The fractal dimension calculated from a power-law fit of SAR data and the fractal
dimension(s) of species that make up that distribution are different quantities.
Species and their population-density distributions are measures that are allo-
cated over a size range (e.g. body size, area) that are in turn embedded in a
surface or volume of habitat. This habitat structure may have a spatial distribu-
tion, which itself displays fractal characteristics (Schmid, 2000 ; Schmidet al.,
2002 ), potentially influencing the spatial distribution of populations in a scale-
invariant manner.
The particle-size distribution (PSD) is one of the fundamental properties
characterizing benthic habitats. Most functions that fit the PSD are based on a
power-law dependence of particle number or mass on particle diameter
(Turcotte, 1986 ). Such power-law dependence has been interpreted as the result
of a fractal distribution often characterized by several scaling domains (Bittelli,
Campbell & Flury, 1999 ). However, because BSDs (body-size distributions), PSDs
and SARs, are measures rather than geometric objects, the measure itself may
display self-similarity.
To understand this concept, we can envisage benthic invertebrates occurring
in two similar-sized areas of a streambed with different species richness and
different densities. We can then equally subdivide the areas and find that the
number of species and their densities are again different. Consequently, this
process reflects an ‘irregular pattern’. These subdivisions could be carried
through to small interstitial pore sizes and more species and individuals will
still be found in one site rather than another. This ‘scaling-down’ process
reflects the distribution and patchiness both of benthic organisms and particles,
and is an example of a measure that is irregular at all scales. When the irregu-
larity is statistically the same at all scales, then the measure is a multifractal.
This phenomenon was first described by Mandelbrot ( 1974 ) for turbulences,
later elaborated as a mathematical tool in an ecological context by several
studies. Empirical applications have been the spatial variability in bioactive
marine sediments (Kroppet al., 1994), chemical soil properties (Kravchenko,
Boast & Bulloc, 1999 ), soil PSDs (Posadaset al., 2001), river networks, stream
order and water discharge (Rodriguez-Iturbe & Rinaldo, 2001 ), canopy structure
(Drake & Weishampel, 2000 ), self-organization and species abundance in a
tropical forest (Manrubia & Sole ́, 1996; Borda-de-A ́guaet al., 2002), zooplankton
biomass distribution (Pascual, Ascioti & Caswell, 1995 ) and copepod behaviour
(Schmitt & Seuront, 2001 ).
Multifractals are then a refinement and a generalization of the fractal proper-
ties that arise naturally from statistically self-similar distributions. Multifractals
describe a local singularity in behaviour of those measures or functions that
display high irregularities. Many physical quantities exhibit a non-trivial scaling
behaviour where multiplicative iterative random processes generate multi-
fractal structures, while additive processes generally produce single fractals.142 P.E. SCHMID AND J. M. SCHMID-ARAYA