##### Document Text Contents

Page 1

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

Cover

title: Differential Equations and Mathematical BiologyChapman &

Hall/CRC Mathematical Biology and Medicine Series

author: Jones, D. S.

publisher:

isbn10 | asin: 1584882964

print isbn13: 9781584882961

ebook isbn13: 9780203009314

language:

subject

publication date:

lcc:

ddc:

subject:

cover

Page i

DIFFERENTIAL EQUATIONS AND MATHEMATICAL BIOLOGY

page_i

Page ii

CHAPMAN & HALL/CRC

Mathematical Biology and Medicine Series

Aims and scope:

This series aims to capture new developments and summarize what is known over the whole spectrum of

mathematical and computational biology and medicine. It seeks to encourage the integration of mathematical,

statistical and computational methods into biology by publishing a broad range of textbooks, reference works

and handbooks. The titles included in the series are meant to appeal to students, researchers and

professionals in the mathematical, statistical and computational sciences, fundamental biology and

bioengineering, as well as interdisciplinary researchers involved in the field. The inclusion of concrete

examples and applications, and programming techniques and examples, is highly encouraged.

Series Editors

Alison M.Etheridge

Department of Statistics

University of Oxford

Louis J.Gross

Department of Ecology and Evolutionary Biology

University of Tennessee

Suzanne Lenhart

Department of Mathematics

University of Tennessee

Philip K.Maini

Page 2

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

Mathematical Institute

University of Oxford

Hershel M.Safer

Informatics Department

Zetiq Technologies, Ltd.

Eberhard O.Voit

Department of Biometry and Epidemiology

Medical University of South Carolina

Proposals for the series should be submitted to one of the series editors above or directly to:

CRC Press UK

23 Blades Court

Deodar Road

London SW15 2NU

UK

page_ii

Page iii

Chapman & Hall/CRC Mathematical Biology and Medicine Series

DIFFERENTIAL EQUATIONS AND MATHEMATICAL BIOLOGY

D.S.JONES

B.D.SLEEMAN

CHAPMAN & HALL/CRC

A CRC Press Company

Boca Raton London NewYork Washington, D.C.

page_iii

Page iv

This edition published in the Taylor & Francis e-Library, 2005.

To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks

please go to www.eBookstore.tandf.co.uk.

Library of Congress Cataloging-in-Publication Data

Jones, D.S. (Douglas Samuel)

Differential equations and mathematical biology/D.S.Jones and B.D.Sleeman.

p. cm.—(Chapman & Hall/CRC mathematical biology and medicine series)

Originally published: London; Boston: Allen & Unwin, 1983.

ISBN 1-58488-296-4 (alk. paper)

1. Biomathematics. 2. Differential equations. I. Sleeman, B.D. II. Title. III. Series.

QH323.5.J65 2003

570′.15′1535–dc21 2002191159

This book contains information obtained from authentic and highly regarded sources. Reprinted material

is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable

efforts have been made to publish reliable data and information, but the author and the publisher cannot

assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic

or mechanical, including photocopying, microfilming, and recording, or by any information storage or

retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for

creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC

for such copying.

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are

used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com

© 2003 by Chapman & Hall/CRC

No claim to original U.S. Government works

ISBN 0-203-00931-2 Master e-book ISBN

Page 124

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

Example 10.1.1

Find the solution of

in x>0 such that u=2y when x=1.

The differential equation for the characteristics is, from (10.1.7),

Therefore the characteristics are

(10.1.12)

The slope of the characteristics is positive in x>0. Also y→−∞ as x→0 and y→∞ as x→∞. The shape of the

characteristics is, consequently, that shown in Figure 10.1.2, the larger C the closer the characteristic is to the

y-axis.

A set of curves that intersects characteristics once and once only is furnished by lines parallel to the x-axis;

another set is provided by lines parallel to the y-axis. Choose those parallel to the y-axis since the initial

values are intimated on x=1. Then, according to (10.1.10),

(10.1.13)

Note that it is always worthwhile considering parallel straight lines first for the ψ curves since they lead to the

simplest form for η. Now

and the given partial differential equation is converted to

page_199

Page 200

FIGURE 10.1.2: The characteristics of (10.1.12).

