SE C O N D E D ITI O N CONSTRUCTION, ANALYSIS, AND INTERPRETATION

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1 Matrix Population Models SE C O N D E D ITI O N CONSTRUCTION, ANALYSIS, AND INTERPRETATION HAL CASWELL

2 Contents Preface Preface to the First Edition xvii xxi 1 Introduction The life cycle: Linking the individual and the population Demography Overview of the book Examples Matlab programs Construction, analysis, and interpretation Mathematical prerequisites Notation Age-Classified Matrix Models The Leslie matrix Projection: the simplest form of analysis A set of questions The Leslie matrix and the life table Survival Reproduction Constructing age-classified matrices Birth-flow populations Birth-pulse populations Assumptions: Projection vs. forecasting History Stage-Classified Life Cycles State variables Zadeh s theory of state State variables in population models

3 viii CONTENTS 3.2 Age as a state variable: When does it fail? Size-dependent vital rates and plastic growth Multiple modes of reproduction Population subdivision and multistate demography Statistical evaluation of state variables Continuous response, continuous or discrete state Discrete state, discrete response Continuous state, discrete response Overview Stage-Classified Matrix Models The life cycle graph The matrix model Metapopulation and multistate models Modelling dispersal Integrodifference equation models Other examples Solution of the projection equation Derivation Derivation Effects of the eigenvalues Ergodicity The Perron-Frobenius theorem Population growth rate: The strong ergodic theorem Imprimitive matrices Reducible matrices Reproductive value Transient dynamics and convergence The damping ratio and convergence The period of oscillation Measuring the distance to the stable stage distribution Population momentum Computation of eigenvalues and eigenvectors Eigenvalues and eigenvectors in Matlab The power method Assumptions revisited Events in the Life Cycle A = T + F The life cycle as a Markov chain The analysis of absorbing chains Lifetime event probabilities Age-specific traits

4 CONTENTS ix Age-specific survival Age-specific fertility Age at first reproduction Net reproductive rate Generation time Age-within-stage distributions Parameter Estimation Identified individuals Observed transition frequencies Mark-recapture methods Inverse methods for time series Regression methods Wood s quadratic programming method A maximum likelihood approach Stage-frequency methods Stable stage-distribution methods Age-classified models Size-classified models Death assemblages Stage-duration distributions The geometric distribution Fixed stage durations Variable stage durations Iterative calculation Negative binomial stage durations Duration distributions compared Multiregional or age-size models The Vandermeer-Moloney algorithms Fertilities in stage-classified models Birth-flow populations Birth-pulse populations Anonymous reproduction Overview Analysis of the Life Cycle Graph The z-transform z-transform solution of difference equations The z-transformed life cycle graph Reduction of the life cycle graph Multistep transitions The characteristic equation Derivation

5 x CONTENTS 7.4 The stable stage distribution Derivation Reproductive value A second interpretation of reproductive value A note on eigenvectors of reducible matrices Partial life cycle analysis Annual organisms Structured Population Models Partial differential equation models Lotka s renewal equation Discretizing Lotka s equation: Don t bother Diffusion models PDE models and matrix models The escalator boxcar train Delay-differential equation models Integrodifference equation models i-state configuration models Choosing a model Sensitivity Analysis Eigenvalue sensitivity Perturbations of matrix elements Sensitivity and age Sensitivities in stage- and size-classified models What about those zeros? Sensitivity to multistep transitions Total derivatives and multiple perturbations Sensitivity to changes in development rate Predictions from sensitivities An overall eigenvalue sensitivity index A third interpretation of reproductive value Elasticity analysis Elasticity and age Elasticities as contributions to λ Elasticities of λ to lower-level parameters Comparative analysis of elasticity patterns Predictions from elasticities Sensitivity or elasticity? Sensitivity analysis of transient dynamics Sensitivity of the damping ratio Sensitivity of the period Sensitivities of eigenvectors

6 CONTENTS xi Sensitivities of scaled eigenvectors Generalized inverses in sensitivity analysis Sensitivity analysis of Markov chains Second Derivatives of Eigenvalues Perturbation analysis of elasticities Life Table Response Experiments Fixed designs One-way designs Factorial designs Random designs and variance decomposition Regression designs Extensions Higher-order terms Other demographic statistics Other demographic models More mechanisms Statistics Prospective and retrospective analyses Evolutionary Demography Fitness Population genetics Quantitative genetics Invasion and ESS analysis Sensitivity, elasticity, and selection Lifetime reproductive success and individual fitness Fitness and reproductive value Statistical Inference Confidence intervals and uncertainty Series approximations Bootstrap standard errors Bootstrap confidence intervals Complex data structures More on the bootstrap Monte Carlo uncertainty analysis The precision of estimates of λ Loglinear analysis of transition matrices One factor Two factors Model selection and AIC

