Differential Equations and Mathematical Biology

by D.S. Jones, Michael Plank, B.D. Sleeman

Overview

Deepen students’ understanding of biological phenomena

Suitable for courses on differential equations with applications to mathematical biology or as an introduction to mathematical biology, Differential Equations and Mathematical Biology, Second Edition introduces students in the physical, mathematical, and biological sciences to fundamental modeling and analytical techniques used to understand biological phenomena. In this edition, many of the chapters have been expanded to include new and topical material.

New to the Second Edition

• A section on spiral waves
• Recent developments in tumor biology
• More on the numerical solution of differential equations and numerical bifurcation analysis
• MATLAB^® files available for download online
• Many additional examples and exercises

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

Product Details

ISBN-13:	9781420083576
Publisher:	Taylor & Francis
Publication date:	11/09/2009
Series:	Chapman & Hall/CRC Mathematical Biology Series , #31
Edition description:	New Edition
Pages:	462
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

D.S. Jones, FRS, FRSE is Professor Emeritus in the Department of Mathematics at the University of Dundee in Scotland.

M.J. Plank is a senior lecturer in the Department of Mathematics and Statistics at the University of Canterbury in Christchurch, New Zealand.

B.D. Sleeman, FRSE is Professor Emeritus in the Department of Applied Mathematics at the University of Leeds in the UK.

