Springer murray j d mathematical biology II spatial models and biomedical applications 3rd ed (springer 2003)

Mathematical Biology
II: Spatial Models and
Biomedical Applications,
Third Edition

J. D. Murray

Springer


Interdisciplinary Applied Mathematics
Volume 18
Editors
S.S. Antman J.E. Marsden
L. Sirovich S. Wiggins
Geophysics and Planetary Sciences
Mathematical Biology
L. Glass, J.D. Murray
Mechanics and Materials
R.V. Kohn
Systems and Control
S.S. Sastry, P.S. Krishnaprasad

Problems in engineering, computational science, and the physical and biological sciences are using increasingly sophisticated mathematical techniques. Thus, the bridge
between the mathematical sciences and other disciplines is heavily traveled. The correspondingly increased dialog between the disciplines has led to the establishment of the
series: Interdisciplinary Applied Mathematics.
The purpose of this series is to meet the current and future needs for the interaction
between various science and technology areas on the one hand and mathematics on the
other. This is done, firstly, by encouraging the ways that mathematics may be applied in
traditional areas, as well as point towards new and innovative areas of applications; and
secondly, by encouraging other scientific disciplines to engage in a dialog with mathematicians outlining their problems to both access new methods and suggest innovative

developments within mathematics itself.
The series will consist of monographs and high-level texts from researchers working on
the interplay between mathematics and other fields of science and technology.


Interdisciplinary Applied Mathematics
Volumes published are listed at the end of the book.

Springer
New York
Berlin
Heidelberg
Hong Kong
London
Milan
Paris
Tokyo


J.D. Murray

Mathematical Biology
II: Spatial Models and
Biomedical Applications
Third Edition

With 298 Illustrations

Springer

J.D. Murray, FRS
Emeritus Professor
University of Oxford and
University of Washington
Box 352420
Department of Applied Mathematics
Seattle, WA 98195-2420
USA
Editors
S.S. Antman
Department of Mathematics
and Institute for Physical Science
and Technology
University of Maryland
College Park, MD 20742-4015
USA


J.E. Marsden
Control and Dynamical Systems
Mail Code 107-81
California Institute of Technology
Pasadena, CA 91125
USA


L. Sirovich
Division of Applied Mathematics
Brown University

Providence, RI 02912
USA


S. Wiggins
School of Mathematics
University of Bristol
Bristol BS8 1TW
UK


Cover illustration: c Alain Pons.
Mathematics Subject Classification (2000): 92B05, 92-01, 92C05, 92D30, 34Cxx
Library of Congress Cataloging-in-Publication Data
Murray, J.D. (James Dickson)
Mathematical biology. II: Spatial models and biomedical applications / J.D. Murray.—3rd ed.
p. cm.—(Interdisciplinary applied mathematics)
Rev. ed. of: Mathematical biology. 2nd ed. c1993.
Includes bibliographical references (p. ).
ISBN 0-387-95228-4 (alk. paper)
1. Biology—Mathematical models. I. Murray, J.D. (James Dickson) Mathematical
biology. II. Title. III. Series.
QH323.5 .M88 2001b
2001020447
570 .1 5118—dc21
ISBN 0-387-95228-4

Printed on acid-free paper.

c 2003 J.D. Murray, c 1989, 1993 Springer-Verlag Berlin Heidelberg.

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA),
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Printed in the United States of America.
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SPIN 10792366

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A member of BertelsmannSpringer Science+Business Media GmbH


To my wife Sheila, whom I married more
than forty years ago and lived happily ever
after, and to our children Mark and Sarah


. . . que se e´ l fuera de su consejo al tiempo de la
general criaci´on del mundo, i de lo que en e´ l se
encierra, i se hall´a ra con e´ l, se huvieran producido
i formado algunas cosas mejor que fueran hechas,
i otras ni se hicieran, u se enmendaran i corrigieran.
Alphonso X (Alphonso the Wise), 1221–1284
King of Castile and Leon (attributed)

If the Lord Almighty had consulted me
before embarking on creation I should
have recommended something simpler.


Preface to the Third Edition

In the thirteen years since the first edition of this book appeared the growth of mathematical biology and the diversity of applications has been astonishing. Its establishment
as a distinct discipline is no longer in question. One pragmatic indication is the increasing number of advertised positions in academia, medicine and industry around the
world; another is the burgeoning membership of societies. People working in the field
now number in the thousands. Mathematical modelling is being applied in every major discipline in the biomedical sciences. A very different application, and surprisingly
successful, is in psychology such as modelling various human interactions, escalation
to date rape and predicting divorce.
The field has become so large that, inevitably, specialised areas have developed
which are, in effect, separate disciplines such as biofluid mechanics, theoretical ecology
and so on. It is relevant therefore to ask why I felt there was a case for a new edition of
a book called simply Mathematical Biology. It is unrealistic to think that a single book
could cover even a significant part of each subdiscipline and this new edition certainly
does not even try to do this. I feel, however, that there is still justification for a book
which can demonstrate to the uninitiated some of the exciting problems that arise in
biology and give some indication of the wide spectrum of topics that modelling can
address.
In many areas the basics are more or less unchanged but the developments during
the past thirteen years have made it impossible to give as comprehensive a picture of the
current approaches in and the state of the field as was possible in the late 1980s. Even
then important areas were not included such as stochastic modelling, biofluid mechanics
and others. Accordingly in this new edition only some of the basic modelling concepts
are discussed—such as in ecology and to a lesser extent epidemiology—but references
are provided for further reading. In other areas recent advances are discussed together
with some new applications of modelling such as in marital interaction (Volume I),

