*Theories of Biological Pattern Formation*(1981), the editor (along with W Jäger) of the Proceedings of

*Modelling of Patterns in Space and Time*(1984), and the editor (along with H G Othmer, P K Maini) of the Proceedings of

*Experimental and Theoretical Advances in Biological Pattern Formation*(1993). Below we give extracts from reviews of various editions of Murray's four books.

**1. Asymptotic Analysis (1974), by James D Murray.**

**1.1. From the introduction.**

This book gives an introduction to the most frequently used methods for obtaining analytical approximations to functions defined by integrals or as solutions of ordinary differential equations. The emphasis throughout is on the practical use of the various techniques discussed. Heuristic reasoning, rather than mathematical rigor, is often used to justify a procedure, or some extension of it. This book is mainly intended for mathematicians and scientists whose primary aim is to get answers to practical problems.

**1.2. Review by: Ll G Chambers.**

*The Mathematical Gazette* **58** (406) (1974), 312.

This book sets out to give a heuristic treatment of asymptotic analysis which will be useful to those who are interested in actually obtaining solutions to problems. There are six chapters. In the first there is a discussion of concepts and definitions of asymptotic expansions, sequences and series, and in the second Watson's lemma and Laplace's method for integrals are considered. It is pointed out that, if a parameter is involved in an integral, the expressions for the asymptotic expression can change discontinuously for continuous changes in the parameter. In the third chapter the author deals with the method of steepest descents, and illustrates the method by finding the first term of the asymptotic expansions for the Hankel and Airy functions. The usefulness of this chapter is consider- ably enhanced by the detailed discussion on the choice of path. The fourth chapter deals with the method of stationary phase and particular attention is paid to the relation with the problem of dispersive wave motion. The fifth chapter, entitled Transform integrals, deals almost entirely with the asymptotic expansions of Fourier transforms for large values of the parameter. The sixth and longest chapter is concerned with a different problem, namely the solution of differential equation ... Whilst on the whole the book is confined to the treatment of the leading term of asymp- totic expansions, it fulfils a want, and can be recommended. The treatment is clear and the principles are set out very well.

**2. Asymptotic Analysis (2nd edition) (1984), by James D Murray.**

**2.1. Review by: P F Hsieh.**

*Mathematical Reviews*MR0740864

**(85m:34085).**

This is the second edition of the author's book [1974]. It contains the basic and elementary subjects of asymptotic methods for functions defined by integrals or as solutions of ordinary differential equations, and can serve as the resource book for readers of more specialized books ... This book is suitable for a semester course for senior or first-year graduate students with some knowledge of complex variables and ordinary differential equations.

**3. Lectures on Nonlinear Differential Equation Models in Biology (1977), by James D Murray.**

**3.1. Review by: P Holgate.**

*Biometrics*

**35**(3) (1979), 705-706.

Although from Fisher's time to the early '60s statistics has been the main point of contact between mathematics and biology, there is an older tradition of applying sets of differential equations as models of biochemical and biophysical processes within an organism. The past decade has seen a tremendous development of interest in this area of research. An interesting mathematical aspect of recent work is that it directs attention to certain non-linear differential equations, and their solution in terms of asymptotic expansions derived from singular perturbation methods. There is a contrast in attitude between workers in this field and those in the statistical and stochastic processes communities, in that here they are concerned with developing mathematical systems which reproduce very accurately the phenomena that are being modelled, rather than merely recapturing what is hoped are the salient features; and that the testing of hypotheses is envisaged in terms of critical laboratory experiments rather than the analysis of data, or the contemplation of broad agreement between theory and observation. J D Murray's book is a valuable and unconventional introduction to the field. The dominant theme is diffusion. ... sufficient detail to be understood by the non-biologist". While this may be strictly true, there is still a feeling of unreality about studying a mathematical analysis designed to resolve several competing biological theories, none of which one understands, and for that reason the book would per- haps most profitably be studied as an adjunct to a course in mathematical biology, given by someone who under-stands both sides, as the author clearly does.

