Ulrich Krause Positive Dynamical Systems in Discrete Time
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De Gruyter Studies in Mathematics
| Edited by Carsten Carstensen, Berlin, Germany Nicola Fusco, Napoli, Italy Fritz Gesztesy, Columbia, Missouri, USA Niels Jacob, Swansea, United Kingdom Karl-Hermann Neeb, Erlangen, Germany
Volume 62
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Ulrich Krause
Positive Dynamical Systems in Discrete Time | Theory, Models, and Applications
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Mathematics Subject Classification 2010 15B48, 37B55, 39B12, 47B65, 47H07, 47N10, 60J10 Author Prof. Dr. Dr. Ulrich Krause Universität Bremen Fachbereich 03 – Mathematik/Informatik Bibliothekstr. 1 28359 Bremen Germany
[email protected]
ISBN 978-3-11-036975-5 e-ISBN (PDF) 978-3-11-036569-6 e-ISBN (EPUB) 978-3-11-039134-3 Set-ISBN 978-3-11-036571-9 ISSN 0179-0986 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Munich/Boston Typesetting: PTP-Berlin, Protago TEX-Production GmbH Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com
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| To Carola and Daniel
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Preface
Positive dynamical systems come into play when relevant variables of a system take on values which are nonnegative in a natural way. This is the case, for example, in fields as biology, demography and economics, where the levels of populations or prices of goods are positive. Positivity comes in also if the formation of averages by weighted means is relevant since weights, for example probabilities, are not negative. This is the case in quite diverse fields ranging from electrical engineering over physics and computer science to sociology. Thereby averaging takes place with respect to signals in a sensor network or in a swarm (of birds or robots) or with respect to velocities of particles or the opinions of people. In the fields mentioned the dynamics is often modeled by difference equations which means that time is treated as discrete. Thus, in reality one meets a huge variety of positive dynamical systems in discrete time. In many cases these systems can be captured by a linear mapping given by a nonnegative matrix. The dynamics (in discrete time) then is given by the powers of the matrix or, equivalently, by the iterates of the linear mapping which maps the positive orthant into itself. A powerful tool then is the Perron–Frobenius Theory of nonnegative matrices (including the asymptotic behavior of powers of those matrices) which has been successful since its inception by O. Perron and G. Frobenius over about hundred years ago. Concerning theory as well as applications there are two insufficient aspects of Perron–Frobenius Theory which later on drove this theory