The Consumer-Resource Relationship Mathematical Modeling [Hardcover]

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.

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. Histol²