andf¼1, 2, 3,...,L. For everyl a number of cells,
N(l)¼ 2 f, are calculated and their measuresi(l) are supplied from the size spectra.
BODY SIZE AND SCALE INVARIANCE 153