Table Of ContentCATASTROPHES
IN NATURE
AND SOCIETH
Mathematical Modeling of Comple Systems
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REM G KHLEBOPROS
Russian Academy of Sciences, Russia
VICTOR A OKHONIN
Toronto University, Canada
ABRAM I FET
Novosibirsk University, Russia
CATASTROPHES
IN NATURE
AND SOCIETY
hMhathematicaal Modeling of Complex Systems
World Scientific
NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Khlebopros, Rem Grigorievich.
Catastrophes in nature and society : mathematical modeling of complex systems / Rem
Khlebopros, Victor Okhonin, Abram Fet.
p. cm.
Includes bibliographical references and index.
ISBN-13 978-981-256-917-2 -- ISBN-10 981-256-917-0
1. Human ecology--Mathematical models. 2. Natural disasters--Mathematical models.
I. Okhonin, Victor. II. Fet, A. I. (Abram Ilyich). III. Title.
GF23.M35 K45 2006
304.201'5118--dc22
2006049232
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2007 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
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Printed in Singapore.
Sandhya - Catastrophes in nature.pmd 1 1/8/2007, 9:15 AM
Dedicated to Prof. Kenneth R. Kenyon of Harvard University
who has worked a miracle.
Rem G. Khlebopros
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Preface
Complex systems, which defy investigation by the classical methods
of mathematical physics, have been among key challenges of science
in the recent century. Studies of complex systems primarily address
understanding of life and its origin. This mystery prompted Ilya
Prigozhine and Manfred Eigen, great scientists and Nobel laureates, to
seek the specific properties of living systems that could let an insight into
possible ways of how life may have begun. They are presumably self-
organization and selection of the most efficient specimen. In fact, the
very idea of self-organizing systems with natural selection originally
comes from economics rather than from biology and belongs to Adam
Smith and Malthus. Malthus’s idea of overpopulation encouraged
Darvin’s discovery of the mechanism that drives the evolution of species.
Self-organization and selection are not specific of living matter but
are common to many inorganic systems. They are found, for instance, in
convection whirls in fluid and gas which one can observe in a river or in
air flows around a hot chimney. However, of special interest are the
living systems, from populations of cells in an organism (e.g., cancer
cells) or pests in a forest to economic and social systems that appear and
disappear in human cultures. In their evolution these systems can
develop the states of stable or unstable equilibrium. A system that arrives
at an unstable state is prone to a catastrophe — invasion of pests, disease,
or crisis — which changes a system dramatically or often drives it to
collapse. That is the reason why the book is called Catastrophes in
Nature and Society.
Classification and prediction of catastrophes is the subject of the
catastrophe theory by Rene Thom. Yet, its topological methods cannot
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viii Catastrophes in Nature and Society: Mathematical Modeling of Complex Systems
provide quantitative predictions so urgent in practical applications. On
the other hand, the traditional approaches of mathematical physics
implying differential equations are inapplicable to complex systems. The
authors of the book approach the subject of complex systems using phase
portrait modeling which is closely related to Newton’s steepest descent
and is thus as classical as differential calculus. Viewed in the context of
phase portraits, the processes that seem disconnected and chaotic at first
sight show up in their consistent intrinsic logic.
The book written in a popular manner presents the results of original
studies by the authors which were partly reported earlier in scientific
journals but many come out for the first time in this publication.
The suggested models of mechanisms behind the evolution of nature
and society are of great theoretical and practical value. Their
understanding and proper use can reduce the risks that threaten our
civilization and eventually bring to harmonic coexistence of economy
and environment instead of their today’s antagonistic confrontation. One
can expect that the book will make for better awareness of readers
willing to participate in solving the vital environmental, economic, and
social problems of our time.
Professor S. Gabuda
Acknowledgements
We wish to express our gratitude to people who contributed to the
preparation of the manuscript.
V.A. Slepkov wrote Chapter 2 on the basis of studies run jointly with
Rem Khlebopros. We also thank him for discussions we profited from
when writing Chapter 9.
V.G. Suhovol’sky was the first to apply the Zipf-Pareto law to check
votes count in elections and wrote the respective section in Chapter 15.
We appreciate his criticism of the remaining part of Chapter 15 which is
of our full responsibility.
V.V. Mezhevikin was very helpful with consulting on biochemical
issues essential for better exposition of Chapter 6 on manmade closed
ecosystems.
V.B. Kashkin furnished global total ozone data and developed the
experimental method as a basis of Chapter 5.
Comments by D. Proy prompted us to make a number of useful
improvements in Chapters 1–3 and 6–8.
The manuscript benefited much from constructive criticism by I.I.
Gitel’zon.
Special thanks go to T.I. Perepelova for her aid in editing and making
the English text.
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