Matrix population models : construction, analysis, and interpretation / Hal Caswell.

Format:

Book

Author/Creator:

Caswell, Hal.

Language:

English

Subjects (All):

Population.

Models, Biological.

Demography.

Biometry.

Population biology--Computer simulation.

Population biology.

Matrices--Data processing.

Matrices.

Medical Subjects:

Population.

Models, Biological.

Demography.

Biometry.

Physical Description:

xxii, 722 pages : illustrations ; 24 cm

Edition:

Second edition.

Place of Publication:

Sunderland, Mass. : Sinauer Associates, [2001]

Summary:

Matrix Population Models, Second Edition includes expanded treatment of stochastic and density-dependent models, sensitivity analysis, and statistical inference, and new chapters on parameter estimation, structured population models, demographic sto-chasticity, and applications of matrix models in conservation biology. This book is an indispensable reference for graduate students and researchers in ecology, population biology, conservation biology, and human demography.

Contents:

1.1 The life cycle: Linking the individual and the population 1

1.3.2 Matlab programs 5

1.4 Construction, analysis, and interpretation 5

1.5 Mathematical prerequisites 5

2 Age-Classified Matrix Models 8

2.1 The Leslie matrix 8

2.2 Projection: the simplest form of analysis 11

2.2.1 A set of questions 18

2.3 The Leslie matrix and the life table 20

2.3.1 Survival 21

2.3.2 Reproduction 22

2.4 Constructing age-classified matrices 22

2.4.1 Birth-flow populations 23

2.4.2 Birth-pulse populations 25

2.5 Assumptions: Projection vs. forecasting 29

2.6 History 31

3 Stage-Classified Life Cycles 35

3.1 State variables 35

3.1.1 Zadeh's theory of state 36

3.1.2 State variables in population models 37

3.2 Age as a state variable: When does it fail? 38

3.2.1 Size-dependent vital rates and plastic growth 39

3.2.2 Multiple modes of reproduction 40

3.2.3 Population subdivision and multistate demography 41

3.3 Statistical evaluation of state variables 41

3.3.1 Continuous response, continuous or discrete state 41

3.3.2 Discrete state, discrete response 43

3.3.3 Continuous state, discrete response 54

4 Stage-Classified Matrix Models 56

4.1 The life cycle graph 56

4.2 The matrix model 59

4.3 Metapopulation and multistate models 62

4.3.1 Modelling dispersal 65

4.3.2 Integrodifference equation models 71

4.4 Solution of the projection equation 72

4.4.1 Derivation 1 74

4.4.2 Derivation 2 75

4.4.3 Effects of the eigenvalues 76

4.5 Ergodicity 79

4.5.1 The Perron-Frobenius theorem 79

4.5.2 Population growth rate: The strong ergodic theorem 84

4.5.3 Imprimitive matrices 87

4.5.4 Reducible matrices 88

4.6 Reproductive value 92

4.7 Transient dynamics and convergence 94

4.7.1 The damping ratio and convergence 95

4.7.2 The period of oscillation 100

4.7.3 Measuring the distance to the stable stage distribution 101

4.7.4 Population momentum 104

4.8 Computation of eigenvalues and eigenvectors 106

4.8.1 Eigenvalues and eigenvectors in Matlab 107

4.8.2 The power method 108

4.9 Assumptions revisited 109

5 Events in the Life Cycle 110

5.1 A = T + F 110

5.1.1 The life cycle as a Markov chain 111

5.1.2 The analysis of absorbing chains 112

5.2 Lifetime event probabilities 115

5.3 Age-specific traits 116

5.3.1 Age-specific survival 118

5.3.2 Age-specific fertility 120

5.3.3 Age at first reproduction 124

5.3.4 Net reproductive rate 126

5.3.5 Generation time 128

5.3.6 Age-within-stage distributions 130

6 Parameter Estimation 133

6.1 Identified individuals 134

6.1.1 Observed transition frequencies 134

6.1.2 Mark-recapture methods 136

6.2 Inverse methods for time series 142

6.2.1 Regression methods 142

6.2.2 Wood's quadratic programming method 144

6.2.3 A maximum likelihood approach 152