growth of cancer tumours (Volume II), temperature-dependent sex determination (Volume I) and wolf territoriality (Volume II). There have been many new and fascinating
developments that I would have liked to include but practical space limitations made
it impossible and necessitated difficult choices. I have tried to give some idea of the
diversity of new developments but the choice is inevitably prejudiced.
As to general approach, if anything it is even more practical in that more emphasis
is given to the close connection many of the models have with experiment, clinical
data and in estimating real parameter values. In several of the chapters it is not yet


viii

Preface to the Third Edition

possible to relate the mathematical models to specific experiments or even biological
entities. Nevertheless such an approach has spawned numerous experiments based as
much on the modelling approach as on the actual mechanism studied. Some of the more
mathematical parts in which the biological connection was less immediate have been
excised while others that have been kept have a mathematical and technical pedagogical
aim but all within the context of their application to biomedical problems. I feel even
more strongly about the philosophy of mathematical modelling espoused in the original
preface as regards what constitutes good mathematical biology. One of the most exciting
aspects regarding the new chapters has been their genuine interdisciplinary collaborative
character. Mathematical or theoretical biology is unquestionably an interdisciplinary
science par excellence.
The unifying aim of theoretical modelling and experimental investigation in the
biomedical sciences is the elucidation of the underlying biological processes that result in a particular observed phenomenon, whether it is pattern formation in development, the dynamics of interacting populations in epidemiology, neuronal connectivity
and information processing, the growth of tumours, marital interaction and so on. I
must stress, however, that mathematical descriptions of biological phenomena are not
biological explanations. The principal use of any theory is in 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. There are numerous examples in the book.
Why use mathematics to study something as intrinsically complicated and ill understood as development, angiogenesis, wound healing, interacting population dynamics, regulatory networks, marital interaction and so on? We suggest that mathematics,
rather theoretical modelling, must be used if we ever hope to genuinely and realistically
convert an understanding of the underlying mechanisms into a predictive science. Mathematics is required to bridge the gap between the level on which most of our knowledge
is accumulating (in developmental biology it is cellular and below) and the macroscopic
level of the patterns we see. In wound healing and scar formation, for example, a mathematical approach lets us explore the logic of the repair process. Even if the mechanisms
were well understood (and they certainly are far from it at this stage) mathematics would
be required to explore the consequences of manipulating the various parameters associated with any particular scenario. In the case of such things as wound healing and
cancer growth—and now in angiogensesis with its relation to possible cancer therapy—
the number of options that are fast becoming available to wound and cancer managers
will become overwhelming unless we can find a way to simulate particular treatment
protocols before applying them in practice. The latter has been already of use in understanding the efficacy of various treatment scenarios with brain tumours (glioblastomas)
and new two step regimes for skin cancer.
The aim in all these applications is not to derive a mathematical model that takes
into account every single process because, even if this were possible, the resulting model
would yield little or no insight on the crucial interactions within the system. Rather the
goal is to develop models which capture the essence of various interactions allowing
their outcome to be more fully understood. As more data emerge from the biological
system, the models become more sophisticated and the mathematics increasingly challenging.


Preface to the Third Edition

ix

In development (by way of example) it is true that we are a long way from being able to reliably simulate actual biological development, in spite of the plethora of
models and theory that abound. Key processes are generally still poorly understood.
Despite these limitations, I feel that exploring the logic of pattern formation is worthwhile, or rather essential, even in our present state of knowledge. It allows us to take
a hypothetical mechanism and examine its consequences in the form of a mathematical model, make predictions and suggest experiments that would verify or invalidate

