In brief:
Dr. Rolfe Leary and I are reviewing applications of
differential, difference, and related equations (integro-
differential, differential-difference, integral, partial
differential) for modeling population dynamics of ecosystems,
especially forest stands. If you have written on this topic or
can inform us of interesting results obtained by others, we would
greatly appreciate receiving a reprint or complete citation from
you.
This review is the first step of a larger project, writing a
text for undergraduate ecologists and foresters. We believe that
ecologists, rather than mathematicians, are better equipped to
convey the beauty and relevance of mathematics to our students.
We hope that our text will be more inspiring than those written
by professional mathematicians. Yet, it will not be just a
cookbook spiced with ecological applications. Our approach to
this project is outlined below.
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We, ecologists, differ greatly in our attitude toward
mathematics. (By mathematics I mean everything above high school
arithmetic, algebra, and cookbooked defensive statistics.) Many
ecologists hate all this differentiation and integration, maybe
because they fail to understand it. For the same reason, others,
like Darwin, hold math in awe. Some of us even dabble with
differential equations, again usually without knowing much about
what we are doing.
Two things unite us. First, many ecologists and practically
all foresters took calculus and were introduced to the elements
of differential equations as college students. Second, very few
of us make use of this, the most powerful tool of science our in
research or practice.
A stupendous experiment has been going on for many years: an
attempt to instill mathematical knowledge in students of biology
and its applied branches such as forestry. This experiment has
been conducted with an awful number of replications and variants
in hundreds of institutions of higher education throughout the
world. It makes sense to pause and look at the results of this
experiment. No statistical analysis is required to see that math
education of ecologists has been a failure.
This problem is not limited to the United States. I received
all my degrees in Moscow, Russia, and can confirm that, although
mathematical training there was more vigorous and extensive, it
was also a complete waste. I retained nothing about calculus from
college. If I know anything about it, it is because later on I
learned calculus anew on my own. My interest was sparked by a
very simple calculus text for votech schools which I happened
across during a boring vacation. While reading and even enjoying
this elementary book, I was amazed that I had forgotten even
introductory concepts. The problem was not with my memory. More
than just passive forgetting was involved. It was as if the brain
actively, though subconsciously, had expunged all those dry
abstract concepts as antithetical to our ecological vocation.
Yet, these concepts, after surmounting the barrier of
initial rejection, appear as the most beautiful and relevant
prerequisites of thinking in any science, ecology including.
Why have mathematicians failed to teach us?
There is a genuine lack of understanding between ecologists
and mathematicians. It appears not only in teaching but in
research as well. A biologist's way of thinking is different from
that of a mathematician and, despite numerous attempts, examples
of productive cooperation between biologists and mathematicians
are as rare as the number of ecologists who mastered math as a
university student. Those who create math and regard it as an end
in itself are not necessarily the best at using it as a means to
an end.
Analysis of this problem deserves a special inquiry, rather
than offhand remarks, because it may help to find better ways of
communication. Basically, there are two explanations. One is that
in principle biology, unlike physics, cannot be explained in
mathematical terms. These areas of knowledge belong to two
inimical and mutually incomprehensible worlds. According to this
explanation, when mathematicians venture to say something about
biology, they are always wrong or, at best, irrelevant. Trofim
Lysenko was the best known champion of this view in the fifties
and Ernst Mayr is at present.
Mayr, an outstanding biologist who upon retirement turned to
the history and philosophy of science, criticizes math in the
course of his struggle with what he calls typological thinking.
Instead, he advocates population thinking that "is a peculiarly
biological concept, alien to the thinking of the physical
scientist" (Mayr 1982, p. 487). Biology and historical sciences,
he believes, deal with systems too complex to be expressed in
mathematical formulas. Only misconceptions" can be produced as a
result of "ill-advised application of mathematics" in biology.
Equally expressive are comments like "It might be mentioned,
incidentally, how misleading it is to refer to mathematics as the
"queen of the sciences"." His thorough knowledge of the history
of science allowed him to state that "even Kant, by 1790, had
abandoned his subservience to mathematics. If the invalidity of
the mathematical ideal of science had not been obvious before, it
certainly became so with the publication of the Origin of
Species" (Mayr 1982, p. 41). Besides these philosophical issues,
Mayr demonstrated, using the argument between Darwin and Lord
Kelvin, that biologists might be better at calculations than
physicists.
Though less knowledgeable and articulate than Mayr, Lysenko
was more effective: he simply prohibited all biologically-related
math. Books on the subject were recalled from all libraries in
the USSR and burned (not publicly). Teachers of this criminal
discipline were exiled, fired, or, after proper public
repentance, forgiven. It is curious that, despite great
differences between the Harvard professor and Stalin's henchman,
both used similar derisive terminology when referring to Mendel's
typological theory of inheritance. Mayr calls it "bean-bag
genetics", while Lysenko dubbed it "peas' laws."
An alternative explanation of our poor comprehension of math
is less radical. It deals with the shortcomings of disciples
rather than disciplines. These weaknesses are relative. They can
be viewed as opposite sides of the assets of the respective
disciplines. Ecologists have a better, though intuitive, grasp of
complicated reality such as ecosystems. But their theories and
explanations sometimes lack coherence and discipline. The strict
rigor and precision, the hallmark of a mathematician, when
applied to ecological problems, too often deteriorates into a
concern about technicalities. These details appear irrelevant to
an ecologist for whom math is not the end, but only a means.