6.2.4 Stage-frequency methods 154

6.3 Stable stage-distribution methods 154

6.3.1 Age-classified models 154

6.3.2 Size-classified models 157

6.3.3 Death assemblages 157

6.4 Stage-duration distributions 159

6.4.1 The geometric distribution 160

6.4.2 Fixed stage durations 160

6.4.3 Variable stage durations 162

6.4.4 Iterative calculation 164

6.4.5 Negative binomial stage durations 164

6.4.6 Duration distributions compared 165

6.5 Multiregional or age-size models 166

6.6 Size categories: The Vandermeer-Moloney algorithms 169

6.7 Fertilities in stage-classified models 171

6.7.1 Birth-flow populations 171

6.7.2 Birth-pulse populations 172

6.7.3 Anonymous reproduction 173

7 Analysis of the Life Cycle Graph 176

7.1 The z-transform 177

7.1.1 z-transform solution of difference equations 177

7.1.2 The z-transformed life cycle graph 178

7.2 Reduction of the life cycle graph 178

7.2.1 Multistep transitions 181

7.3 The characteristic equation 181

7.3.1 Derivation 184

7.4 The stable stage distribution 185

7.4.1 Derivation 187

7.5 Reproductive value 187

7.5.1 A second interpretation of reproductive value 189

7.5.2 A note on eigenvectors of reducible matrices 190

7.6 Partial life cycle analysis 190

7.7 Annual organisms 192

8 Structured Population Models 194

8.1 Partial differential equation models 194

8.1.1 Lotka's renewal equation 196

8.1.2 Discretizing Lotka's equation: Don't bother 197

8.1.3 Diffusion models 198

8.1.4 PDE models and matrix models 198

8.1.5 The escalator boxcar train 199

8.2 Delay-differential equation models 201

8.3 Integrodifference equation models 202

8.4 i-state configuration models 202

8.5 Choosing a model 204

9 Sensitivity Analysis 206

9.1 Eigenvalue sensitivity 208

9.1.1 Perturbations of matrix elements 208

9.1.2 Sensitivity and age 211

9.1.3 Sensitivities in stage- and size-classified models 213

9.1.4 What about those zeros? 215

9.1.5 Sensitivity to multistep transitions 217

9.1.6 Total derivatives and multiple perturbations 218

9.1.7 Sensitivity to changes in development rate 220

9.1.8 Predictions from sensitivities 224

9.1.9 An overall eigenvalue sensitivity index 224

9.1.10 A third interpretation of reproductive value 225

9.2 Elasticity analysis 226

9.2.1 Elasticity and age 227

9.2.2 Elasticities as contributions to [lambda] 229

9.2.3 Elasticities of [lambda] to lower-level parameters 232

9.2.4 Comparative analysis of elasticity patterns 233

9.2.5 Predictions from elasticities 240

9.2.6 Sensitivity or elasticity? 243

9.3 Sensitivity analysis of transient dynamics 244

9.3.1 Sensitivity of the damping ratio 244

9.3.2 Sensitivity of the period 247

9.4 Sensitivities of eigenvectors 247

9.4.1 Sensitivities of scaled eigenvectors 250

9.5 Generalized inverses in sensitivity analysis 251

9.6 Sensitivity analysis of Markov chains 251

9.7 Second Derivatives of Eigenvalues 254

9.7.1 Perturbation analysis of elasticities 256

10 Life Table Response Experiments 258

10.1 Fixed designs 260

10.1.1 One-way designs 260

10.1.2 Factorial designs 263

10.2 Random designs and variance decomposition 269

10.3 Regression designs 273

10.4 Extensions 274

10.4.1 Higher-order terms 274

10.4.2 Other demographic statistics 275

10.4.3 Other demographic models 275

10.4.4 More mechanisms 276

10.4.5 Statistics 277

10.5 Prospective and retrospective analyses 277

11 Evolutionary Demography 279

11.1 Fitness 280

11.1.1 Population genetics 281

11.1.2 Quantitative genetics 282

11.1.3 Invasion and ESS analysis 291

11.2 Sensitivity, elasticity, and selection 295

11.3 Lifetime reproductive success and individual fitness 295

11.4 Fitness and reproductive value 297

12 Statistical Inference 299

12.1 Confidence intervals and uncertainty 300

12.1.1 Series approximations 300

12.1.2 Bootstrap standard errors 304

