Details

The Consumer-Resource Relationship


The Consumer-Resource Relationship

Mathematical Modeling
1. Aufl.

von: Claude Lobry

139,99 €

Verlag: Wiley
Format: EPUB
Veröffentl.: 06.08.2018
ISBN/EAN: 9781119543992
Sprache: englisch
Anzahl Seiten: 274

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Beschreibungen

<p>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.</p> <p>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.</p>
<p>Preface<b> </b>ix</p> <p><b>Chapter 1. History of the Predator–Prey Model</b> <b>1</b></p> <p>1.1. The logistic model 1</p> <p>1.1.1. Notations, terminology 2</p> <p>1.1.2. Growth with feedback and resource 4</p> <p>1.1.3. Another interpretation of the logistic equation: the interference between individuals 9</p> <p>1.1.4. (r, α)-model or (r,K)-model? 11</p> <p>1.1.5. Historical notes and criticisms 14</p> <p>1.2. The Lotka–Volterra <i>predator–prey</i> <i>model</i> 14</p> <p>1.2.1. The model 14</p> <p>1.2.2. Model analysis 15</p> <p>1.2.3. Phase portrait and simulations 19</p> <p>1.2.4. Historical notes and criticisms 20</p> <p>1.3. The Gause model 24</p> <p>1.3.1. The model 24</p> <p>1.3.2. Model simulations 26</p> <p>1.3.3. Historical notes and criticisms 29</p> <p>1.4. The Rosenzweig–MacArthur model 31</p> <p>1.4.1. The model 31</p> <p>1.4.2. Analysis and simulations 32</p> <p>1.4.3. Historical remarks and criticisms 35</p> <p>1.5. The “ratio-dependent” model 38</p> <p>1.5.1. Model analysis and simulations 38</p> <p>1.5.2. Historical notes and criticisms 41</p> <p>1.6. Conclusion 42</p> <p><b>Chapter 2. The Consumer–Resource Model 43</b></p> <p>2.1. The general model 43</p> <p>2.1.1. General assumptions on the model 44</p> <p>2.1.2. Properties 45</p> <p>2.2. The “resource-dependent” model 47</p> <p>2.2.1. Development of the Rosenzweig–MacArthur model 47</p> <p>2.2.2. Analysis of the RMA model 52</p> <p>2.2.3. Variants of the RMA model 59</p> <p>2.3. The Arditi–Ginzburg “ratio-dependent” model 65</p> <p>2.3.1. Development of the “RC-dependent” and “ratio-dependent” model 65</p> <p>2.3.2. Analysis of RC and ratio-dependent models 68</p> <p>2.3.3. Simulations of the ratio-dependent model 77</p> <p>2.4. Historical and bibliographical remarks 83</p> <p><b>Chapter 3. Competition 87</b></p> <p>3.1. Introduction 87</p> <p>3.2. The two-species competition Volterra model 89</p> <p>3.2.1. Population 2 wins the competition 89</p> <p>3.2.2. Population 1 wins the competition 90</p> <p>3.2.3. Coexistence of both populations 91</p> <p>3.2.4. Conditional exclusion 92</p> <p>3.2.5. Interference 93</p> <p>3.3. Competition and the Rosenzweig–MacArthur model 93</p> <p>3.3.1. Equilibria of the competition RMA model 94</p> <p>3.3.2. The exclusion theorem at equilibrium 96</p> <p>3.3.3. The exclusion theorem and the Volterra model 99</p> <p>3.4. Competition with RC and ratio-dependent models 100</p> <p>3.4.1. Characteristics at equilibrium 100</p> <p>3.4.2. Growth thresholds and equilibria of model [3.10] 102</p> <p>3.4.3. Stability of coexistence equilibria 106</p> <p>3.4.4. Criticism of RC and ratio-dependent competition models 109</p> <p>3.4.5. Simulations 110</p> <p>3.5. Coexistence through periodic solutions 119</p> <p>3.5.1. Self-oscillating pair (x, y) 119</p> <p>3.5.2. Adding a competitor 121</p> <p>3.6. Historical and bibliographical remarks 123</p> <p><b>Chapter 4. “Demographic Noise” and “Atto-fox” Problem 125</b></p> <p>4.1. The “atto-fox” problem 125</p> <p>4.2. The RMA model with small yield 126</p> <p>4.2.1. Notations, terminology 128</p> <p>4.2.2. The “constrained system” 130</p> <p>4.2.3. Phase portrait of [4.3] when Π<sub>δ</sub> crosses the parabola “far away” from the peak 132</p> <p>4.2.4. Phase portrait when Π<sub>δ</sub> crosses the parabola “close” to the peak 139</p> <p>4.3. The RC-dependent model with small yield 148</p> <p>4.4. The persistence problem in population dynamics 151</p> <p>4.4.1. Demographic noise and the atto-fox problem 153</p> <p>4.4.2. Sensibility of atto-fox phenomena 159</p> <p>4.4.3. About the very unlikely nature of canard values 163</p> <p>4.5. Historical and bibliographical remarks 165</p> <p><b>Chapter 5. Mathematical Supplement: “Canards” of Planar Systems</b> <b>169</b></p> <p>5.1. Planar slow–fast vector fields 169</p> <p>5.1.1. Concerning <i>orders of magnitude</i> 169</p> <p>5.1.2. First approximation: the constrained system 172</p> <p>5.1.3. Constrained trajectories 173</p> <p>5.1.4. Constrained trajectories and “real trajectories” 175</p> <p>5.2. Bifurcation of planar vector fields 183</p> <p>5.2.1. System equivalence 184</p> <p>5.2.2. Andronov–Hopf bifurcation 186</p> <p>5.3. Bifurcation of a slow–fast vector field 190</p> <p>5.3.1. A surprising Andronov–Hopf bifurcation 190</p> <p>5.3.2. The particular case: p=0 193</p> <p>5.3.3. Some terminology 201</p> <p>5.3.4. Back to the initial model 202</p> <p>5.3.5. The general case p ≠ 0 204</p> <p>5.4. Bifurcation delay 212</p> <p>5.4.1. Another surprising simulation 212</p> <p>5.4.2. One more surprise 216</p> <p>5.4.3. The Shiskova–Neishtadt theorem 219</p> <p>5.5. Historical and bibliographical remarks 220</p> <p><b>Appendices.225</b></p> <p>Appendix 1. Differential Equations and Vector Fields 227</p> <p>Appendix 2. Planar Vector Field 235</p> <p>Appendix 3. Discontinuous Planar Vector Fields 241</p> <p>Bibliography 253</p> <p>Index 259</p> <p> </p>
<strong>Claude Lobry</strong>, Professeur retraité, Université de Nice. <p><strong>Jérôme Harmand</strong>, D.R. INRA. <p><strong>Alain Rapaport</strong>, D.R. INRA. <p><strong>Tewfik Sari</strong>, Professeur Université de Mulhouse et IRSTEA.

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