Preface
The cow is back. This time she is cylindrical, not spherical. Still no
legs,
udders, or head, but the torso is a more
realistic shape.
Following on the hooves of Consider a
Spherical Cow (Harte, J. 1988.
Consider a spherical cow: a
course in environmental problem solving.
Sausalito, CA: University Science Books)
called COW-1 herein, COW-2
will teach you additional modeling skills of
use in environmental
science. Some of these skills are at roughly
the same mathematical level
as Chapters II and III of COW-1,
whereas others are more advanced.
Although the emphasis here, as in COW-1,
is on analytic approaches to
squeezing information from models, I also
include spread-sheet
methods for simulating the behavior of models
constructed from
differential equations. You will get the most
out of COW-2 if you have
worked through COW-1, but I have tried
to make COW-2 self-
contained.
I have assumed that you, the reader, are
acquainted with both
differential and integral calculus, though
the relationship need not be
lustful. Indeed, only a passionate dislike
(or deep and unyielding fear)
of mathematics will disqualify you from
proceeding further. Some past
exposure to matrices will also help; for
those who need to do some
remedial reading on calculus or matrix
algebra, I highly recommend
Clifford Swartz's excellent Used Math
(Swartz, C. E. 1993. Used math: for
the first two years of
college science. College Park, MD: AAPT Press).
I have also assumed that you are ready to
seek out deeper insights
into ways of modeling the complexities of
nature, but the underlying
goal is the same as in COW-1: to teach
ways of stripping away
inessential detail and capturing with
mathematics the essentials of a
complex system.
Why do I want you to learn to use
mathematics? Can't we just talk
about environmental problems? There is
another reason in addition to
the often-expressed, and correct, argument
that mathematics seems to
be the "language that nature
speaks" and therefore facilitates the
understanding of nature. Mathematics is a
kind of global language that
not only needs no translation from nation to
nation but also bridges the
disciplines. Economists and scientists often
confuse each other (and
sometimes themselves) using everyday language
to describe how
things
depend upon other things; the reason is that in the two
academic traditions the same word may be used in different senses.
(Consider, for example, the terms equity,
parity, derivative, and stock.)
Mathematics, however, has a way of cutting through such confusion.
Just as the "Tools of the Trade"
chapter of COW-1 was structured
around the topics of steady-state box models,
thermodynamics,
chemical equilibrium theory, and
non-steady-state box models, COW-2
is structured around the central themes of
probability, optimization,
scaling, differential equations, stability,
and feedback. Why these
themes? Consider such problems as (a)
determining how some shift in
land-use practice (e.g., conversion of
tropical forest to grazing land)
might result in species extinction or climate
alteration in the region, or
(b) determining whether exposure to a trace
substance released from a
factory poses a serious health threat to
people living nearby. In
problems such as these, themes that affect
our understanding and
analysis of the problem can be extracted as
follows:
1. Much of our information about these situations comes to us in the
form of data derived from some sampling
scheme. We need to know
something about probability and statistics to
assess how representative
the data are.
2. We may discover that the activity leading
to the environmental
damage does bring some benefit to society,
but that if the activity is too
intense, then society is the loser. We need
to know something about
optimization to estimate where the balance
point between societal
benefits and societal costs lies.
3. We may be able to determine something
about the effects of a
toxic substance on adults but not on
children, so we need to know
something about how to scale our knowledge
from big people to little
people. Or, to estimate species loss, we may
be able to use existing data
on the rate of local species extinctions from
deforestation, provided, we
can "scale up" to larger regions.
4. We will want to predict not only the
eventual long-term-averaged
climate conditions that will result from the
loss of forest but also the
way in which climatic conditions will change
over time in the shorter
term as they approach that eventual state.
The mathematical tools
needed to predict that climate trajectory
over time are differential
equations. We need to know how to set up the
appropriate differential
equation for the problem at hand and how to
find approximate
solutions to that equation.
5. As we alter a complex system, for example
by cutting down trees,
thresholds of instability may be crossed.
Before the threshold is
reached, the system can assimilate the stress
and maintain some sort of
equilibrium, but if the stress is too great,
then the system may undergo
a dramatic response. We need to know how to estimate whether such a
threshold exists and where it lies.
6.
Climate changes induced by altered land-use practices can further
alter ecosystems and degrade ecosystem
services of benefit to
society. These feedbacks may further alter
human behavior as people
seek to compensate for the lost ecosystem
services. We need to be able
to estimate the magnitude and consequences of
such feedback effects.
The pervasiveness of these themes is well known to environmental
scientists in fields such as climatology,
ecology, hydrology, toxicology,
and atmospheric chemistry. Despite this
understanding among
scientists, public policy debates often
neglect such critical linkages and
issues. The public is easily misled by
twisted probabilistic reasoning.
Misuse of optimization methods can lead to
suboptimization (doing
better and better something that would not
have been done at all if the
boundary of analysis had been enlarged). Analytic methods, insights,
institutions, adaptations, and solutions
appropriate at one scale (e.g.,
experimental plots in ecology, individual
firms or power plants in
industry, a grid square in global climate
modeling) are often naively
and mistakenly extrapolated to other scales.
Incorrect conclusions
about the distribution and fate of emitted
substances usually can be
spotted and corrected with the aid of
approximation methods for
studying solutions to differential equations.
Instabilities and feedbacks,
both between and within human and biophysical
systems, are often
ignored in political, social, and even
scientific analysis, leading society
to be inadequately prepared for the possible
abruptness and intensity
of resulting changes.
At the beginning of each of the five thematic
sections in this book is
a general treatment of the relevant
mathematical concepts. The core of
the book is 25 fully worked-out problems. As
in COW-1, each problem
statement is posed more like a research
question than a standard
homework exercise. Homework exercises follow
each worked-out
problem solution and range from relatively
straightforward exercises
that probe readers' understanding of the
concepts and methods to
more difficult and open-ended research
suggestions. The Appendix
summarizes many useful mathematical notations
and formulae you are
likely
to encounter.