The Consumer-Resource Relationship

Mathematical Modeling
1. Auflage 2018
ISBN: 978-1-78630-044-7
Verlag: Wiley

Buch, Englisch, 274 Seiten, Format (B × H): 163 mm x 239 mm, Gewicht: 558 g

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.

Autoren/Hrsg.

Lobry, Claude

Inhaltsverzeichnis

Preface ix

Chapter 1. History of the Predator–Prey Model 1

1.1. The logistic model 1

1.1.1. Notations, terminology 2

1.1.2. Growth with feedback and resource 4

1.1.3. Another interpretation of the logistic equation: the interference between individuals 9

1.1.4. (r, a)-model or (r,K)-model? 11

1.1.5. Historical notes and criticisms 14

1.2. The Lotka–Volterra predator–prey model 14

1.2.1. The model 14

1.2.2. Model analysis 15

1.2.3. Phase portrait and simulations 19

1.2.4. Historical notes and criticisms 20

1.3. The Gause model 24

1.3.1. The model 24

1.3.2. Model simulations 26

1.3.3. Historical notes and criticisms 29

1.4. The Rosenzweig–MacArthur model 31

1.4.1. The model 31

1.4.2. Analysis and simulations 32

1.4.3. Historical remarks and criticisms 35

1.5. The “ratio-dependent” model 38

1.5.1. Model analysis and simulations 38

1.5.2. Historical notes and criticisms 41

1.6. Conclusion 42

Chapter 2. The Consumer–Resource Model 43

2.1. The general model 43

2.1.1. General assumptions on the model 44

2.1.2. Properties 45

2.2. The “resource-dependent” model 47

2.2.1. Development of the Rosenzweig–MacArthur model 47

2.2.2. Analysis of the RMA model 52

2.2.3. Variants of the RMA model 59

2.3. The Arditi–Ginzburg “ratio-dependent” model 65

2.3.1. Development of the “RC-dependent” and “ratio-dependent” model 65

2.3.2. Analysis of RC and ratio-dependent models 68

2.3.3. Simulations of the ratio-dependent model 77

2.4. Historical and bibliographical remarks 83

Chapter 3. Competition 87

3.1. Introduction 87

3.2. The two-species competition Volterra model 89

3.2.1. Population 2 wins the competition 89

3.2.2. Population 1 wins the competition 90

3.2.3. Coexistence of both populat

Über Autor(innen)

Claude Lobry is a former Professor at both the University of Bordeaux and the University of Nice in France. His present research interests are the mathematical theory of "nonstandard" differential systems and mathematical modeling of population dynamics.