the model; even the latter casts light on the biology. The very process of constructing
a mathematical model can be useful in its own right. Not only must we commit to a
particular mechanism, but we are also forced to consider what is truly essential to the
process, the central players (variables) and mechanisms by which they evolve. We are
thus involved in constructing frameworks on which we can hang our understanding. The
model equations, the mathematical analysis and the numerical simulations that follow
serve to reveal quantitatively as well as qualitatively the consequences of that logical
structure.
This new edition is published in two volumes. Volume I is an introduction to the
field; the mathematics mainly involves ordinary differential equations but with some
basic partial differential equation models and is suitable for undergraduate and graduate
courses at different levels. Volume II requires more knowledge of partial differential
equations and is more suitable for graduate courses and reference.
I would like to acknowledge the encouragement and generosity of the many people who have written to me (including a prison inmate in New England) since the appearance of the first edition of this book, many of whom took the trouble to send me
details of errors, misprints, suggestions for extending some of the models, suggesting
collaborations and so on. Their input has resulted in many successful interdisciplinary
research projects several of which are discussed in this new edition. I would like to
thank my colleagues Mark Kot and Hong Qian, many of my former students, in particular Patricia Burgess, Julian Cook, Trac´e Jackson, Mark Lewis, Philip Maini, Patrick
Nelson, Jonathan Sherratt, Kristin Swanson and Rebecca Tyson for their advice or careful reading of parts of the manuscript. I would also like to thank my former secretary
Erik Hinkle for the care, thoughtfulness and dedication with which he put much of the
manuscript into LATEX and his general help in tracking down numerous obscure references and material.
I am very grateful to Professor John Gottman of the Psychology Department at the
University of Washington, a world leader in the clinical study of marital and family interactions, with whom I have had the good fortune to collaborate for nearly ten years.
Without his infectious enthusiasm, strong belief in the use of mathematical modelling,
perseverance in the face of my initial scepticism and his practical insight into human interactions I would never have become involved in developing with him a general theory
of marital interaction. I would also like to acknowledge my debt to Professor Ellworth
C. Alvord, Jr., Head of Neuropathology in the University of Washington with whom I
have collaborated for the past seven years on the modelling of the growth and control of
brain tumours. As to my general, and I hope practical, approach to modelling I am most
indebted to Professor George F. Carrier who had the major influence on me when I went

to Harvard on first coming to the U.S.A. in 1956. His astonishing insight and ability to
extract the key elements from a complex problem and incorporate them into a realistic


x

Preface to the Third Edition

and informative model is a talent I have tried to acquire throughout my career. Finally,
although it is not possible to thank by name all of my past students, postdoctorals, numerous collaborators and colleagues around the world who have encouraged me in this
field, I am certainly very much in their debt.
Looking back on my involvement with mathematics and the biomedical sciences
over the past nearly thirty years my major regret is that I did not start working in the
field years earlier.
Bainbridge Island, Washington
January 2002

J.D. Murray


Preface to the First Edition

Mathematics has always benefited from its involvement with developing sciences. Each
successive interaction revitalises and enhances the field. Biomedical science is clearly
the premier science of the foreseeable future. For the continuing health of their subject,
mathematicians must become involved with biology. With the example of how mathematics has benefited from and influenced physics, it is clear that if mathematicians do
not become involved in the biosciences they will simply not be a part of what are likely
to be the most important and exciting scientific discoveries of all time.
Mathematical biology is a fast-growing, well-recognised, albeit not clearly defined,
subject and is, to my mind, the most exciting modern application of mathematics. The

increasing use of mathematics in biology is inevitable as biology becomes more quantitative. The complexity of the biological sciences makes interdisciplinary involvement
essential. For the mathematician, biology opens up new and exciting branches, while for
the biologist, mathematical modelling offers another research tool commensurate with
a new powerful laboratory technique but only if used appropriately and its limitations
recognised. However, the use of esoteric mathematics arrogantly applied to biological problems by mathematicians who know little about the real biology, together with
unsubstantiated claims as to how important such theories are, do little to promote the
interdisciplinary involvement which is so essential.
Mathematical biology research, to be useful and interesting, must be relevant biologically. The best models show how a process works and then predict what may follow. If these are not already obvious to the biologists and the predictions turn out to be
right, then you will have the biologists’ attention. Suggestions as to what the governing
mechanisms are may evolve from this. Genuine interdisciplinary research and the use
of models can produce exciting results, many of which are described in this book.
No previous knowledge of biology is assumed of the reader. With each topic discussed I give a brief description of the biological background sufficient to understand
the models studied. Although stochastic models are important, to keep the book within
reasonable bounds, I deal exclusively with deterministic models. The book provides a
toolkit of modelling techniques with numerous examples drawn from population ecology, reaction kinetics, biological oscillators, developmental biology, evolution, epidemiology and other areas.
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 book also illustrates some of the pitfalls of indiscriminate, naive or un-


xii

Preface to the First Edition

informed use of models. I hope the reader will acquire a practical and realistic view
of biological modelling and the mathematical techniques needed to get approximate
quantitative solutions and will thereby realise the importance of relating the models and
results to the real biological problems under study. If the use of a model stimulates
experiments—even if the model is subsequently shown to be wrong—then it has been
successful. Models can provide biological insight and be very useful in summarising,
interpreting and interpolating real data. I hope the reader will also learn that (certainly

