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Mathematical ecology is the application of mathematics to describe and understand ecosystems. There are two main approaches. One is to describe natural communities and induce statistical patterns or relationships which should generally occur. However, this book is devoted entirely to introducing the student to the second approach: to study deterministic mathematical models and, on the basis of mathematical results on the models, to look for the same patterns or relationships in nature. This book is a compromise between three competing desiderata. It seeks to: maximize the generality of the models; constrain the models to "behave" realistically, that is, to exhibit stability and other features; and minimize the difficulty of presentations of the models. The ultimate goal of the book is to introduce the reader to the general mathematical tools used in building realistic ecosystem models. Just such a model is presented in Chapter Nine. The book should also serve as a stepping-stone both to advanced mathematical works like Stability of Biological Communities by Yu. M. Svirezhev and D. O. Logofet (Mir, Moscow, 1983) and to advanced modeling texts like Freshwater Ecosystems by M. Straskraba and A. H. Gnauch (Elsevier, Amsterdam, 1985).
Sommario
One-An Introduction to Dynamical Systems as Models.- 1.1 Ecosystem Development in Terms of Ecology.- 1.2 State Space, or How to Add Apples and Oranges.- 1.3 Dynamical Systems as Treasure Hunts.- Two-Simple Difference Equation Models.- 2.1 Predator-Prey Difference Equation Dynamical Systems.- 2.2 Probabilistic Limit Cycles.- Three-Formalizing the Notion of Stability.- 3.1 The Concept of Ecosystem Stability.- 3.2 The Relation of Difference and Differential Equations.- 3.3 Limit Cycles.- 3.4 Lyapunov Theory.- 3.5 The Trapping of Trajectories.- Four-Introduction to Ecosystem Models.- 4.1 Brewing Beer and Yeast Population Dynamics.- 4.2 Attractor Trajectories.- 4.3 Derivatives of System Functions.- 4.4 The Linearization Theorem.- 4.5 The Hurwitz Stability Test.- Five-Introduction to Ecosystem Models.- 5.1 The Community Matrix.- 5.2 Predator-Prey Equations and Generalizations Thereof.- 5.3 Signed Digraphs.- 5.4 Qualitative Stability of Linear Systems.- Six-Qualitative Stability of Ecosystem Models.- 6.1 Qualitative Results in Modeling.- 6.2 Holistic Ecosystem Models.- 6.3 Holistic Ecosystem Models with Attractor Trajectories.- Seven-The Behavior of Models with Attractor Regions.- 7.1 Attractor Regions.- 7.2 The Lorenz Model.- 7.3 Elementary Ecosystem Models with Chaotic Dynamics.- Eight-Holistic Ecosystem Models with Attractor Regions.- 8.1 An Attractor Region Theorem.- 8.2 An Example.- Nine-Sequencing Energy Flow Models to Account for Shortgrass Prairie Energy Dynamics.- 9.1 Energy Flow and Accumulation Modeling.- 9.2 Accumulation Modeling.- 9.3 Estimating Energy Flows.- 9.4 Equations and Trajectories.- 9.5 Stability.
