and PSD are similar between the Oberer Seebach (SB) and Afon Mynach (MY) for
larger species and particle sizes. A distinct departure occurred between the two
streams for smaller, more common species and particles. Further evidence for a
close link between habitat- and body-size is given by recent experiments con-
ducted in a Japanese mountain stream. This study revealed that body sizes of
invertebrates tended to decline with decreasing crevice size, thus, strongly
underlining the effect of fractal habitat complexity on body-size distribution
(Taniguchi & Tokeshi, 2004 ; see alsoTownsend & Thompson, this volume).
Multifractal species-area relationships
The species-area relationship (SAR) is one of the most widely studied patterns in
community ecology and is often expressed as a single power-law curve.
However, the SAR should not be characterized by a single curve that assigns
one species richness value to each sampled area, as different samples of equal
area differ in species number. Therefore, the SAR should be given as the relation-
ship between area and its mean number of species,.was calculated for
non-overlapping but equally shaped sampling areas taken over the sampling
period. In order to estimate the mean number of species in an increasing area of
sizel ¼k^2 (wherek¼1, 2, 3,...!l maximum area), 10^6 Monte-Carlo random-
izations without replications of the data matrix were conducted for eachk-set.
By randomly pooling samples of the observed data sets taken at arbitrary
sampling positions and/or occasions, it was possible to obtain samples incorpo-
rating invertebrate species over different developmental stages, body-size
ranges and densities. The species-area curves were obtained by averaging
species richness of those randomly pooled sample sites for each area of sizel.
Some authors have suggested that the SAR following a power-law curve holds
at all spatial scales and implies self-similarity in the distribution and abundance
of species (e.g. Harteet al., 1999). A problem with this monofractal assumption is
that scale invariance of SARs is not maintained over all spatial scales, but that
there are different scaling domains underlying SARs over different spatial
scales. Thus, irregularities in the species-abundance distribution may be statis-
tically the same across areas resulting in a spectrum of fractal subsets. To test for
different scaling domains underlying the SARs of the stream communities,
the method of Re ́nyi’s generalized dimension (Dq) was used for the species-
abundance data. This method follows Eqs.(8.7) to (8.11)outlined in Fractal proper-
ties of size-structured communities (see above), where in the current context,pi
is defined as the relative abundance of speciesiin an areal.
The relationships between mean species richness and sampled area is dis-
played as a power-law function for each of the three streams (Fig.8.8; Table8.4).
The mean values for the parameterz(Table8.4) differed significantly between
stream communities (ANCOVA:F2,41¼54.82,P<0.001) due to a higherz-value
in the LL. No differences in slopes were found between the SB and MY
BODY SIZE AND SCALE INVARIANCE 159