image: Some animal behaviors are highly determined by abiotic environmental variables; others are influenced relatively little. Model predictions are in color; observations are black circles. The R2 value can be interpreted as the proportion of variability in the data that is explained by abiotic environmental variables. (From "The Mathematics of Animal Behavior: An Interdisciplinary Dialogue", by Shandelle M. Henson and James L. Hayward, Notices of the American Mathematical Society, November 2010.) view more
Credit: James L. Hayward and Shandelle Henson
Providence, RI---It was a mathematician, Joseph Fourier (1768-1830), who coined the term "greenhouse effect". That this term, so commonly used today to describe human effects on the global climate, originated with a mathematician points to the insights that mathematics can offer into environmental problems. Three articles in the November 2010 issue of the Notices of the American Mathematical Society examine ways in which mathematics can contribute to understanding environmental and ecological issues.
"Earthquakes and Weatherquakes: Mathematics and Climate Change", by Martin E. Walter (University of Colorado)
Data about earthquakes indicates that there are thousands of small earthquakes that do no damage, and there are just a few very strong earthquakes that do a great deal of damage. A striking fact emerges from the data: Over a sufficiently long period of time, the sum of the "intensity" of all earthquakes of a given Richter scale magnitude is the same for any point on the Richter scale. So for example the total intensity of the 100,000 magnitude-3 quakes that occur over the course of a year is the same as the intensity of a single magnitude-8 trembler. Put another way, there is no preferred size or scale of earthquakes. This is an empirical fact that can be easily translated into mathematical terms, by noting that the data for earthquakes follows what is known as a power law. The author uses the example of earthquakes to formulate a hypothesis about "weatherquakes"---extreme weather events like hurricanes and tornadoes. As in the case of earthquakes, he suggests, there is no preferred size or scale for the intensity of weatherquakes. That is, weatherquake phenomena also follow a power law. Taking the mathematics a few steps further, the author examines what would happen to the distribution of extreme weather events if the global climate heated up. The finding is worrisome: As temperatures rise, the most intense weatherquakes would increase in number.
"Environmental Problems, Uncertainty, and Mathematical Modeling", by John W. Boland, Jerzy A. Filar, and Phil G. Howlett (all three authors affiliated with the Institute for Sustainable Systems and Technologies at the University of South Australia)
This article examines some special characteristics shared by many models of environmental phenomena: 1) the relevant variables (e.g., levels of persistent contamination in a lake) are not known precisely but evolve over time with some degree of randomness; 2) both the short-term behavior (day-by-day interaction of toxins in the lake) and longer-term behavior (cumulative effects of repeated winter freezes) are important; and 3) the system is subject to outside influences from human behavior, such as industrial pollution and environmental regulations. Concerning the latter characteristic, the article discusses ideas from a branch of mathematics called control theory, which studies how systems are affected when they are strategically influenced from the outside. Interventions for environmental problems can influence ecological systems dramatically but are often neglected in development planning. Control theory offers methods for determining an appropriate level of intervention and for evaluating its effects. One example from the article looks at the use of solar panels to run a desalination plant. A model using ideas from control theory can guide optimal use of the plant in the sense of maximizing the expected volume of fresh water produced.
"The Mathematics of Animal Behavior: An Interdisciplinary Dialogue", by Shandelle M. Henson and James L. Hayward (both authors at Andrews University, Michigan)
The two authors, one an applied mathematician and the other a biologist, teamed up to model aspects of gull behavior in a wildlife preserve in Washington state. The article is structured in an unusual way, as a sort of conversation between the two researchers describing their work together. Before the two began collaborating, the biologist collected reams of data on gull behavior; his biology colleagues teased him, "Don't you know how to sample?" But the applied mathematician was delighted to have such complete data. She and the biologist constructed a model representing a group of gulls as they "loaf". For gulls the term "loafing" refers to a collection of behaviors---such as sleeping, sitting, standing, resting, preening, and defecating---during which the birds are immobile. Loafing is of practical importance because it often conflicts with human interests. The model constructed by Henson and Hayward fit beautifully with the data and also produced predictions about how the number of birds loafing in a given location changed over time. For example, the loafing model correctly predicted that the lowest numbers of gulls would occur at high tide on days corresponding to tidal nodes. This is contrary to previously published assertions, based on data averaging, that the lowest numbers occur near low tide. Their work also showed that it is not always necessary to base models of animal group dynamics on behavior of the individual animals. As Henson puts it, "You wouldn't use quantum models to study the classical dynamics of a falling apple." Similarly, you don't always need to use a collection of individual-based simulations to study the dynamics of a group behavior.
Advance copies of these articles are available to reporters at the following web sites:
http://www.ams.org/staff/jackson/fea-walter.pdf
http://www.ams.org/staff/jackson/fea-boland.pdf
http://www.ams.org/staff/jackson/fea-henson.pdf
On October 12, 2010, they will be publicly posted on the Notices web site, http://www.ams.org/notices.
For specific questions, please contact the authors of the articles. General inquiries may be directed to:
Mike Breen and Annette Emerson
AMS Public Awareness Office
Email: paoffice@ams.org
Telephone: 401-455-4000
Founded in 1888 to further mathematical research and scholarship, the more than 30,000-member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.
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