Dispersal, Individual Movement and Spatial Ecology: A Mathematical Perspective

Cover......Page 1
Dispersal, Individual Movement and Spatial Ecology......Page 4
Copyright......Page 5
Numerical Study of Pest Population Size at Various Diffusion Rates......Page 6
Appendix 1......Page 7
Appendix 2......Page 10
Contents......Page 14
Stochastic Optimal Foraging Theory......Page 16
1 Introduction......Page 17
2 Some Preliminary Assumptions of the Model......Page 18
3 Calculation of  and <|l|>......Page 20
4 Discrete Space Calculation......Page 24
5 Lévy Random Searchers......Page 27
6 Search Diffusivity......Page 29
6.1 Characterizing First-Passage-Time Diffusivity......Page 30
6.2 Characterizing Search Dynamics Diffusivity......Page 34
7 Search in Heterogeneous Landscapes: Distributions of Starting Points......Page 36
8 Discussion......Page 42
References......Page 43
1.1 Lévy Walks......Page 47
1.2 Correlated Random Walks and Composite Strategies......Page 48
1.3 Determining Movement Processes from Observational Data......Page 49
2.1 Discrete Time Sampling......Page 50
2.2 Identification of Turning Points......Page 51
3.1 The Composite Correlated Random Walk Model......Page 53
3.2 Fitting a Composite Exponential Distribution......Page 54
3.3 Testing Step Length Data......Page 57
3.4 Testing Uniformity of Turning Angles......Page 60
4 Discussion......Page 61
References......Page 62
1 Introduction......Page 67
2 Underlying Mechanisms: The Key to Prediction and Understanding......Page 68
2.1 Serial Correlation......Page 69
2.2 Random Reorientation at Cues Left by Correlated Random Walkers......Page 70
2.3 Lévy Walks as by Products of Advantageous Foraging Behaviours......Page 71
3 Translating Observations Taken at Small Spatiotemporal Scales into Expected Patterns at Greater Scales......Page 73
4 Enlarging the Framework of Lévy Walk Search Theory......Page 74
4.1 Balancing the Demands of Foraging and Safety from Predation......Page 76
4.2 Red Queen Dynamics......Page 78
4.3 Intermittent Searches......Page 79
4.4 Optimizing the Encounter Rate in Biological Interactions......Page 80
5.1 Opening the Lévy Gates......Page 81
5.2 Lévy Walks in Collective Motions: How the Blind Could Lead the Blind......Page 82
5.3 Mathematical Challenges......Page 84
References......Page 85
1 Introduction......Page 90
2.1 A Summary of the Levels of Description......Page 93
2.2 The Fokker-Planck and Smoluchowski Equations......Page 96
2.3 Interacting Particles, Liouville\'s Equation and Reduced Descriptions......Page 97
3.1 The Pearson Random Walk......Page 99
3.2 The General Evolution Equation for Space-Jump or Kangaroo Processes......Page 101
3.3 The Evolution of Spatial Moments for General Kernels......Page 103
3.4 The Effects of Long Waits or Large Jumps......Page 105
3.5 Biased Jumps Dependent on Gradients or Internal Dynamics......Page 107
3.6 Aggregation in Reinforced Random Walks......Page 109
4.1 The General Velocity-Jump Process......Page 114
4.2 The Telegraph Process......Page 117
4.3 Reduction of the VJ Process to a Diffusion Process......Page 120
4.4 The Role of Internal Dynamics......Page 124
4.5 Macroscopic Descriptions of Eukaryotic Cell Movement......Page 129
5 Discussion......Page 133
References......Page 134
1 Introduction......Page 139
2 Continuum vs. Agent-Based Models......Page 141
2.1 Agent-Based Modelling......Page 142
3 Hybrid Modelling: Theoretical Framework......Page 144
3.1 A Position-Based Hybrid Model......Page 146
4 Hybrid Modelling: Numerical Implementation......Page 147
4.1 Spatial Matching in Numerical Simulations......Page 148
4.2 Other Aspects of Numerical Simulations......Page 150
5.1 The Keller-Segel Model......Page 151
5.2 Hybrid Models of Chemotaxis......Page 153
5.3 Analysis of the Dropout......Page 157
6 Discussion......Page 163
References......Page 165
1 Introduction......Page 168
2.1 Population Dynamics......Page 170
2.2 Individual Movement......Page 172
2.2.1 Resource-Dependent Settling......Page 173
2.2.3 Prospect-Dependent Settling......Page 174
2.2.4 Nondimensionalization......Page 175
3 Effects of Movement Rules on Persistence......Page 176
4.1 Resource-Dependent Settling......Page 177
4.2 Time-Dependent Settling......Page 180
4.3 Prospect-Dependent Settling......Page 181
5 Discussion......Page 182
References......Page 183
1.2 Mathematical Modelling......Page 186
2 Transport Equations......Page 188
