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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
Better known as the "predator-prey relationship," the consumer-resource relationship means the situation where a single species of organisms consumes for survival and reproduction. For example, Escherichia coli consumes glucose, cows consume grass, cheetahs consume baboons; these three very different situations, the first concerns the world of bacteria and the resource is a chemical species, the second concerns mammals and the resource is a plant, and in the final case the consumer and the resource are mammals, have in common the fact of consuming.
In a chemostat, microorganisms generally consume (abiotic) minerals, but not always, bacteriophages consume bacteria that constitute a biotic resource. 'The Chemostat' book dealt only with the case of abiotic resources. Mathematically this amounts to replacing in the two equation system of the chemostat the decreasing function by a general increasing then decreasing function. This simple change has greatly enriched the theory. This book shows in this new framework the problem of competition for the same resource.
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Table of Contents
Cover
Preface
1 History of the Predator–Prey Model
1.1. The logistic model
1.2. The Lotka–Volterra predator–prey model
1.3. The Gause model
1.4. The Rosenzweig–MacArthur model
1.5. The “ratio-dependent” model
1.6. Conclusion
2 The Consumer–Resource Model
2.1. The general model
2.2. The “resource-dependent” model
2.3. The Arditi–Ginzburg “ratio-dependent” model
2.4. Historical and bibliographical remarks
3 Competition
3.1. Introduction
3.2. The two-species competition Volterra model
3.3. Competition and the Rosenzweig–MacArthur model
3.4. Competition with RC and ratio-dependent models
3.5. Coexistence through periodic solutions
3.6. Historical and bibliographical remarks
4 “Demographic Noise” and “Atto-fox” Problem
4.1. The “atto-fox” problem
4.2. The RMA model with small yield
4.3. The RC-dependent model with small yield
4.4. The persistence problem in population dynamics
4.5. Historical and bibliographical remarks
5 Mathematical Supplement: “Canards” of Planar Systems
5.1. Planar slow–fast vector fields
5.2. Bifurcation of planar vector fields
5.3. Bifurcation of a slow–fast vector field
5.4. Bifurcation delay
5.5. Historical and bibliographical remarks
Appendices
Appendix 1: Differential Equations and Vector Fields
A1.1. Existence and uniqueness theorem. Notations
A1.2. Vector fields
A1.3. Euler’s method
A1.4. Equilibria
Appendix 2: Planar Vector Field
A2.1. Typology of equilibria
A2.2. The method of isoclines
A2.3. Poincaré-Bendixson theory
A2.4. Differential inequalities
Appendix 3: Discontinuous Planar Vector Fields
A3.1. Definition of a discontinuous vector field
A3.2. Discontinuous vector field orbit continuation rules
A3.3. Bibliographic comment
Bibliography
Index
End User License Agreement
Guide
Cover
Table of Contents
Begin Reading
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G1
G2
e1
Chemostat and Bioprocesses Set
coordinated byClaude Lobry
Volume 2
The Consumer–Resource Relationship
Mathematical Modeling
Claude Lobry
First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd27-37 St George’s RoadLondon SW19 4EUUKwww.iste.co.uk
John Wiley & Sons, Inc.111 River StreetHoboken, NJ 07030USAwww.wiley.com
© ISTE Ltd 2018
The rights of Claude Lobry to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2018940976
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78630-044-7
Preface
This book is intended for students or researchers with an engineering school or undergraduate degree and those with backgrounds in mathematics, who share an interest in ecological theory. To have a mathematical background means having come across certain mathematical objects but it does not necessarily mean that familiarity with their practice has been preserved. As we expect to raise interest in readers who want to know the mathematical reasons that explain certain phenomena but do not necessarily intend to practice the mathematics in question themselves, we tried, as much as possible, to avoid certain technical elaborations which would discourage such readers. Following that same spirit, we have illustrated all results by means of numerous simulations. To the reader who wishes to further explore the mathematical aspects, we suggest avenues for further research in the bibliographic comments that come with each chapter.
Population dynamics is the mathematical study of certain models of the evolution of population sizes proposed by ecological theory. This is a very broad topic. Two major classes of models can be distinguished: deterministic and stochastic models. This book will address only deterministic models except for one small case that we will identify further in the text.