From (10.1.13), x=η and so that

has to be solved. From

(10.1.14)

is deduced

The arbitrary function F(ξ) of ξ is used rather than a constant because of the partial derivative in (10.1.14).

Page 125

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

It is required that u=2y when x=1, i.e., u=2ξ+η+ln η when η=1 or u=2ξ+1 when η=1, and this is to hold

for all ξ. Therefore

whence

Replacing ξ, η by x, y via (10.1.13) we obtain

page_200

Page 201

This solution is valid everywhere in x>0, because all the characteristics intersect x=1 and every point in x>0

lies on some characteristic.

Example 10.1.2

Find the solution of

in 0<x<y such that u=2x on y=3x.

The characteristics satisfy

whence their equation is ln y=−ln x+constant, which may be simplified to

Each characteristic in the first quadrant is therefore the branch of a hyperbola (Figure 10.1.3). The

intersecting curves may again be chosen as straight lines and to illustrate the fact that they need not be

parallel to the coordinate axes we select them to be parallel to y=x. (What would be the objection to making

them parallel to y=−x?) Thus the substitution

(10.1.15)

is made.

With this change of variable

and the partial differential equation is transformed to

FIGURE 10.1.3: The characteristic hyperbolae.

page_201

Page 202

Now, if y is eliminated from (10.1.15),

so that . However, x and ξ must both be positive in the first quadrant so that the upper

sign must be selected, i.e.,

Hence we are led to

Page 248

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html [17/09/2009 11:56:24]

nodes, 117

rest state, 145–146

of spherical tumours, 349–351

Standard boundary conditions, 69

Starling’s law, 154

Stoichiometric factor, 99

Subharmonic resonance, 134

Sulphuric acid, 98

Surface tension, 98, 342–343

Survival, 187

Symbolic computation, 71–79

Synapses, 92

System of first-order equations, 138

Systole, 84, 148

T

TAF (tumour angionesis factors), 341

Taylor’s theorem, 149

Taylor expansion, 114, 323

Taylor series, 166, 284

Temperature gradient, 100

Test determinant, 49–52

Theorems

Cayley-Hamilton, 57

comparison, 246–258

De Moivre, 30

existence, 25

Existence Theorem I, 109–112, 135–138

Existence Theorem II, 138

Existence Theorem III, 139–140

Fourier integral, 228

Gauss divergence, 265

Hodgkin-Huxley model, 94–96

Kelvin’s cable, 95

Mendelian theory of genetics, 83

Poincaré-Bendixon, 131, 162–165

Taylor, 149

uniqueness, 25, 43–45, 109, 210–211

Thoracic surgery, 88

Threshold, 159–160

epidemics, 356

heart beat cycle, 148–150

membrane potential, 93

nerve impulse transmission, 160

Trajectories, 114–115

computing, 140

for constant fishing rate, 186

critical point, 119–120

epidemics, 356–357

for logistic growth, 185

shapes of, 124–126

three-dimensional, 322

Transient, 40

Transmembrane current, 94

Travelling waves, 97

classification of, 240

determining, 240–243

in nerve impulse transmission, 167–169

in phase plane analysis of Fisher’s equation, 176–177

qualitative behaviour of, 169–172

Page 249

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html [17/09/2009 11:56:24]

and simple evolutionary equations, 239–240

Trigger waves, 100

Trigonometric functions, 34

Tuckwell, H.C., 104

Tumours, 339–341

angionesis factors (TAF), 341

avascular, 341–342

growth, 341–342

malignant, 341

mathematical model, 342–345

radius, 345, 350

spherical, 345–349

stability, 349–351

surface tension, 349

vascular, 340–341

Turing, A.M., 282, 296

Turing diffusion driven instability, 282

Turing parameter space, 291

U

Undetermined coefficients method, 31–35

Uniqueness theorem, 25, 43–45

elliptic partial differential equations, 210–211

ordinary differential equations, 109–112

Unstable focus, 120

Unstable limit cycle, 130

Unstable node, 117

page_389

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

Cover

title: Differential Equations and Mathematical BiologyChapman &

Hall/CRC Mathematical Biology and Medicine Series

author: Jones, D. S.