2.1 Movement in an Oriented Environment......Page 189
2.2 Environmental Distributions......Page 191
3.1 Motivation of the Parabolic Limit......Page 192
3.2 Parabolic Limit in an Oriented Landscape......Page 194
4 The Hyperbolic Scaling......Page 196
5 The Moment Approach......Page 200
5.1 Moment Closure......Page 204
5.2 Fast Flux Relaxation......Page 205
6.1 Relationships Between Limit Equations......Page 206
6.2 Assumptions Behind Limit Equations......Page 207
7 Examples and Applications......Page 208
7.1.1 Isotropic Diffusion: The Pearson Walk......Page 209
7.1.2 Anisotropic Diffusion Example......Page 210
7.1.4 Application to Seismic Line Following......Page 213
7.2 Unidirectional Environments......Page 216
7.2.1 Anisotropic Diffusion-Drift Example......Page 217
7.3 Singular Distributions......Page 218
7.3.1 Strictly Bidirectional: Degenerate Diffusion......Page 219
7.3.2 Strictly Unidirectional: Relation to ODEs......Page 220
8 Discussion......Page 221
9.1 Unimodal von Mises Distribution......Page 224
10.1 Simulations of Transport Model......Page 227
References......Page 228
1 Introduction......Page 232
2 The Lagrangian vs. the Eulerian Approach in the Modelling of Zooplankton Dynamics......Page 235
3 Modelling and Scaling the Zooplankton Functional Response......Page 236
3.1 Defining the Zooplankton Functional Response in Real Ecosystems......Page 237
3.2 Emergence of a Sigmoid (Holling Type III) Overall Zooplankton Functional Response......Page 240
4 The Role of Intra-population Variability of Zooplankton in Population Persistence......Page 245
4.1 Describing the Intra-population Variability of Zooplankton Grazers......Page 246
4.2 Analysis of the Model of the Intra-population Variability of Zooplankton......Page 249
5 Discussion and Conclusions......Page 254
References......Page 262
1 Introduction......Page 268
2 Is Further Better?......Page 270
3 The Critical Range-Shift Speed......Page 272
4 Numerical Approaches......Page 276
5 Analytic Approximations......Page 278
6 A Simple Example......Page 280
7 A Simplifying Approximation......Page 283
8.1 Gaussian Distribution......Page 285
8.2 Laplace Distribution......Page 287
8.3 Cauchy Distribution......Page 288
8.4 Modified Bessel Distribution......Page 289
9 Discussion......Page 290
References......Page 293
2 A Competition-Diffusion Model with Annual Cycles and Random Extreme Events......Page 299
2.2 Annual Cycles of Growth, Harvesting and Diffusion......Page 300
2.3 Random Extreme Events and Assisted Long-Distance Transport......Page 301
2.4 Numerical Simulations I......Page 302
3 A Competition-Diffusion Model with Infected Invader......Page 304
3.1 Local Dynamics with Infection......Page 305
3.3 Numerical Simulations II......Page 306
References......Page 308
1 Introduction......Page 312
2.1 The Single-Species Model......Page 313
2.3 Competition: Two Competing Species......Page 315
2.4 Competition: More Than Two Competing Species......Page 321
2.5 Predation......Page 322
2.7 Mutualism......Page 325
2.8 A Simple Food Web......Page 326
2.9 Heterogeneity......Page 330
3 Discussion......Page 331
References......Page 332
1 Speciation Theory and Nonlocal Reaction-Diffusion Equations......Page 336
2.1 Equation (3)......Page 340
2.2 Equation (4)......Page 341
2.3 Essential Spectrum of the Operator Linearized about a Wave......Page 342
3 Nonlinear Stability......Page 344
3.1 An Abstract Theorem on Stability of Stationary Solutions......Page 346
3.2 Stability of the Homogeneous Solution......Page 348
3.3 Stability of Waves......Page 350
4 Discussion......Page 353
References......Page 355
1 Introduction......Page 359
1.1 Spatial Heterogeneity and the Effect of Diffusion......Page 361
1.2 Goals and the Road Map......Page 364
2 Numerical Integration on Coarse Grids: The Problem Outline......Page 365
2.1 The Method......Page 367
3 Simulation Data......Page 368
3.1 Estimating Pest Population Size on Coarse Grids: Numerical Test Cases......Page 369
4 The Impact of the Diffusion Rates on the Accuracy of Numerical Integration......Page 371
4.1 Uniform Grid......Page 373
4.2 The Analysis of the Grid Step Size for Ecological Distributions......Page 374
4.3 Arbitrary Location of the Peak on a Uniform Coarse Grid......Page 376
5 Nonuniform Grid......Page 379
6 Discussion and Conclusions......Page 381
References......Page 387
LECTURE NOTES IN MATHEMATICS......Page 389