| Author: |
J. D. Murray
|
| Published in: | Springer |
| Release Year: | 2003 |
| ISBN: | 0-387-95228-4 |
| Pages: | 839 |
| Edition: | Third Edition |
| File Size: | 12 MB |
| File Type: | |
| Language: | English |
Description of Mathematical Biology II Spatial Models and Biomedical Applications,
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 ma-
jor 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 the 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 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 angiogenesis 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.
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 worth-while, 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, Trace 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 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, postdoctoral, 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.
jor 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 the 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 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 angiogenesis 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.
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 worth-while, 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, Trace 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 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, postdoctoral, 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.
Content of Mathematical Biology II Spatial Models and Biomedical Applications,
1. Multi-Species Waves and Practical Applications 1
1.1 Intuitive Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Waves of Pursuit and Evasion in Predator–Prey Systems . . . . . . . 5
1.3 Competition Model for the Spatial Spread of the Grey Squirrel
in Britain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Spread of Genetically Engineered Organisms . . . . . . . . . . . . . 18
1.5 Travelling Fronts in the Belousov–Zhabotinskii Reaction . . . . . . . 35
1.6 Waves in Excitable Media . . . . . . . . . . . . . . . . . . . . . . . 41
1.7 Travelling Wave Trains in Reaction Diffusion Systems with
Oscillatory Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.8 Spiral Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.9 Spiral Wave Solutions of λ–ω Reaction Diffusion Systems . . . . . . 61
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2. Spatial Pattern Formation with Reaction Diffusion Systems 71
2.1 Role of Pattern in Biology . . . . . . . . . . . . . . . . . . . . . . . 71
2.2 Reaction Diffusion (Turing) Mechanisms . . . . . . . . . . . . . . . 75
2.3 General Conditions for Diffusion-Driven Instability:
Linear Stability Analysis and Evolution of Spatial Pattern . . . . . . . 82
2.4 Detailed Analysis of Pattern Initiation in a Reaction Diffusion
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.5 Dispersion Relation, Turing Space, Scale and Geometry Effects
in Pattern Formation Models . . . . . . . . . . . . . . . . . . . . . . 103
2.6 Mode Selection and the Dispersion Relation . . . . . . . . . . . . . . 113
2.7 Pattern Generation with Single-Species Models: Spatial
Heterogeneity with the Spruce Budworm Model . . . . . . . . . . . . 120
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. Animal Coat Patterns and Other Practical Applications of Reaction
Diffusion Mechanisms 141
3.1 Mammalian Coat Patterns—‘How the Leopard Got Its Spots’ . . . . . 142
3.2 Teratologies: Examples of Animal Coat Pattern Abnormalities . . . . 156
3.3 A Pattern Formation Mechanism for Butterfly Wing Patterns . . . . . 161
3.4 Modelling Hair Patterns in a Whorl in Acetabularia . . . . . . . . . . 180
4. Pattern Formation on Growing Domains: Alligators and Snakes 192
4.1 Stripe Pattern Formation in the Alligator: Experiments . . . . . . . . 193
4.2 Modelling Concepts: Determining the Time of Stripe Formation . . . 196
4.3 Stripes and Shadow Stripes on the Alligator . . . . . . . . . . . . . . 200
4.4 Spatial Patterning of Teeth Primordia in the Alligator:
Background and Relevance . . . . . . . . . . . . . . . . . . . . . . . 205
