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
Click to access Life’sUniversalScalingLaws.pdf
A General Model for the Origin of Allometric Scaling Laws in Biology
Geoffrey B. West, James H. Brown, Brian J. Enquist
Click to access West_Brown_Enquist_1997.pdf
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
Click to access OntogeneticGrowth.pdf
Plants on a different scale
Lars O. Hedin
Click to access nature_news_views_06.pdf
The Fourth Dimension of Life: Fractal Geometry and Allometric Scaling of Organisms
Geoffrey B. West, James H. Brown, Brian J. Enquist
Click to access S1999_West.pdf
TOWARD A METABOLIC THEORY OF ECOLOGY
JAMES H. BROWN,
with JAMES F. GILLOOL Y, ANDREW P. ALLEN, VAN M. SA V AGE, AND GEOFFREY B. WEST
Click to access Brown_JH_MA.pdf
Complexity and Transdisciplinarity; Science for the 21st Century(?)!
GEOFFREY WEST
Click to access Geoffrey%20West.pdf
Scaling Laws in Complex Systems
Click to access ma_scaling_laws.pdf
A General Model for the Origin of Allometric Scaling Laws in Biology
Geoffrey B. West, James H. Brown,* Brian J. Enquist
Click to access Science-1997-West.pdf
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
https://dspace.unm.edu/bitstream/handle/1928/1656/science2001.pdf?sequence=2&isAllowed=y
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
Click to access pone.0013541.pdf
URBAN DYNAMIC LAWS AND OUR DEGREES OF FREEDOM FOR DEVELOPMENT
Francisco J. Martínez
Click to access Francisco-Martinez_Urban-dynamic-laws.pdf
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
Click to access Zipf_Dittmar.pdf
Self-similarity and power laws
Click to access komulainen.pdf
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
Click to access Fractal-Nature.pdf
Metabolic Rate and Kleiber’s Law
https://universe-review.ca/R10-35-metabolic.htm
Patterns in Nature
http://www.patternsinnature.org/Book/PowerLaws.html
Zipf, Power-laws, and Pareto – a ranking tutorial
Lada A. Adamic
http://www.labs.hp.com/research/idl/papers/ranking/ranking.html
The Power of Power Laws
http://www.the-scientist.com/?articles.view/articleNo/14689/title/The-Power-of-Power-Laws/
Re-examination of the 3/4-law of Metabolism
P. S. DODDS, D. H. ROTHMAN- AND J. S. WEITZ
Click to access Dodds%20et%20al%202001.pdf
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
Click to access W020090624623546294020.pdf
Is West, Brown and Enquist’s model of allometric scaling mathematically correct and biologically relevant?
J. KOZLOWSKI and M. KONARZEWSK
Evidence against universal metabolic allometry.
Folmer Bokma
Click to access bokma2003u.pdf
An evaluation of two controversial metabolic theories of ecology
ASSESSING SCALING RELATIONSHIPS: USES, ABUSES, AND ALTERNATIVES
Karl J. Niklas1, and Sean T. Hammond
http://www.journals.uchicago.edu/doi/pdfplus/10.1086/677238
Network Allometry
Click to access network%20allometry.pdf
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
Viewpoints which Matter 