12.1.3 Bootstrap confidence intervals 306

12.1.4 Complex data structures 309

12.1.5 More on the bootstrap 315

12.1.6 Monte Carlo uncertainty analysis 319

12.1.7 The precision of estimates of [lambda] 322

12.2 Loglinear analysis of transition matrices 326

12.2.1 One factor 327

12.2.2 Two factors 330

12.2.3 Model selection and AIC 332

12.2.4 Presentation of loglinear analyses 334

12.3 Randomization tests 335

12.3.1 The randomization test procedure 337

12.3.2 Types of data 338

12.3.3 Examples of randomization tests 338

12.3.4 Advantages of randomization tests 343

12.3.5 Implementation 345

13 Periodic Environments 346

13.1 Periodic matrix products 347

13.1.2 Eigenvalues and eigenvectors 349

13.1.3 Matrices don't commute, and why that matters 354

13.1.4 Sensitivity analysis of periodic matrix models 356

13.2 Annual organisms 361

13.2.1 Periodic matrix models for annuals 362

13.3 Other approaches

to periodic environments 368

13.3.1 Classification by season of birth 368

13.3.2 Discrete Fourier analysis 368

13.4 Deterministic, aperiodic environments 369

13.4.1 Weak ergodicity 369

14 Environmental Stochasticity 377

14.1 Formulation of stochastic models 377

14.1.1 Models for the environment 378

14.1.2 Linking the environment and the vital rates 381

14.1.3 Projecting the population 382

14.2 Stochastic ergodic theorems 382

14.2.1 Stage distributions 382

14.2.2 Stochastic reproductive value 384

14.2.3 Sufficient conditions for stochastic ergodicity 384

14.2.4 An overview of ergodic results 386

14.3 Stochastic population growth 387

14.3.1 The Lewontin-Cohen model 387

14.3.2 Beyond iid processes 392

14.3.3 Ergodic properties of random matrix products 393

14.3.4 Growth of the mean 394

14.3.5 Which growth rate is relevant? 395

14.3.6 Calculating the stochastic growth rate 396

14.3.7 Calculation of the variance [sigma superscript 2] 399

14.3.8 Scalar and matrix models compared 400

14.4 Sensitivity and elasticity analyses 401

14.4.1 From numerical simulations 402

14.4.2 From Tuljapurkar's approximation 407

14.4.3 Sensitivity of log [lambda subscript s] to variability 408

14.5 Examples of stochastic models 409

14.5.1 Striped bass: Variability in recruitment 409

14.5.2 Clams: Parametric distributions of recruitment 410

14.5.3 Stochastic models from sequence of matrices 415

14.5.4 Markov chain models for the environment 419

14.5.5 Random selection of matrix elements 430

14.5.6 Applications of Tuljapurkar's approximation 435

14.5.7 Some suggestions 435

14.6 Evolution in stochastic environments 436

14.6.1 Fitness and ESS in stochastic environments 436

14.6.2 An example: Delayed reproduction 437

14.6.3 Life history studies 440

14.7 Sensitivity: Stochastic and deterministic models 443

14.8 Extinction in stochastic environments 443

14.8.1 A model for quasi-extinction 444

14.8.2 Sensitivity analysis 447

14.9 Short-term stochastic forecasts 449

15 Demographic Stochasticity 452

15.1 Stochastic simulations 453

15.1.1 Assumptions, essential and otherwise 456

15.1.2 Simulating individuals 456

15.1.3 A computationally efficient alternative 458

15.1.4 Bad luck, or something worse? 462

15.1.5 Time-varying and density-dependent models 464

15.2 The Galton-Watson branching process 464

15.2.1 Probability generating functions 466

15.2.2 Population projection 467

15.2.3 Projection of moments 470

15.2.4 Limit theorems and asymptotic dynamics 471

15.2.5 Extinction 472

15.2.6 Quasi-stationary distributions 475

15.2.7 Extinction, effective population size, and elasticity 475

15.3 Multitype branching processes 478

15.3.1 From matrix models to branching processes 479

15.3.2 Mean and covariance of offspring production 483

15.4 Analysis of multitype branching processes 486

15.4.1 Population projection 486

15.4.2 Projection of moments 486