4.5 Biology of Tooth Initiation . . . . . . . . . . . . . . . . . . . . . . . 207
4.6 Modelling Tooth Primordium Initiation: Background . . . . . . . . . 213
4.7 Model Mechanism for Alligator Teeth Patterning . . . . . . . . . . . 215
4.8 Results and Comparison with Experimental Data . . . . . . . . . . . 224
4.9 Prediction Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.10 Concluding Remarks on Alligator Tooth Spatial Patterning . . . . . . 232
4.11 Pigmentation Pattern Formation on Snakes . . . . . . . . . . . . . . . 234
4.12 Cell-Chemotaxis Model Mechanism . . . . . . . . . . . . . . . . . . 238
4.13 Simple and Complex Snake Pattern Elements . . . . . . . . . . . . . 241
4.14 Propagating Pattern Generation with the Cell-Chemotaxis System . . 248
5. Bacterial Patterns and Chemotaxis 253
5.1 Background and Experimental Results . . . . . . . . . . . . . . . . . 253
5.2 Model Mechanism for E. coli in the Semi-Solid Experiments . . . . . 260
5.3 Liquid Phase Model: Intuitive Analysis of Pattern Formation . . . . . 267
5.4 Interpretation of the Analytical Results and Numerical Solutions . . . 274
5.5 Semi-Solid Phase Model Mechanism for S. typhimurium . . . . . . . 279
5.6 Linear Analysis of the Basic Semi-Solid Model . . . . . . . . . . . . 281
5.7 Brief Outline and Results of the Nonlinear Analysis . . . . . . . . . . 287
5.8 Simulation Results, Parameter Spaces and Basic Patterns . . . . . . . 292
5.9 Numerical Results with Initial Conditions from the Experiments . . . 297
5.10 Swarm Ring Patterns with the Semi-Solid Phase Model Mechanism . 299
5.11 Branching Patterns in Bacillus subtilis . . . . . . . . . . . . . . . . . 306
6. Mechanical Theory for Generating Pattern and Form in Development 311
6.1 Introduction, Motivation and Background Biology . . . . . . . . . . . 311
6.2 Mechanical Model for Mesenchymal Morphogenesis . . . . . . . . . 319
6.3 Linear Analysis, Dispersion Relation and Pattern
Formation Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
6.4 Simple Mechanical Models Which Generate Spatial Patterns with
Complex Dispersion Relations . . . . . . . . . . . . . . . . . . . . . 334
6.5 Periodic Patterns of Feather Germs . . . . . . . . . . . . . . . . . . . 345
6.6 Cartilage Condensations in Limb Morphogenesis
and Morphogenetic Rules . . . . . . . . . . . . . . . . . . . . . . . . 350
6.7 Embryonic Fingerprint Formation . . . . . . . . . . . . . . . . . . . 358
6.8 Mechanochemical Model for the Epidermis . . . . . . . . . . . . . . 367
6.9 Formation of Microvilli . . . . . . . . . . . . . . . . . . . . . . . . . 374
6.10 Complex Pattern Formation and Tissue Interaction Models . . . . . . 381
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
7. Evolution, Morphogenetic Laws, Developmental Constraints
and Teratologies 396
7.1 Evolution and Morphogenesis . . . . . . . . . . . . . . . . . . . . . 396
7.2 Evolution and Morphogenetic Rules in Cartilage Formation in the
Vertebrate Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
7.3 Teratologies (Monsters) . . . . . . . . . . . . . . . . . . . . . . . . . 407
7.4 Developmental Constraints, Morphogenetic Rules and
the Consequences for Evolution . . . . . . . . . . . . . . . . . . . . 411
8. A Mechanical Theory of Vascular Network Formation 416
8.1 Biological Background and Motivation . . . . . . . . . . . . . . . . . 416
8.2 Cell–Extracellular Matrix Interactions for Vasculogenesis . . . . . . . 417
8.3 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
8.4 Analysis of the Model Equations . . . . . . . . . . . . . . . . . . . . 427
8.5 Network Patterns: Numerical Simulations and Conclusions . . . . . . 433
9. Epidermal Wound Healing 441
9.1 Brief History of Wound Healing . . . . . . . . . . . . . . . . . . . . 441
9.2 Biological Background: Epidermal Wounds . . . . . . . . . . . . . . 444
9.3 Model for Epidermal Wound Healing . . . . . . . . . . . . . . . . . 447
9.4 Nondimensional Form, Linear Stability and Parameter Values . . . . . 450
9.5 Numerical Solution for the Epidermal Wound Repair Model . . . . . 451
