The Consumer-Resource Relationship Mathematical Modeling

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.

Jérôme Harmand, D.R. INRA.

Alain Rapaport, D.R. INRA.

Tewfik Sari, Professeur Université de Mulhouse et IRSTEA.

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, α)-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 populations 91

3.2.4. Conditional exclusion 92

3.2.5. Interference 93

3.3. Competition and the Rosenzweig–MacArthur model 93

3.3.1. Equilibria of the competition RMA model 94

3.3.2. The exclusion theorem at equilibrium 96

3.3.3. The exclusion theorem and the Volterra model 99

3.4. Competition with RC and ratio-dependent models 100

3.4.1. Characteristics at equilibrium 100

3.4.2. Growth thresholds and equilibria of model [3.10] 102

3.4.3. Stability of coexistence equilibria 106

3.4.4. Criticism of RC and ratio-dependent competition models 109

3.4.5. Simulations 110

3.5. Coexistence through periodic solutions 119

3.5.1. Self-oscillating pair (x, y) 119

3.5.2. Adding a competitor 121

3.6. Historical and bibliographical remarks 123

Chapter 4. “Demographic Noise” and “Atto-fox” Problem 125

4.1. The “atto-fox” problem 125

4.2. The RMA model with small yield 126

4.2.1. Notations, terminology 128

4.2.2. The “constrained system” 130

4.2.3. Phase portrait of [4.3] when Π_δ crosses the parabola “far away” from the peak 132

4.2.4. Phase portrait when Π_δ crosses the parabola “close” to the peak 139

4.3. The RC-dependent model with small yield 148

4.4. The persistence problem in population dynamics 151

4.4.1. Demographic noise and the atto-fox problem 153

4.4.2. Sensibility of atto-fox phenomena 159

4.4.3. About the very unlikely nature of canard values 163

4.5. Historical and bibliographical remarks 165

Chapter 5. Mathematical Supplement: “Canards” of Planar Systems 169

5.1. Planar slow–fast vector fields 169

5.1.1. Concerning orders of magnitude 169

5.1.2. First approximation: the constrained system 172

5.1.3. Constrained trajectories 173

5.1.4. Constrained trajectories and “real trajectories” 175

5.2. Bifurcation of planar vector fields 183

5.2.1. System equivalence 184

5.2.2. Andronov–Hopf bifurcation 186

5.3. Bifurcation of a slow–fast vector field 190

5.3.1. A surprising Andronov–Hopf bifurcation 190

5.3.2. The particular case: p=0 193

5.3.3. Some terminology 201

5.3.4. Back to the initial model 202

5.3.5. The general case p ≠ 0 204

5.4. Bifurcation delay 212

5.4.1. Another surprising simulation 212

5.4.2. One more surprise 216

5.4.3. The Shiskova–Neishtadt theorem 219

5.5. Historical and bibliographical remarks 220

Appendices.225

Appendix 1. Differential Equations and Vector Fields 227

Appendix 2. Planar Vector Field 235

Appendix 3. Discontinuous Planar Vector Fields 241

Bibliography 253

Index 259

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