Many ecological phenomena involve space as well as time and arise from a combination of random and deterministic processes. Such phenomena include the effects of habitat fragmentation, which is a common result of human activity and a major problem in biological conservation. Reaction-diffusion models provide one approach to describing how random movements and deterministic interactions between individuals combine to influence the dynamics of populations and the structure of ecological communities. Spatial Ecology via Reaction-Diffusion Equations addresses the problem of modeling spatial effects in ecology and population dynamics using reaction-diffusion models.* Provides broad coverage of a rapidly expanding area of research for ecologists and applied mathematicians.* Provides a unified and coherent account of methods developed to study spatial ecology via reaction-diffusion models.* Provides the reader with the tools needed to construct and interpret models.* Includes specific applications of both the models and the methods described.Spatial Ecology via Reaction-Diffusion Equations provides a practical introduction to the subject for graduate students and researchers working in spatial modeling from mathematics, statistics, ecology, geography and biology.
Many ecological phenomena may be modelled using apparently random processes involving space (and possibly time). Such phenomena are classified as spatial in their nature and include all aspects of pollution. This book addresses the problem of modelling spatial effects in ecology and population dynamics using reaction-diffusion models.* Rapidly expanding area of research for biologists and applied mathematicians* Provides a unified and coherent account of methods developed to study spatial ecology via reaction-diffusion models* Provides the reader with the tools needed to construct and interpret models* Offers specific applications of both the models and the methods* Authors have played a dominant role in the field for yearsEssential reading for graduate students and researchers working with spatial modelling from mathematics, statistics, ecology, geography and biology.|Viele Probleme aus der Ökologie, darunter zum Beispiel die Ausbreitung von Schadstoffen, lassen sich modellieren, wenn man auf scheinbar zufällige Prozesse mit einer räumlichen und gegebenenfalls auch einer zeitlichen Dimension zurückgreift. Dieses Buch zeigt Ihnen, wie man räumliche Effekte in der Ökologie und Populationsdynamik mit sogenannten Reaktions-Diffusions-Modellen beschreiben kann. Sie lernen, Modelle aufzubauen und die Resultate zu interpretieren. Die Umsetzung der Theorie in die Praxis wird anhand spezieller Anwendungsbeispiele demonstriert.
Viele Probleme aus der Ökologie, darunter zum Beispiel die Ausbreitung von Schadstoffen, lassen sich modellieren, wenn man auf scheinbar zufällige Prozesse mit einer räumlichen und gegebenenfalls auch einer zeitlichen Dimension zurückgreift. Dieses Buch zeigt Ihnen, wie man räumliche Effekte in der Ökologie und Populationsdynamik mit so genannten Reaktions-Diffusions-Modellen beschreiben kann. Sie lernen, Modelle aufzubauen und die Resultate zu interpretieren. Die Umsetzung der Theorie in die Praxis wird anhand spezieller Anwendungsbeispiele demonstriert.
Preface.
Series Preface.
1 Introduction.
1.1 Introductory Remarks.
1.2 Nonspatial Models for a Single Species.
1.3 Nonspatial Models For Interacting Species.
1.4 Spatial Models: A General Overview.
1.5 Reaction-Diffusion Models.
1.6 Mathematical Background.
2 Linear Growth Models for a Single Species: Averaging Spatial Effects Via Eigenvalues.
2.1 Eigenvalues, Persistence, and Scaling in Simple Models.
2.2 Variational Formulations of Eigenvalues: Accounting for Heterogeneity.
2.3 Effects of Fragmentation and Advection/Taxis in Simple Linear Models.
2.4 Graphical Analysis in One Space Dimension.
2.5 Eigenvalues and Positivity.
2.6 Connections with Other Topics and Models.
Appendix.
3 Density Dependent Single-Species Models.
3.1 The Importance of Equilibria in Single Species Models.
3.2 Equilibria and Stability: Sub- and Supersolutions.
3.3 Equilibria and Scaling: One Space Dimension.
3.4 Continuation and Bifurcation of Equilibria.
3.5 Applications and Properties of Single Species Models.
3.6 More General Single Species Models.
Appendix.
4 Permanence.
4.1 Introduction.
4.2 Definition of Permanence.
4.3 Techniques for Establishing Permanence.
4.4 Invasibility Implies Coexistence.
4.5 Permanence in Reaction-Diffusion Models for Predation.
4.6 Ecological Permanence and Equilibria.
Appendix.