9.6 Travelling Wave Solutions for the Epidermal Model . . . . . . . . . . 454
9.7 Clinical Implications of the Epidermal Wound Model . . . . . . . . . 461
9.8 Mechanisms of Epidermal Repair in Embryos . . . . . . . . . . . . . 468
9.9 Actin Alignment in Embryonic Wounds: A Mechanical Model . . . . 471
9.10 Mechanical Model with Stress Alignment of the Actin
Filaments in Two Dimensions . . . . . . . . . . . . . . . . . . . . . 482
10. Dermal Wound Healing 491
10.1 Background and Motivation—General and Biological . . . . . . . . . 491
10.2 Logic of Wound Healing and Initial Models . . . . . . . . . . . . . . 495
10.3 Brief Review of Subsequent Developments . . . . . . . . . . . . . . 500
10.4 Model for Fibroblast-Driven Wound Healing: Residual Strain and
Tissue Remodelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
10.5 Solutions of the Model Equations and Comparison with
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
10.6 Wound Healing Model of Cook (1995) . . . . . . . . . . . . . . . . . 511
10.7 Matrix Secretion and Degradation . . . . . . . . . . . . . . . . . . . 515
10.8 Cell Movement in an Oriented Environment . . . . . . . . . . . . . . 518
10.9 Model System for Dermal Wound Healing with Tissue Structure . . . 521
10.10 One-Dimensional Model for the Structure of Pathological Scars . . . 526
10.11 Open Problems in Wound Healing . . . . . . . . . . . . . . . . . . . 530
10.12 Concluding Remarks on Wound Healing . . . . . . . . . . . . . . . . 533
11. Growth and Control of Brain Tumours 536
11.1 Medical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 538
11.2 Basic Mathematical Model of Glioma Growth and Invasion . . . . . . 542
11.3 Tumour Spread In Vitro: Parameter Estimation . . . . . . . . . . . . . 550
11.4 Tumour Invasion in the Rat Brain . . . . . . . . . . . . . . . . . . . . 559
11.5 Tumour Invasion in the Human Brain . . . . . . . . . . . . . . . . . 563
11.6 Modelling Treatment Scenarios: General Comments . . . . . . . . . . 579
11.7 Modelling Tumour Resection in Homogeneous Tissue . . . . . . . . . 580
11.8 Analytical Solution for Tumour Recurrence After Resection . . . . . 584
11.9 Modelling Surgical Resection with Brain Tissue Heterogeneity . . . . 588
11.10 Modelling the Effect of Chemotherapy on Tumour Growth . . . . . . 594
11.11 Modelling Tumour Polyclonality and Cell Mutation . . . . . . . . . . 605
12. Neural Models of Pattern Formation 614
12.1 Spatial Patterning in Neural Firing with a
Simple Activation–Inhibition Model . . . . . . . . . . . . . . . . . . 614
12.2 A Mechanism for Stripe Formation in the Visual Cortex . . . . . . . . 622
12.3 A Model for the Brain Mechanism Underlying Visual
Hallucination Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 627
12.4 Neural Activity Model for Shell Patterns . . . . . . . . . . . . . . . . 638
12.5 Shamanism and Rock Art . . . . . . . . . . . . . . . . . . . . . . . . 655
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
13. Geographic Spread and Control of Epidemics 661
13.1 Simple Model for the Spatial Spread of an Epidemic . . . . . . . . . 661
13.2 Spread of the Black Death in Europe 1347–1350 . . . . . . . . . . . 664
13.3 Brief History of Rabies: Facts and Myths . . . . . . . . . . . . . . . 669
13.4 The Spatial Spread of Rabies Among Foxes I: Background and
Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
13.5 The Spatial Spread of Rabies Among Foxes II:
Three-Species (SIR) Model . . . . . . . . . . . . . . . . . . . . . . . 681
13.6 Control Strategy Based on Wave Propagation into a
Nonepidemic Region: Estimate of Width of a Rabies Barrier . . . . . 696
13.7 Analytic Approximation for the Width of the Rabies
Control Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700
13.8 Two-Dimensional Epizootic Fronts and Effects of Variable Fox
Densities: Quantitative Predictions for a Rabies Outbreak
in England . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
13.9 Effect of Fox Immunity on the Spatial Spread of Rabies . . . . . . . . 710
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
14. Wolf Territoriality, Wolf–Deer Interaction and Survival 722
