Growth and diffusion phenomena. Mathematical frameworks and applications.

*(English)*Zbl 0788.92001
Texts in Applied Mathematics. 14. Berlin: Springer-Verlag. xix, 455 p. (1994).

A major purpose of this book is an “attempt to amalgamate some of the many advances made by people working in numerous diverse disciplines on topics related to growth and diffusion phenomena”.

After the introduction (Chapter 1), Chapters 2 and 3 mostly deal with an enormous variety of “one species” growth ODE models with time-invariant parameters. Chapters 4 and 5 consider time-dependent parameters. Chapter 6 deals with fixed and distributed time delays. Chapter 7 considers growth and spatial diffusion using PDE models, but the treatment is, contrary to other chapters, a bit sketchy. Chapter 8 refers briefly to topics under current research.

Competence in elementary calculus would suffice to follow the careful and (sometimes excessively) detailed mathematical treatment of the models, although occasionally Laplace transforms and complex integrals are used. Models are profusely illustrated with real examples and data mainly from biology, demography and geography, and innovation and technology transfer, but also from economics, agriculture, engineering, ecology, and other areas.

This is probably the most comprehensive set of “one species” growth models and applications one can find in a single text, together with the appropriate references. It stresses the unity provided by mathematical modelling. It explains how to fit models to data. Usually, it explains the rationale behind the models, but often uses a model just because it fits available data well.

I believe it is a valuable reference text for which the author and the Springer’s series “Texts in applied mathematics” should be commended. However, an advanced course corresponding to its contents for a science or engineering degree could be boring and of questionable usefulness.

After the introduction (Chapter 1), Chapters 2 and 3 mostly deal with an enormous variety of “one species” growth ODE models with time-invariant parameters. Chapters 4 and 5 consider time-dependent parameters. Chapter 6 deals with fixed and distributed time delays. Chapter 7 considers growth and spatial diffusion using PDE models, but the treatment is, contrary to other chapters, a bit sketchy. Chapter 8 refers briefly to topics under current research.

Competence in elementary calculus would suffice to follow the careful and (sometimes excessively) detailed mathematical treatment of the models, although occasionally Laplace transforms and complex integrals are used. Models are profusely illustrated with real examples and data mainly from biology, demography and geography, and innovation and technology transfer, but also from economics, agriculture, engineering, ecology, and other areas.

This is probably the most comprehensive set of “one species” growth models and applications one can find in a single text, together with the appropriate references. It stresses the unity provided by mathematical modelling. It explains how to fit models to data. Usually, it explains the rationale behind the models, but often uses a model just because it fits available data well.

I believe it is a valuable reference text for which the author and the Springer’s series “Texts in applied mathematics” should be commended. However, an advanced course corresponding to its contents for a science or engineering degree could be boring and of questionable usefulness.

Reviewer: C.A.Braumann (Evora)

##### MSC:

92B05 | General biology and biomathematics |

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

92D25 | Population dynamics (general) |

35Q80 | Applications of PDE in areas other than physics (MSC2000) |

34A99 | General theory for ordinary differential equations |