quantitative features that are intriguingly similar. These properties can be
conveniently grouped under the heading of scale invariance. By scale invariance
is meant a hierarchical organization that often results in power-law behaviour
over a wide range of values of some control parameters such as species size
(Stanleyet al., 2000). These empirical scaling relations often exhibit power-law
behaviour and are essential to describe and understand the distribution of
species populations and, consequently, the emergent properties of commun-
ities. Particularly, non-integer power laws have become an intensively studied
area covering all fields of the biological sciences with large overlap into the field
of statistical physics. The latent causes of power-law behaviour often relate to
fractal structures or the recursive dynamics of systems (Gisiger, 2001 ).
The recognition of fractal properties as a framework from which non-integer
power laws derive, and their implications for other ecological characteristics,
have intensified in ecology (Margalef, 1996 , 1997; Kunin, 1998 ; Harte, Kinzig &
Green, 1999 ; Schmid, 2000 ). If the same scaling law applies to an object or
measure at any arbitrary positions, we can define a homogeneous fractal (statis-
tically self-similar), while different scaling laws found at different positions of
the object yield inhomogeneous, statistically self-affine fractals (Schmid, 2000 ).
Fractal geometry serves as a mathematical framework that can quantify the
relationships between the many physical and biological phenomena in ecolog-
ical systems, including patterns of habitat, behavioural patterns of individuals
and size-frequency spectra (Schmid, 2000 ; Schmid, Tokeshi & Schmid-Araya,
2002 ).
More intensively studied ecological patterns, which can display scale-invariance
and fractal properties, include species-abundance distributions (SADs) and
species-area relationships (SARs). SADs of taxonomic assemblages have been
fitted to statistical, niche-assembly and dispersal-assembly models (e.g. Tokeshi,
1999 ; Hubbell, 2001 ), while SARs have been described by many statistical
models, with no consensus whether individual mechanisms contribute to the
generation of different types of SARs observed in nature (e.g. Rosenzweig, 1995 ;
Lennon, Kunin & Hartley, 2002 ). The shapes of species-area curves have been
discussed extensively and are most commonly fitted to either a power law or a
double-logarithmic transformed data set (Tokeshi, 1999 ). The power function is
given asS¼cAz, whereSis the species richness within areaA,cis a constant
dependent on spatial scale andzis the scale-free regression exponent. The log-
log transformation that follows a power-law relationship may imply statistical
self-similarity, and fractal models have been proposed to predict SARs for plants
at the level of assemblages (Harteet al.,1999) and species (Lennonet al., 2002).
However, single fractal dimensions (monofractals) for size-related distribu-
tions or SARs do not provide comprehensive quantitative information on the
species and the inherent population-size structure of communities (Schmid,
Tokeshi & Schmid-Araya, 2002 ; Borda-de-A ́gua, Hubbell & McAllister, 2002 ).
BODY SIZE AND SCALE INVARIANCE 141