14.1 Introduction and Wolf Ecology . . . . . . . . . . . . . . . . . . . . . 722
14.2 Models for Wolf Pack Territory Formation:
Single Pack—Home Range Model . . . . . . . . . . . . . . . . . . . 729
14.3 Multi-Wolf Pack Territorial Model . . . . . . . . . . . . . . . . . . . 734
14.4 Wolf–Deer Predator–Prey Model . . . . . . . . . . . . . . . . . . . . 745
14.5 Concluding Remarks on Wolf Territoriality and Deer Survival . . . . 751
14.6 Coyote Home Range Patterns . . . . . . . . . . . . . . . . . . . . . . 753
14.7 Chippewa and Sioux Intertribal Conflict c1750–1850 . . . . . . . . . 754
Appendix
A. General Results for the Laplacian Operator in Bounded Domains 757
Bibliography 761
Index 791
1.1 Intuitive Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Waves of Pursuit and Evasion in Predator–Prey Systems . . . . . . . 5
1.3 Competition Model for the Spatial Spread of the Grey Squirrel
in Britain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 Spread of Genetically Engineered Organisms . . . . . . . . . . . . . 18
1.5 Travelling Fronts in the Belousov–Zhabotinskii Reaction . . . . . . . 35
1.6 Waves in Excitable Media . . . . . . . . . . . . . . . . . . . . . . . 41
1.7 Travelling Wave Trains in Reaction Diffusion Systems with
Oscillatory Kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.8 Spiral Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
1.9 Spiral Wave Solutions of λ–ω Reaction Diffusion Systems . . . . . . 61
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
2. Spatial Pattern Formation with Reaction Diffusion Systems 71
2.1 Role of Pattern in Biology . . . . . . . . . . . . . . . . . . . . . . . 71
2.2 Reaction Diffusion (Turing) Mechanisms . . . . . . . . . . . . . . . 75
2.3 General Conditions for Diffusion-Driven Instability:
Linear Stability Analysis and Evolution of Spatial Pattern . . . . . . . 82
2.4 Detailed Analysis of Pattern Initiation in a Reaction Diffusion
Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
2.5 Dispersion Relation, Turing Space, Scale and Geometry Effects
in Pattern Formation Models . . . . . . . . . . . . . . . . . . . . . . 103
2.6 Mode Selection and the Dispersion Relation . . . . . . . . . . . . . . 113
2.7 Pattern Generation with Single-Species Models: Spatial
Heterogeneity with the Spruce Budworm Model . . . . . . . . . . . . 120
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. Animal Coat Patterns and Other Practical Applications of Reaction
Diffusion Mechanisms 141
3.1 Mammalian Coat Patterns—‘How the Leopard Got Its Spots’ . . . . . 142
3.2 Teratologies: Examples of Animal Coat Pattern Abnormalities . . . . 156
3.3 A Pattern Formation Mechanism for Butterfly Wing Patterns . . . . . 161
3.4 Modelling Hair Patterns in a Whorl in Acetabularia . . . . . . . . . . 180
4. Pattern Formation on Growing Domains: Alligators and Snakes 192
4.1 Stripe Pattern Formation in the Alligator: Experiments . . . . . . . . 193
4.2 Modelling Concepts: Determining the Time of Stripe Formation . . . 196
4.3 Stripes and Shadow Stripes on the Alligator . . . . . . . . . . . . . . 200
4.4 Spatial Patterning of Teeth Primordia in the Alligator:
Background and Relevance . . . . . . . . . . . . . . . . . . . . . . . 205
4.5 Biology of Tooth Initiation . . . . . . . . . . . . . . . . . . . . . . . 207
4.6 Modelling Tooth Primordium Initiation: Background . . . . . . . . . 213
4.7 Model Mechanism for Alligator Teeth Patterning . . . . . . . . . . . 215
4.8 Results and Comparison with Experimental Data . . . . . . . . . . . 224
4.9 Prediction Experiments . . . . . . . . . . . . . . . . . . . . . . . . . 228
4.10 Concluding Remarks on Alligator Tooth Spatial Patterning . . . . . . 232
4.11 Pigmentation Pattern Formation on Snakes . . . . . . . . . . . . . . . 234
4.12 Cell-Chemotaxis Model Mechanism . . . . . . . . . . . . . . . . . . 238
4.13 Simple and Complex Snake Pattern Elements . . . . . . . . . . . . . 241
4.14 Propagating Pattern Generation with the Cell-Chemotaxis System . . 248
5. Bacterial Patterns and Chemotaxis 253
