Caswell, Hal Sensitivity analysis. Matrix methods in demography and ecology. (English) Zbl 1425.91005 Demographic Research Monographs. Cham: Springer (ISBN 978-3-030-10533-4/hbk; 978-3-030-10534-1/ebook). xviii, 299 p., open access (2019). Reviewed book uses matrix formulations generalized beyond projections to age-structured and stage-structured populations, to linear and nonlinear dynamics, to time-invariant and time-varying vital rates, and to multistate models that combine age and stage information. It’s known that the matrix formulation provides easily computable outcomes at the level of the individual and the population. The mathematical connection of matrix models and the theory of finite-state Markov chains make it possible to go beyond expected outcomes to calculate variances and higher moments and to take full advantage of the stochasticity of demographic events at the individual level. Because of matrix calculus permits easy differentiation of scalar-, vector-, and matrix-valued functions of scalar-, vector-, and matrix-valued arguments, this book demonstrate an application of these methods to demographic problems. Initially was analyzed linear models for population growth, longevity, and reproduction. When the rates are also time-invariant, these models lead to a stable age or stage structure and exponential growth. Then analyzes the sensitivity of population growth rate from three directions: differentiation of the characteristic equation, eigenvalue perturbation theory, and matrix calculus. Focusing on longevity, presented the sensitivity analysis of life expectancy, variance in longevity, and life disparity. Considering the important concept of individual stochasticity (stochastic outcomes of probabilistic transitions in the life cycle), explores its effects on longevity, net reproductive rate, birth intervals, and age at reproduction. Also some aspects of time variation are introduced, including the powerful vec-permutation matrix method to describe temporally varying environments. Understanding that a first step in the construction of any demographic model is the choice of the individual state (i-state) variables that capture the relevant information about individuals, presents the sensitivity analysis of the models, using the vec-permutation method to construct multistate models and matrix calculus to differentiate the results. The sensitivity analysis of transient dynamics, i.e.short-term population growth and structure may differ in important ways from the growth and structure implied by stable population theory. So these differences was explored for cases where the vital rates may be fixed, varying, or even nonlinear. Periodic models appear in a variety of guises: as matrix products describing periodic (e.g., seasonal) environmental variation and as matrix products describing distinct processes embedded within an apparently single projection matrix and in the construction of multistate matrix models. The goal was to describe the sensitivity of some overall outcome product, to changes in parameters affecting each component of the matrix. Population growth in stochastic environments and the problem of decomposing differences in stochastic growth rates into components due to the environment and to the vital rates was analysed. This required a combination of the first-order approximate decomposition known as life table response experiment (LTRE) analysis with the more specialized Kitagawa-Keyfitz decomposition and has potential implications far beyond the stochastic environment case. At the end of the book analyzes nonlinear models, including density-dependent models, frequency-dependent models, nonlinear models for subsidized populations, and a nonlinear approach to the sensitivity of the stable structure and the reproductive value of linear models. In conclusion the author discuss the Markov chain models taking a more mathematical approach to the sensitivity analysis of Markov chains, including some aspects that have yet to find wide demographic application. Particularly, analyzes discrete-time chains, and ergodic chains that include no absorbing states. Also presents the sensitivity analysis of continuous-time absorbing Markov chains. Reviewer: Fatima T. Adylova (Tashkent) Cited in 4 Documents MSC: 91-02 Research exposition (monographs, survey articles) pertaining to game theory, economics, and finance 92-02 Research exposition (monographs, survey articles) pertaining to biology 91D20 Mathematical geography and demography 92D25 Population dynamics (general) 92D40 Ecology 60J20 Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.) 60J28 Applications of continuous-time Markov processes on discrete state spaces 15A99 Basic linear algebra Keywords:sensitivity analysis; matrix methods; demography; ecology Software:Matlab; Human Mortality; R PDFBibTeX XMLCite \textit{H. Caswell}, Sensitivity analysis. Matrix methods in demography and ecology. Cham: Springer (2019; Zbl 1425.91005) Full Text: DOI Link