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Spatial Ecology via Reaction-Diffusion Equations




Spatial Ecology via Reaction-Diffusion Equations

Robert Stephen Cantrell, Chris Cosner

ISBN: 978-0-471-49301-3 September 2003 428 Pages


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 years
Essential reading for graduate students and researchers working with spatial modelling from mathematics, statistics, ecology, geography and biology.

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.


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.


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.


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.


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.


7 Nonmonotone Systems.

7.1 Introduction.

7.2 Predator Mediated Coexistence.

7.3 Three Species Competition.

7.4 Three Trophic Level Models.



"…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)