at this stage) there is usually no ‘right’ model: producing similar temporal or spatial patterns to those experimentally observed is only a first step and does not imply the model
mechanism is the one which applies. Mathematical descriptions are not explanations.
Mathematics can never provide the complete solution to a biological problem on its
own. Modern biology is certainly not at the stage where it is appropriate for mathematicians to try to construct comprehensive theories. A close collaboration with biologists is
needed for realism, stimulation and help in modifying the model mechanisms to reflect
the biology more accurately.
Although this book is titled mathematical biology it is not, and could not be, a
definitive all-encompassing text. The immense breadth of the field necessitates a restricted choice of topics. Some of the models have been deliberately kept simple for
pedagogical purposes. The exclusion of a particular topic—population genetics, for
example—in no way reflects my view as to its importance. However, I hope the range
of topics discussed will show how exciting intercollaborative research can be and how
significant a role mathematics can play. 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. The material presented does not purport to be the latest developments in the various fields, many of which are constantly expanding. The already
lengthy list of references is by no means exhaustive and I apologise for the exclusion of
many that should be included in a definitive list.
With the specimen models discussed and the philosophy which pervades the book,
the reader should be in a position to tackle the modelling of genuinely practical problems with realism. From a mathematical point of view, the art of good modelling relies
on: (i) a sound understanding and appreciation of the biological problem; (ii) a realistic
mathematical representation of the important biological phenomena; (iii) finding useful solutions, preferably quantitative; and what is crucially important; (iv) a biological
interpretation of the mathematical results in terms of insights and predictions. The mathematics is dictated by the biology and not vice versa. Sometimes the mathematics can
be very simple. Useful mathematical biology research is not judged by mathematical
standards but by different and no less demanding ones.
The book is suitable for physical science courses at various levels. The level of
mathematics needed in collaborative biomedical research varies from the very simple to
the sophisticated. Selected chapters have been used for applied mathematics courses in
the University of Oxford at the final-year undergraduate and first-year graduate levels. In
the U.S.A. the material has also been used for courses for students from the second-year
undergraduate level through graduate level. It is also accessible to the more theoretically
oriented bioscientists who have some knowledge of calculus and differential equations.

I would like to express my gratitude to the many colleagues around the world who
have, over the past few years, commented on various chapters of the manuscript, made


Preface to the First Edition

xiii

valuable suggestions and kindly provided me with photographs. I would particularly
like to thank Drs. Philip Maini, David Lane, and Diana Woodward and my present
graduate students who read various drafts with such care, specifically Daniel Bentil,
Meghan Burke, David Crawford, Michael Jenkins, Mark Lewis, Gwen Littlewort, Mary
Myerscough, Katherine Rogers and Louisa Shaw.
Oxford
January 1989

J.D. Murray


This page intentionally left blank


Table of Contents

CONTENTS, VOLUME II
Preface to the Third Edition

vii

Preface to the First Edition

xi

1.

1
1
5

2.

Multi-Species Waves and Practical Applications
1.1
Intuitive Expectations . . . . . . . . . . . . . . . . . . . . . .
1.2
Waves of Pursuit and Evasion in Predator–Prey Systems . . .
1.3
Competition Model for the Spatial Spread of the Grey Squirrel
in Britain . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Spread of Genetically Engineered Organisms . . . . . . . . .
1.5
Travelling Fronts in the Belousov–Zhabotinskii Reaction . . .
1.6
Waves in Excitable Media . . . . . . . . . . . . . . . . . . .
1.7
Travelling Wave Trains in Reaction Diffusion Systems with
Oscillatory Kinetics . . . . . . . . . . . . . . . . . . . . . . .
1.8
Spiral Waves . . . . . . . . . . . . . . . . . . . . . . . . . .

1.9
Spiral Wave Solutions of λ–ω Reaction Diffusion Systems . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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49
54
61
67

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71

71
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82

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90

Spatial Pattern Formation with Reaction Diffusion Systems
2.1
Role of Pattern in Biology . . . . . . . . . . . . . . . . . . . .
2.2
Reaction Diffusion (Turing) Mechanisms . . . . . . . . . . . .
2.3
General Conditions for Diffusion-Driven Instability:
Linear Stability Analysis and Evolution of Spatial Pattern . . . .
2.4
Detailed Analysis of Pattern Initiation in a Reaction Diffusion
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5
Dispersion Relation, Turing Space, Scale and Geometry Effects
in Pattern Formation Models . . . . . . . . . . . . . . . . . . .
2.6
Mode Selection and the Dispersion Relation . . . . . . . . . . .
2.7
Pattern Generation with Single-Species Models: Spatial
Heterogeneity with the Spruce Budworm Model . . . . . . . . .

. . . 103
. . . 113
. . . 120


xvi

Contents, Volume II

2.8

Spatial Patterns in Scalar Population Interaction Diffusion
Equations with Convection: Ecological Control Strategies . . . . . . . 125
2.9
Nonexistence of Spatial Patterns in Reaction Diffusion Systems:
General and Particular Results . . . . . . . . . . . . . . . . . . . . . 130
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.

