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**Mathematical models in biology.**
*(English)*
Zbl 0674.92001

The Random House/BirkhĂ¤user Mathematics Series. New York: Random House. xvii, 586 p. DM 105.50 (1988).

This is an introduction to the interplay of two disciplines, mathematics and biology. Models on topics ranging from population biology to physiology and from cellular to molecular biology are drawn from both classical literature and current research. These are accompanied by a detailed explanation of mathematical techniques such as difference equations, qualitative methods in ordinary differential equations, linear stability theory, bifurcations, and an introduction to partial differential equations. The book is subdivided into three parts, dealing respectively with discrete, continuous and multivariable models. Parallels, distinctions, and interrelationships are highlighted where applicable.

Part I, “Discrete Processes in Biology”, begins with discrete difference equations models, which are fairly elementary, requiring little mathematical sophistication. Part II, “Continuous Processes and Ordinary Differential Equations”, introduces ordinary differential equations and presents a selection of models, ranging from the molecular to the population level. Phase-plane methods and quantitative solutions, applications of continuous models to population dynamics, and models for molecular events are presented. In Part III, “Spatially Distributed Systems and Partial Differential Equation Models”, the emphasis is on formulating, understanding, and interpreting partial differential equation models, like models for development and pattern formation in biological systems.

A feature of this book is that mathematical techniques and biological applications are developed side by side. Common underlying themes such as “steady states”, “stability” and “parameter variations” reappear in a variety of contexts and in a diverse collection of models. The emphasis throughout is on qualitative methods and graphical techniques rather than on long calculations. Other special features include examples and exercises drawn from the research literature, separation of long calculations in boxes to ensure a smooth flow of text suggestions for further reading on related topics, and bibliographies at the end of chapters.

Part I, “Discrete Processes in Biology”, begins with discrete difference equations models, which are fairly elementary, requiring little mathematical sophistication. Part II, “Continuous Processes and Ordinary Differential Equations”, introduces ordinary differential equations and presents a selection of models, ranging from the molecular to the population level. Phase-plane methods and quantitative solutions, applications of continuous models to population dynamics, and models for molecular events are presented. In Part III, “Spatially Distributed Systems and Partial Differential Equation Models”, the emphasis is on formulating, understanding, and interpreting partial differential equation models, like models for development and pattern formation in biological systems.

A feature of this book is that mathematical techniques and biological applications are developed side by side. Common underlying themes such as “steady states”, “stability” and “parameter variations” reappear in a variety of contexts and in a diverse collection of models. The emphasis throughout is on qualitative methods and graphical techniques rather than on long calculations. Other special features include examples and exercises drawn from the research literature, separation of long calculations in boxes to ensure a smooth flow of text suggestions for further reading on related topics, and bibliographies at the end of chapters.

Reviewer: T.Postelnicu

### MSC:

92-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to biology |

92B05 | General biology and biomathematics |