Dynamics of Complex Interconnected Biological Systems

344 Pages·1990·5.826 MB·English

by  Michael A. B. Deakin (auth.), Thomas L. Vincent, Alistair I. Mees, Leslie S. Jennings (eds.)

MATHEMATICAL
MODELING
No.6
Edited by
William F. Lucas, Claremont Graduate School
Maynard Thompson, Indiana University
Thomas L. Vincent Alistair I. Mees
Leslie S . Jennings
Editors
Dynamics of
Complex Interconnected
Biological Systems
With 91 Illustrations
Birkhauser
Boston . Basel . Berlin
Thomas L. Vincent Alistair I. Mees
Department of Aerospace and Mathematics Department
Mechanical Engineering University of Western Australia
University of Arizona Nedlands 6009
Tucson, AZ 85721 Western Australia
USA Australia
Leslie S. Jennings
Mathematics Department
University of Western Australia
Nedlands 6009
Western Australia
Australia
Library of Congress Cataloging-in-Publication Data
Dynamics of complex interconnected biological systems / Thomas L.
Vincent, Alistair I. Mees, Leslie S. Jennings, editors.
p. cm. - (Mathematical modeling ; no. 6)
Proceedings of a workshop held in Albany, Western Australia, Jan.
1-5, 1989 and sponsored by the Dept. of Industry, Technology and
Commerce (Australia) and the National Science Foundation (USA).
I. Biological systems-Mathematical models-Congresses. 2. Game
theory-Congresses. 3. Chaotic behavior in systems-Mathematical
models-Congresses. I. Vincent, Thomas L. n. Mees, A.I.
m. Jennings, Leslie S. (Leslie Stephen) IV. Australia. Dept. of
Industry, Technology and Commerce. V. National Science Foundation
(U.S.) VI. Series: Mathematical modeling (Boston, Mass.) ; no. 6.
QH323.5.D96 1990
574.01'I-dc20 90-581
ISBN-13: 978-1-4684-6786-4 e-1SBN-13: 978-1-4684-6784-0
DOl: 10.1007/978-1-4684·6784·0
© Birkhiiuser Boston, 1990
Softcover reprint of the hardcover 1st edition1990
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system. or
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U.S.A.
Camera-ready copy prepared from the editors' TEX file.
9 8 7 6 5 432 I
v
PREFACE
This volume contains the proceedings of the U.S. Australia workshop
on Complex Interconnected Biological Systems held in Albany, Western
Australia January 1-5, 1989. The workshop was jointly sponsored by the
Department of Industry, Trade and Commerce (Australia), and the Na
tional Science Foundation (USA) under the US-Australia agreement.
Biological systems are typically hard to study mathematically. This is
particularly so in the case of systems with strong interconnections, such as
ecosystems or networks of neurons. In the past few years there have been
substantial improvements in the mathematical tools available for study
ing complexity. Theoretical advances include substantially improved un
derstanding of the features of nonlinear systems that lead to important
behaviour patterns such as chaos. Practical advances include improved
modelling techniques, and deeper understanding of complexity indicators
such as fractal dimension.
Game theory is now playing an increasingly important role in under
standing and describing evolutionary processes in interconnected systems.
The strategies of individuals which affect each other's fitness may be incor
porated into models as parameters. Strategies which have the property of
evolutionary stabilty result from particular parameter values which may be
determined using game theoretic methods. Since the main feature of living
systems is that they evolve, it seems appropriate that any model used to
describe such systems should have this feature as well. Evolutionary game
theory should lead the way in the development of such methods.
The workshop brought together researchers in Australia and the USA
who had worked on these problems or on methodologies which would be
suitable for solving them. The participants included applied mathemati
cians, control theorists, mathematical biologists, and biologists. Each par
ticipant was invited to give an informal presentation in his or her field of
expertise as related to the overall theme. The formal papers (contained
in this volume) were written after the workshop so that the authors could
take into account the workshop discussions, and relate their work to that
of other participants. To further encourage this exchange, each paper con
tained in this volume was reviewed by two other participants who then
wrote formal comments. These comments, with the author's reply in some
cases, are appended to each paper. We feel that these comments and replies
form a very valuable part of this volume in that they give the reader a share
VI PREFACE
in the workshop experience. They also provide a thread which helps join
many of the papers together. Indeed, the reader may wish to peruse the
comments first.
The papers are grouped according to the the classifications Modelling,
Tools, and Games. Part I, Modelling, includes papers dealing with gen
eral problems of modelling; 3 papers on models of specific biosystems; and
2 papers on less detailed but broader models which attempt to capture
features of classes of biological phenomena. Part II, Tools, presents re