7 xii CONTENTS Presentation of loglinear analyses Randomization tests The randomization test procedure Types of data Examples of randomization tests Advantages of randomization tests Implementation Periodic Environments Periodic matrix products Notation Eigenvalues and eigenvectors Matrices don t commute, and why that matters Sensitivity analysis of periodic matrix models Annual organisms Periodic matrix models for annuals Other approaches to periodic environments Classification by season of birth Discrete Fourier analysis Deterministic, aperiodic environments Weak ergodicity Environmental Stochasticity Formulation of stochastic models Models for the environment Linking the environment and the vital rates Projecting the population Stochastic ergodic theorems Stage distributions Stochastic reproductive value Sufficient conditions for stochastic ergodicity An overview of ergodic results Stochastic population growth The Lewontin-Cohen model Beyond iid processes Ergodic properties of random matrix products Growth of the mean Which growth rate is relevant? Calculating the stochastic growth rate Calculation of the variance σ Scalar and matrix models compared Sensitivity and elasticity analyses From numerical simulations

8 CONTENTS xiii From Tuljapurkar s approximation Sensitivity of log λ s to variability Examples of stochastic models Striped bass: Variability in recruitment Clams: Parametric distributions of recruitment Stochastic models from sequence of matrices Markov chain models for the environment Random selection of matrix elements Applications of Tuljapurkar s approximation Some suggestions Evolution in stochastic environments Fitness and ESS in stochastic environments An example: Delayed reproduction Life history studies Stochastic and deterministic sensitivity Extinction in stochastic environments A model for quasi-extinction Sensitivity analysis Short-term stochastic forecasts Demographic Stochasticity Stochastic simulations Assumptions, essential and otherwise Simulating individuals A computationally efficient alternative Bad luck, or something worse? Time-varying and density-dependent models The Galton-Watson branching process Probability generating functions Population projection Projection of moments Limit theorems and asymptotic dynamics Extinction Quasi-stationary distributions Extinction, effective population size, and elasticity Multitype branching processes From matrix models to branching processes Mean and covariance of offspring production Analysis of multitype branching processes Population projection Projection of moments Limit theorems and asymptotic dynamics Extinction probability

9 xiv CONTENTS Extinction probability and reproductive value Elasticities of extinction probability and of λ Subcritical multitype branching processes Branching processes in random environments Assumptions revisited Density-Dependent Models Model construction Types of density dependence Examples Asymptotic dynamics and invariant sets Finding equilibria Stability and instability Local stability of equilibria The Jury criteria Bifurcation diagrams Bifurcations of equilibria: A field guide bifurcations bifurcations:theflipbifurcation Complex conjugate pairs: The Hopf bifurcation Supercritical and subcritical bifurcations Chaos Lyapunov exponents and quantitative unpredictability Routes to chaos Chaotic power spectra Transient dynamics Reactivity and resilience of stable equilibria Unstable equilibria Strange repellers and chaotic transients Effects of random perturbations Multiple attractors and qualitative unpredictability Tribolium: Modelsandexperiments Perturbation analysis and evolution Sensitivity analysis of equilibria Invasion and evolution Stochasticity and density dependence Two-Sex Models Sexual dimorphism in the vital rates Dominance, sex ratio, and the marriage squeeze

10 CONTENTS xv 17.3 Two-sex models A simple two-sex model The birth and fertility functions Frequency and density dependence The equilibrium population structure Stability of population structure Nussbaum s global stability theorem Competition for mates Numerical results: Competition and instability The birth matrix-mating rule model More detailed models of mating Frequency and density dependence combined Extinction and the sex ratio Conservation and Management Conservation Assessment Diagnosis Prescription Prognosis Conservation conclusions Pest control Reducing population size Extermination Halting invasion Some examples of pest control Harvesting Optimal harvesting Overview Concluding Remarks The most important task Testing models A complete demographic analysis Directions for research A The Basics of Matrix Algebra 653 A.1 Motivation A.2 Definitions A.3 Operations A.3.1 Addition A.3.2 Scalar multiplication A.3.3 The transpose and the adjoint

11 xvi CONTENTS A.3.4 The trace A.3.5 Scalar product A.3.6 Matrix multiplication A.3.7 The Kronecker and Hadamard products A.4 Matrix inversion A.4.1 The identity matrix A.4.2 Inversion and the solution of algebraic equations A.4.3 A useful fact about homogeneous systems A.5 Determinants A.5.1 Properties of determinants A.6 Eigenvalues and eigenvectors A.6.1 Eigenvectors A.6.2 Left eigenvectors A.6.3 The characteristic equation A.6.4 Finding the eigenvectors A.6.5 Complications A.6.6 Linear independence of eigenvectors A.6.7 Left and right eigenvectors A.6.8 Computation of eigenvalues and eigenvectors A.7 Similarity A.7.1 Properties of similar matrices A.8 Norms of vectors and matrices A.9 Suggested reading Bibliography 669 Copyright Permissions 711 Index 713

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