Reproductive value, the stable stage distribution, and the sensitivity of the population growth rate to changes in vital rates

Caswell, Hal

doi:10.4054/demres.2010.23.19

Cited by 36 publications

(47 citation statements)

References 28 publications

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“…The underlying theory is presented in detail by Magnus and Neudecker (1985); for an introductory account see Abadir and Magnus (2005). The methods have been applied to demography in a series of papers (Caswell 2006(Caswell , 2007(Caswell , 2008(Caswell , 2010(Caswell , 2012Caswell and Shyu 2012;van Raalte and Caswell 2013;Engelman, Caswell, and Agree 2014). If y is a n × 1 vector function of the m × 1 vector x, then the sensitivity of y to x is the n × m Jacobian matrix written as…”

Section: Matrix Calculus Notationmentioning

confidence: 99%

The sensitivity analysis of population projections

Caswell¹,

Gassen²

2015

DemRes

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BACKGROUND Population projections using the cohort component method can be written as time-varying matrix population models. The matrices are parameterized by schedules of mortality, fertility, immigration, and emigration over the duration of the projection. A variety of dependent variables are routinely calculated (the population vector, various weighted population sizes, dependency ratios, etc.) from such projections. OBJECTIVEOur goal is to derive and apply theory to compute the sensitivity and the elasticity (proportional sensitivity) of any projection outcome to changes in any of the parameters, where those changes are applied at any time during the projection interval. METHODSWe use matrix calculus to derive a set of equations for the sensitivity and elasticity of any vector valued outcome ξ(t) at time t to any perturbation of a parameter vector θ(s) at any time s. RESULTSThe results appear in the form of a set of dynamic equations for the derivatives that are integrated in parallel with the dynamic equations for the projection itself. We show results for single-sex projections and for the more detailed case of projections including age distributions for both sexes. We apply the results to a projection of the population of Spain, from 2012 to 2052, prepared by the Instituto Nacional de Estadística, and determine the sensitivity and elasticity of (1) total population, (2) the school-age population, (3) the population subject to dementia, (4) the total dependency ratio, and (5) the economic support ratio. Caswell & Sánchez Gassen: The sensitivity analysis of population projections CONCLUSIONS Writing population projections in matrix form makes sensitivity analysis possible. Such analyses are a powerful tool for the exploration of how detailed aspects of the projection output are determined by the mortality, fertility, and migration schedules that underlie the projection.

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Section: Matrix Calculus Notationmentioning

confidence: 99%

The sensitivity analysis of population projections

Caswell¹,

Gassen²

2015

DemRes

Self Cite

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“…However, the computations are possible using matrix calculus, a formalism that permits consistent calculation of the derivatives of scalar-, vector-or matrix-valued functions of scalar-, vector-, or matrix-valued arguments Neudecker 1985, 1988). See Abadir and Magnus (2005) for an introduction and Caswell (2006, 2007, 2008, 2009a, b, 2010, 2011a, b), Jenouvrier et al (2010), Klepac andCaswell (2010), andStrasser et al (2010) for ecological applications. A brief outline of the methods is provided in "Online Resource B".…”

Section: Perturbation Analysismentioning

confidence: 99%

“…(20) is the dominant eigenvalue λ. Let w and v be the right and left eigenvectors corresponding to λ, scaled so that (Caswell 2010). In this model, the vital rates are functions only of stage; the phenotype is blind to the age of the individual.…”

Section: Population Growth Rate and Selection Gradientsmentioning

confidence: 99%

Matrix models and sensitivity analysis of populations classified by age and stage: a vec-permutation matrix approach

Caswell

2011

Theor Ecol

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126

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Matrix population models in which individuals are classified by both age and stage can be constructed using the vec-permutation matrix. The resulting age-stage models can be used to derive the age-specific consequences of a stage-specific life history or to describe populations in which the vital rates respond to both age and stage. I derive a general formula for the sensitivity of any output (scalar, vector, or matrix-valued) of the model, to any vector of parameters, using matrix calculus. The matrices describing age-stage dynamics are almost always reducible; I present results giving conditions under which population growth is ergodic from any initial condition. As an example, I analyze a published stage-specific model of Scotch broom (Cytisus scoparius), an invasive perennial shrub. Sensitivity analysis of the population growth rate finds that the selection gradients on adult survival do not always decrease with age but may increase over a range of ages. This may have implications for the evolution of senescence in stage-classified populations. I also derive and analyze the joint distribution of age and stage at death and present a sensitivity analysis of Electronic supplementary material The online version of this article

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“…where w and v are the right and left eigenvectors corresponding to λ, scaled so that v T w = 1 (Caswell 2010). …”

Section: Linear Modelsmentioning

confidence: 99%

Sensitivity analysis of periodic matrix population models

Caswell

Shyu

2012

Theoretical Population Biology

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Periodic matrix models are frequently used to describe cyclic temporal variation (seasonal or interannual) and to account for the operation of multiple processes (e.g., demography and dispersal) within a single projection interval. In either case, the models take the form of periodic matrix products. The perturbation analysis of periodic models must trace the effects of parameter changes, at each phase of the cycle, on output variables that are calculated over the entire cycle. Here, we apply matrix calculus to obtain the sensitivity and elasticity of scalar-, vector-, or matrix-valued output variables. We apply the method to linear models for periodic environments (including seasonal harvest models), to vec-permutation models in which individuals are classified by multiple criteria, and to nonlinear models including both immediate and delayed density dependence. The results can be used to evaluate management strategies and to study selection gradients in periodic environments.

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Reproductive value, the stable stage distribution, and the sensitivity of the population growth rate to changes in vital rates

Cited by 36 publications

References 28 publications

The sensitivity analysis of population projections

The sensitivity analysis of population projections

Matrix models and sensitivity analysis of populations classified by age and stage: a vec-permutation matrix approach

Sensitivity analysis of periodic matrix population models

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