PCL0.9). Predictedm-values close to 1 would imply that local communities are
continuously reassembled by metacommunities similar to results shown for pela-
gic copepod assemblages (Hubbell, 2001 ). However, the highly insignificant fits of
the ZSM to the observed stream community data demonstrate that the ZSM does
not predict patterns in benthic systems dominated by environmental and species-
mediated resource heterogeneity. Differences between species in response to
resource supply and discharge variations refutes the central assumption of
Hubbell’s UTB that species are demographically similar (Schmid, Schmid-
Araya & Tokeshi, unpublished; Dornelas, Connolly & Hughes, 2006 ). Thus, at
the community level aspects of resource apportionment and their interspecific
variations play a much more prominent role in streams than the dispersal-
mediated processes predicted by the ZSM. It remains to be tested if and under
what circumstances, particular taxonomic assemblages are more likely to fol-
low neutral rather than niche-apportionment assumptions in benthic systems.
Evidence to date supports the notion that species-rich and abundant assem-
blages, such as larval chironomids, assemble and reassemble stochastically in
stream environments, significantly fitting to random niche-assortment and
apportionment models (Tokeshi, 1993 ; Schmid, 1997 ; Fesl, 2002 ).Fractal properties of size-structured communities
Power-law scaling can help to reveal if fractal characteristics govern size- and
scale-related distribution patterns. Fractals are particularly useful models to help
to understand the multiscale nature of natural phenomena. The fractal dimen-
sion,D, is the basic parameter for describing structures that display self-similarity
or scale invariance over a wide range of resolutions. Scale invariance can be the
result of iterative diffusion processes of mass in the habitat, which repeat them-
selves at different scales of observation (Johnson, Tempelman & Patil, 1995 ).
A method often applied to estimate self-similarity is the capacity dimension,
introduced by Kolmogorov ( 1959 ). Thus, this fractal dimension (Dk) defines a
partition of the fractal into equally sized lattice cells with edge sizel as:Dk¼liml! 0 logNðlÞ
logð 1 =lÞ( 8 : 3 )whereN(l) denotes the minimum number of cells required for covering the
mathematical set. In nature, however, the limit cannot be obtained and, instead,
a scaling region of a specified size rangel must be used as:NðlÞ/clD l! 0 ( 8 : 4 )
whereN(l)arethenumberofcellsofsizel needed to enclose the object,cis a
proportionality constant ofd-dimensional length ofl,andDis the Hausdorff-
Besicovitch dimension. Random (natural) fractals define these real distributions,
which have natural upper- and lower-scale limits, and separate them from infinite,152 P.E. SCHMID AND J. M. SCHMID-ARAYA