A MATHEMATICAL MODEL APPLIED TO A STUDY OF THE EVOLUTION OF SPECIES

This paper presents a mathematical model that is believed to be applicable to the solution of certain fundamental problems of evolution. Its use is illustrated by the study of a particular problem. Attention is focused upon the nature of morphological change at a low taxonomic level including analyses of the mechanism and modes of these changes. The data that modern experimental and analytical methodology can provide are as yet insufficient for comprehensive answers to all phases of problems of this type and severe restrictions are imposed when, as in-the illustration, investigations are limited to part of a single anatomical system of extinct organisms. Nevertheless, much of interest and significance may be gained by carefully planned exploitation of the data that are available. Certain taxonomic problems have proved especially amenable to solution by statistical procedures but, mostly, results are either supplementary or complementary to those achieved by other methods of study. There are, however, many problems of evolution that cannot be treated adequately by classical approaches or by application of the statistical procedures now commonly used by students whose data are primarily morphological. Among these are problems concerned with the intricate, small morphological modifications that occur in the evolution of subspecies and species. By use of the method outlined here we hope to detect and describe such changes, thus adding to an understanding of the nature of differences separating small taxonomic units, and also, more importantly, to draw inferences concerning factors that have operated to produce these changes. The characters treated in numerical studies, such as the one considered here, are necessarily dimensions. A serious difficulty in many analyses of the nature of change stems from the tendency to treat modifications of each dimension, as if they were independent of other modifications, whereas intimate interdependence actually exists. Much information, on the other hand, is lost if various changing dimensions are treated together in a way that masks the importance of individual contributions to the whole. Most profitable would be a search for some unifying principle under which each changing part can be studied in relation to its effects on other changing parts, whether these be closely related or partially or completely independent. Such a principle, involving the concept of closely related groups of measurements, has been developed and demonstrated in work, as yet unpublished, by the junior author. We may hope by its use as demonstrated here to approach a little more closely to an organismic basis for analysis of change in evolution, and to provide more definite bases for determining how and why particular changes came about, than has been possible in the past. The problem considered here concerns three closely related species of early Permian reptiles of the Family Captorhinidae, whose essentially contemporaneous existence implies a not remote common ancestor. The acceptance of this implication is the basis for inquiry concerning changes that have probably taken place in the evolution of descendant species, although the ancestral stock is unknown. Mr. William Kruskal, a member of the Committee on Statistics of the University of Chicago, has given much time