**3.2. Review by: Fred L Bookstein.**

*Human Biology* **50** (4) (1978), 562-564.

No one seems to be bridging the crevasse between mathematical and organismic biology. D'Arcy Thompson's On Growth and Form , the unique masterpiece acknowledged by both fields, was the last major effort at modelling of whole organisms; it invoked old-fashioned geometry as its primary mathematics. Since then, mathematical biology has gone off in directions incompatible with organismic concerns. The elements of models are now drawn from information theory, algebra, and differential analysis quantitative or qualitative. Notwithstanding the hybrid re- searches, like R. A. Fishers on travelling waves in genetic propagation, in the main the field is badly split between two spirits: that of the physiologist and biochemist, who attempt to reproduce reliably observed curves using systems of model equations with a few parameters; and that of the "biological topologists," notably René Thom and the late C. H. Waddington, who attempt to describe the principles underlying the very presence of reliable patterning across organisms. One camp studies how biological systems work, while the other attempts to understand how it is that the first camp has something to study - why systems happen to work at all. Mr Murray s competences lie squarely on one side of the crevasse, although he finds the problems of the other side as intriguing as any intelligent layman should. The book before us deals, in five chapters, with enzyme kinetics, facilitated diffusion (by ligands across membranes), diffusion processes in chemical communication, homogeneous temporal oscillators and nonlinear wave phenomena based in those oscillators. ... From the deterministic study of nonlinear differential equations can emerge biochemical models of reproducible quantitative phenomena; but the biological aspect of these systems can be studied, at best, by analogy, while chemical mechanization of macroscopic phenomena is premature. It is that reproducibility itself which is the object of biological study, not the particular values or curves at which systems stabilize. Mr Murray is not interested in these essentially biological concerns, but only in the behaviour of the mathematical systems. One concludes from these Lectures that this branch of mathematical biology will remain far more mathematical than biological for a long time to come.

**4. Mathematical biology (1989), by James D Murray.**

**4.1. Review by: David Griffel.**

*The Mathematical Gazette*

**75**(472) (1991), 240-243.

This fat book is full of goodies. Flicking through the pages shows a stimulating mixture of graphs, equations, biological diagrams and photographs (my favourite shows an adolescent Valais goat with its front half black and its rear half white). The topics discussed include population dynamics (of course), the kinetics of enzyme reactions, chemical oscillations and waves, wavelike movements in fish, the spread of epidemics, and pattern formation in many contexts, including the coats of mammals (this is where the goat comes in), butterfly wings, sea shells, and visual hallucinations. You can enjoy the book without any prior knowledge of biology; it is written for mathematicians, by an applied mathematician who has devoted many years to serious biological research. Each biological topic is introduced carefully and in some depth. But you need to know plenty of mathematics, at the level of advanced undergraduate courses on mathematical techniques. Readers should be familiar with (or prepared to read up about) the phase plane, matrix methods for linear ordinary differential equations, asymptotic methods, elasticity theory, Laplace transforms, and so on. The method of matched expansions for singular perturbation problems is explained from scratch, in the context of the kinetics of enzyme reactions, but it would probably help to have met it before. The author is always careful to give references for mathematical techniques as well as to the biological literature.

**4.2. Review by: Karel Kuna.**

*Folia Geobotanica & Phytotaxonomica* **30** (1) (1995), 101-102.