We often casually say that sizes of plants or animals are
normally distributed. This statement would not go unnoticed by a
mathematician. "Do you really mean that some rabbits have a
negative weight?" - he or she would ask. A more serious
predicament arises when we try to discuss our research. "Your
problem is poorly defined" (or "ill-posed") is the most common
response by a mathematician. As a rule, the cooperation stops
here. In rare cases when it continues, a mathematician using
impeccably correct methods will produce an outlandish answer,
while an ecologist employing hopelessly faulty reasoning (and
elusive intuition) will produce something more sensible.
The difference in the level of development of ecology and
math does not foster understanding and cooperation. Ecosystems
might be extremely complex and their explanation may require very
sophisticated math, much of which is yet to be discovered.
However, the present extent of our knowledge of ecological
complexity is limited. From a mathematician's viewpoint, our
greatest achievements so far have been the comprehension of
exponentiation (as is the case with natural selection) and
elementary combinatorics (Mendel's genetics). This disparity
between developmental levels makes cooperation tenuous.
Regardless of the reasons and explanations for this lack of
mutual understanding, we must recognize that it does exist and
that we should do something about it.
What should we do?
If we have failed to learn from mathematicians, we should
try something else: teach ourselves. Ecologically-inspired math
is too important to be confined to esoteric graduate courses. It
should be taught right at the beginning of our university
education (and even prior) by mathematically-minded ecologists.
Mathematicians will remain indispensable as consultants. They are
at their best when they teach future mathematicians rather than
biologists and engineers.
To start with, we need to write a good text for our
undergraduates. Actually, many texts should be written in order
to create the competitive environment needed to facilitate the
evolution of our mathematical education. While there are decent
books on mathematical ecology for graduate biology students,
undergraduate texts are commonly written by mathematicians. Our
text will be devoted to undergraduate ecological mathematics
rather than mathematical ecology. In other words, the stress will
be on teaching math using ecology as a spring board. We will
consider tree growth and animal behavior only to expose the
beauty and utility of logical trees and mathematical animals such
as derivatives, integrals, and absolute truth.
To apply math, future ecologists will need to master more
than just a few useful tricks such as the Taylor series. Math is
a method for expressing ideas - our own. This method is
distinguished from others by an unsurpassed discipline of
thinking. The problem is to develop mathematical imagery and the
ability to discern the substance of the studied process. To be
successful, teaching math should not be restricted to utilitarian
ends. We will be able to apply math only when we admire it for
its own sake.
Methods
To convey the relevance of math and make it attractive to
future ecologists, the text will:
1. Start with problems such as growth and then proceed to the
exposition of mathematical methods for their description. After
all, math itself originated from the need to solve practical
problems.
2. When appropriate, introduce mathematical ideas in the context
of the history of biology. For example, after Malthus found an
expression for biotic potential (the exponent), the stage was set
to reveal the opposite force, environmental resistance. This was
done mathematically by Verhulst and verbally (among many other
things) by Darwin and Wallace. This example, the starting point
for mathematical biology, naturally leads to presenting other
milestones such as the interaction of two populations (Lotka,
Volterra), and the contribution of ongoing adaptation to this
interaction.
3. Skip some derivations and techniques. We cannot expect to
master them in an undergraduate course anyway. Solutions to some
complicated problems will require the cooperation of professional
mathematicians. Still, we do not want to compile a cookbook like
some of the math texts for engineers. We will retain a sample of
simple proofs, which often are the most beautiful.
4. Teach ways to express vague ecological hypotheses and
assumptions in the form of mathematical models, which are often
differential and difference equations. Demonstrate their
relevance to our work. It is not just interesting, it is
startling when manipulation with equations opens your eyes to the
existence of a biological phenomenon that is not only new, but
contrary to the expected. Students of plant growth are accustomed
to seeing smooth S-shaped curves that instill the idea that
growth is continuous. Yet, through mathematical analysis we have
realized that this smooth appearance masks a sharp transition
from growth to reproduction. Maximal reproduction in a
predictable environment is possible only when this transition is
instantaneous. Even when environmental vagarities smooth this
transition, the onset of reproduction is always immediate.
5. Reconsider the order of exposition. A linear sequence, common
to many boring novels of the past, may not always be the best way
to convey the subject. Because their pertinence to ecology is the
most obvious, we may try to start with differential equations.
The magic of differentiation and integration will be revealed
when students need it to solve an equation.
6. Equip students with a more balanced picture of the world than
that inculcated by the mathematics of linear systems such as
regression techniques and Fourier analysis. "The mathematical
intuition so developed ill equips the student to confront the
bizarre behaviour exhibited by the simplest of discrete nonlinear
systems" (May 1976, p. 467). The text will show how deterministic
chaos arises from simple difference equations which are often
used in ecology to describe growth. This chaotic behavior implies
that not all fluctuations in population density are caused by the
environment and that long term predictions are not possible.
7. Our main concern will be to keep the text exciting. Texts not
only instill knowledge. They also create a certain emotional
aura. We should expect that students will soon forget most of the
course. We are struggling not only to fill their heads with
mathematical knowledge. It is equally important that students
acquire and retain a positive feeling toward math. This emotional
residue will decide whether our students will purge their minds
of math as soon as possible, or will be eager to recall some of
it, apply it, and try to learn more on their own.
We would greatly appreciate your comments, suggestions,
examples, reprints, and references related to this project.
Boris Zeide
Department of Forestry, University of Arkansas, Monticello
AR 71656. Phone: 501-460-1648. Internet: zeide@uamont.edu
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