Mathematics and the environment
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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.