This volume of biomathematic texts covers a wide range of methods used in mathematical modelling of living systems. Twenty chapters of the book deal with continuous and discrete population models, reaction kinetics, modelling of biological oscillators, modelling based on diffusion, wave phenomena, spatial pattern formation, epidemic models. Some of the advanced mathematics used is discussed in three appendices: phase plane analysis, means for assessing the roots of a polynomial (this is useful for stability analysis of differential equations), Hopf's bifurcation theorem with proof and examples of use. (Hopf's theorem gives the necessary conditions for the existence of periodic solutions of the system of ordinary differential equations.) I would describe the book generally as a survey of modelling by deterministic dynamical systems. It can be used as a toolkit of techniques or as a learning text. ... Throughout the book the exposition is quite clear, and each chapter is followed by a few exercises. No previous knowledge of biology is needed, but the level of mathematics required is sometimes demanding. Nevertheless, basic knowledge of ordinary differential equations and linear algebra is sufficient for informative reading. Murray's book can be an excellent guide to an exciting field of biomathematics for people who are interested in deterministic modelling. Stochastic approaches (together with some other topics - e.g. population genetics) are not considered. Of course, deterministic models are too coarse in comparison with Nature and should be used and interpreted carefully. Let me conclude by quoting the Preface. "... the use of esoteric mathematics arrogantly applied to biological problems by mathematicians who know little about real biology, together with unsubstantial claims as to how important such theories are, does little to promote the interdisciplinary involvement which is so essential."

**4.3. Review by: Jonathan G Bell.**

*SIAM Review* 32 (3) (1990), 487-489.

It has been fashionable in recent years to include in the mathematical curriculum a course on mathematical modelling, and to have biological topics included in the course. It has been less common to devote a whole course to mathematical biology. Part of the reason has been that in the past, few references have been available in textbook form to draw from at the upper undergraduate and beginning graduate level. ... Murray's new book is ... sweeping in scope, though topics of classic centrality in mathematical biology, such as neurobiology and genetics, are not well represented. This lack reflects partly the author's tastes, and partly the breadth of each of the latter subjects taken on their own. Murray, of course, is a leading biomathematician whose interests lie particularly in developmental biology, and this topic is strongly represented in the book under review. ... There is a wealth of material here, only a fraction of which could be covered in a one semester course, but I think the book would be fun to use at this level. ... I like the author's choice of topics and general style, and look forward to trying the book out on a class.

**4.4. Review by: J France.**

*Journal of the Royal Statistical Society. Series D (The Statistician) ***40** (3), Special Issue: Survey Design, Methodology and Analysis (2) (1991), 344-345.

An impressive cover, showing a magnificent yellow leopard imperiously gazing at you from a green background, sets the scene for this handsome book. 'Mathematical Biology', with its 292 figures, is a rich and beautifully illustrated textbook on a subject which represents an exciting modern application of mathematics. The emphasis throughout the book is on the application of mathematical models in helping to understand the underlying mechanisms involved in a biological process. ... This book therefore contains much useful material for biologists and mathematicians alike, at both undergraduate and research stages. Parts of the book are suitable for biological and mathematical science courses at various levels and it provides an invaluable reference work for practising modellers and biomathematicians. It is also essential reading for all those traditional applied mathematicians and biometricians who require a little cultural broadening. The book demonstrates admirably that there is much more to applied mathematics than classical studies of physical problems, and that there is much more to biomathematics than traditional biometrics, rooted in the philosophy of empiricism and steeped in the doctrines of experimental design and analysis and the obfuscations of stochastic modelling. It refreshingly illustrates the considerable potential of dynamic, deterministic, mechanistic modelling as an aid to advancing biological concepts and thereby increasing understanding in the life sciences.

**4.5. Review by: Jonathan Bell.**

*Mathematical Reviews* MR1007836 **(90g:92001).**

This book covers a diversity of topics from mathematical biology. ... The book is well written and its level is appropriate for a beginning graduate class. It has a number of exercises at the end of each chapter.

**5. Mathematical Biology (Second Edition) (1993), by James D Murray.**

**5.1. Review by: Gabriela Kirlinger.**

*Mathematical Reviews*MR1239892

**(94j:92002).**

The readers of this book are intended to be biologists as well as mathematicians. No previous knowledge of biology is assumed, but a brief description of the biological background of each topic is given. This book contains 20 chapters and several appendices on phase plane analysis, Routh-Hurwitz conditions, Hopf bifurcation and limit cycles as well as many others topics. The author considers mathematical biology to be the most exciting modern application of mathematics. He believes that mathematical biology research, to be useful and interesting, must be relevant biologically. Thus the emphasis throughout the book is on the practical application of mathematical models in helping to unravel the underlying mechanisms involved in the biological processes. The main purpose of the book is to present some of the basic and, to a large extent, generally accepted theoretical frameworks for a variety of biological models. Some of the main topics are population ecology, reaction kinetics, reaction diffusion mechanisms, travelling waves, special pattern formation, evolution and epidemiology.

