1749Living entities are embedded in and constituted by, networksat any level of organization, from cells to ecosystems (e.g.Ulanowicz, 1986; Pahl-Wostl, 1995; Strogatz, 2001; Barabásiand Oltvai, 2004). The structure and dynamics of thesenetworks emerge as a result of the processes whereby energy,materials and information are acquired, stored, distributed andtransformed. Biological networks typically consist of a largenumber of non-identical elements whose interaction areusually localized, although their effects are not and whoseemergence, maintenance and dynamics represent a challengeto understanding let alone prediction (e.g. Weng et al., 1999;Levin, 1998, 1999, 2002). Biological networks represent themost complex physical system in the universe and yet, as mostcomplex systems they can be described by simple relationships(West, 1999; Brown et al., 2000). These relationships are ofthe formY = βxα, (1)where Y is some response or dependent variable, x representsan independent or explanatory variable, β is a normalizationconstant and α is the scaling exponent. Depending on the valueof the exponent these relationships are called allometric (α≠1)or isometric (α=1). The functional form of the relationship inEquation 1 is also called a power-law relationship, where somequantity can be expressed as some power of another. Power-laws are ubiquitous in physical and social systems where theymost commonly arise as probability or frequency distributions,of the form f(x)=βxα, different from the usual exponential orGaussian distributions. For example power-law distributionsdescribe phenomena such as the frequency of earthquakesof different magnitudes (the Gutenberg-Richter law), thedistribution of income among individuals (Pareto’s law) andthe rank-frequency distribution of words in natural languagesand city sizes (Zipf’s law). Power-laws are well-known tobiologists in the form of bivariate relationships of power-lawtype, called scaling relationships (e.g. Peters, 1983; Niklas,1994; Wiesenfeld, 2001; Brown and West, 2000; Brown et al.,2002; Chave and Levin, 2003) by which molecular,physiological, ecological and life history attributes relate tosome attribute of organisms raised to a power as in Equation1. Although the history of the term scaling in biology probablyhas deep roots in time, its use has been associated withrelationship where the independent variable is the size of anorganism (Calder, 1983, 1984; Peters, 1983; Schmidt-Nielsen,1984). For the sake of consistency we will retain the use ofscaling as related to relationships involving body size and willThe Journal of Experimental Biology 208, 1749-1769Published by The Company of Biologists 2005doi:10.1242/jeb.01588Scaling relationships (where body size features as theindependent variable) and power-law distributions arecommonly reported in ecological systems. In this reviewwe analyze scaling relationships related to energyacquisition and transformation and power-laws related tofluctuations in numbers. Our aim is to show howindividual level attributes can help to explain and predictpatterns at the level of populations that can propagateat upper levels of organization. We review similarrelationships also appearing in the analysis of aquaticecosystems (i.e. the biomass spectra) in the context ofecological invariant relationships (i.e. independent of size)such as the ‘energetic equivalence rule’ and the ‘linearbiomass hypothesis’. We also discuss some power-lawdistributions emerging in the analysis of numbers andfluctuations in ecological attributes as they point toregularities that are yet to be integrated with traditionalscaling relationships and which we foresee as an excitingarea of future research.Key words: scaling, power-law, metabolism, complexity.SummaryIntroductionReviewScaling and power-laws in ecological systemsPablo A. Marquet1,2,*, Renato A. Quiñones3, Sebastian Abades1, Fabio Labra1, Marcelo Tognelli1,Matias Arim1and Marcelo Rivadeneira11Center for Advanced Studies in Ecology and Biodiversity (CASEB) and Departamento de Ecología, Facultad deCiencias Biológicas, Pontificia Universidad Católica de Chile, casilla 114-D, Santiago, Chile,2Santa Fe Institute,1399 Hyde Park Road, Santa Fe, NM 87501, USA and 3Centro de Investigación Oceanográfica en el Pacífico Sur-Oriental (COPAS) and departamento de Ocenografía, Universidad de Concepción, Casilla 160-C, Concepción, Chile*Author for correspondence (e-mail: [email protected])Accepted 14 March 2005THE JOURNAL OF EXPERIMENTAL BIOLOGY1750differentiate them from power-law distributions as definedabove.This special issue is devoted to explore the consequences oforganismal size as affecting biological processes. Most of thepapers in this special issue have addressed scaling relationshipswhere the response variable is an individual level attribute,such as metabolic rate, life span and running speed, and wherethe independent variable is body size. However, as pointed outabove, scaling relationships are common at higher levels oforganizations as well, such as at the level of populations,communities and ecosystems, and they are usually anallometric function of body size (Peters, 1983; Calder, 1984;Schmidt-Nielsen, 1984; Bonner, 1988; Brown, 1995). This factunderscores the importance of body size at all levels oforganization, and opens the way for synthesis and integrationacross levels. In fact, it has been a pressing challenge forecologists and evolutionary biologists to develop a conceptualand quantitative framework bringing together disciplinestraditionally viewed as distinct, such as physiology, ecology,biogeography and macroevolution (e.g. Brown and Maurer,1987, 1989; Ricklefs, 1987; Brown, 1995, 1999; Marquet andTaper, 1998) and much of this quest for a synthetic frameworkhas been based on empirical statistical patterns relating bodysize with physiological, ecological and evolutionary traits (e.g.Lawton, 1990; Blackburn et al., 1993a; Brown et al., 1993;Brown, 1995). In what follows we review some of thisrelationships as they emerge at the population, community andecosystem levels, and emphasize their connections as well asfuture developments.Our main focus in this review paper will be scalingrelationships were body size features as the independentvariable, however we will restrict ourselves to scalingrelationships related to energy acquisition and transformationprimarily at the level of populations. Our aim will be to showhow individual level attributes