1.1. Preface.

This book arose out of a series of lectures given at the University of Tennessee at Knoxville in the spring of 1977. It is a pleasure to acknowledge the hospitality of the Department of Mathematics at the University of Tennessee during 1976-77 when the author was a visiting Professor there. The purpose of this book is to bring to a wide audience an up-to-date account of the developments in multiparameter spectral theory in Hilbert space. Chapter one is introductory and is intended to give a background and motivation for the material contained in subsequent chapters. Chapter two sets down the basic concepts and ideas required for a proper understanding of the theory developed in chapters three, four and five. It is mainly concerned with the concept of tensor products of Hilbert spaces and the spectral properties of linear operators in such spaces. Most of the theorems contained in this chapter are given without proof but nevertheless adequate references are included in which complete proofs may be found. In chapter three multiparameter spectral theory is developed for the case of bounded operators and this is generalised to include unbounded operators in chapters four and five. Chapter six deals with a certain abstract relation arising in multiparameter spectral theory and is analogous to the integral equations and relations well known in the study of boundary value problems for ordinary differential equations. Chapters seven and eight exploit the theory in application to coupled operator systems and to polynomial bundles. Finally chapter nine reviews the material of the previous chapters, points out open problems and indicates paths of new investigations. Over the years the author has benefited from collaboration and guidance from a number of colleagues. In particular it is a pleasure to acknowledge the guidance and stimulation from my colleague and former teacher Felix Arscott who first aroused my interest in multiparameter spectral theory. I also wish to acknowledge the influence and collaboration of Patrick Browne (who read the entire manuscript and made a number of suggestions for improving certain sections), Anders Källström and Gary Roach whose contributions are significant in much of the theory developed here. I would also like to thank Julie my wife for her sustained encouragement during the writing of this book. She not only prepared the index and list of references but also contributed to the style and layout of the work. Finally I would like to express my appreciation to Mrs Norah Thompson who so skilfully typed the entire manuscript.

1.2. An Introduction.

Multiparameter spectral theory like its one parameter counterpart, spectral theory of linear operators, which is the subject of a vast and active literature, has its roots in the classic problem of solving boundary value problems for partial differential equations via the method of separation of variables. In the standard case the separation technique leads to the study of systems of ordinary differential equations coupled via spectral parameters (i.e. separation constants) in only a non-essential manner. For example the problem of vibration of a rectangular membrane with fixed boundary leads to a pair of Sturm-Liouville eigenvalue problems for ordinary differential equations which are separate not only as regards their independent variables but also in regard to the spectral parameters as well. The same problem posed for the circular membrane leads to only mild parametric coupling. This is a kind of triangular situation. The parameter in the angular equation must be adjusted for periodicity and the resulting values substituted in the radial equation leading to the study of various Bessel functions. The multiparameter situation arises in full if we pursue this class of problems a little further. Take for example the vibration problem of an elliptic membrane with clamped boundary. It is appropriate here to use elliptic coordinates. Application of the separation of variables method; leads to the study of eigenvalue problems for a pair of ordinary differential equations both of which contain the same two spectral parameters. This is then a genuine two-parameter eigenvalue problem. The ordinary differential equations which arise are Mathieu equations whose solutions are expressible in terms of Mathieu functions. Other problems of this type give rise to two or three parameter eigenvalue problems and their resolution lies to a large extent in the properties of the "higher" special functions of mathematical physics, e.g. Lamé functions, spheroidal wave functions, paraboloidal wave functions, ellipsoidal wave functions etc. We refer to the encyclopaedic work of Erdélyi et al [A Erdélyi, W Magnus, F Oberhettinger and F G Tricomi (eds.) H Bateman, Higher transcendental functions Vols I, II, III (1953)] and also the book of Arscott [F M Arscott, Periodic differential equations (1964)] for an account of these functions. Many of these special functions possess as yet unrevealed secrets even though they have been studied vigorously over the past fifty or so years. It is perhaps not so surprising then that multiparameter spectral theory per se has been rather neglected over the years despite the fact that it arose almost as long ago as the classic work of Sturm and Liouville regarding one-parameter eigenvalue problems particularly oscillation theory.

2.1. From the Publisher.

This textbook shows how first-order ordinary differential equations (ODEs) are used to model the growth of a population, the administration of drugs, and the mechanism by which living cells divide. The authors present linear ODEs with constant coefficients, extend the theory to systems of equations, model biological phenomena, and offer solutions to first-order autonomous systems of nonlinear differential equations using the Poincaré phase plane. They also analyse the heartbeat, nerve impulse transmission, chemical reactions, and predator-prey problems. After covering partial differential equations and evolutionary equations, the book discusses diffusion processes, the theory of bifurcation, and chaotic behaviour. It concludes with problems of tumour growth and the spread of infectious diseases.

