An introduction to delay differential equations with applications to the life sciences.

*(English)*Zbl 1227.34001
Texts in Applied Mathematics 57. New York, NY: Springer (ISBN 978-1-4419-7645-1/hbk; 978-1-4614-2697-4/pbk; 978-1-4419-7646-8/ebook). xi, 172 p. (2011).

This book gives a first introduction to delay differential equations that is intended for mathematics students. Thanks to the emphasis on applications to life sciences, it can be recommended also to scientists from this discipline who wish to get a deeper understanding of the theoretical aspects for this widely used class of models.

Indeed, the strength of this monograph consists in giving an overview of necessary mathematical concepts for the treatment of delay differential equations without requiring too much mathematical background and in illustrating these concepts and related dynamical phenomena in a wealth of examples, mostly from biology. In this way, also, various aspects of mathematical modeling are covered by this monograph.

Requiring only some background from basic calculus and the theory of ordinary differential equations, the book provides an introduction to basic concepts from the theory of dynamical systems in chapter 4 and to several topics from the theory of real and complex analysis (like the theorem of Arzela-Ascoli and Gronwall’s lemma) in an appendix.

Central theorems that would need a bigger mathematical machinery, such as, in particular, theorems on linearized stability or Hopf-bifurcation, are stated without proof, referring to the classical textbooks of Hale and Lunel or Diekman et al. The book is written in the style of lecture notes and contains a large number of exercises that also elaborate on details in the proofs that have been omitted, and on further aspects of various examples from applications.

Indeed, the strength of this monograph consists in giving an overview of necessary mathematical concepts for the treatment of delay differential equations without requiring too much mathematical background and in illustrating these concepts and related dynamical phenomena in a wealth of examples, mostly from biology. In this way, also, various aspects of mathematical modeling are covered by this monograph.

Requiring only some background from basic calculus and the theory of ordinary differential equations, the book provides an introduction to basic concepts from the theory of dynamical systems in chapter 4 and to several topics from the theory of real and complex analysis (like the theorem of Arzela-Ascoli and Gronwall’s lemma) in an appendix.

Central theorems that would need a bigger mathematical machinery, such as, in particular, theorems on linearized stability or Hopf-bifurcation, are stated without proof, referring to the classical textbooks of Hale and Lunel or Diekman et al. The book is written in the style of lecture notes and contains a large number of exercises that also elaborate on details in the proofs that have been omitted, and on further aspects of various examples from applications.

Reviewer: Matthias Wolfrum (Berlin)

##### MSC:

34-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordinary differential equations |

34Kxx | Functional-differential equations (including equations with delayed, advanced or state-dependent argument) |

92-XX | Biology and other natural sciences |