Growth and Form in Nature: Power Laws and Fractals

Growth and Form in Nature: Power Laws and Fractals


There are several instances of power laws found in nature and in society.  Some of the well known ones are:

  • City Sizes (Zipf’s Law)
  • Firm Sizes
  • Stock Market Movements
  • Income and Wealth (Pareto’s Law)
  • Metabolic Rate and Body Mass (Kleiber’s Law-3/4 or Rubner’s Law-2/3)


Power laws and Scaling in Biology

After 1997 paper by West et all, many publications have analyzed  empirical evidence as to what the correct exponent is and what is the fundamental theoretical basis for power law.

West found 3/4 as exponent, others have reported 1/4, 2/3, 4/5 etc.

Animals and Mammals follow 3/4 exponent.  Plants follow 2/3.

The Metabolic Theory of Ecology

Scaling in biology has a rich and important history. Typically body mass, or some other parameter relating to organism size, is related to anatomical, physiological, and ecological parameters across species. Quite remarkably, diverse organisms, from tiny microbes to the earth’s largest organisms are found to fall along a common slope, with a high degree of variance explained. The beauty of such scaling ‘‘laws’’ has been the generality in biotic organization that they suggest, and the challenge (for ecologists) has often been interpreting their mechanistic bases and ecological consequences.

Scaling laws have thus far inspired scientists in at least three major areas. First, scaling laws may illuminate biology that is otherwise shrouded. For example, if scaling relationships can account for variation in a parameter of interest, the residual variation may be much more easily examined because the major influence of some trait, say, body size, is removed. Second, some scientists have taken an interest in ‘‘the exponent’’—essentially the exponential scaling values that produce the allometric relationship. What are the precise values of these exponents? Are they all from a family of particular values (quarter powers) for many different biological relationships? This area seeks to define the generality of patterns in nature and to explore the empirical robustness of the relationships. Third, from a mechanistic perspective, if scaling laws are mechanistic and truly general, then this suggests some underlying common biological process that forms the structure and function of species and ultimately generates biological diversity. The mechanistics of scaling from metabolism and the currently favored fractal network model of resource acquisition and allocation may allow scientists to understand the laws of how life diversified and is constrained. Perhaps more importantly, such a mechanistic understanding should allow the successful prediction of evolutionary trends, responses of organisms to global change, and other basic and applied biological problems.

The Ecological Society of America’s MacArthur Award winner, James H. Brown, working together with colleagues for over a decade on scaling in biology, has arrived at an outline for a metabolic theory of ecology—a proposal for a unifying theory employing one of the most fundamental aspects of biology, metabolism. This metabolic theory incorporates body size, temperature (metabolic kinetics described by the Boltzmann factor), and resource ratios of the essential elements of life (stoichiometry). Indeed, this bold and visionary proposal is likely to inspire ecologists and provoke much discussion. My goal in assembling this Forum was to work toward a balanced discussion of the power and logic of the metabolic theory of ecology. I have asked both junior and senior scientists to evaluate the ideas presented in the metabolic theory and to go beyond the listing of strong and weak points. As such, this collection of commentaries should be viewed neither as a celebration of the theory nor as a roast of Jim Brown. It should, however, serve as a springboard for future research and refinements of the metabolic theory.

Several themes and axes of admiration and agitation emerge from the forum. The focus on metabolism, and metabolic rate in particular, is an advance that most agree is the fundamental basis for the processes of acquisition of resources from the environment and, ultimately, survival and reproduction of organisms. The combination of size, temperature, and nutrients has compelling predictive power in explaining life-history traits, population parameters, and even broader-scale ecosystem processes. The key point here is that Brown et al. are making a direct link between factors that affect the functioning of individuals and the complex role that those individuals play in communities and ecosystems. Although what we have before us is a proposal for a unified theory of ‘‘biological processing of energy and materials’’ in ecosystems, Brown et al. embrace the unexplained variation and acknowledge other areas of ecology that may not be subject to metabolic laws.

The commentaries presented in this Forum are unanimous in their admiration of Brown et al.’s broad theoretical proposal and its clear predictions. Yet, points of discussion abound and range widely: What really is the correct exponent? Does the scale at which scaling is applied affect its explanatory power? Are the laws really based on mechanism or phenomena? How does the addition of temperature and resource limitation enhance the power of scaling relationships? And, is scaling up from the metabolic rate and body mass of organisms to population dynamics, community structure, and ecosystem processes possible? This Forum ends with Brown’s response to the commentaries. Although there will be continued debate over the correct exponent, the data at hand from the broadest taxonomic groups support quarter powers. There is general agreement over the issue of scale and the fact that, depending on the scale of interest, metabolic theory may have more or less to offer. Finally, nutrient stoichiometry is the most recent addition to metabolic theory, and all agree that further research and refinement will determine the role for such nutrient ratios in the ecological scaling. The benefits of a metabolic theory of ecology are clear. The authors of this Forum have outlined some of the future challenges, and tomorrow’s questions will evaluate these theses.


