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First published online April 26, 2005
Journal of Experimental Biology 208, 1749-1769 (2005)
Published by The Company of Biologists 2005
doi: 10.1242/jeb.01588
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Scaling and power-laws in ecological systems

Pablo A. Marquet1,2,*, Renato A. Quiñones3, Sebastian Abades1, Fabio Labra1, Marcelo Tognelli1, Matias Arim1 and Marcelo Rivadeneira1

1 Center for Advanced Studies in Ecology and Biodiversity (CASEB) and Departamento de Ecología, Facultad de Ciencias Biológicas, Pontificia Universidad Católica de Chile, casilla 114-D, Santiago, Chile
2 Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
3 Centro de Investigación Oceanográfica en el Pacífico Sur-Oriental (COPAS) and departamento de Ocenografía, Universidad de Concepción, Casilla 160-C, Concepción, Chile



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  		Fig. 1. Population density scaling for intertidal invertebrates found in a protected (inside) and unprotected (outside) site in central Chile. The presence (outside) or absence (inside) of human exploitation is associated with strong changes in community composition and dominance as shown below each graph (after Durán and Castilla, 1989Go and Marquet et al., 1990Go). Density (individuals m-2); body mass (Mb) (g).

 


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  		Fig. 2. Population density scaling for primary (open symbols) and secondary (filled symbol) consumer species of mammals. The slope of the relationships is -0.73 (not different from -3/4) for primary consumers and -0.99 (not different from -1) for secondary consumers (data from Damuth, 1993). Density (individuals km-2); Mb (kg).

 


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  		Fig. 3. Relationship landmass area and maximum (filled symbol) and minimum (open symbol) body sizes of mammalian (A, after Marquet and Taper 1998Go; land mass area, km2; body size, g) and snake (B, data from Boback and Guyer 2003Go; land mass area, km2; body size, cm) species inhabiting them.

 


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  		Fig. 4. Power-law relationship relating number of species and body size in South American mammals.

 


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  		Fig. 5. Constant population energy use in plant species (after Enquist et al., 1998Go). Population energy use (xylem flux in l m-2 d-1); Mb (g).

 


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  		Fig. 6. Example of a unnormalized biomass size spectrum from Sprules et al. (1991Go) and compiled from their sampling of Lake Michigan, showing the component trophic groups. (After Thiebaux and Dickie, 1993Go.) Biomass (g m-2); Mb (g).

 


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  		Fig. 7. Normalized biomass size-spectra in carbon units from several stations in the New England Seamounts Area (Northwest Atlantic). Size range: 1.6x10-9 to 1.33x103 µg C individual-1 (from bacteria to meso-zooplankton). Depth range: 0-400 m. (After Quiñones et al., 2003Go.)

 


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  		Fig. 8. Schematic representation of the equivalent body-size spectra of: (A) specific production, expressed as log P/B by trophic positional grouping. (B) Biomass density of a given enviroment. (After Boudreau et al., 1991Go.) Biomass (Mb) is expressed as energy equivalents of body mass.

 


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  		Fig. 9. Scaling of population growth rates in the BBS data set. (A) Distribution of population growth rates s{equiv}log[N(t+1)/N(t)] across all species in the data set. The growth rate s is calculated by log transforming the ratio of species abundances in successive years. Abundances are taken as the total number of individuals of a particular species counted within each survey route. (After Keitt and Stanley, 1998Go.) (B) Probability density p(s|N) of the growth rate s for all bird species in the BBS database for different initial population size classes. The distribution represents all annual growth rates observed in the 31 year period 1966-1996. Data are shown for three different bins of initial sizes (circles, 100<N(t)<101; squares, 101<N(t)<102.4; diamonds, 102.4<N(t)<,103.8). The solid lines are exponential fits to the empirical data close to the peak. (C) Scaled probability density pscal{equiv}{sigma}p(s|N) as a function of the scaled growth rate s scal{equiv}[s-<s>]/{sigma} for all species and years in the survey. The values were re-scaled using the measured values of <s> and {sigma}. Notice that all the data collapse upon the universal curve pscal {equiv}(-|r scal|). (After Keitt et al., 2002Go.)

 

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© The Company of Biologists Ltd 2005