2.2. Preface.

In recent years, mathematics has made a considerable impact as a tool with which to model and understand biological phenomena. In return biology has confronted the mathematician with a variety of challenging problems which have stimulated developments in the theory of non-linear differential equations. This book is the outcome of the need to introduce undergraduates of mathematics, the physical and biological sciences to some of these developments. It is primarily directed to university students who are interested in modelling and the application of mathematics to biological and physical situations.

Chapter 1 is introductory, showing how the study of first-order ordinary differential equations may be used to model the growth of a population, monitoring the administration of drugs and the mechanism by which living cells divide. In Chapter 2, a fairly comprehensive account of a linear ordinary differential equation with constant coefficients is given while Chapter 3 extends the theory to systems of equations. Such equations arise frequently in the discussion of the biological models encountered throughout the text. Chapter 4 is devoted to modelling biological phenomena and in particular includes (i) physiology of the heart beat cycle, (ii) blood flow, (iii) the transmission of electrochemical pulses in the nerve, (iv) the Belousov-Zhabotinskii chemical reaction and (v) predator-prey models.

Nearly all the biological models described in Chapter 4 have special solutions which arise as solutions to first-order autonomous systems of non-linear differential equations. Chapter 5 gives an account of such systems through the use of the Poincaré phase plane.

With the knowledge of differential equations developed thus far, we are in a position to begin an analysis of the heart beat, nerve impulse transmission, chemical reactions and predator-prey problems. These are the subjects of Chapters 6-9.

In order to gain a deeper insight into biological models, it is necessary to have a knowledge of partial differential equations. These are the subject of Chapters 10 and 11. In particular, a number of the models discussed in Chapter 4 involve processes of diffusion (Chapter 12), and the evolutionary equations considered in Chapter 11 are basic for an understanding of these processes. A special feature of Chapter 12 is a treatment of pattern formation in developmental biology based on Turing's famous idea of diffusion driven instabilities. The theory of bifurcation and chaotic behaviour is playing an increasing role in fundamental problems of biological modelling. An introduction to these topics is contained in Chapter 13. Chapter 14 models and studies problems of growth of solid avascular tumours. Again differential equations play a fundamental part. However, a new feature here is that we encounter moving boundary problems. The book concludes in Chapter 15 with a discussion of epidemics and the spread of infectious diseases, modelled via various differential equations.

As an encouragement to further study, some of the chapters have notes indicating sources of material as well as references to additional literature. Each chapter has a set of exercises which either illustrate some of the ideas discussed or require readers to develop and test models of their own.

2.3. Review by: Hal L Smith.
SIAM Review 46 (1) (2004), 183-184.

Mathematical biology is enjoying a lot of attention these days and the inevitable consequence is a steady stream of introductory texts. ... The book under review has a somewhat broader aim. To understand the authors' intentions, a quote from the preface is illuminating: "In recent years, mathematics has made a considerable impact as a tool with which to model and understand biological phenomena. In return biology has confronted the mathematician with a variety of challenging problems which have stimulated developments in the theory of non-linear differential equations. This book is the outcome of the need to introduce undergraduates of mathematics, the physical and biological sciences to some of these developments. It is primarily directed to university students who are interested in modelling and the application of mathematics to biological and physical situations." The authors suggest that the book may serve a multipurpose role: (1) as a text for a first ODE course, or (2) as a course in biological modelling for students of math and the physical sciences, or (3) as a course in differential equations models of biology for life sciences students. ... Despite the authors' contention that the book could be used as a course on modelling in biology, it is really more accurate to say that applications to biology are presented. ... A strength of the present book is its concise coverage of a broad range of topics in differential equations that are useful in the analysis of mathematical models in biology and the application of some of these topics to a diverse collection of such models. It is truly remarkable how much material is squeezed into this slim book's 400 pages.

