Spatial ecology via reaction-diffusion equations / Robert Stephen Cantrell and Chris Cosner.

Format:

Book

Author/Creator:

Cantrell, Robert Stephen.

Contributor:

Cosner, Chris.

Alumni and Friends Memorial Book Fund.

Series:

Wiley series in mathematical and computational biology

Language:

English

Subjects (All):

Spatial ecology--Mathematical models.

Spatial ecology.

Reaction-diffusion equations.

Computational Biology--methods.

Medical Subjects:

Computational Biology--methods.

Physical Description:

xiii, 411 pages : illustrations ; 25 cm.

Place of Publication:

Chichester, West Sussex, England ; Hoboken, NJ : J. Wiley, [2003]

Summary:

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.

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.

Contents:

1.2 Nonspatial Models for a Single Species 3

1.3 Nonspatial Models For Interacting Species 8

1.3.1 Mass-Action and Lotka-Volterra Models 8

1.3.2 Beyond Mass-Action: The Functional Response 9

1.4 Spatial Models: A General Overview 12

1.5 Reaction-Diffusion Models 19

1.5.1 Deriving Diffusion Models 19

1.5.2 Diffusion Models Via Interacting Particle Systems: The Importance of Being Smooth 24

1.5.3 What Can Reaction-Diffusion Models Tell Us? 28

1.5.4 Edges, Boundary Conditions, and Environmental Heterogeneity 30

1.6 Mathematical Background 33

1.6.1 Dynamical Systems 33

1.6.2 Basic Concepts in Partial Differential Equations: An Example 45

1.6.3 Modern Approaches to Partial Differential Equations: Analogies with Linear Algebra and Matrix Theory 50

1.6.4 Elliptic Operators: Weak Solutions, State Spaces, and Mapping Properties 53

1.6.5 Reaction-Diffusion Models as Dynamical Systems 72

1.6.6 Classical Regularity Theory for Parabolic Equations 76

1.6.7 Maximum Principles and Monotonicity 78

2 Linear Growth Models for a Single Species: Averaging Spatial Effects Via Eigenvalues 89

2.1 Eigenvalues, Persistence, and Scaling in Simple Models 89

2.1.1 An Application: Species-Area Relations 91

2.2 Variational Formulations of Eigenvalues: Accounting for Heterogeneity 92

2.3 Effects of Fragmentation and Advection/Taxis in Simple Linear Models 102

2.3.1 Fragmentation 102

2.3.2 Advection/Taxis 104

2.4 Graphical Analysis in One Space Dimension 107

2.4.1 The Best Location for a Favorable Habitat Patch 107

2.4.2 Effects of Buffer Zones and Boundary Behavior 112

2.5 Eigenvalues and Positivity 117

2.5.1 Advective Models 119

2.5.2 Time Periodicity 123

2.5.3 Additional Results on Eigenvalues and Positivity 125

2.6 Connections with Other Topics and Models 126

2.6.1 Eigenvalues, Solvability, and Multiplicity 126

2.6.2 Other Model Types: Discrete Space and Time 127

3 Density Dependent Single-Species Models 141

3.1 The Importance of Equilibria in Single Species Models 141

3.2 Equilibria and Stability: Sub- and Supersolutions 144

3.2.1 Persistence and Extinction 144

3.2.2 Minimal Patch Sizes 146

3.2.3 Uniqueness of Equilibria 148

3.3 Equilibria and Scaling: One Space Dimension 151

3.3.1 Minimum Patch Size Revisited 151

3.4 Continuation and Bifurcation of Equilibria 159

3.4.1 Continuation 159

3.4.2 Bifurcation Results 164

3.4.3 Discussion and Conclusions 173

3.5 Applications and Properties of Single Species Models 175

3.5.1 How Predator Incursions Affect Critical Patch Size 175

3.5.2 Diffusion and Allee Effects 178

3.5.3 Properties of Equilibria 182

3.6 More General Single Species Models 185

4 Permanence 199

4.1.1 Ecological Overview 199

4.1.2 ODE Models as Examples 202

4.1.3 A Little Historical Perspective 211

4.2 Definition of Permanence 213

4.2.1 Ecological Permanence 214

4.2.2 Abstract Permanence 216

4.3 Techniques for Establishing Permanence 217

4.3.1 Average Lyapunov Function Approach 218

4.3.2 Acyclicity Approach 219

4.4 Invasibility Implies Coexistence 220

4.4.1 Acyclicity and an ODE Competition Model 221

4.4.2 A Reaction-Diffusion Analogue 224

4.4.3 Connection to Eigenvalues 228

4.5 Permanence in Reaction-Diffusion Models for Predation 231

4.6 Ecological Permanence and Equilibria 239

4.6.1 Abstract Permanence Implies Ecological Permance 239

4.6.2 Permanence Implies the Existence of a Componentwise Positive Equilibrium 240

5 Beyond Permanence: More Persistence Theory 245

5.2 Compressivity 246

5.3 Practical Persistence 252

5.4 Bounding Transient Orbits 261

5.5 Persistence in Nonautonomous Systems 265

5.6 Conditional Persistence 278

5.7 Extinction Results 284

6 Spatial Heterogeneity in Reaction-Diffusion Models 295

6.2 Spatial Heterogeneity within the Habitat Patch 305

6.2.1 How Spatial Segregation May Facilitate Coexistence 308

6.2.2 Some Disparities Between Local and Global Competition 312

6.2.3 Coexistence Mediated by the Shape of the Habitat Patch 316

6.3 Edge Mediated Effects 318

6.3.1 A Note About Eigenvalues 319

6.3.2 Competitive Reversals Inside Ecological Reserves Via External Habitat Degradation: Effects of Boundary Conditions 321

6.3.3 Cross-Edge Subsidies and the Balance of Competition in Nature Preserves 329

6.3.4 Competition Mediated by Pathogen Transmission 335

6.4 Estimates and Consequences 340

7 Nonmonotone Systems 351

7.2 Predator Mediated Coexistence 356

7.3 Three Species Competition 364

7.3.1 How Two Dominant Competitors May Mediate the Persistence of an Inferior Competitor 364

7.3.2 The May-Leonard Example Revisited 373

7.4 Three Trophic Level Models 378.

Notes:

Includes bibliographical references (pages [395]-408).

Local Notes:

Acquired for the Penn Libraries with assistance from the Alumni and Friends Memorial Book Fund.

ISBN:

0471493015

9780471493013

OCLC:

52418304

Online:

Publisher description