Theory of nonlinear age-dependent population dynamics.

*(English)*Zbl 0555.92014The simplest model for age-dependent growth of a single population is the linear first order hyperbolic partial differential equation known as the McKendrick-von Foerster model. It can be solved via the method of characteristics. A nonlinear version of this model, wherein the fertility and mortality coefficients depend on the total population, was developed by Gurtin and MacCamy and modified by many other investigators. Again using the method of characteristics, the equation can be converted to two nonlinear integral equations for the population distribution and the total population; these can be analysed by functional analytic techniques.

The author has been the leading proponent of studying the above equations using the theory of nonlinear semigroups of operators. That is, solutions are regarded as maps from \(L^ 1\), the space of initial age distributions, to \(L^ 1\) and form a strongly continuous nonlinear semigroup. The partial differential equation can then be regarded as an abstract evolution equation. This approach, which many others are now pursuing, leads to the most general formulation of the problem of studying age-dependent growth and allows a rich variety of mathematical theory to be applied to the existence and analysis of properties of solutions.

This book is a complete discussion of the theory as stated above, and very carefully develops all the concepts and results needed. There are five chapters: 1. Statement of the problem, 2. Basic theory of solutions, 3. The nonlinear semigroup associated with the solutions. 4. Equilibrium solutions and their stability, and 5. Biological population models. The twenty page list of references is very complete and extremely valuable to the researcher.

It is not an easy book to read because of the author’s penchant for verifying every estimate or inequality step-by-step. But it is a much needed book since it will provide the beginning researcher with the tools and the right framework (in the reviewer’s opinion) to study this very rich and complicated model for population growth.

The author has been the leading proponent of studying the above equations using the theory of nonlinear semigroups of operators. That is, solutions are regarded as maps from \(L^ 1\), the space of initial age distributions, to \(L^ 1\) and form a strongly continuous nonlinear semigroup. The partial differential equation can then be regarded as an abstract evolution equation. This approach, which many others are now pursuing, leads to the most general formulation of the problem of studying age-dependent growth and allows a rich variety of mathematical theory to be applied to the existence and analysis of properties of solutions.

This book is a complete discussion of the theory as stated above, and very carefully develops all the concepts and results needed. There are five chapters: 1. Statement of the problem, 2. Basic theory of solutions, 3. The nonlinear semigroup associated with the solutions. 4. Equilibrium solutions and their stability, and 5. Biological population models. The twenty page list of references is very complete and extremely valuable to the researcher.

It is not an easy book to read because of the author’s penchant for verifying every estimate or inequality step-by-step. But it is a much needed book since it will provide the beginning researcher with the tools and the right framework (in the reviewer’s opinion) to study this very rich and complicated model for population growth.

Reviewer: D.A.SĂˇnchez

##### MSC:

92D25 | Population dynamics (general) |

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

35G10 | Initial value problems for linear higher-order PDEs |

35K25 | Higher-order parabolic equations |

47H20 | Semigroups of nonlinear operators |

47D03 | Groups and semigroups of linear operators |