**6. Mathematical Biology 1: An Introduction. Third Edition (2002); Mathematical Biology II: Spatial Models and Biomedical Applications. Third Edition (2003).**

**6.1. Review by: Leah Edelstein-Keshet.**

*SIAM Review*

**46**(1) (2004), 143-147.

Now into its third edition, Murray's book has grown to such an extent that it has undergone binary fission, spawning twins (here after nicknamed affectionately MBI and MBII). Together, these volumes total some 1362 pages, nearly double the size of the first edition. I leafed through MBI page by page, comparing chapters and sections to the corresponding parts of the first edition. I noted with pleasure that many up dates had been incorporated throughout the material; for example, population growth figures had been updated and corrected (to 2100), and much more detailed biological background with actual facts had been included in section 1.1 on continuous growth models. Nice touches of humour ("Great is the power of steady misinterpretation" Darwin, p. 4) and a good discussion of the pitfalls of curve fitting were new. My (now middle-aged) eyes appreciated the much clearer and more modern font used through out the books. This makes the text clearer and more readable. Many of the figures have also been improved, enlarged, and formatted better than in the first edition. ... MBII [is] subtitled "Spatial Models and Biomedical Applications." Many of the original chapters on the topics of PDEs, pat terns, and models for embryonic development that I had remembered from the first edition are included in this volume. But it quickly became apparent that the older material had been considerably augmented with new chapters, new topics, and a variety of works, based on research publications of the past decade. In this volume it becomes clear that compiling the third edition was a "labour of love." The book has a significantly different feel from the original first edition. Considerable scholar ship and research have gone into the many historical and background passages, and the author's enjoyment in setting down these histories, anecdotes, and curiosities is evident. ... In this volume, one can see more clearly the underlying strategy employed in constructing this two-volume survey of the field of mathematical biology: "Write about what you know best." Indeed, MBII is to some extent a "Collected Works of the J. D. Murray School." As such, it forms an impressive legacy and displays true mastery of genuinely interesting and nontrivial mathematics, coupled with solid applied mathematics techniques, motivated by actual biomedical and biological problems. Applied mathematicians will find plenty of problems here to which analytic, numeric, and approximation methods can be applied. Aspiring modellers will see many examples of the art of modelling and learn some good philosophical approaches. (Take, for example, these quotes: "The final arbiter of... biological relevance ... must lie with further ... experiments... . The interaction between experimental investigation and theoretical modelling . .. resulted in better understanding of the underlying biology," p. 234; "We argue that mathematics is required to bridge the gap between the level on which most of our knowledge is accumulating ... and the macroscopic level ... which is of primary concern," p. 533.) ... In summary, I recommend the new and expanded third edition to any serious young student interested in mathematical biology who already has a solid basis in applied mathematics. Personally, I'd not trade in my copy of the older first edition with its (comparatively) sleek and compact form, but, to my mind, the third edition volumes rank among the best graduate-level texts currently on the market.

**6.2. Review by: Peter Saunders.**

*The Mathematical Gazette* **90** (518) (2006), 373-374.

