Consumer-Resource Relationship

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.

• Preface ixChapter 1. History of the Predator–Prey Model 11.1. The logistic model 11.1.1. Notations, terminology 21.1.2. Growth with feedback and resource 41.1.3. Another interpretation of the logistic equation: the interference between individuals 91.1.4. (r, α)-model or (r,K)-model? 111.1.5. Historical notes and criticisms 141.2. The Lotka–Volterra predator–prey model 141.2.1. The model 141.2.2. Model analysis 151.2.3. Phase portrait and simulations 191.2.4. Historical notes and criticisms 201.3. The Gause model 241.3.1. The model 241.3.2. Model simulations 261.3.3. Historical notes and criticisms 291.4. The Rosenzweig–MacArthur model 311.4.1. The model 311.4.2. Analysis and simulations 321.4.3. Historical remarks and criticisms 351.5. The “ratio-dependent” model 381.5.1. Model analysis and simulations 381.5.2. Historical notes and criticisms 411.6. Conclusion 42Chapter 2. The Consumer–Resource Model 432.1. The general model 432.1.1. General assumptions on the model 442.1.2. Properties 452.2. The “resource-dependent” model 472.2.1. Development of the Rosenzweig–MacArthur model 472.2.2. Analysis of the RMA model 522.2.3. Variants of the RMA model 592.3. The Arditi–Ginzburg “ratio-dependent” model 652.3.1. Development of the “RC-dependent” and “ratio-dependent” model 652.3.2. Analysis of RC and ratio-dependent models 682.3.3. Simulations of the ratio-dependent model 772.4. Historical and bibliographical remarks 83Chapter 3. Competition 873.1. Introduction 873.2. The two-species competition Volterra model 893.2.1. Population 2 wins the competition 893.2.2. Population 1 wins the competition 903.2.3. Coexistence of both populations 913.2.4. Conditional exclusion 923.2.5. Interference 933.3. Competition and the Rosenzweig–MacArthur model 933.3.1. Equilibria of the competition RMA model 943.3.2. The exclusion theorem at equilibrium 963.3.3. The exclusion theorem and the Volterra model 993.4. Competition with RC and ratio-dependent models 1003.4.1. Characteristics at equilibrium 1003.4.2. Growth thresholds and equilibria of model [3.10] 1023.4.3. Stability of coexistence equilibria 1063.4.4. Criticism of RC and ratio-dependent competition models 1093.4.5. Simulations 1103.5. Coexistence through periodic solutions 1193.5.1. Self-oscillating pair (x, y) 1193.5.2. Adding a competitor 1213.6. Historical and bibliographical remarks 123Chapter 4. “Demographic Noise” and “Atto-fox” Problem 1254.1. The “atto-fox” problem 1254.2. The RMA model with small yield 1264.2.1. Notations, terminology 1284.2.2. The “constrained system” 1304.2.3. Phase portrait of [4.3] when Πδ crosses the parabola “far away” from the peak 1324.2.4. Phase portrait when Πδ crosses the parabola “close” to the peak 1394.3. The RC-dependent model with small yield 1484.4. The persistence problem in population dynamics 1514.4.1. Demographic noise and the atto-fox problem 1534.4.2. Sensibility of atto-fox phenomena 1594.4.3. About the very unlikely nature of canard values 1634.5. Historical and bibliographical remarks 165Chapter 5. Mathematical Supplement: “Canards” of Planar Systems 1695.1. Planar slow–fast vector fields 1695.1.1. Concerning orders of magnitude 1695.1.2. First approximation: the constrained system 1725.1.3. Constrained trajectories 1735.1.4. Constrained trajectories and “real trajectories” 1755.2. Bifurcation of planar vector fields 1835.2.1. System equivalence 1845.2.2. Andronov–Hopf bifurcation 1865.3. Bifurcation of a slow–fast vector field 1905.3.1. A surprising Andronov–Hopf bifurcation 1905.3.2. The particular case: p=0 1935.3.3. Some terminology 2015.3.4. Back to the initial model 2025.3.5. The general case p ≠ 0 2045.4. Bifurcation delay 2125.4.1. Another surprising simulation 2125.4.2. One more surprise 2165.4.3. The Shiskova–Neishtadt theorem 2195.5. Historical and bibliographical remarks 220Appendices.225Appendix 1. Differential Equations and Vector Fields 227Appendix 2. Planar Vector Field 235Appendix 3. Discontinuous Planar Vector Fields 241Bibliography 253Index 259