The consumer-resource relationship : mathematical modeling / Claude Lobry.

Format:

Book

Author/Creator:

Lobry, C. (Claude), author.

Language:

English

Subjects (All):

Population biology--Mathematical models.

Population biology.

Predation (Biology)--Mathematical models.

Predation (Biology).

Physical Description:

1 online resource (279 pages)

Edition:

1st ed.

Place of Publication:

London : ISTE ; Hoboken, NJ : Wiley, [2018]

Summary:

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.

Contents:

Cover

Half-Title Page

Title Page

Copyright Page

Contents

Preface

1. History of the Predator-Prey Model

1.1. The logistic model

1.1.1. Notations, terminology

1.1.2. Growth with feedback and resource

1.1.3. Another interpretation of the logistic equation: the interference between individuals

1.1.4. (r, α)-model or (r,K)-model?

1.1.5. Historical notes and criticisms

1.2. The Lotka-Volterra predator-prey model

1.2.1. The model

1.2.2. Model analysis

1.2.3. Phase portrait and simulations

1.2.4. Historical notes and criticisms

1.3. The Gause model

1.3.1. The model

1.3.2. Model simulations

1.3.3. Historical notes and criticisms

1.4. The Rosenzweig-MacArthur model

1.4.1. The model

1.4.2. Analysis and simulations

1.4.3. Historical remarks and criticisms

1.5. The "ratio-dependent" model

1.5.1. Model analysis and simulations

1.5.2. Historical notes and criticisms

1.6. Conclusion

2. The Consumer-Resource Model

2.1. The general model

2.1.1. General assumptions on the model

2.1.2. Properties

2.2. The "resource-dependent" model

2.2.1. Development of the Rosenzweig-MacArthur model

2.2.2. Analysis of the RMA model

2.2.3. Variants of the RMA model

2.3. The Arditi-Ginzburg "ratio-dependent" model

2.3.1. Development of the "RC-dependent" and "ratio-dependent" model

2.3.2. Analysis of RC and ratio-dependent models

2.3.3. Simulations of the ratio-dependent model

2.4. Historical and bibliographical remarks

3. Competition

3.1. Introduction

3.2. The two-species competition Volterra model

3.2.1. Population 2 wins the competition

3.2.2. Population 1 wins the competition

3.2.3. Coexistence of both populations

3.2.4. Conditional exclusion

3.2.5. Interference

3.3. Competition and the Rosenzweig-MacArthur model.

3.3.1. Equilibria of the competition RMA model

3.3.2. The exclusion theorem at equilibrium

3.3.3. The exclusion theorem and the Volterra model

3.4. Competition with RC and ratio-dependent models

3.4.1. Characteristics at equilibrium

3.4.2. Growth thresholds and equilibria of model [3.10]

3.4.3. Stability of coexistence equilibria

3.4.4. Criticism of RC and ratio-dependent competition models

3.4.5. Simulations

3.5. Coexistence through periodic solutions

3.5.1. Self-oscillating pair (x, y)

3.5.2. Adding a competitor

3.6. Historical and bibliographical remarks

4. "Demographic Noise" and the "Atto-fox" Problem

4.1. The "atto-fox" problem

4.2. The RMA model with small yield

4.2.1. Notations, terminology

4.2.2. The "constrained system"

4.2.3. Phase portrait of [4.3] when Πδ crosses the parabola "far away" from the peak

4.2.4. Phase portrait when Πδ crosses the parabola "close" to the peak

4.3. The RC-dependent model with small yield

4.4. The persistence problem in population dynamics

4.4.1. Demographic noise and the atto-fox problem

4.4.2. Sensibility of atto-fox phenomena

4.4.3. About the very unlikely nature of canard values

4.5. Historical and bibliographical remarks

5. Mathematical Supplement: "Canards" of Planar Systems

5.1. Planar slow-fast vector fields

5.1.1. Concerning orders of magnitude

5.1.2. First approximation: the constrained system

5.1.3. Constrained trajectories

5.1.4. Constrained trajectories and "real trajectories"

5.2. Bifurcation of planar vector fields

5.2.1. System equivalence

5.2.2. Andronov-Hopf bifurcation

5.3. Bifurcation of a slow-fast vector field

5.3.1. A surprising Andronov-Hopf bifurcation

5.3.2. The particular case: p=0

5.3.3. Some terminology

5.3.4. Back to the initial model

5.3.5. The general case p ≠ 0.

5.4. Bifurcation delay

5.4.1. Another surprising simulation

5.4.2. One more surprise

5.4.3. The Shiskova-Neishtadt theorem

5.5. Historical and bibliographical remarks

Appendices

Appendix 1: Differential Equations and Vector Fields

Appendix 2: Planar Vector Field

Appendix 3: Discontinuous Planar Vector Fields

Bibliography

Index

Other titles from iSTE in Chemical Engineering

EULA.

Notes:

Description based on print version record.

ISBN:

9781119543992

1119543991

9781119544029

1119544025

9781119544012

1119544017

OCLC:

1048796400