Preface

Summary

This book is intended as an introduction to methods that are useful for studying population phenomena. The models are presented in terms of difference equations. Experience has shown that this approach facilitates communicating the derivation of models and statements of results about them to scientists who do not have a strong mathematical background. However, in most cases of difference equations the mathematician must exert greater analytical effort because many of the features of calculus are not available in this setting. Important models that do involve extensive use of calculus are presented in exercises. In most cases, the exercises are fronts for presenting models more detailed than those derived and studied in the text.

The material is graded in terms of mathematical difficulty. The earlier chapters involve elementary difference equations, and later chapters involve topics requiring more mathematical preparation. First, models of total population and population age structure are derived and studied. Next, models of random population events are presented in terms of Markov chains. The final two chapters deal with mathematical methods used to uncover qualitative behavior of more complicated difference equations. For example, the material on geographically distributed populations eventually involves nonlinear diffusion equations. In each case, the chapter begins with a simple model, usually of some historical interest, that motivates the primary goals of the chapter.

The approach taken here evolved over many years from sets of lectures presented at New York University, the Courant Institute of Mathematical Sciences, and the University of Utah, and many students and colleagues participated in and contributed to the topics covered.