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Differential equations and mathematical biology. (English) Zbl 1020.92001

Chapman & Hall/CRC Mathematical Biology and Medicine Series. Boca Raton, FL: Chapman & Hall/CRC. xii, 390 p. (2003).
Old age and routine has crept in on the marriage of mathematics and physics. But a new sweetheart for mathematics is around, biology. This can be seen by an increasing number of books and papers that address more and more areas from biology with increasingly sophisticated mathematics. In fact, the book under discussion belongs, so to speak, to the first generation, since it appeared first in 1983 with George Allen and Unwin publishers, see the review Zbl 0504.92002. The introduction has been rewritten slightly, without referring to the “first edition” nor to the changes introduced. The changes are more up to date references and the new fashionable field “bifurcation and chaos” instead of the then fashionable area of catastrophe theory. References to Mathematica are also included and the illustrations have been redone.
The stated aim of this book is to introduce undergraduates of mathematics, physical or biological sciences to applications of nonlinear ordinary or partial differential equations. In the authors opinion this book could be used as a base for a course on differential equations (Chapters 1, 2, 3, 5, 10, 11), or a course on biological modelling, or finally a course on differential equation models in biology (Chapters 1, 2, 3, 5, 10, 11, 12, 13,…). Consequently the book gives some introduction into the theory of ordinary and partial differential equations. The contents of the mathematics chapters are:
Chapter 1: Population models, separable equations, first order linear equations. Chapter 2: Linear equations with constant coefficients. Chapter 3: Linear systems with constant coefficients. Chapter 5: Nonlinear systems in \({\mathbb{R}}^2\), existence and uniqueness. Chapter 10: Partial differential equations of first order in two variables and their characteristics. Equations of second order in two variables and their classification. Chapter 11: The heat equation, separation of variables, maximum principles.
The biological part of this book is very much determined by the research interests of the authors: application of nonlinear partial differential equations. Thus chapter 4 lays the ground for later chapters by introducing models for the heartbeat (chapter 6), the Hodgkin-Huxley model and Fitz Hugh-Nagumo equations (chapter 7), the Belousov-Zhabotinskij reaction (chapter 8) and predator prey models (chapter 9). In chapters 12 and 13, diffusion, pattern formation, bifurcation and chaos are treated. Growth of tumors and epidemics are treated in the last two chapters as in the “first edition”. The book has an extensive index, though the list of references is too short for a field that has been expanding greatly in recent years. The illustrations and plots have been improved but in the referees opinion there are still too few.
In judging the overall impression of the book one will have to answer the questions: Who will benefit from it and how well have the goals of the authors been achieved. This book aims at undergraduates, but leaves out elementary material like compartment models or phase plane analysis. Instead the students are exposed to reaction diffusion equations and seemingly ad hoc approximations in heavy doses. It is supposed to be used as a base for a course in differential equations. Anyone who has ever taught such a course can easily point out necessary material which is not covered. The material is not presented in a systematic way and proper definitions are missing. Try for example the notions of stability or chaos. Even the existence theorems fall short, leaving out the continuation or continuous dependence on initial conditions. That the not so mathematically inclined biologist will despair with such a mass of tricky mathematics should be clear to anyone teaching “biological modelling”. So this book will most likely be used as a source, a sort of quarry, for one or the other model in this area.

MSC:

92B05 General biology and biomathematics
92-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology
35Q92 PDEs in connection with biology, chemistry and other natural sciences
34-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations

Citations:

Zbl 0504.92002
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