4.

5.

6.

Animal Coat Patterns and Other Practical Applications of Reaction
Diffusion Mechanisms
3.1
Mammalian Coat Patterns—‘How the Leopard Got Its Spots’ . .

3.2
Teratologies: Examples of Animal Coat Pattern Abnormalities .
3.3
A Pattern Formation Mechanism for Butterfly Wing Patterns . .
3.4
Modelling Hair Patterns in a Whorl in Acetabularia . . . . . . .

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Pattern Formation on Growing Domains: Alligators and Snakes
4.1
Stripe Pattern Formation in the Alligator: Experiments . . . . . .
4.2
Modelling Concepts: Determining the Time of Stripe Formation .
4.3
Stripes and Shadow Stripes on the Alligator . . . . . . . . . . . .
4.4
Spatial Patterning of Teeth Primordia in the Alligator:
Background and Relevance . . . . . . . . . . . . . . . . . . . . .
4.5
Biology of Tooth Initiation . . . . . . . . . . . . . . . . . . . . .
4.6
Modelling Tooth Primordium Initiation: Background . . . . . . .
4.7
Model Mechanism for Alligator Teeth Patterning . . . . . . . . .
4.8
Results and Comparison with Experimental Data . . . . . . . . .

4.9
Prediction Experiments . . . . . . . . . . . . . . . . . . . . . . .
4.10 Concluding Remarks on Alligator Tooth Spatial Patterning . . . .
4.11 Pigmentation Pattern Formation on Snakes . . . . . . . . . . . . .
4.12 Cell-Chemotaxis Model Mechanism . . . . . . . . . . . . . . . .
4.13 Simple and Complex Snake Pattern Elements . . . . . . . . . . .
4.14 Propagating Pattern Generation with the Cell-Chemotaxis System

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141
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156
161
180

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Bacterial Patterns and Chemotaxis
5.1
Background and Experimental Results . . . . . . . . . . . . . . . .
5.2
Model Mechanism for E. coli in the Semi-Solid Experiments . . . .
5.3
Liquid Phase Model: Intuitive Analysis of Pattern Formation . . . .
5.4
Interpretation of the Analytical Results and Numerical Solutions . .
5.5
Semi-Solid Phase Model Mechanism for S. typhimurium . . . . . .
5.6
Linear Analysis of the Basic Semi-Solid Model . . . . . . . . . . .
5.7
Brief Outline and Results of the Nonlinear Analysis . . . . . . . . .
5.8
Simulation Results, Parameter Spaces and Basic Patterns . . . . . .
5.9
Numerical Results with Initial Conditions from the Experiments . .
5.10 Swarm Ring Patterns with the Semi-Solid Phase Model Mechanism

5.11 Branching Patterns in Bacillus subtilis . . . . . . . . . . . . . . . .

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253
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267
274
279
281
287
292
297
299
306

Mechanical Theory for Generating Pattern and Form in Development
311
6.1
Introduction, Motivation and Background Biology . . . . . . . . . . . 311


Contents, Volume II

xvii

Mechanical Model for Mesenchymal Morphogenesis . . . . . . .
Linear Analysis, Dispersion Relation and Pattern
Formation Potential . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
Simple Mechanical Models Which Generate Spatial Patterns with
Complex Dispersion Relations . . . . . . . . . . . . . . . . . . .
6.5
Periodic Patterns of Feather Germs . . . . . . . . . . . . . . . . .
6.6
Cartilage Condensations in Limb Morphogenesis
and Morphogenetic Rules . . . . . . . . . . . . . . . . . . . . . .
6.7
Embryonic Fingerprint Formation . . . . . . . . . . . . . . . . .
6.8
Mechanochemical Model for the Epidermis . . . . . . . . . . . .
6.9
Formation of Microvilli . . . . . . . . . . . . . . . . . . . . . . .
6.10 Complex Pattern Formation and Tissue Interaction Models . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 319

6.2
6.3

7.

8.

9.

Evolution, Morphogenetic Laws, Developmental Constraints
and Teratologies
7.1
Evolution and Morphogenesis . . . . . . . . . . . . . . . . . . .
7.2
Evolution and Morphogenetic Rules in Cartilage Formation in the
Vertebrate Limb . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3
Teratologies (Monsters) . . . . . . . . . . . . . . . . . . . . . . .
7.4
Developmental Constraints, Morphogenetic Rules and
the Consequences for Evolution . . . . . . . . . . . . . . . . . .
A Mechanical Theory of Vascular Network Formation
8.1
Biological Background and Motivation . . . . . . . . . . .
8.2
Cell–Extracellular Matrix Interactions for Vasculogenesis .
8.3
Parameter Values . . . . . . . . . . . . . . . . . . . . . .
8.4
Analysis of the Model Equations . . . . . . . . . . . . . .
8.5
Network Patterns: Numerical Simulations and Conclusions

