Providence, RI---It was a mathematician, Joseph Fourier (1768-1830),who coined the term "greenhouse effect". That this term, so commonlyused today to describe human effects on the global climate, originatedwith a mathematician points to the insights that mathematics can offerinto environmental problems. Three articles in the November 2010issue of the Notices of the American Mathematical Society examine waysin which mathematics can contribute to understanding environmental andecological issues.
"Earthquakes and Weatherquakes: Mathematics and Climate Change", byMartin E. Walter (University of Colorado)
Data about earthquakes indicates that there are thousands of smallearthquakes that do no damage, and there are just a few very strongearthquakes that do a great deal of damage. A striking fact emergesfrom the data: Over a sufficiently long period of time, the sum of the"intensity" of all earthquakes of a given Richter scale magnitude isthe same for any point on the Richter scale. So for example the totalintensity of the 100,000 magnitude-3 quakes that occur over the courseof a year is the same as the intensity of a single magnitude-8trembler. Put another way, there is no preferred size or scale ofearthquakes. This is an empirical fact that can be easily translatedinto mathematical terms, by noting that the data for earthquakesfollows what is known as a power law. The author uses the example ofearthquakes to formulate a hypothesis about "weatherquakes"---extremeweather events like hurricanes and tornadoes. As in the case ofearthquakes, he suggests, there is no preferred size or scale for theintensity of weatherquakes. That is, weatherquake phenomena alsofollow a power law. Taking the mathematics a few steps further, theauthor examines what would happen to the distribution of extremeweather events if the global climate heated up. The finding isworrisome: As temperatures rise, the most intense weatherquakes wouldincrease in number.
"Environmental Problems, Uncertainty, and Mathematical Modeling", byJohn W. Boland, Jerzy A. Filar, and Phil G. Howlett (all three authorsaffiliated with the Institute for Sustainable Systems and Technologiesat the University of South Australia)
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.)
(Photo Credit: James L. Hayward and Shandelle Henson)
This article examines some special characteristics shared by manymodels of environmental phenomena: 1) the relevant variables (e.g.,levels of persistent contamination in a lake) are not known preciselybut evolve over time with some degree of randomness; 2) both theshort-term behavior (day-by-day interaction of toxins in the lake) andlonger-term behavior (cumulative effects of repeated winter freezes)are important; and 3) the system is subject to outside influences fromhuman behavior, such as industrial pollution and environmentalregulations. Concerning the latter characteristic, the articlediscusses ideas from a branch of mathematics called control theory,which studies how systems are affected when they are strategicallyinfluenced from the outside. Interventions for environmental problemscan influence ecological systems dramatically but are often neglectedin development planning. Control theory offers methods fordetermining an appropriate level of intervention and for evaluatingits effects. One example from the article looks at the use of solarpanels to run a desalination plant. A model using ideas from controltheory can guide optimal use of the plant in the sense of maximizingthe 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 AndrewsUniversity, Michigan)
The two authors, one an applied mathematician and the other abiologist, teamed up to model aspects of gull behavior in a wildlifepreserve in Washington state. The article is structured in an unusualway, as a sort of conversation between the two researchers describingtheir work together. Before the two began collaborating, thebiologist collected reams of data on gull behavior; his biologycolleagues teased him, "Don't you know how to sample?" But theapplied mathematician was delighted to have such complete data. Sheand the biologist constructed a model representing a group of gulls asthey "loaf". For gulls the term "loafing" refers to a collection ofbehaviors---such as sleeping, sitting, standing, resting, preening,and defecating---during which the birds are immobile. Loafing is ofpractical importance because it often conflicts with human interests.The model constructed by Henson and Hayward fit beautifully with thedata and also produced predictions about how the number of birdsloafing in a given location changed over time. For example, theloafing model correctly predicted that the lowest numbers of gullswould occur at high tide on days corresponding to tidal nodes. Thisis contrary to previously published assertions, based on dataaveraging, that the lowest numbers occur near low tide. Their workalso showed that it is not always necessary to base models of animalgroup dynamics on behavior of the individual animals. As Henson putsit, "You wouldn't use quantum models to study the classical dynamicsof a falling apple." Similarly, you don't always need to use acollection of individual-based simulations to study the dynamics of agroup behavior.
Source: American Mathematical Society