publisher:

isbn10 | asin: 1584882964

print isbn13: 9781584882961

ebook isbn13: 9780203009314

language:

subject

publication date:

lcc:

ddc:

subject:

cover

Page i

DIFFERENTIAL EQUATIONS AND MATHEMATICAL BIOLOGY

page_i

Page ii

CHAPMAN & HALL/CRC

Mathematical Biology and Medicine Series

Aims and scope:

This series aims to capture new developments and summarize what is known over the whole spectrum of

mathematical and computational biology and medicine. It seeks to encourage the integration of mathematical,

statistical and computational methods into biology by publishing a broad range of textbooks, reference works

and handbooks. The titles included in the series are meant to appeal to students, researchers and

professionals in the mathematical, statistical and computational sciences, fundamental biology and

bioengineering, as well as interdisciplinary researchers involved in the field. The inclusion of concrete

examples and applications, and programming techniques and examples, is highly encouraged.

Series Editors

Alison M.Etheridge

Department of Statistics

University of Oxford

Louis J.Gross

Department of Ecology and Evolutionary Biology

University of Tennessee

Suzanne Lenhart

Department of Mathematics

University of Tennessee

Philip K.Maini

Page 2

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

Mathematical Institute

University of Oxford

Hershel M.Safer

Informatics Department

Zetiq Technologies, Ltd.

Eberhard O.Voit

Department of Biometry and Epidemiology

Medical University of South Carolina

Proposals for the series should be submitted to one of the series editors above or directly to:

CRC Press UK

23 Blades Court

Deodar Road

London SW15 2NU

UK

page_ii

Page iii

Chapman & Hall/CRC Mathematical Biology and Medicine Series

DIFFERENTIAL EQUATIONS AND MATHEMATICAL BIOLOGY

D.S.JONES

B.D.SLEEMAN

CHAPMAN & HALL/CRC

A CRC Press Company

Boca Raton London NewYork Washington, D.C.

page_iii

Page iv

This edition published in the Taylor & Francis e-Library, 2005.

To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks

please go to www.eBookstore.tandf.co.uk.

Library of Congress Cataloging-in-Publication Data

Jones, D.S. (Douglas Samuel)

Differential equations and mathematical biology/D.S.Jones and B.D.Sleeman.

p. cm.—(Chapman & Hall/CRC mathematical biology and medicine series)

Originally published: London; Boston: Allen & Unwin, 1983.

ISBN 1-58488-296-4 (alk. paper)

1. Biomathematics. 2. Differential equations. I. Sleeman, B.D. II. Title. III. Series.

QH323.5.J65 2003

570′.15′1535–dc21 2002191159

This book contains information obtained from authentic and highly regarded sources. Reprinted material

is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable

efforts have been made to publish reliable data and information, but the author and the publisher cannot

assume responsibility for the validity of all materials or for the consequences of their use.

Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic

or mechanical, including photocopying, microfilming, and recording, or by any information storage or

retrieval system, without prior permission in writing from the publisher.

The consent of CRC Press LLC does not extend to copying for general distribution, for promotion, for

creating new works, or for resale. Specific permission must be obtained in writing from CRC Press LLC

for such copying.

Direct all inquiries to CRC Press LLC, 2000 N.W. Corporate Blvd., Boca Raton, Florida 33431.

Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are

used only for identification and explanation, without intent to infringe.

Visit the CRC Press Web site at www.crcpress.com

© 2003 by Chapman & Hall/CRC

No claim to original U.S. Government works

ISBN 0-203-00931-2 Master e-book ISBN

Page 124

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

Example 10.1.1

Find the solution of

in x>0 such that u=2y when x=1.

The differential equation for the characteristics is, from (10.1.7),

Therefore the characteristics are

(10.1.12)

The slope of the characteristics is positive in x>0. Also y→−∞ as x→0 and y→∞ as x→∞. The shape of the

characteristics is, consequently, that shown in Figure 10.1.2, the larger C the closer the characteristic is to the

y-axis.

A set of curves that intersects characteristics once and once only is furnished by lines parallel to the x-axis;

another set is provided by lines parallel to the y-axis. Choose those parallel to the y-axis since the initial

values are intimated on x=1. Then, according to (10.1.10),

(10.1.13)

Note that it is always worthwhile considering parallel straight lines first for the ψ curves since they lead to the

simplest form for η. Now

and the given partial differential equation is converted to

page_199

Page 200

FIGURE 10.1.2: The characteristics of (10.1.12).