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Epidermal Wound Healing
9.1
Brief History of Wound Healing . . . . . . . . . . . . . . . .
9.2
Biological Background: Epidermal Wounds . . . . . . . . . .
9.3
Model for Epidermal Wound Healing . . . . . . . . . . . . .
9.4
Nondimensional Form, Linear Stability and Parameter Values .
9.5
Numerical Solution for the Epidermal Wound Repair Model .
9.6
Travelling Wave Solutions for the Epidermal Model . . . . . .
9.7
Clinical Implications of the Epidermal Wound Model . . . . .
9.8
Mechanisms of Epidermal Repair in Embryos . . . . . . . . .
9.9
Actin Alignment in Embryonic Wounds: A Mechanical Model
9.10 Mechanical Model with Stress Alignment of the Actin
Filaments in Two Dimensions . . . . . . . . . . . . . . . . .

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10. Dermal Wound Healing
491
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xviii

Contents, Volume II

10.2
10.3
10.4

Logic of Wound Healing and Initial Models . . . . . . . . . . . .
Brief Review of Subsequent Developments . . . . . . . . . . . .
Model for Fibroblast-Driven Wound Healing: Residual Strain and
Tissue Remodelling . . . . . . . . . . . . . . . . . . . . . . . . .
10.5 Solutions of the Model Equations and Comparison with
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Wound Healing Model of Cook (1995) . . . . . . . . . . . . . . .
10.7 Matrix Secretion and Degradation . . . . . . . . . . . . . . . . .
10.8 Cell Movement in an Oriented Environment . . . . . . . . . . . .
10.9 Model System for Dermal Wound Healing with Tissue Structure .
10.10 One-Dimensional Model for the Structure of Pathological Scars .
10.11 Open Problems in Wound Healing . . . . . . . . . . . . . . . . .
10.12 Concluding Remarks on Wound Healing . . . . . . . . . . . . . .
11. Growth and Control of Brain Tumours
11.1 Medical Background . . . . . . . . . . . . . . . . . . . . . .
11.2 Basic Mathematical Model of Glioma Growth and Invasion . .
11.3 Tumour Spread In Vitro: Parameter Estimation . . . . . . . . .
11.4 Tumour Invasion in the Rat Brain . . . . . . . . . . . . . . . .
11.5 Tumour Invasion in the Human Brain . . . . . . . . . . . . .
11.6 Modelling Treatment Scenarios: General Comments . . . . . .
11.7 Modelling Tumour Resection in Homogeneous Tissue . . . . .
11.8 Analytical Solution for Tumour Recurrence After Resection .
11.9 Modelling Surgical Resection with Brain Tissue Heterogeneity

11.10 Modelling the Effect of Chemotherapy on Tumour Growth . .
11.11 Modelling Tumour Polyclonality and Cell Mutation . . . . . .
12. Neural Models of Pattern Formation
12.1 Spatial Patterning in Neural Firing with a
Simple Activation–Inhibition Model . . . . . . . . . .
12.2 A Mechanism for Stripe Formation in the Visual Cortex
12.3 A Model for the Brain Mechanism Underlying Visual
Hallucination Patterns . . . . . . . . . . . . . . . . . .
12.4 Neural Activity Model for Shell Patterns . . . . . . . .
12.5 Shamanism and Rock Art . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13. Geographic Spread and Control of Epidemics
13.1 Simple Model for the Spatial Spread of an Epidemic . . . . .
13.2 Spread of the Black Death in Europe 1347–1350 . . . . . . .
13.3 Brief History of Rabies: Facts and Myths . . . . . . . . . . .
13.4 The Spatial Spread of Rabies Among Foxes I: Background and
Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . .
13.5 The Spatial Spread of Rabies Among Foxes II:
Three-Species (SIR) Model . . . . . . . . . . . . . . . . . . .

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Contents, Volume II

Control Strategy Based on Wave Propagation into a
Nonepidemic Region: Estimate of Width of a Rabies Barrier .
13.7 Analytic Approximation for the Width of the Rabies
Control Break . . . . . . . . . . . . . . . . . . . . . . . . . .
13.8 Two-Dimensional Epizootic Fronts and Effects of Variable Fox
Densities: Quantitative Predictions for a Rabies Outbreak
in England . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.9 Effect of Fox Immunity on the Spatial Spread of Rabies . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xix

13.6

14. Wolf Territoriality, Wolf–Deer Interaction and Survival
14.1 Introduction and Wolf Ecology . . . . . . . . . . . . . . . . .
14.2 Models for Wolf Pack Territory Formation:
Single Pack—Home Range Model . . . . . . . . . . . . . . .
14.3 Multi-Wolf Pack Territorial Model . . . . . . . . . . . . . . .
14.4 Wolf–Deer Predator–Prey Model . . . . . . . . . . . . . . . .
14.5 Concluding Remarks on Wolf Territoriality and Deer Survival
14.6 Coyote Home Range Patterns . . . . . . . . . . . . . . . . . .
14.7 Chippewa and Sioux Intertribal Conflict c1750–1850 . . . . .

