Petrovskii / Li Exactly Solvable Models of Biological Invasion

Erscheinungsjahr 2005
ISBN: 978-1-4200-3496-7
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 232 Seiten

Reihe: Chapman & Hall/CRC Mathematical & Computational Biology

Much of our current knowledge on biological invasion was derived from field studies, but many recent advances relied heavily on mathematics and computing, particularly mathematical modeling. While numerical simulations are clearly a useful approach, they have some serious drawbacks. Approximations errors and the number of parameter values can have a significant impact on the simulation results, the extent of which often remains obscure. Such difficulties do not arise, however, when the problem can be solved analytically.

Exactly Solvable Models of Biological Invasion demonstrates the advantages and methods of obtaining exact solutions of partial differential equations that describe nonlinear problems encountered in the study of invasive species spread. With emphasis on PDEs of diffusion-reaction type, the authors present a comprehensive collection of exactly solvable models and a unified, self-contained description of the relevant mathematical methods. In doing so, they also provide new insight into important issues such as the impact of the Allee effect, the impact of predation, and the interplay between different modes of species dispersal. Full calculation details make this presentation accessible to biologists as well as applied mathematicians, and a range of ecological examples and applications demonstrate the utility of exact methods in practice.

Exact solutions provide an immediate, complete description of system dynamics for a wide class of initial conditions and serve as a convenient tool for testing numerical algorithms and codes used in more specialized studies. This book lays the groundwork for bringing the power of exactly solvable models to bear on real-world ecological problems.

Zielgruppe

Mathematicians, biological scientists, and ecological researchers

Autoren/Hrsg.

Petrovskii, Sergei V.

Li, Bai-Lian

Fachgebiete

• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Computeranwendungen in der Mathematik
• Mathematik | Informatik Mathematik Numerik und Wissenschaftliches Rechnen Angewandte Mathematik, Mathematische Modelle
• Mathematik | Informatik Mathematik Mathematische Analysis Differentialrechnungen und -gleichungen

Inhaltsverzeichnis

INTRODUCTION
Why exactly solvable models are important
Intra- and inter-species interactions and local population dynamics
Basic mechanisms of species transport
Biological invasion: main facts and constituting examples
MODELS OF BIOLOGICAL INVASION
Diffusion-reaction equations
Integral-difference models
Space-discrete models
Stochastic models
Concluding remarks
BASIC METHODS AND RELEVANT EXAMPLES
The Cole-Hopf transformation and the Burgers equation as a paradigm
Further application of the Cole-Hopf transformation
Method of piecewise linear approximation
Exact solutions of a generalized Fisher equation
More about ansatz
SINGLE-SPECIES MODELS
Impact of advection and migration
Accelerating population waves
The problem of critical aggregation
DENSITY-DEPENDENT DIFFUSION
The Aronson-Newman solution and its generalization
Stratified diffusion and the Allee effect
MODELS OF INTERACTING POPULATIONS
Exact solution for a diffusive predator-prey system
Migration waves in a resource-consumer system
SOME ALTERNATIVE AND COMPLEMENTARY APPROACHES
Wave speed and the eigenvalue problem
Convergence of the initial conditions
Convergence and the paradox of linearization
Application of the comparison principle
ECOLOGICAL EXAMPLES AND APPLICATIONS
Invasion of Japanese beetle in the United States
Mount St. Helens recolonization and the impact of predation
Stratified diffusion and rapid plant invasion
APPENDIX: BASIC BACKGROUND MATHEMATICS
Ordinary differential equations and their solutions
Phase plane and stability analysis
Diffusion equation

References
Index