sults and methods useful in analysis of the apparently very diverse range
of biological systems which have been modelled. There are 4 papers on
complex behaviour of simple systems, including the question of whether
simple models can be built from complex data; and 3 papers on other
tools and methods. Part III, Games, has 2 papers on evolutionary games,
and 2 papers on game-theoretic aspects of human intervention in biological
systems.
We would like to acknowledge the financial support of DITC and NSF
which made the workshop possible. Ms Cathy Nicotina provided invaluable
assistance in organising the conference. Cynthia Belonogoff supervised the
typesetting using the 'IE;X system.
Thomas L. Vincent
Alistair I. Mees
Leslie S. Jennings
Tucson, Arizona
Perth, Western Australia
August 1989
vii
PARTICIPANTS
The name, postal address and hence affiliation, telephone number,
facsimile number and E-mail address if available are included for com
pleteness. Note that E-mail addresses sometimes have to be rearranged
for local computer systems.
Participants from the United States of America
Professor Tom Vincent
Department of Aerospace and Mechanical Engineering, University of Ari
zona, Tucson, AZ 85721, USA.
Telephone: [phone] Fax: [phone]
E-mail: Bitnet% "vincentltl@arizrvax"
Professor Joel Brown
Department of Biological Science, University of Illinios at Chicago, Chicago,
IL 60680, USA.
Telephone: [phone] Fax: [phone]
E-mail: Bitnet% "u30119@uicvm"
Professor Yosef Cohen
Department of Fisheries and Wildlife, University of Minnesota, St. Paul,
MN 55108, USA.
Telephone: [phone] Fax: [phone]
E-mail: Bitnet% "sbpq@uminn1"
Professor Walt Grantham
Department of Mechanical and Materials Engineering, Washington State
University, Pullman WA 99164-2920, USA.
Telephone: [phone] Fax: [phone]
E-mail: Bitnet% "grantham@wsuvm1"
Professor Bob McKelvey
Department of Mathematical Sciences, University of Montana, Missoula,
MT 59812, USA.
Telephone: [phone] Fax: [phone]
Professor Paul Rapp
Department of Physiology and Biochemistry, The Medical College of Penn
sylvania, 3300 Henry Avenue, Philadelphia, PA 19129, USA.
Telephone: [phone] Fax: [phone]
E-mail: Bitnet%"P _RAPP@BRYNMAWR
viii PARTICIPANTS
Australian Participants
Professor Alistair Mees
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
Dr Nick Caputi
Fisheries Department, Western Australia Marine Research Laboratory, P.O.
Box 20, North Beach, WA 6020, Australia.
Telephone: [phone] Fax: [phone]
Dr Micheal Deakin
Department of Mathematics, Monash University, Melbourne, VIC 3217,
Australia.
Telephone: [phone] Fax: [phone]
Dr Phil Diamond
Mathematics Department, University of Queensland, St. Lucia, QLD 4067,
Australia.
Telephone: Fax: [phone]
E-mail: [email protected]
Dr Norm Hall
Fisheries Department, Western Australia Marine Research Laboratory, P.O.
Box 20, North Beach, WA 6020, Australia.
Telephone: [phone] Fax: [phone]
Dr Sean McElwain
Department of Mathematics, University of Newcastle, Newcastle, NSW
2308, Australia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
Dr Mary Myerscough
School of Chemistry, Macquarie University, North Ryde, NSW 2113, Aus
tralia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
Dr Mike Fisher
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
PARTICIPANTS IX
Mr J ames Glover
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
Dr Les Jennings
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
Mr Kevin Judd
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
Associate Professor Tony Pakes
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
Associate Professor Kok Lay Teo
Department of Mathematics, The University of Western Australia, Ned
lands, WA 6009, Australia.
Telephone: [phone] Fax: [phone]
E-mail: [email protected]
xi
CONTENTS
PREFACE ............................................................. v
PARTICIPANTS ..................................................... vii
PART I MODELLING
MODELLING BIOLOGICAL SYSTEMS
Michael A.B. Deakin ................................................... 2
A LENGTH-STRUCTURED MODEL OF THE WESTERN ROCK
LOBSTER FISHERY OF WESTERN AUSTRALIA
N.G. Hall, R.S. Brown, N. Caputi ..................................... 17
LEGUMES AT LOGGERHEADS: MODELLING COMPETITION
BETWEEN TWO STRAINS OF SUB-CLOVER
Anthony G. Pakes .................................................... 40
TWO DIMENSIONAL PATTERN FORMATION IN A CHEMO
TACTIC SYSTEM
M.R. Meyerscough, P.K. Maini, J.D. Murray, K.H. Winters ............ 65
MATHEMATICAL MODELLING OF THE CONTROL OF BLOOD
GLUCOSE LEVELS IN DIABETICS BY INSULIN INFUSION
M.E. Fisher .......................................................... 84
PART II TOOLS
MODELLING COMPLEX SYSTEMS
Alistair 1. Mees ...................................................... 104
DETECTING FOLDS IN CHAOTIC PROCESSES BY MAPPING
THE CONVEX HULL
J. Glover ............................................................ 125
CHAOS IN COMPLEX SYSTEMS
Kevin Judd .......................................................... 139
A CHAOTIC SYSTEM: DISCRETIZATION AND CONTROL
Walter J. Grantham, Amit M. Athalye ............................... 155