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title: Differential Equations and Mathematical BiologyChapman &
Hall/CRC Mathematical Biology and Medicine Series

author: Jones, D. S.
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isbn10 | asin: 1584882964
print isbn13: 9781584882961

ebook isbn13: 9780203009314
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DIFFERENTIAL EQUATIONS AND MATHEMATICAL BIOLOGY

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CHAPMAN & HALL/CRC
Mathematical Biology and Medicine Series
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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.

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

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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


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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).

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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


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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.
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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

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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

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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

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