5.1 Background and Experimental Results . . . . . . . . . . . . . . . . . 253
5.2 Model Mechanism for E. coli in the Semi-Solid Experiments . . . . . 260
5.3 Liquid Phase Model: Intuitive Analysis of Pattern Formation . . . . . 267
5.4 Interpretation of the Analytical Results and Numerical Solutions . . . 274
5.5 Semi-Solid Phase Model Mechanism for S. typhimurium . . . . . . . 279
5.6 Linear Analysis of the Basic Semi-Solid Model . . . . . . . . . . . . 281
5.7 Brief Outline and Results of the Nonlinear Analysis . . . . . . . . . . 287
5.8 Simulation Results, Parameter Spaces and Basic Patterns . . . . . . . 292
5.9 Numerical Results with Initial Conditions from the Experiments . . . 297
5.10 Swarm Ring Patterns with the Semi-Solid Phase Model Mechanism . 299
5.11 Branching Patterns in Bacillus subtilis . . . . . . . . . . . . . . . . . 306
6. Mechanical Theory for Generating Pattern and Form in Development 311
6.1 Introduction, Motivation and Background Biology . . . . . . . . . . . 311
6.2 Mechanical Model for Mesenchymal Morphogenesis . . . . . . . . . 319
6.3 Linear Analysis, Dispersion Relation and Pattern
Formation Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
6.4 Simple Mechanical Models Which Generate Spatial Patterns with
Complex Dispersion Relations . . . . . . . . . . . . . . . . . . . . . 334
6.5 Periodic Patterns of Feather Germs . . . . . . . . . . . . . . . . . . . 345
6.6 Cartilage Condensations in Limb Morphogenesis
and Morphogenetic Rules . . . . . . . . . . . . . . . . . . . . . . . . 350
6.7 Embryonic Fingerprint Formation . . . . . . . . . . . . . . . . . . . 358
6.8 Mechanochemical Model for the Epidermis . . . . . . . . . . . . . . 367
6.9 Formation of Microvilli . . . . . . . . . . . . . . . . . . . . . . . . . 374
6.10 Complex Pattern Formation and Tissue Interaction Models . . . . . . 381
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394
7. Evolution, Morphogenetic Laws, Developmental Constraints
and Teratologies 396
7.1 Evolution and Morphogenesis . . . . . . . . . . . . . . . . . . . . . 396
7.2 Evolution and Morphogenetic Rules in Cartilage Formation in the
Vertebrate Limb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 402
7.3 Teratologies (Monsters) . . . . . . . . . . . . . . . . . . . . . . . . . 407
7.4 Developmental Constraints, Morphogenetic Rules and
the Consequences for Evolution . . . . . . . . . . . . . . . . . . . . 411
8. A Mechanical Theory of Vascular Network Formation 416
8.1 Biological Background and Motivation . . . . . . . . . . . . . . . . . 416
8.2 Cell–Extracellular Matrix Interactions for Vasculogenesis . . . . . . . 417
8.3 Parameter Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
8.4 Analysis of the Model Equations . . . . . . . . . . . . . . . . . . . . 427
8.5 Network Patterns: Numerical Simulations and Conclusions . . . . . . 433
9. Epidermal Wound Healing 441
9.1 Brief History of Wound Healing . . . . . . . . . . . . . . . . . . . . 441
9.2 Biological Background: Epidermal Wounds . . . . . . . . . . . . . . 444
9.3 Model for Epidermal Wound Healing . . . . . . . . . . . . . . . . . 447
9.4 Nondimensional Form, Linear Stability and Parameter Values . . . . . 450
9.5 Numerical Solution for the Epidermal Wound Repair Model . . . . . 451
9.6 Travelling Wave Solutions for the Epidermal Model . . . . . . . . . . 454
9.7 Clinical Implications of the Epidermal Wound Model . . . . . . . . . 461
9.8 Mechanisms of Epidermal Repair in Embryos . . . . . . . . . . . . . 468
9.9 Actin Alignment in Embryonic Wounds: A Mechanical Model . . . . 471
9.10 Mechanical Model with Stress Alignment of the Actin
Filaments in Two Dimensions . . . . . . . . . . . . . . . . . . . . . 482
10. Dermal Wound Healing 491
10.1 Background and Motivation—General and Biological . . . . . . . . . 491
10.2 Logic of Wound Healing and Initial Models . . . . . . . . . . . . . . 495
10.3 Brief Review of Subsequent Developments . . . . . . . . . . . . . . 500
10.4 Model for Fibroblast-Driven Wound Healing: Residual Strain and
Tissue Remodelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
