Mathematical Modeling in Ecology: A Workbook for StudentsMathematical 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). |
Contents
| 1 | |
Chapter TwoSimple Difference Equation Models | 20 |
Chapter ThreeFormalizing the Notion of Stability | 36 |
Chapter FourIntroduction to Ecosystem Models | 43 |
Chapter FiveIntroduction to Ecosystem Models | 101 |
Chapter SixQualitative Stability of Ecosystem Models | 128 |
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Common terms and phrases
2-cycles 2+sin(t arbitrary trajectory asymptotically approach attractor region attractor trajectory auto-regulation autotroph autotroph compartment basin of attraction biomass calculation Chapter characteristic polynomial clay-colored sparrow coefficients community matrix Consider constant trajectory defined detritus compartment detritus donation difference equation dynamical double-approximation procedure ecology ecosystem model energy conversion efficiency energy flow equation dynamical system existence finite boundary food web grasshopper holistic ecosystem model Hurwitz test hypersurface initial L₁ level cylinder level set limit cycle linear approximation matrix linear dynamical system Linearization Theorem Lyapunov function Lyapunov Theorem mathematical multinomial n-dimensional space negative nonnegative orthant points positive number positive orthant predation community predation interactions predation loops predator-prey model prey problem qualitative radius rate of change reach the finite SD(A shown in Fig sign stable signed digraph space-time sparrow stability Suppose system functions trajectory starting Trajectory Trapping Theorem two-dimensional typical trajectory vertices x(t+At x₁ x₂ yeast zero