Library of Congress Cataloging-in-Publication Data Smith, Hal L. Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems / Hal L. Smith. p. cm.-(Mathematical surveys and monographs, ISSN 0076-5376; v. 41) Includes bibliographical references and index. ISBN 0-8218-0393-X 1. Differentiable dynamical systems. 2. Monotonic functions. I. Title. II. Series: Mathematical surveys and monographs; no. 41. QA614.8.S63 1995 515 / .352-dc20 94-48032 CIP AMS softcover ISBN 978-0-8218-4487-8 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given.

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The chemostat is a basic piece of laboratory apparatus, yet it has occupied an increasingly central role in ecological studies. The ecological environment created by a chemostat is one of the few completely controlled experimental systems for testing microbial growth and competition. As a tool in biotechnology, the chemostat plays an important role in bioprocessing. This book presents the theory of the chemostat as a model for larger ecological problems such as food chains, competition along a gradient, competition in the presence of an inhibitor, and the effects of time varying inputs. Models which take account of size structure, variable yields, and diffusion are also considered. The basic phenomena are modelled and analysed using the dynamical systems approach. Directions for research and open problems are discussed. Six appendices provide an elementary description of the necessary mathematical tools. Teachers, researchers, and students in applied mathematics, chemical engineering and ecology will find this book a welcome resource.

Preface ix §1.3. Invariant sets §1.4. Exercises Chapter 2. Compact Attractors §2.1. Compact attractors of individual sets §2.2. Compact attractors of classes of sets §2.3. A sufficient condition for asymptotic smoothness §2.4. α-limit sets of total trajectories §2.5. Invariant sets identified through Lyapunov functions §2.6. Discrete semiflows induced by weak contractions §2.7. Exercises Chapter 3. Uniform Weak Persistence §3.1. Persistence definitions §3.2. An SEIRS epidemic model in patchy host populations §3.3. Nonlinear matrix models: Prolog §3.4. The May-Leonard example of cyclic competition §3.5. Exercises v vi Contents Chapter 4. Uniform Persistence §4.1. From uniform weak to uniform persistence §4.2. From uniform weak to uniform persistence: Discrete case §4.3. Application to a metered endemic model of SIR type §4.4. From uniform weak to uniform persistence for time-set R + §4.5. Persistenceà la Baron von Münchhausen §4.6. Navigating between alternative persistence functions §4.7. A fertility reducing endemic with two stages of infection §4.8. Exercises Chapter 5. The Interplay of Attractors, Repellers, and Persistence §5.1. An attractor of points facilitates persistence §5.2. Partition of the global attractor under uniform persistence §5.3. Repellers and dual attractors §5.4. The cyclic competition model of May and Leonard revisited §5.5. Attractors at the brink of extinction §5.6. An attractor under two persistence functions §5.7. Persistence of bacteria and phages in a chemostat §5.8. Exercises Chapter 6. Existence of Nontrivial Fixed Points via Persistence §6.1. Nontrivial fixed points in the global compact attractor §6.2. Periodic solutions of the Lotka-Volterra predator-prey model §6.3. Exercises Chapter 7. Nonlinear Matrix Models: Main Act §7.1. Forward invariant balls and compact attractors of bounded sets §7.2. Existence of nontrivial fixed points §7.3. Uniform persistence and persistence attractors §7.4. Stage persistence §7.5. Exercises Chapter 8. Topological Approaches to Persistence §8.1. Attractors and repellers §8.2. Chain transitivity and the Butler-McGehee lemma §8.3. Acyclicity implies uniform weak persistence §8.4. Uniform persistence in a food chain Contents vii §8.5. The metered endemic model revisited §8.6. Nonlinear matrix models (epilog): Biennials §8.7. An endemic with vaccination and temporary immunity §8.8. Lyapunov exponents and persistence for ODEs and maps §8.9. Exercises Chapter 9. An SI Endemic Model with Variable Infectivity §9.1. The model §9.2. Host persistence and disease extinction §9.3. Uniform weak disease persistence §9.4. The semiflow §9.5. Existence of a global compact attractor §9.6. Uniform disease persistence §9.7. Disease extinction and the disease-free equilibrium §9.8. The endemic equilibrium §9.9. Persistence as a crossroad to global stability §9.10. Measure-valued distributions of infection-age Chapter 10. Semiflows Induced by Semilinear Cauchy Problems §10.1. Classical, integral, and mild solutions §10.2. Semiflow via Lipschitz condition and contraction p...

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