Table of Contents

Preface to the First Edition xiii

Preface to the Second Edition xv

1 Introduction 1

1.1 Population growth 1

1.2 Administration of drugs 4

1.3 Cell division 9

1.4 Differential equations with separable variables 11

1.5 Equations of homogeneous type 14

1.6 Linear differential equations of the first order 16

1.7 Numerical solution of first-order equations 19

1.8 Symbolic computation in MATLAB? 24

1.9 Notes 27

2 Linear Ordinary Differential Equations with Constant Coefficients 33

2.1 Introduction 33

2.2 First-order linear differential equations 35

2.3 Linear equations of the second order 36

2.4 Finding the complementary function 37

2.5 Determining a particular integral 41

2.6 Forced oscillations 50

2.7 Differential equations of order n 52

2.8 Uniqueness 55

3 Systems of Linear Ordinary Differential Equations 61

3.1 First-order systems of equations with constant coefficients 61

3.2 Replacement of one differential equation by a system 64

3.3 The general system 66

3.4 The fundamental system 68

3.5 Matrix notation 72

3.6 Initial and boundary value problems 77

3.7 Solving the inhomogeneous differential equation 82

3.8 Numerical solution of linear boundary value problems 84

4 Modelling Biological Phenomena 91

4.1 Introduction 91

4.2 Heartbeat 91

4.3 Nerve impulse transmission 94

4.4 Chemical reactions 100

4.5 Predator-prey models 106

4.6 Notes 109

5 First-Order Systems of Ordinary Differential Equations 115

5.1 Existence and uniqueness 115

5.2 Epidemics 118

5.3 The phase plane and the Jacobian matrix 119

5.4 Local stability 121

5.5 Stability 128

5.6 Limit cycles 133

5.7 Forced oscillations 139

5.8 Numerical solution of systems of equations 143

5.9 Symbolic computation on first-order systems of equations and higher-order equations 147

5.10 Numerical solution of nonlinear boundary value problems 149

5.11 Appendix: existence theory 153

6 Mathematics of Heart Physiology 163

6.1 The local model 163

6.2 The threshold effect 166

6.3 The phase plane analysis and the heartbeat model 168

6.4 Physiological considerations of the heartbeat cycle 171

6.5 A model of the cardiac pacemaker 173

6.6 Notes 175

7 Mathematics of Nerve Impulse Transmission 177

7.1 Excitability and repetitive firing 177

7.2 Travelling waves 185

7.3 Qualitative behaviour of travelling waves 187

7.4 Piecewise linear model 190

7.5 Notes 194

8 Chemical Reactions 197

8.1 Wavefronts for the Belousov-Zhabotinskii reaction 197

8.2 Phase plane analysis of Fisher's equation 198

8.3 Qualitative behaviour in the general case 199

8.4 Spiral waves and λ - ω systems 204

8.5 Notes 207

9 Predator and Prey 211

9.1 Catching fish 211

9.2 The effect of fishing 213

9.3 The Volterra-Lotka model 215

10 Partial Differential Equations 223

10.1 Characteristics for equations of the first order 223

10.2 Another view of characteristics 230

10.3 Linear partial differential equations of the second order 232

10.4 Elliptic partial differential equations 235

10.5 Parabolic partial differential equations 239

10.6 Hyperbolic partial differential equations 239

10.7 The wave equation 240

10.8 Typical problems for the hyperbolic equation 245

10.9 The Euler-Darboux equation 250

10.10 Visualisation of solutions 251

11 Evolutionary Equations 259

11.1 The heat equation 259

11.2 Separation of variables 262

11.3 Simple evolutionary equations 269

11.4 Comparison theorems 277

11.5 Notes 289

12 Problems of Diffusion 293

12.1 Diffusion through membranes 293

12.2 Energy and energy estimates 299

12.3 Global behaviour of nerve impulse transmissions 304

12.4 Global behaviour in chemical reactions 308

12.5 Turing diffusion driven instability and pattern formation 311

12.6 Finite pattern forming domains 321

12.7 Notes 325

13 Bifurcation and Chaos 329

13.1 Bifurcation 329

13.2 Bifurcation of a limit cycle 334

13.3 Discrete bifurcation and period-doubling 336

13.4 Chaos 342

13.5 Stability of limit cycles 346

13.6 The Poincar? plane 350

13.7 Averaging 355

14 Numerical Bifurcation Analysis 367

14.1 Fixed points and stability 367

14.2 Path-following and bifurcation analysis 370

14.3 Following stable limit cycles 376

14.4 Bifurcation in discrete systems 378

14.5 Strange attractors and chaos 380

14.6 Stability analysis of partial differential equations 384

14.7 Notes 385

15 Growth of Tumours 389

15.1 Introduction 389

15.2 Mathematical Model I of tumour growth 392

15.3 Spherical tumour growth based on Model I 395

15.4 Stability of tumour growth based on Model I 399

15.5 Mathematical Model II of tumour growth 401

15.6 Spherical tumour growth based on Model II 404

15.7 Stability of tumour growth based on Model II 406

15.8 Notes 407

16 Epidemics 411

16.1 The Kermack-McKendrick model 411

16.2 Vaccination 413

16.3 An incubation model 414

16.4 Spreading in space 418

Answers to Selected Exercises 427

Index 439

Editorial Reviews

"… Much progress by these authors and others over the past quarter century in modeling biological and other scientific phenomena make this differential equations textbook more valuable and better motivated than ever. … The writing is clear, though the modeling is not oversimplified. Overall, this book should convince math majors how demanding math modeling needs to be and biologists that taking another course in differential equations will be worthwhile. The coauthors deserve congratulations as well as course adoptions."
—SIAM Review, Sept. 2010, Vol. 52, No. 3

"… Where this text stands out is in its thoughtful organization and the clarity of its writing. This is a very solid book … The authors succeed because they do a splendid job of integrating their treatment of differential equations with the applications, and they don’t try to do too much. … Each chapter comes with a collection of well-selected exercises, and plenty of references for further reading."
—MAA Reviews, April 2010

Praise for the First Edition
"A strength of [this book] is its concise coverage of a broad range of topics. … It is truly remarkable how much material is squeezed into the slim book’s 400 pages."
—SIAM Review, Vol. 46, No. 1

"It is remarkable that without the classical scheme (definition, theorem, and proof) it is possible to explain rather deep results like properties of the Fitz–Hugh–Nagumo model … or the Turing model. … This feature makes the reading of this text pleasant business for mathematicians. … [This book] can be recommended for students of mathematics who like to see applications, because it introduces them to problems on how to model processes in biology, and also for theoretically oriented students of biology, because it presents constructions of mathematical models and the steps needed for their investigations in a clear way and without references to other books."
—EMS Newsletter

"The title precisely reflects the contents of the book, a valuable addition to the growing literature in mathematical biology from a deterministic modeling approach. This book is a suitable textbook for multiple purposes. … Overall, topics are carefully chosen and well balanced. …The book is written by experts in the research fields of dynamical systems and population biology. As such, it presents a clear picture of how applied dynamical systems and theoretical biology interact and stimulate each other—a fascinating positive feedback whose strength is anticipated to be enhanced by outstanding texts like the work under review."
—Mathematical Reviews, Issue 2004g

From the Publisher