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

Hal Smith
Publication Date: 
Number of Pages: 
Texts in Applied Mathematics 57
BLL Rating: 

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
William J. Satzer
, on

This text is designed to be an introduction to the theory of differential equations with delay for advanced undergraduates and beginning graduate students. It incorporates a few significant applications to biology and medicine. Students are expected to have a strong background in ordinary differential equations. In practice, this means a good grasp of existence and uniqueness theorems and their proofs, experience with stability analysis, and a basic understanding of flows and other aspects of the dynamical systems approach. Although applications are an important part of the book, the theoretical aspects of the subject are handled very rigorously.

Although delay differential equations have been a subject of research interest for at least forty years, there have been very few attempts to produce an introductory textbook, and none with the kind of applications described here. The student needs a fair amount of mathematical maturity to make progress, in part because complex analysis plays an important and subtle role in the analysis of the characteristic equations arising from linearization around equilibria. The author provides an appendix to sketch in the needed background in real and complex analysis, but many students will find it tough going.

The author begins with the simplest possible differential equation with delay:

u'(t) = – u(t – τ)

where τ > 0 is the delay and prescribed initial data are: u(t) = 1 for –τ ≤ t ≤ 0.

He uses this equation to introduce the complications that come along with delay equations, and it has the advantage of being amenable to explicit solution.

From there the book moves on to existence and uniqueness theorems for differential delay equations, linear systems and linearization, Hopf bifurcation analysis, and semidynamical systems induced by delay equations. All in this in barely a hundred pages.

The application that is covered in greatest detail is a mathematical model of bacteriophage predation on bacteria in a chemostat where the delay comes from an assumed fixed latent period for the phage inside the cell. Biologically, the main issues are the persistence and extinction of the bacteria and phages. Mathematically, the major element is a Hopf bifurcation from the coexistence equilibrium.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

BLL — The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

1 Introduction.-The Simplest Delay Equation.-Delayed Negative Feedback: A Warm-Up.- Existence of Solutions.- Linear Systems and Linearization.- Semidynamical Systems and Delay Equations.- Hopf Bifurcation.- Distributed Delay Equations and the Linear Chain Trick.- Phage and Bacteria in a Chemostat.-References.- Index.