Sensitivity Analysis : : Matrix Methods in Demography and Ecology.

• Superior document: Demographic Research Monographs
• Author: Caswell, Hal.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2019.; ©2019.
• Year of Publication: 2019
• Edition: 1st ed.
• Language: English
• Series: Demographic Research Monographs
• Physical Description: 1 online resource (308 pages)

Table of Contents

Intro -- Preface -- Bibliography -- Acknowledgements -- Contents -- Part I Introductory and Methodological -- 1 Introduction: Sensitivity Analysis - What and Why? -- 1.1 Introduction -- 1.2 Sensitivity, Calculus, and Matrix Calculus -- 1.3 Some Issues -- 1.3.1 Prospective and Retrospective Analyses: Sensitivity and Decomposition -- 1.3.2 Uncertainty Propagation -- 1.3.3 Why Not Just Simulate? -- 1.3.4 Sensitivity and Identifying Targets for Intervention -- 1.3.5 The Dream of Easy Interpretation -- 1.4 The Importance of Change -- Bibliography -- 2 Matrix Calculus and Notation -- 2.1 Introduction: Can It Possibly Be That Simple? -- 2.2 Notation and Matrix Operations -- 2.2.1 Notation -- 2.2.2 Operations -- 2.2.3 The Vec Operator and Vec-Permutation Matrix -- 2.2.4 Roth's Theorem -- 2.3 Defining Matrix Derivatives -- 2.4 The Chain Rule -- 2.5 Derivatives from Differentials -- 2.5.1 Differentials of Scalar Function -- 2.5.2 Differentials of Vectors and Matrices -- 2.6 The First Identification Theorem -- 2.6.1 The Chain Rule and the First IdentificationTheorem -- 2.7 Elasticity -- 2.8 Some Useful Matrix Calculus Results -- 2.9 LTRE Decomposition of Demographic Differences -- 2.10 A Protocol for Sensitivity Analysis -- Bibliography -- Part II Linear Models -- 3 The Sensitivity of Population Growth Rate: Three Approaches -- 3.1 Introduction -- 3.2 Hamilton's Equation for Age-Classified Populations -- 3.2.1 Effects of Changes in Mortality -- 3.2.2 Effects of Changes in Fertility -- 3.2.3 History and Perspectives -- 3.3 Stage-Classified Populations: Eigenvalue Perturbations -- 3.3.1 Age-Classified Models as a Special Case -- 3.3.2 Sensitivity to Lower-Level DemographicParameters -- 3.3.3 History -- 3.4 Growth Rate Sensitivity via Matrix Calculus -- 3.5 Second Derivatives of Population Growth Rate -- 3.6 Conclusion -- Bibliography.

4 Sensitivity Analysis of Longevity and Life Disparity -- 4.1 Introduction -- 4.2 Life Expectancy in Age-Classified Populations -- 4.2.1 Derivation -- 4.3 A Markov Chain Model for the Life Cycle -- 4.3.1 A Markov Chain Formulation of the Life Cycle -- 4.3.2 Occupancy Times -- 4.3.3 Longevity -- 4.3.4 Age or Stage at Death -- 4.3.5 Life Lost and Life Disparity -- 4.4 Sensitivity Analysis -- 4.4.1 Sensitivity of the Fundamental Matrix -- 4.4.2 Sensitivity of Life Expectancy -- 4.4.3 Generalizing the Keyfitz-Pollard Formula -- 4.4.4 Sensitivity of the Variance of Longevity -- 4.4.5 Sensitivity of the Distribution of Age at Death -- 4.4.6 Sensitivity of Life Disparity -- 4.5 A Time-Series LTRE Decomposition: Life Disparity -- 4.6 Conclusion -- Bibliography -- 5 Individual Stochasticity and Implicit Age Dependence -- 5.1 Introduction -- 5.1.1 Age and Stage, Implicit and Explicit -- 5.1.2 Individual Stochasticity and Heterogeneity -- 5.1.3 Examples -- 5.2 Markov Chains -- 5.2.1 An Absorbing Markov Chain -- 5.2.2 Occupancy Times and the Fundamental Matrix -- 5.2.3 Sensitivity of the Fundamental Matrix -- 5.3 From Stage to Age -- 5.3.1 Variance in Occupancy Time -- 5.3.2 Longevity and Life Expectancy -- 5.3.3 Variance in Longevity -- 5.3.4 Cohort Generation Time -- 5.4 The Net Reproductive Rate -- 5.4.1 Net Reproductive Rate in Periodic Environments -- 5.4.2 Sensitivity of the Net Reproductive Rate -- 5.4.3 Invasion Exponents, Selection Gradients, and R0 -- 5.4.4 Beyond R0: Individual Stochasticity in Lifetime Reproduction -- 5.5 Variable and Stochastic Environments -- 5.5.1 A Model for Variable Environments -- 5.5.2 The Fundamental Matrix -- 5.5.3 Longevity in a Variable Environment -- 5.5.3.1 Variance in Longevity -- 5.5.4 A Time-Varying Example: Lomatium bradshawii -- 5.6 The Importance of Individual Stochasticity -- 5.7 Discussion.