The mathematical theory of deterministic models of population dynamics alone covers a considerably sized field that can be more or less described in increasing order of mathematical complexity:
1) growth of a single species;
2) interaction of two species (predation or competition relation);
3) interaction of more than two species and more than two trophic levels (predation, competition, mutualism, etc.);
4) models including migration between two sites or more;
5) spatialized model described by partial differential equations;
6) models including delays;
7) etc.
As expected, these are the simplest models that were the first to have raised the concern of mathematicians, such as Fibonacci (1170–1245), Euler (1707–1783) and Verhults (1804–1849). Later, at the beginning of the 20th Century, great interest was expressed about point two, a topic in which famous personalities such as Lotka, Volterra, Gause and Kolmogorov would distinguish themselves and whose work would be commented on, improved, and clarified in the 1950-70s by a whole legion of scientists, ecologists and mathematicians, so numerous that it is impossible to name them all. This is the specific field that is covered in this book from the perspective of the predator–prey relationship (consumer–resource).
We may wonder what is the point of writing a book solely dedicated to this simple topic that seems far outdated now? Here is the reason.
The thing is that within a few years, two events occurred: the first in the field of qualitative theory of differential equations and the second in ecological theory, which leads us to reconsider a number of basic questions.
– The first event, at the turn of the 1970s, occurred when E. Benoît, J.-L. Callot, F. Diener and M. Diener [BEN 81] brought forward, concerning one of the most classical equations of physics, the Van der Pol equation, involving a new type of solution which had been overlooked by research, maybe due to its very high instability. These solutions, which are oddly called “canard” solutions, can be found in a system of differential equations from the moment there are two time scales, which is indeed the case in the predator–prey relationship.
– The second event, a little less than 10 years later, was the questioning by theoretical ecology of the vision of the predator–prey relationship as depending on the overall concentration of prey by a vision where it depends on the amount of prey available to every predator: this is the
ratio-dependent
model by R. Arditi and L. Ginzburg [ARD 89].
These two events by themselves justify the review of these traditional questions but there is an additional reason to explain it: the possibilities for simulation offered by personal computers. When theory tells us that solutions converge towards an equilibrium or a periodic solution, it is informative to observe how this convergence occurs in practice. Therefore, we will see that models with very reasonable parameters produce solutions that, prior to reaching an equilibrium, can take on values as unreasonable as 10−24 which, if one unit represents a population of 106 individuals (e.g. foxes), means that we are talking about a 10−18-th of individuals, or more precisely of an “atto-fox” [MOL 91] which is obviously absurd. Nonetheless, the difficulty begins significantly before this small portion of individual. As a matter of fact, it is not possible to model the evolution of a population with a small number of individuals based on differential equations: probabilistic models have to be used. We will not address probabilistic models but we will carefully outline the limits of validity of our deterministic models, which will compel us to discuss a little bit about random processes.
The reader might also wonder why this book is part of a series dedicated to the chemostat.
Continuous culture devices are used to observe and control the evolution of a large number of interacting species. In microbiology, when we observe the competition between two species as in Hansell and Hubbell’s famous experiments [HAN 80], the order of magnitude of the number of individuals in the populations is very high, one of the highest one might come to observe in an ecosystem. Consequently, when the aim is to find the relevance of deterministic models of population dynamics, it is most certainly in microbial ecosystems that it may be found.
Chapter 1 is devoted to the description, in the order of their appearance, of the most famous models, Verhulst, Lotka Volterra, Gause, Rosenzweig–MacArthur and Arditi–Ginzburg. Chapter 2 deals specifically with the “predator–prey” relationship (or “consumer–resource”) including a comparison of the properties of “resource-dependent” models on the one hand, and “ratio-dependent” on the other hand. In Chapter 3, we will address the issue of the competition for a resource and we will examine in particular what researchers have agreed to refer to since Hardin [HAR 60] as the competitive exclusion principle. Chapter 4 is dedicated to the “atto-fox” problem that we have just mentioned, thus the limits of our deterministic models.
At the present time, there is no didactic work on the “canard” theory addressing the audience that we wish to reach out to. That is why, for those readers who would like to explore this issue further than the heuristic arguments that we present in previous chapters, we have written a fifth chapter called Mathematical Supplements. In the Appendices we recall the basic vocabulary of the theory of differential equations and we give a few details on discontinuous right-hand differential equations.
Acknowledgments
The author wishes to thank Jérôme Harmand, Alain Rapaport and Tewfik Sari for a long and warm collaboration, without which this book would not have been possible.
Claude LOBRYMay 2018
