Mathematics in population biology.

*(English)*Zbl 1054.92042
Princeton Series in Theoretical and Computational Biology. Princeton, NJ: Princeton University Press (ISBN 0-691-09291-5/pbk). xviii, 543 p. (2003).

This is not just another textbook on population biology mathematical modeling. It takes a different look, matching model derivation from first biological principles with mathematical rigor and, most of all, seeking maximum generality. This is a remarkable long overdue feature most texts in modeling tend to forget. Let us give just an example (many could be given). Most books will talk about the logistic growth model and conclude that population growth follows a sigmoid curve, as if nature has to obey the logistic rule of growth. This book considers the likelihood that nature growth dynamics may not be logistic and looks for sufficient qualitative conditions for a general ODE population growth model to exhibit sigmoidal growth. It still treats the logistic and other specific models as well, which are derived from biological principles as approximate models under certain environmental and biological conditions.

Part 1 is on basic population growth models (a deep and extensive study, even including seasonality and effects of toxicants in aquatic populations). Part 2 contains a detailed treatment on stage transitions and demographics. Part 3 is on host-parasite population growth and epidemiology of infectious diseases. Unlike many others, this book makes no concessions on mathematical rigor and presents mathematical derivations for almost all the results. Some chapters require advanced calculus knowledge, although the reader can build up the necessary requirements from the book itself. In fact, Part IV – Toolbox – contains a thorough digested minicourse on ordinary differential equations, integral equations, and an introduction to measure and integration and to convex analysis. All chapters have exercises, some with solutions. Persistence theory and time-scale methods are techniques often used.

The book is quite appropriate for a graduate or an advanced undergraduate course. The author prefers to deal with aspects that others do not, and so the book does not cover subjects as relevant as multispecies (except for some limiting cases and host-parasite models), stochastic, and population genetics models. This lack of comprehensiveness may not recommend it as the sole main textbook, but it should definitely be in the main bibliography. A researcher in population dynamics modeling will find it very inspiring and illuminating and would certainly profit from its reading.

Part 1 is on basic population growth models (a deep and extensive study, even including seasonality and effects of toxicants in aquatic populations). Part 2 contains a detailed treatment on stage transitions and demographics. Part 3 is on host-parasite population growth and epidemiology of infectious diseases. Unlike many others, this book makes no concessions on mathematical rigor and presents mathematical derivations for almost all the results. Some chapters require advanced calculus knowledge, although the reader can build up the necessary requirements from the book itself. In fact, Part IV – Toolbox – contains a thorough digested minicourse on ordinary differential equations, integral equations, and an introduction to measure and integration and to convex analysis. All chapters have exercises, some with solutions. Persistence theory and time-scale methods are techniques often used.

The book is quite appropriate for a graduate or an advanced undergraduate course. The author prefers to deal with aspects that others do not, and so the book does not cover subjects as relevant as multispecies (except for some limiting cases and host-parasite models), stochastic, and population genetics models. This lack of comprehensiveness may not recommend it as the sole main textbook, but it should definitely be in the main bibliography. A researcher in population dynamics modeling will find it very inspiring and illuminating and would certainly profit from its reading.

Reviewer: Carlos A. Braumann (Evora)

##### MSC:

92D25 | Population dynamics (general) |

92D40 | Ecology |

92-02 | Research exposition (monographs, survey articles) pertaining to biology |

92D30 | Epidemiology |