Mathematical Essays on Growth and the Emergence of Form

Charles R. Crawford, a former associate professor of computer science at
York University, is a computer programming and mathematics consultant in
Toronto.

This collection of essays covers several technical topics in the mathematical modeling of natural growth processes and problems in pure mathematics that arise from those models. The increased interest since the early 1960s among biologists in modeling complex ecosystems stimulated interest among mathematicians. Since then, joint research, based largely on pioneering work of Vito Volterra (1860-1940), has led to considerable progress as indicated in these essays. The book requires a background in mathematical physics, since the approaches to biological modeling presented here are similar to those used in physics.

Unlike many collections of mathematical essays, this one includes some historical background. It includes a translation of a paper by Volterra in which he presents his model of a system “where the individuals of various species, living together, devour one another.” Such models are often presented in simplified form in undergraduate mathematics texts as “predator/prey ecosystems” — e.g., Forsythe, Malcolm, and Moler’s Computer Methods fir Mathematical Computations (Englewood Cliffs, N.J.: Prentice-Hall, 1977, p. 151). It is very useful to have the original work on these models more accessible to undergraduates. A second translation included in the collection is “Principles of Zoological Philosophy,” by J. von Goethe. This work describes the conflict between two researchers on methods of biological research — Baron Cuvier, a “differentiator,” and Geoffrey de Saint-Hilaire, “concerned with analogies... and... hidden relationships.” This older (1830) essay deals with a more philosophical issue than Volterra’s and thus has a less direct connection with others in the collection. These translations, however, gathered as a chapter entitled “The Historical Record,” act as a welcome balance to the usual anti-historical style of writing in mathematics.