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734
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751
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754

Appendix
A. General Results for the Laplacian Operator in Bounded Domains

757

Bibliography

761

Index

791


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Table of Contents (continued)

CONTENTS, VOLUME I
J.D. Murray: Mathematical Biology, I: An Introduction
Preface to the Third Edition

vii

Preface to the First Edition

xi

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Models for Interacting Populations
3.1
Predator–Prey Models: Lotka–Volterra Systems . . . . . . . . . . . .
3.2
Complexity and Stability . . . . . . . . . . . . . . . . . . . . . . . .

79
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3.

Continuous Population Models for Single Species
1.1
Continuous Growth Models . . . . . . . . . . . . . . . . . . . .
1.2
Insect Outbreak Model: Spruce Budworm . . . . . . . . . . . .
1.3
Delay Models . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4

Linear Analysis of Delay Population Models: Periodic Solutions
1.5
Delay Models in Physiology: Periodic Dynamic Diseases . . . .
1.6
Harvesting a Single Natural Population . . . . . . . . . . . . .
1.7
Population Model with Age Distribution . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discrete Population Models for a Single Species
2.1
Introduction: Simple Models . . . . . . . . . . .
2.2
Cobwebbing: A Graphical Procedure of Solution
2.3
Discrete Logistic-Type Model: Chaos . . . . . .
2.4
Stability, Periodic Solutions and Bifurcations . .
2.5
Discrete Delay Models . . . . . . . . . . . . . .
2.6
Fishery Management Model . . . . . . . . . . .
2.7
Ecological Implications and Caveats . . . . . . .
2.8
Tumour Cell Growth . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . .

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xxii

Contents, Volume I

3.3
3.4

Realistic Predator–Prey Models . . . . . . . . . . . . .
Analysis of a Predator–Prey Model with Limit Cycle
Periodic Behaviour: Parameter Domains of Stability . .
3.5
Competition Models: Competitive Exclusion Principle
3.6
Mutualism or Symbiosis . . . . . . . . . . . . . . . .
3.7
General Models and Cautionary Remarks . . . . . . .
3.8
Threshold Phenomena . . . . . . . . . . . . . . . . .
3.9
Discrete Growth Models for Interacting Populations . .
3.10 Predator–Prey Models: Detailed Analysis . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Temperature-Dependent Sex Determination (TSD)
4.1
Biological Introduction and Historical Asides on the Crocodilia .
4.2
Nesting Assumptions and Simple Population Model . . . . . . .
4.3
Age-Structured Population Model for Crocodilia . . . . . . . .
4.4
Density-Dependent Age-Structured Model Equations . . . . . .
4.5
Stability of the Female Population in Wet Marsh Region I . . . .
4.6
Sex Ratio and Survivorship . . . . . . . . . . . . . . . . . . . .
4.7
Temperature-Dependent Sex Determination (TSD) Versus
Genetic Sex Determination (GSD) . . . . . . . . . . . . . . . .
4.8
Related Aspects on Sex Determination . . . . . . . . . . . . . .

Exercise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Modelling the Dynamics of Marital Interaction: Divorce Prediction
and Marriage Repair
5.1
Psychological Background and Data:
Gottman and Levenson Methodology . . . . . . . . . . . . . . .
5.2
Marital Typology and Modelling Motivation . . . . . . . . . . .
5.3
Modelling Strategy and the Model Equations . . . . . . . . . .
5.4
Steady States and Stability . . . . . . . . . . . . . . . . . . . .
5.5
Practical Results from the Model . . . . . . . . . . . . . . . . .
5.6
Benefits, Implications and Marriage Repair Scenarios . . . . . .

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Reaction Kinetics
6.1
Enzyme Kinetics: Basic Enzyme Reaction . . . . . .

6.2
Transient Time Estimates and Nondimensionalisation
6.3
Michaelis–Menten Quasi-Steady State Analysis . . .
6.4
Suicide Substrate Kinetics . . . . . . . . . . . . . .
6.5
Cooperative Phenomena . . . . . . . . . . . . . . .
6.6
Autocatalysis, Activation and Inhibition . . . . . . .
6.7
Multiple Steady States, Mushrooms and Isolas . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Biological Oscillators and Switches
218
7.1
Motivation, Brief History and Background . . . . . . . . . . . . . . . 218
7.2
Feedback Control Mechanisms . . . . . . . . . . . . . . . . . . . . . 221


Contents, Volume I

Oscillators and Switches with Two or More Species:
General Qualitative Results . . . . . . . . . . . . . .
7.4

Simple Two-Species Oscillators: Parameter Domain
Determination for Oscillations . . . . . . . . . . . .
7.5
Hodgkin–Huxley Theory of Nerve Membranes:
FitzHugh–Nagumo Model . . . . . . . . . . . . . .
7.6
Modelling the Control of Testosterone Secretion and
Chemical Castration . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . .

xxiii

7.3

8.