While mathematical biology is now an established discipline, it is not a subject in the same sense that mathematical physics is. It does not possess a relatively small set of fundamental laws and equations from which large numbers of applications flow, and it probably never will. Instead, mathematicians and physicists, almost always working closely with biologists, are tackling a wide range of problems using whatever techniques seem appropriate. A book on mathematical biology cannot follow the pattern we would expect in physics: basic theory leading on to applications. It cannot even be a sort of encyclopaedia of the subject, because there is far too much to cover. So Murray has selected a number of areas that he considers especially interesting or important, naturally including those to which he himself has contributed. He supplies the necessary biological (and, where appropriate, physical and chemical) background to allow someone new to the field to understand the problems; and while he expects a reasonable mathematical sophistication from his reader, his calculations are given with sufficient detail and explanation for them not be too hard for most of his intended readers to follow. ... I find myself a bit out of sympathy with this book. To explain why, let me quote from the preface, where Murray writes: 'I must stress, however, that mathematical descriptions of biological phenomena are not biological explanations. The principal use of any theory is its predictions and, even though different models might be able to create similar spatiotemporal behaviours, they are mainly distinguished by the different experiments they suggest and, of course, how closely they relate to the real biology.' Those of us who, like Murray, have been in the field long enough to remember some of what used to pass for mathematical biology, will understand why he feels it necessary to insist that we tackle real problems and look for results that actually say something about the real world and can be tested. All the same, my own view is that science is at least as much about explanation as it is about prediction and that theory can make fundamental contributions to biology just as it has to physics. ... Mathematical Biology provides a good way in to the field and a useful reference for those of us already there. It may attract more mathematicians to work in biology by showing them that there is real work to be done. What I missed, however, was the excitement that comes when you realise that there are some very deep questions in biology and that mathematics has an important role to play in tackling them.

**6.3. Review by: Trachette L Jackson.**

*Mathematical Reviews* MR1908418 **(2004b:92003).**

*Mathematical Reviews* MR1952568 **(2004b:92001).**

Thirteen years after the first edition [1989], the two-part third edition of this book made its much anticipated appearance. Part I: An Introduction, focuses on the use of ordinary differential equations to model physiological, ecological and medical phenomena. This book is timely, up-to-date, and features many new subjects including the use and abuse of fractals, temperature-dependent sex determination, dynamics of marital interactions, and HIV pathogenesis. Each of the fourteen chapters thoroughly tackles a subject by considering numerous examples. All sections begin with biological background, and at times historical context, progress to mathematical model development and end with the biological implications of the mathematical approach. This book introduces the field of mathematical biology to the reader and presents models which capture the essence of various interactions allowing their outcome to be more fully understood. Natural systems often behave in a way that reflects an underlying spatial pattern. This second volume of the third edition of Murray's *Mathematical biology* focuses on partial differential equations (spatial models) and their application to the biomedical sciences. Reaction-diffusion and mechanical mechanisms for generating spatial patterns in biology are two major areas of emphasis. Examples are taken from mammalian coat patterns, stripe and tooth formation in alligators, pigmentation patterns on snakes, bacterial chemotaxis, vascular network formation, and wound healing. There are also interesting chapters on the geographic spread of epidemics and the growth and control of brain tumors. Each chapter deals with its particular topic in great detail, usually focusing on one biological example and the associated mathematical model and results. This volume is not an introductory text and requires knowledge of partial differential equations, making it extremely useful in graduate courses and for reference.

**7. The mathematics of marriage: dynamic nonlinear models (2002), by John M Gottman, James D Murray, Catherine C Swanson, Rebecca Tyson and Kristin R Swanson.**

**7.1. Review by: John Adam.**

*Mathematical Reviews*MR1959242

**(2003k:00008).**

The authors write, "We are motivated by trying to understand why some couples divorce, but others do not, and why, among those who remain married, some are happy and some are miserable with one another. There are high levels of divorce in today's world ... . We began this modeling about seven years ago attempting to model only one phenomenon: the fact that in our laboratory we could predict from one variable which couples would eventually divorce and which would stay married É" I was rather surprised by the title of the book when I received it; I had not really thought of describing marriage in mathematical terms (as my wife will gratefully attest), though of course there is no reason why this should not be a valid level of description for the theoretical social scientist. This book is the result of a great deal of work by an interdisciplinary group of scientists, and I think deserves to be seen by a wider audience than the title might suggest.