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**Mathematical models in the social, management and life sciences.**
*(English)*
Zbl 0528.00002

Mathematics and its Applications. Chichester: Ellis Horwood Ltd.; New York - Chichester - Brisbane: Halsted Press (John Wiley & Sons). 287 p. $ 38.95 (1980).

Summary: This book, designed for use in an undergraduate course on mathematical models, is unusual in requiring (at least in principle) a minimum of mathematical background. The book might be used at the freshman level, perhaps even the high school level.

The attempt to minimize mathematical requirements, however, presents the authors with a dilemma, since not many convincing models can be presented and analyzed on such a flimsy foundation. The following approach is taken to handle this problem: first a “case study” is presented, this is followed immediately by development of relevant mathematical techniques, which are then used to model the case study and to analyze the model; finally, further case studies amenable to similar techniques are discussed.

Thus the chapters are organized around mathematical techniques, rather than problem areas. Chapter titles include: Chapter 2. Sequences and series; Chapter 3. Limits and continuity; Chapter 4. Differential calculus; Chapter 5. First order differential equations; Chapter 8. Vectors and matrices; Chapter 10. Theory of games; and (inevitably) Chapter 11. Catastrophe theory.

The models are related either to problems from economics and business (the size of firms, inventory management, theory of advertising, supply and demand, economic growth, etc.) or to biological or medical questions (drug concentration, logistic population growth, predator prey model, competition of species, epidemics, animal aggression, etc.)

No doubt there is a demand for material of this kind – the biological and social sciences are becoming progressively more mathematical, while many students entering these fields continue to be poorly prepared in mathematics. The question is whether meeting the demand by such superficial presentation may do more harm than good. In this book, for example, the definition of the derivative on p. 63 is followed by a five page summary of the differential calculus; differential equations appear on p. 82, and on p. 83 the definite integral sign first appears, unheralded and undefined! (There is a four page appendix on integration but the reader is not referred to it.) On p. 86 a convolution integral appears. What is the unsophisticated student to make of all this?

The book gives the student little practice in the actual process of modelling, and provides little criticism of the models presented. For a text in which the modelling process is treated more seriously and critically, see E. A. Bender’s book [An introduction to mathematical modelling. New York etc.: John Wiley (1978; Zbl 0388.93008)].

The attempt to minimize mathematical requirements, however, presents the authors with a dilemma, since not many convincing models can be presented and analyzed on such a flimsy foundation. The following approach is taken to handle this problem: first a “case study” is presented, this is followed immediately by development of relevant mathematical techniques, which are then used to model the case study and to analyze the model; finally, further case studies amenable to similar techniques are discussed.

Thus the chapters are organized around mathematical techniques, rather than problem areas. Chapter titles include: Chapter 2. Sequences and series; Chapter 3. Limits and continuity; Chapter 4. Differential calculus; Chapter 5. First order differential equations; Chapter 8. Vectors and matrices; Chapter 10. Theory of games; and (inevitably) Chapter 11. Catastrophe theory.

The models are related either to problems from economics and business (the size of firms, inventory management, theory of advertising, supply and demand, economic growth, etc.) or to biological or medical questions (drug concentration, logistic population growth, predator prey model, competition of species, epidemics, animal aggression, etc.)

No doubt there is a demand for material of this kind – the biological and social sciences are becoming progressively more mathematical, while many students entering these fields continue to be poorly prepared in mathematics. The question is whether meeting the demand by such superficial presentation may do more harm than good. In this book, for example, the definition of the derivative on p. 63 is followed by a five page summary of the differential calculus; differential equations appear on p. 82, and on p. 83 the definite integral sign first appears, unheralded and undefined! (There is a four page appendix on integration but the reader is not referred to it.) On p. 86 a convolution integral appears. What is the unsophisticated student to make of all this?

The book gives the student little practice in the actual process of modelling, and provides little criticism of the models presented. For a text in which the modelling process is treated more seriously and critically, see E. A. Bender’s book [An introduction to mathematical modelling. New York etc.: John Wiley (1978; Zbl 0388.93008)].

Reviewer: Colin W. Clark (M.R. 81b:00009)

### MSC:

00A06 | Mathematics for nonmathematicians (engineering, social sciences, etc.) |

93A30 | Mathematical modelling of systems (MSC2010) |

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

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

93-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to systems and control theory |