Consider a Cylindrical Cow

More Adventures in Environmental Problem Solving

John Harte
University of California, Berkeley


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-


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.