10.5 Solutions of the Model Equations and Comparison with
Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
10.6 Wound Healing Model of Cook (1995) . . . . . . . . . . . . . . . . . 511
10.7 Matrix Secretion and Degradation . . . . . . . . . . . . . . . . . . . 515
10.8 Cell Movement in an Oriented Environment . . . . . . . . . . . . . . 518
10.9 Model System for Dermal Wound Healing with Tissue Structure . . . 521
10.10 One-Dimensional Model for the Structure of Pathological Scars . . . 526
10.11 Open Problems in Wound Healing . . . . . . . . . . . . . . . . . . . 530
10.12 Concluding Remarks on Wound Healing . . . . . . . . . . . . . . . . 533
11. Growth and Control of Brain Tumours 536
11.1 Medical Background . . . . . . . . . . . . . . . . . . . . . . . . . . 538
11.2 Basic Mathematical Model of Glioma Growth and Invasion . . . . . . 542
11.3 Tumour Spread In Vitro: Parameter Estimation . . . . . . . . . . . . . 550
11.4 Tumour Invasion in the Rat Brain . . . . . . . . . . . . . . . . . . . . 559
11.5 Tumour Invasion in the Human Brain . . . . . . . . . . . . . . . . . 563
11.6 Modelling Treatment Scenarios: General Comments . . . . . . . . . . 579
11.7 Modelling Tumour Resection in Homogeneous Tissue . . . . . . . . . 580
11.8 Analytical Solution for Tumour Recurrence After Resection . . . . . 584
11.9 Modelling Surgical Resection with Brain Tissue Heterogeneity . . . . 588
11.10 Modelling the Effect of Chemotherapy on Tumour Growth . . . . . . 594
11.11 Modelling Tumour Polyclonality and Cell Mutation . . . . . . . . . . 605
12. Neural Models of Pattern Formation 614
12.1 Spatial Patterning in Neural Firing with a
Simple Activation–Inhibition Model . . . . . . . . . . . . . . . . . . 614
12.2 A Mechanism for Stripe Formation in the Visual Cortex . . . . . . . . 622
12.3 A Model for the Brain Mechanism Underlying Visual
Hallucination Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 627
12.4 Neural Activity Model for Shell Patterns . . . . . . . . . . . . . . . . 638
12.5 Shamanism and Rock Art . . . . . . . . . . . . . . . . . . . . . . . . 655
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 659
13. Geographic Spread and Control of Epidemics 661
13.1 Simple Model for the Spatial Spread of an Epidemic . . . . . . . . . 661
13.2 Spread of the Black Death in Europe 1347–1350 . . . . . . . . . . . 664
13.3 Brief History of Rabies: Facts and Myths . . . . . . . . . . . . . . . 669
13.4 The Spatial Spread of Rabies Among Foxes I: Background and
Simple Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 673
13.5 The Spatial Spread of Rabies Among Foxes II:
Three-Species (SIR) Model . . . . . . . . . . . . . . . . . . . . . . . 681
13.6 Control Strategy Based on Wave Propagation into a
Nonepidemic Region: Estimate of Width of a Rabies Barrier . . . . . 696
13.7 Analytic Approximation for the Width of the Rabies
Control Break . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 700
13.8 Two-Dimensional Epizootic Fronts and Effects of Variable Fox
Densities: Quantitative Predictions for a Rabies Outbreak
in England . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
13.9 Effect of Fox Immunity on the Spatial Spread of Rabies . . . . . . . . 710
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
14. Wolf Territoriality, Wolf–Deer Interaction and Survival 722
14.1 Introduction and Wolf Ecology . . . . . . . . . . . . . . . . . . . . . 722
14.2 Models for Wolf Pack Territory Formation:
Single Pack—Home Range Model . . . . . . . . . . . . . . . . . . . 729
14.3 Multi-Wolf Pack Territorial Model . . . . . . . . . . . . . . . . . . . 734
14.4 Wolf–Deer Predator–Prey Model . . . . . . . . . . . . . . . . . . . . 745
14.5 Concluding Remarks on Wolf Territoriality and Deer Survival . . . . 751
14.6 Coyote Home Range Patterns . . . . . . . . . . . . . . . . . . . . . . 753
14.7 Chippewa and Sioux Intertribal Conflict c1750–1850 . . . . . . . . . 754
Appendix
A. General Results for the Laplacian Operator in Bounded Domains 757
Bibliography 761
Index 791
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