2.4. Review by: Robert E O'Malley, Jr.
SIAM Review 52 (3) (2010), 586-587.

While they were colleagues in Dundee, the distinguished applied analysts Douglas Jones and Brian Sleeman published a book in 1983 with Allen & Unwin with the same title as this one. The outline of the new book is nearly the same, except that a chapter on "Catastrophe Theory and Biological Phenomena" has been replaced by successful new chapters on "Bifurcation and Chaos" and "Numerical Bifurcation Analysis," while more computational approaches and the use of MATLAB have been added throughout. Much progress by these authors and others over the past quarter century in modelling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. The intended audience is broad and, in contrast with many texts on modelling in biology, the differential equations cover age is quite sophisticated. ... Overall, this book should convince math majors how demanding math modelling needs to be and biologists that taking an other course in differential equations will be worthwhile. The co-authors deserve congratulations as well as course adoptions.

2.5. Review by: Jian Hong Wu.
Mathematical Reviews MR1967145 (2004g:92003).

The title precisely reflects the contents of the book, a valuable addition to the growing literature in mathematical biology from a deterministic modelling approach. This book is a suitable textbook for multiple purposes: for applied qualitative theory of differential equations and for mathematical biology, and at multiple levels: for senior undergraduate students majoring in applied mathematics, for graduate students in applied mathematics, and for students in theoretical ecology interested in mathematical modelling. ... Overall, topics are carefully chosen and well balanced. I would like to see a chapter on delay differential equations with applications, but this may be just a personal preference. The book is written by experts in the research fields of dynamical systems and population biology. As such, it presents a clear picture of how applied dynamical systems and theoretical biology interact and stimulate each other - a fascinating positive feedback whose strength is anticipated to be enhanced by outstanding texts like the work under review.

2.6. Review by: William J Satzer.
Mathematical Association of America, https://www.maa.org/press/maa-reviews/differential-equations-and-mathematical-biology

This is the second edition of a book in the Chapman and Hall/CRC Mathematical and Computational Biology series. It is primarily about differential equations - ordinary and partial - with applications to biology. The authors have devised the text to serve three separate, partially overlapping purposes: a basic course in differential equations, a course in biological modelling for students of mathematics and the physical sciences, and a course in modelling with differential equations for students in the life sciences. There are no surprises in either the treatment of differential equations or the biological applications. The topics and approach to differential equations are standard, and the nearly all the applications have also been treated in comparable books. Where this text stands out is in its thoughtful organization and the clarity of its writing. This is a very solid book, not at all flashy. Illustrations, for example, are black and white and mostly line drawings. The authors succeed because they do a splendid job of integrating their treatment of differential equations with the applications, and they don't try to do too much.

The basics of ordinary differential equations are treated in Chapters 1, 2, 3, and 5. No linear algebra background is assumed. Chapter 4 introduces the process of modelling biological phenomena, beginning with a model for the heartbeat. Here the authors note, "... the real test of the model is that it not only agrees qualitatively with the biological process but has the ability to suggest new experiments and bring deeper insight into the biological situation." Although this is true of all modelling endeavours, it deserves special emphasis here because mathematical biology is still a relatively immature field. Differential equations provide one class of tools; many others remain to be developed. Specific applications are treated next: heart physiology, nerve impulse transmission, chemical reactions, and predator-prey interactions. A very nice introduction to partial differential equations follows in Chapters 10 and 11. The authors emphasize the importance of diffusion processes in biology and present several examples in Chapter 12; these include Turing's approach to pattern formation arising from diffusion-driven instability.

The first edition of this book had a chapter on catastrophe theory. In the second edition, this has been replaced by two chapters on bifurcation, chaos and numerical bifurcation analysis. The book concludes with two more application chapters; one describing models of tumour growth and the other epidemics. Each chapter comes with a collection of well-selected exercises, and plenty of references for further reading.

2.7. Review by: jmil.
European Mathematical Society,
https://euro-math-soc.eu/review/differential-equations-and-mathematical-biology

The book is written with two aims: firstly, to be an introduction both to ordinary and partial differential equations; secondly, to present main ideas on how to model deterministic (and mostly continuous) processes in biology, physiology and ecology. The style of writing is subordinated to these purposes. It is remarkable that without the classical scheme (definition, theorem and proof) it is possible to explain rather deep results like properties of the Fitz-Hugh-Nagumo model of nerve impulse transmission or the Turing model of pattern formation. This feature makes the reading of this text pleasant business for mathematicians also. There exists a similar book written by J D Murray (Mathematical Biology), which contains more biological models. In comparison with it, the book under review is also a textbook on differential equations. It can be recommended for students of mathematics who like to see applications, because it introduces them to problems on how to model processes in biology, and also for theoretically oriented students of biology, because it presents constructions of mathematical models and the steps needed for their investigations in a clear way and without references to other books.