Metabolism provides a basis for using first principles of physics, chemistry, and biology to link the biology of individual organisms to the ecology of populations, communities, and ecosystems. Metabolic rate, the rate at which organisms take up, transform, and expend energy and materials, is the most fundamental biological rate. We have developed a quantitative theory for how metabolic rate varies with body size and temperature. Metabolic theory predicts how metabolic rate, by setting the rates of resource uptake from the environment and resource allocation to survival, growth, and reproduction, controls ecological processes at all levels of organization from individuals to the biosphere. Examples include:

(1) life history attributes, including development rate, mortality rate, age at maturity, life span, and population growth rate;

(2) population interactions, including carrying capacity, rates of competition and predation, and patterns of species diversity;

(3) ecosystem processes, including rates of biomass production and respiration and patterns of trophic dynamics.

Data compiled from the ecological literature strongly support the theoretical predictions. Eventually, metabolic theory may provide a conceptual foundation for much of ecology, just as genetic theory provides a foundation for much of evolutionary biology.



Key Terms

  • Power Laws
  • Multi-scale
  • Fractals
  • Allometric Scaling Laws
  • Kleiber Law
  • Metabolic Ecology
  • Zipf Distribution
  • allometry
  • biogeochemical cycles
  • body size
  • development
  • ecological interactions
  • ecological theory
  • metabolism
  • population growth
  • production
  • stoichiometry
  • temperature
  • trophic dynamics




Key Sources of Research:


The Origin of Universal Scaling Laws in Biology

Geoffrey B. West



Life’s Universal Scaling Laws

Geoffrey B. West and James H. Brown’sUniversalScalingLaws.pdf



A General Model for the Origin of Allometric Scaling Laws in Biology

Geoffrey B. West, James H. Brown, Brian J. Enquist



Power Laws in Economics: An Introduction

Xavier Gabaix




The origin of allometric scaling laws in biology from genomes to ecosystems: towards a quantitative unifying theory of biological structure and organization

Geoffrey B. West, James H. Brown




A general model for ontogenetic growth

Geoffrey B. West, James H. Brown & Brian J. Enquist




Plants on a different scale

Lars O. Hedin




The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms

Geoffrey B. West, James H. Brown, Brian J. Enquist










Complexity and Transdisciplinarity; Science for the 21st Century(?)!





Scaling Laws in Complex Systems




A General Model for the Origin of Allometric Scaling Laws in Biology

Geoffrey B. West, James H. Brown,* Brian J. Enquist




Effects of Size and Temperature on Metabolic Rate

James F. Gillooly,1* James H. Brown,1,2 Geoffrey B. West,2,3 Van M. Savage,2,3 Eric L. Charnov




Growth, innovation, scaling, and the pace of life in cities

Luís M. A. Bettencourt, Jose ́ Lobo, Dirk Helbing, Christian Kuhnert, and Geoffrey B. West




Urban Scaling and Its Deviations: Revealing the Structure of Wealth, Innovation and Crime across Cities

Lu ́ıs M. A. Bettencourt1,2*, Jose ́ Lobo3, Deborah Strumsky4, Geoffrey B. West1,2





Francisco J. Martínez




Allometric Scaling Laws and the Derivation of the Scaling Exponent

Marcel Grunert



Cities, Markets, and Growth: The Emergence of Zipf’s Law

Jeremiah Dittmar

August 10, 2011



Self-similarity and power laws



The fractal nature of nature: power laws, ecological complexity and biodiversity

James H. Brown1,2*, Vijay K. Gupta3, Bai-Lian Li1, Bruce T. Milne1, Carla Restrepo1 and Geoffrey B. West



Metabolic Rate and Kleiber’s Law



Patterns in Nature



Zipf, Power-laws, and Pareto – a ranking tutorial

Lada A. Adamic



The Power of Power Laws



Re-examination of the 3/4-law of Metabolism




Fifth dimension of life and the 4/5 allometric scaling law for human brain

Ji-Huan He, Juan Zhang



Lack of Evidence for 3/4 Scaling of Metabolism in Terrestrial Plants

Hai-Tao LI1*, Xing-Guo HAN2 and Jian-Guo WU



Is West, Brown and Enquist’s model of allometric scaling mathematically correct and biologically relevant?




Evidence against universal metabolic allometry.

Folmer Bokma



An evaluation of two controversial metabolic theories of ecology




Karl J. Niklas1, and Sean T. Hammond



􏱂􏱅Network Allometry



A critical understanding of the fractal model of metabolic scaling

José Guilherme Chaui-Berlinck




Allometric scaling of metabolic rate from molecules and mitochondria to cells and mammals

Geoffrey B. West*†‡, William H. Woodruff*§, and James H. Brown

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Author: Mayank Chaturvedi

You can contact me using this email mchatur at the rate of AOL.COM. My professional profile is on

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