Exactly solvable models of biological invasion /

Bibliographic Details
Main Author:	Petrovskii, Sergei V (Author)
Other Authors:	Li, Bai-Lian
Format:	Book
Language:	English
Published:	Boca Raton : Chapman and Hall/CRC, 2006.
Series:	Chapman & Hall/CRC mathematical biology and medicine series.
Subjects:	Biological invasions > Mathematical models. Invasions biologiques > Modèles mathématiques.
Online Access:	CRCnetBASE Table of contents Publisher description

Table of Contents:

1.1 Why exactly solvable models are important 1
1.2 Intra- and inter-species interactions and local population dynamics 5
1.3 Basic mechanisms of species transport 11
1.4 Biological invasion: main facts and constituting examples 14
2 Models of biological invasion 17
2.1 Diffusion-reaction equations 17
2.2 Integral-difference models 24
2.3 Space-discrete models 30
2.4 Stochastic models 39
3 Basic methods and relevant examples 45
3.1 The Cole-Hopf transformation and the Burgers equation as a paradigm 46
3.1.1 * Exact solutions for a forced Burgers equation 50
3.2 Further application of the Cole-Hopf transformation 56
3.3 Method of piecewise linear approximation 60
3.3.1 Exact solution for a population with logistic growth 60
3.3.2 Exact solution for a population with a strong Allee effect 63
3.4 Exact solutions of a generalized Fisher equation 68
3.4.1 Ansatz 69
3.4.2 * The Ablowitz-Zeppetella method 71
3.5 More about ansatz 74
4 Single-species models 81
4.1 Impact of advection and migration 82
4.1.1 Advection 84
4.1.2 Density-dependent migration 85
4.1.3 General case 88
4.2 Accelerating population waves 89
4.2.1 Self-similar exact solution 93
4.3 The problem of critical aggregation 102
4.3.1 Practical stability concept 105
4.3.2 * The Wilhelmsson "blow-up" solution 111
5 Density-dependent diffusion 117
5.1 The Aronson-Newman solution and its generalization 117
5.1.1 A general case 120
5.2 Stratified diffusion and the Allee effect 126
6 Models of interacting populations 137
6.1 Exact solution for a diffusive predator-prey system 137
6.1.1 * Properties of the local system 140
6.1.2 Exact solution and its properties 143
6.1.3 * Formal derivation of the exact solution 149
6.2 Migration waves in a resource-consumer system 154
7 Some alternative and complementary approaches 159
7.1 Wave speed and the eigenvalue problem 160
7.2 Convergence of the initial conditions 163
7.3 Convergence and the paradox of linearization 165
7.4 Application of the comparison principle 168
8 Ecological examples and applications 171
8.1 Invasion of Japanese beetle in the United States 172
8.2 Mount St. Helens recolonization and the impact of predation 178
8.3 Stratified diffusion and rapid plant invasion 187
9 Appendix: Basic background mathematics 195
9.1 Ordinary differential equations and their solutions 195
9.2 Phase plane and stability analysis 198
9.3 Diffusion equation 200.