15.4.3 Limit theorems and asymptotic dynamics 491

15.4.4 Extinction probability 493

15.4.5 Extinction probability and reproductive value 497

15.4.6 Elasticities of extinction probability and of [lambda] 497

15.4.7 Subcritical multitype branching processes 499

15.5 Branching processes in random environments 501

15.6 Assumptions revisited 502

16 Density-Dependent Models 504

16.1 Model construction 505

16.1.1 Types of density dependence 505

16.2 Asymptotic dynamics and invariant sets 513

16.2.1 Finding equilibria 518

16.3 Stability and instability 519

16.4 Local stability of equilibria 519

16.4.1 The Jury criteria 522

16.5 Bifurcation diagrams 525

16.6 Bifurcations of equilibria: A field guide 526

16.6.1 +1 bifurcations 528

16.6.2 -1 bifurcations: The flip bifurcation 533

16.6.3 Complex conjugate pairs: The Hopf bifurcation 533

16.6.4 Supercritical and subcritical bifurcations 537

16.7 Chaos 538

16.7.1 Lyapunov exponents and quantitative unpredictability 542

16.7.2 Routes to chaos 543

16.7.3 Chaotic power spectra 546

16.8 Transient dynamics 548

16.8.1 Reactivity and resilience of stable equilibria 548

16.8.2 Unstable equilibria 550

16.8.3 Strange repellers and chaotic transients 551

16.8.4 Effects of random perturbations 551

16.9 Multiple attractors and qualitative unpredictability 552

16.10 Tribolium: Models and experiments 553

16.11 Perturbation analysis and evolution 557

16.11.1 Sensitivity analysis of equilibria 559

16.11.2 Invasion and evolution 560

16.12 Stochasticity and density dependence 565

17 Two-Sex Models 568

17.1 Sexual dimorphism in the vital rates 568

17.2 Dominance, sex ratio, and the marriage squeeze 570

17.3 Two-sex models 571

17.3.1 A simple two-sex model 572

17.3.2 The birth and fertility functions 574

17.3.3 Frequency and density dependence 576

17.3.4 The equilibrium population structure 576

17.3.5 Stability of population structure 578

17.3.6 Nussbaum's global stability theorem 581

17.4 Competition for mates 581

17.4.1 Numerical results: Competition and instability 583

17.5 The birth matrix-mating rule model 585

17.6 More detailed models of mating 587

17.7 Frequency and density dependence combined 588

17.8 Extinction and the sex ratio 589

18 Conservation and Management 591

18.1 Conservation 592

18.1.1 Assessment 592

18.1.2 Diagnosis 601

18.1.3 Prescription 604

18.1.4 Prognosis 619

18.1.5 Conservation conclusions 629

18.2 Pest control 629

18.2.1 Reducing population size 630

18.2.2 Extermination 631

18.2.3 Halting invasion 632

18.2.4 Some examples of pest control 634

18.3 Harvesting 640

18.3.1 Optimal harvesting 642

19.1 The most important task 646

19.2 Testing models 648

19.3 A complete demographic analysis 649

19.4 Directions for research 651

A The Basics of Matrix Algebra 653

A.1 Motivation 653

A.3 Operations 655

A.3.1 Addition 655

A.3.2 Scalar multiplication 655

A.3.3 The transpose and the adjoint 655

A.3.4 The trace 656

A.3.5 Scalar product 656

A.3.6 Matrix multiplication 656

A.3.7 The Kronecker and Hadamard products 658

A.4 Matrix inversion 659

A.4.1 The identity matrix 659

A.4.2 Inversion and the solution of algebraic equations 659

A.4.3 A useful fact about homogeneous systems 660

A.5 Determinants 660

A.5.1 Properties of determinants 662

A.6 Eigenvalues and eigenvectors 662

A.6.1 Eigenvectors 662

A.6.2 Left eigenvectors 663

A.6.3 The characteristic equation 663

A.6.4 Finding the eigenvectors 664

A.6.5 Complications 665

A.6.6 Linear independence of eigenvectors 665

A.6.7 Left and right eigenvectors 666

A.6.8 Computation of eigenvalues and eigenvectors 666

A.7 Similarity 666

A.7.1 Properties of similar matrices 667

A.8 Norms of vectors and matrices 667.

Notes:

Includes bibliographical references (pages [669]-710) and index.

ISBN:

0878930965

OCLC:

44619483