© 2011

An Introduction to Delay Differential Equations with Applications to the Life Sciences


Part of the Texts in Applied Mathematics book series (TAM, volume 57)

Table of contents

  1. Front Matter
    Pages i-xi
  2. Hal Smith
    Pages 1-11
  3. Hal Smith
    Pages 13-24
  4. Hal Smith
    Pages 25-39
  5. Hal Smith
    Pages 41-59
  6. Hal Smith
    Pages 87-118
  7. Hal Smith
    Pages 131-147
  8. Back Matter
    Pages 171-172

About this book


This book is intended to be an introduction to Delay Differential Equations for upper level undergraduates or beginning graduate mathematics students who have a good background in ordinary differential equations and would like to learn about the applications. It may also be of interest to applied mathematicians, computational scientists, and engineers. It focuses on key tools necessary to understand the applications literature involving delay equations and to construct and analyze mathematical models. Aside from standard well-posedness results for the initial value problem, it focuses on stability of equilibria via linearization and Lyapunov functions and on Hopf bifurcation. It contains a brief introduction to abstract dynamical systems focused on those generated by delay equations, introducing limit sets and their properties. Differential inequalities play a significant role in applications and are treated here, along with an introduction to monotone systems generated by delay equations. The book contains some quite recent results such as the Poincare-Bendixson theory for monotone cyclic feedback systems, obtained by Mallet-Paret and Sell. The linear chain trick for a special family of infinite delay equations is treated. The book is distinguished by the wealth of examples that are introduced and treated in detail. These include the delayed logistic equation, delayed chemostat model of microbial growth, inverted pendulum with delayed feedback control, a gene regulatory system, and an HIV transmission model. An entire chapter is devoted to the interesting dynamics exhibited by a chemostat model of bacteriophage parasitism of bacteria. The book has a large number of exercises and illustrations. Hal Smith is a Professor at the School of Mathematical and Statistical Sciences at Arizona State University. 


Delay differential equations functional differential equations time delay

Authors and affiliations

  1. 1., School of Mathematical and Statistical SArizona State UniversityTempeUSA

About the authors

Hal Smith is a Professor at the School of Mathematical and Statistical Sciences at Arizona State University.

Bibliographic information

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From the reviews:

“This text is designed to be an introduction to the theory of differential equations with delay for advanced undergraduates and beginning graduate students.” (William J. Satzer, The Mathematical Association of America, November, 2010)

“This textbook would serve as an excellent reference text to help a mathematics faculty develop an introductory mathematics course on DDEs … . This textbook clearly places the study of DDEs firmly within the standard curriculum of a graduate program in mathematics and well within the scope of a strong final-year undergraduate mathematics major. … this text will have a positive impact on biomathematics education and be a welcome addition. We certainly find it so.” (John G. Milton and Michael C. Mackey, Mathematical Reviews, Issue 2011 k)

“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 that wish to get a deeper understanding of the theoretical aspects for this widely used class of models.” (Matthias Wolfrum, Zentralblatt MATH, Vol. 1227, 2012)