From (10.1.13), x=η and so that

has to be solved. From

(10.1.14)

is deduced

The arbitrary function F(ξ) of ξ is used rather than a constant because of the partial derivative in (10.1.14).

Page 125

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html[17/09/2009 11:56:24]

It is required that u=2y when x=1, i.e., u=2ξ+η+ln η when η=1 or u=2ξ+1 when η=1, and this is to hold

for all ξ. Therefore

whence

Replacing ξ, η by x, y via (10.1.13) we obtain

page_200

Page 201

This solution is valid everywhere in x>0, because all the characteristics intersect x=1 and every point in x>0

lies on some characteristic.

Example 10.1.2

Find the solution of

in 0<x<y such that u=2x on y=3x.

The characteristics satisfy

whence their equation is ln y=−ln x+constant, which may be simplified to

Each characteristic in the first quadrant is therefore the branch of a hyperbola (Figure 10.1.3). The

intersecting curves may again be chosen as straight lines and to illustrate the fact that they need not be

parallel to the coordinate axes we select them to be parallel to y=x. (What would be the objection to making

them parallel to y=−x?) Thus the substitution

(10.1.15)

is made.

With this change of variable

and the partial differential equation is transformed to

FIGURE 10.1.3: The characteristic hyperbolae.

page_201

Page 202

Now, if y is eliminated from (10.1.15),

so that . However, x and ξ must both be positive in the first quadrant so that the upper

sign must be selected, i.e.,

Hence we are led to

Page 248

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html [17/09/2009 11:56:24]

nodes, 117

rest state, 145–146

of spherical tumours, 349–351

Standard boundary conditions, 69

Starling’s law, 154

Stoichiometric factor, 99

Subharmonic resonance, 134

Sulphuric acid, 98

Surface tension, 98, 342–343

Survival, 187

Symbolic computation, 71–79

Synapses, 92

System of first-order equations, 138

Systole, 84, 148

T

TAF (tumour angionesis factors), 341

Taylor’s theorem, 149

Taylor expansion, 114, 323

Taylor series, 166, 284

Temperature gradient, 100

Test determinant, 49–52

Theorems

Cayley-Hamilton, 57

comparison, 246–258

De Moivre, 30

existence, 25

Existence Theorem I, 109–112, 135–138

Existence Theorem II, 138

Existence Theorem III, 139–140

Fourier integral, 228

Gauss divergence, 265

Hodgkin-Huxley model, 94–96

Kelvin’s cable, 95

Mendelian theory of genetics, 83

Poincaré-Bendixon, 131, 162–165

Taylor, 149

uniqueness, 25, 43–45, 109, 210–211

Thoracic surgery, 88

Threshold, 159–160

epidemics, 356

heart beat cycle, 148–150

membrane potential, 93

nerve impulse transmission, 160

Trajectories, 114–115

computing, 140

for constant fishing rate, 186

critical point, 119–120

epidemics, 356–357

for logistic growth, 185

shapes of, 124–126

three-dimensional, 322

Transient, 40

Transmembrane current, 94

Travelling waves, 97

classification of, 240

determining, 240–243

in nerve impulse transmission, 167–169

in phase plane analysis of Fisher’s equation, 176–177

qualitative behaviour of, 169–172

Page 249

cover

file:///G|/SMILEY/1584882964__gigle.ws/1584882964_html/files/__joined.html [17/09/2009 11:56:24]

and simple evolutionary equations, 239–240

Trigger waves, 100

Trigonometric functions, 34

Tuckwell, H.C., 104

Tumours, 339–341

angionesis factors (TAF), 341

avascular, 341–342

growth, 341–342

malignant, 341

mathematical model, 342–345

radius, 345, 350

spherical, 345–349

stability, 349–351

surface tension, 349

vascular, 340–341

Turing, A.M., 282, 296

Turing diffusion driven instability, 282

Turing parameter space, 291

U

Undetermined coefficients method, 31–35

Uniqueness theorem, 25, 43–45

elliptic partial differential equations, 210–211

ordinary differential equations, 109–112

Unstable focus, 120

Unstable limit cycle, 130

Unstable node, 117

page_389