5 Beyond Permanence: More Persistence Theory.
5.1 Introduction.
5.2 Compressivity.
5.3 Practical Persistence.
5.4 Bounding Transient Orbits.
5.5 Persistence in Nonautonomous Systems.
5.6 Conditional Persistence.
5.7 Extinction Results.
Appendix.
6 Spatial Heterogeneity in Reaction-Diffusion Models.
6.1 Introduction.
6.2 Spatial Heterogeneity within the Habitat Patch.
6.3 Edge Mediated Effects.
6.4 Estimates and Consequences.
Appendix.
7 Nonmonotone Systems.
7.1 Introduction.
7.2 Predator Mediated Coexistence.
7.3 Three Species Competition.
7.4 Three Trophic Level Models.
Appendix.
References.
Index.
"...particularly attractive and useful for graduate students and other researchers who are interested in studying applications of reaction-diffusion theory to spatial ecology." ( Mathematical Reviews , Issue 2007a)
"...I would recommend this book to anyone who wants a well supported journey into the modern theory of partial differential equations and dynamic systems..." ( The Mathematical Gazette , March 2005)
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Inhaltsverzeichnis
Preface.
Series Preface.
1 Introduction.
1.1 Introductory Remarks.
1.2 Nonspatial Models for a Single Species.
1.3 Nonspatial Models For Interacting Species.
1.4 Spatial Models: A General Overview.
1.5 Reaction-Diffusion Models.
1.6 Mathematical Background.
2 Linear Growth Models for a Single Species: Averaging Spatial Effects Via Eigenvalues.
2.1 Eigenvalues, Persistence, and Scaling in Simple Models.
2.2 Variational Formulations of Eigenvalues: Accounting for Heterogeneity.
2.3 Effects of Fragmentation and Advection/Taxis in Simple Linear Models.
2.4 Graphical Analysis in One Space Dimension.
2.5 Eigenvalues and Positivity.
2.6 Connections with Other Topics and Models.
Appendix.
3 Density Dependent Single-Species Models.
3.1 The Importance of Equilibria in Single Species Models.
3.2 Equilibria and Stability: Sub- and Supersolutions.
3.3 Equilibria and Scaling: One Space Dimension.
3.4 Continuation and Bifurcation of Equilibria.
3.5 Applications and Properties of Single Species Models.
3.6 More General Single Species Models.
Appendix.
4 Permanence.
4.1 Introduction.
4.2 Definition of Permanence.
4.3 Techniques for Establishing Permanence.
4.4 Invasibility Implies Coexistence.
4.5 Permanence in Reaction-Diffusion Models for Predation.
4.6 Ecological Permanence and Equilibria.
Appendix.
5 Beyond Permanence: More Persistence Theory.
5.1 Introduction.
5.2 Compressivity.
5.3 Practical Persistence.
5.4 Bounding Transient Orbits.
5.5 Persistence in Nonautonomous Systems.
5.6 Conditional Persistence.
5.7 Extinction Results.
Appendix.
6 Spatial Heterogeneity in Reaction-Diffusion Models.
6.1 Introduction.
6.2 Spatial Heterogeneity within the Habitat Patch.
6.3 Edge Mediated Effects.
6.4 Estimates and Consequences.
Appendix.
7 Nonmonotone Systems.
7.1 Introduction.
7.2 Predator Mediated Coexistence.
7.3 Three Species Competition.
7.4 Three Trophic Level Models.
Appendix.
References.
Index.
Klappentext
Viele Probleme aus der Ökologie, darunter zum Beispiel die Ausbreitung von Schadstoffen, lassen sich modellieren, wenn man auf scheinbar zufällige Prozesse mit einer räumlichen und gegebenenfalls auch einer zeitlichen Dimension zurückgreift. Dieses Buch zeigt Ihnen, wie man räumliche Effekte in der Ökologie und Populationsdynamik mit sogenannten Reaktions-Diffusions-Modellen beschreiben kann. Sie lernen, Modelle aufzubauen und die Resultate zu interpretieren. Die Umsetzung der Theorie in die Praxis wird anhand spezieller Anwendungsbeispiele demonstriert.