9.

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BZ Oscillating Reactions
8.1
Belousov Reaction and the Field–K¨or¨os–Noyes (FKN) Model
8.2
Linear Stability Analysis of the FKN Model and Existence
of Limit Cycle Solutions . . . . . . . . . . . . . . . . . . . .

8.3
Nonlocal Stability of the FKN Model . . . . . . . . . . . . .
8.4
Relaxation Oscillators: Approximation for the
Belousov–Zhabotinskii Reaction . . . . . . . . . . . . . . . .
8.5
Analysis of a Relaxation Model for Limit Cycle Oscillations
in the Belousov–Zhabotinskii Reaction . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Perturbed and Coupled Oscillators and Black Holes
9.1
Phase Resetting in Oscillators . . . . . . . . . . . . . . . .
9.2
Phase Resetting Curves . . . . . . . . . . . . . . . . . . . .
9.3
Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . .
9.4
Black Holes in Real Biological Oscillators . . . . . . . . . .
9.5
Coupled Oscillators: Motivation and Model System . . . . .
9.6
Phase Locking of Oscillations: Synchronisation in Fireflies .
9.7
Singular Perturbation Analysis: Preliminary Transformation
9.8
Singular Perturbation Analysis: Transformed System . . . .
9.9
Singular Perturbation Analysis: Two-Time Expansion . . . .
9.10 Analysis of the Phase Shift Equation and Application
to Coupled Belousov–Zhabotinskii Reactions . . . . . . . .

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10. Dynamics of Infectious Diseases
10.1 Historical Aside on Epidemics . . . . . . . . . . . . . . . . .
10.2 Simple Epidemic Models and Practical Applications . . . . .
10.3 Modelling Venereal Diseases . . . . . . . . . . . . . . . . . .
10.4 Multi-Group Model for Gonorrhea and Its Control . . . . . . .
10.5 AIDS: Modelling the Transmission Dynamics of the Human
Immunodeficiency Virus (HIV) . . . . . . . . . . . . . . . . .
10.6 HIV: Modelling Combination Drug Therapy . . . . . . . . . .
10.7 Delay Model for HIV Infection with Drug Therapy . . . . . .
10.8 Modelling the Population Dynamics of Acquired Immunity to
Parasite Infection . . . . . . . . . . . . . . . . . . . . . . . .

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xxiv

Contents, Volume I

10.9
10.10
10.11
10.12

Age-Dependent Epidemic Model and Threshold Criterion .
Simple Drug Use Epidemic Model and Threshold Analysis
Bovine Tuberculosis Infection in Badgers and Cattle . . .
Modelling Control Strategies for Bovine Tuberculosis
in Badgers and Cattle . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11. Reaction Diffusion, Chemotaxis, and Nonlocal Mechanisms
11.1 Simple Random Walk and Derivation of the Diffusion Equation
11.2 Reaction Diffusion Equations . . . . . . . . . . . . . . . . . . .
11.3 Models for Animal Dispersal . . . . . . . . . . . . . . . . . . .
11.4 Chemotaxis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.5 Nonlocal Effects and Long Range Diffusion . . . . . . . . . . .
11.6 Cell Potential and Energy Approach to Diffusion
and Long Range Effects . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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12. Oscillator-Generated Wave Phenomena
12.1 Belousov–Zhabotinskii Reaction Kinematic Waves . . . . . . .

12.2 Central Pattern Generator: Experimental Facts in the Swimming
of Fish . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12.3 Mathematical Model for the Central Pattern Generator . . . . .
12.4 Analysis of the Phase Coupled Model System . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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13. Biological Waves: Single-Species Models
13.1 Background and the Travelling Waveform . . . . . . . . . . .
13.2 Fisher–Kolmogoroff Equation and Propagating Wave Solutions
13.3 Asymptotic Solution and Stability of Wavefront Solutions
of the Fisher–Kolmogoroff Equation . . . . . . . . . . . . . .
13.4 Density-Dependent Diffusion-Reaction Diffusion Models
and Some Exact Solutions . . . . . . . . . . . . . . . . . . .
13.5 Waves in Models with Multi-Steady State Kinetics:
Spread and Control of an Insect Population . . . . . . . . . .
13.6 Calcium Waves on Amphibian Eggs: Activation Waves
on Medaka Eggs . . . . . . . . . . . . . . . . . . . . . . . .
13.7 Invasion Wavespeeds with Dispersive Variability . . . . . . .
13.8 Species Invasion and Range Expansion . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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14. Use and Abuse of Fractals
14.1 Fractals: Basic Concepts and Biological Relevance . . .
14.2 Examples of Fractals and Their Generation . . . . . . .
14.3 Fractal Dimension: Concepts and Methods of Calculation
14.4 Fractals or Space-Filling? . . . . . . . . . . . . . . . . .

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