A Appendix: Derivations -- A.1 Variance in Occupancy Times -- A.2 Life Expectancy -- A.3 Variance in Longevity -- A.4 Net Reproductive Rate -- A.5 Cohort Generation Time -- A.5.1 Sensitivity of Generation Time -- Bibliography -- 6 AgeStage-Classified Models -- 6.1 Introduction -- 6.2 Model Construction -- 6.3 Sensitivity Analysis -- 6.4 Examples -- 6.4.1 Population Growth Rate and Selection Gradients -- 6.4.2 Distributions of Age and Stage at Death -- 6.4.2.1 Perturbation Analysis -- 6.5 Discussion -- 6.5.1 Reducibility and Ergodicity -- 6.5.2 A Protocol for AgeStage-Classified Models -- A Appendix: Population Growth and Reducible Matrices -- Bibliography -- Part III Time-Varying and Stochastic Models -- 7 Transient Population Dynamics -- 7.1 Introduction -- 7.2 Time-Invariant Models -- 7.3 Sensitivity of What? Choosing Dependent Variables -- 7.4 Elasticity Analysis -- 7.5 Sensitivity of Time-Varying Models -- 7.6 Sensitivity of Subsidized Populations -- 7.7 Sensitivity of Nonlinear Models -- 7.8 Sensitivity of Population Projections -- 7.9 Discussion -- Bibliography -- 8 Periodic Models -- 8.1 Introduction -- 8.1.1 Perturbation Analysis -- 8.2 Linear Models -- 8.2.1 A Simple Harvest Model -- 8.3 Multistate Models -- 8.4 Nonlinear Models and Delayed Density Dependence -- 8.4.1 Averages -- 8.4.2 A Nonlinear Example -- 8.5 LTRE Decomposition Analysis -- 8.6 Discussion -- Bibliography -- 9 LTRE Decomposition of the Stochastic Growth Rate -- 9.1 Introduction -- 9.2 Decomposition with Derivatives -- 9.3 Kitagawa and Keyfitz: Decomposition Without Derivatives -- 9.4 Stochastic Population Growth -- 9.4.1 Environment-Specific Sensitivities -- 9.5 LTRE Decomposition Analysis for logλs -- 9.5.1 Case 1: Vital Rates Differ, Environments Identical -- 9.5.2 Case 2: Vital Rates Identical, Environments Differ -- 9.5.3 Case 3: Vital Rates and Environments Differ.

9.6 An Example: Fire and an Endangered Plant -- 9.6.1 The Stochastic Fire Environment -- 9.6.2 LTRE Analysis -- 9.7 Discussion -- Bibliography -- Part IV Nonlinear Models -- 10 Sensitivity Analysis of Nonlinear Demographic Models -- 10.1 Introduction -- 10.2 Density-Dependent Models -- 10.2.1 Linearizations Around Equilibria -- 10.2.2 Sensitivity of Equilibrium -- 10.2.3 Dependent Variables: Beyond -- 10.2.4 Reactivity and Transient Dynamics -- 10.2.5 Elasticity Analysis -- 10.2.6 Continuous-Time Models -- 10.3 Environmental Feedback Models -- 10.4 Subsidized Populations and Competition for Space -- 10.4.1 Density-Independent Subsidized Populations -- 10.4.2 Linear Subsidized Models with Competitionfor Space -- 10.4.3 Density-Dependent Subsidized Models -- 10.5 Stable Structure and Reproductive Value -- 10.5.1 Stable Structure -- 10.5.2 Reproductive Value -- 10.5.3 Sensitivity of the Dependency Ratio -- 10.5.4 Sensitivity of Mean Age and Related Quantities -- 10.5.5 Sensitivity of Variance in Age -- 10.6 Frequency-Dependent Two-Sex Models -- 10.6.1 Sensitivity of the Population Structure -- 10.6.2 Population Growth Rate in Two-Sex Models -- 10.6.3 The Birth Matrix-Mating Rule Model -- 10.7 Sensitivity of Population Cycles -- 10.7.1 Sensitivity of the Population Vector -- 10.7.2 Sensitivity of Weighted Densities and TimeAverages -- 10.7.3 Sensitivity of Temporal Variance in Density -- 10.7.4 Periodic Dynamics in Periodic Environments -- 10.8 Dynamic Environmental Feedback Models -- 10.9 Stage-Structured Epidemics -- 10.10 Moments of Longevity in Nonlinear Models -- 10.11 Summary -- References -- Part V Markov Chains -- 11 Sensitivity Analysis of Discrete Markov Chains -- 11.1 Introduction -- 11.2 Absorbing Chains -- 11.2.1 Occupancy: Visits to Transient States -- 11.2.2 Time to Absorption -- 11.2.3 Number of States Visited Before Absorption.

11.2.4 Multiple Absorbing States and Probabilities of Absorption -- 11.2.5 The Quasistationary Distribution -- 11.3 Life Lost Due to Mortality -- 11.4 Ergodic Chains -- 11.4.1 The Stationary Distribution -- 11.4.2 The Fundamental Matrix -- 11.4.3 The First Passage Time Matrix -- 11.4.4 Mixing Time and the Kemeny Constant -- 11.4.5 Implicit Parameters and Compensation -- 11.5 Species Succession in a Marine Community -- 11.5.1 Biotic Diversity -- 11.5.2 The Kemeny Constant and Ecological Mixing -- 11.6 Discussion -- A Appendix A: Proofs -- A.1 Derivatives of the Moments of Occupancy Times -- A.2 Derivatives of the Moments of Time to Absorption -- B Appendix B: Marine Community Matrix -- References -- 12 Sensitivity Analysis of Continuous Markov Chains -- 12.1 Introduction -- 12.1.1 Absorbing Markov Chains -- 12.2 Occupancy Time in Transient States -- 12.3 Longevity: Time to Absorption -- 12.4 Multiple Absorbing States and Probabilities of Absorption -- 12.5 The Embedded Chain: Discrete Transitions Within a Continuous Process -- 12.6 An Example: A Model of Disease Progression -- 12.6.1 Sensitivity Results -- 12.6.2 Sensitivity of the Embedded Chain -- 12.7 Discussion -- References.