A Note on the vec Operator Applied to Unbalanced Block-Structured Matrices

Caswell, Hal; Daalen, Silke van

doi:10.1155/2016/4590817

Cited by 8 publications

(8 citation statements)

References 8 publications

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“…A new way to obtain these derivatives is based on Eqs. and of Caswell and van Daalen (). Define the matrices P i and Q i , of dimension

ω s \times s

and

s \times ω s

, respectively,

{boldP}_{i} = (\begin{matrix} 0_{s false(i - 1 false) \times s} \\ I_{s} \\ 0_{s false(normalω - i false) \times s} \end{matrix}) {boldQ}_{i} = (\begin{matrix} 0_{s \times false(i - 1 false) s} & I_{s} & 0_{s \times false(normalω - i false) s} \end{matrix}) .

Then

\frac{d vec U}{d vec {boldU}_{i}} = \frac{d vec F}{d vec {boldF}_{i}} = {boldQ}_{i}^{⊤} \otimes {boldP}_{i} .

Again, note that this term is independent of the dependent variable and the parameters being perturbed.…”

Section: Sensitivity Analysis Of Age × Stage‐classified Modelsmentioning

confidence: 97%

See 1 more Smart Citation

Age × stage‐classified demographic analysis: a comprehensive approach

Caswell

Vries

Hartemink

et al. 2018

Ecological Monographs

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This paper presents a comprehensive theory for the demographic analysis of populations in which individuals are classified by both age and stage. The earliest demographic models were age classified. Ecologists adopted methods developed by human demographers and used life tables to quantify survivorship and fertility of cohorts and the growth rates and structures of populations. Later, motivated by studies of plants and insects, matrix population models structured by size or stage were developed. The theory of these models has been extended to cover all the aspects of age‐classified demography and more. It is a natural development to consider populations classified by both age and stage. A steady trickle of results has appeared since the 1960s, analyzing one or another aspect of age × stage‐classified populations, in both ecology and human demography. Here, we use the vec‐permutation formulation of multistate matrix population models to incorporate age‐ and stage‐specific vital rates into demographic analysis. We present cohort results for the life table functions (survivorship, mortality, and fertility), the dynamics of intra‐cohort selection, the statistics of longevity, the joint distribution of age and stage at death, and the statistics of life disparity. Combining transitions and fertility yields a complete set of population dynamic results, including population growth rates and structures, net reproductive rate, the statistics of lifetime reproduction, and measures of generation time. We present a complete analysis of a hypothetical model species, inspired by poecilogonous marine invertebrates that produce two kinds of larval offspring. Given the joint effects of age and stage, many familiar demographic results become multidimensional, so calculations of marginal and mixture distributions are an important tool. From an age‐classified point of view, stage structure is a form of unobserved heterogeneity. From a stage‐classified point of view, age structure is unobserved heterogeneity. In an age × stage‐classified model, variance in demographic outcomes can be partitioned into contributions from both sources. Because these models are formulated as matrices, they are amenable to a complete sensitivity analysis. As more detailed and longer longitudinal studies are developed, age × stage‐classified demography will become more common and more important.

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“…A new way to obtain these derivatives is based on Eqs. and of Caswell and van Daalen (). Define the matrices P i and Q i , of dimension

ω s \times s

and

s \times ω s

, respectively,

{boldP}_{i} = (\begin{matrix} 0_{s false(i - 1 false) \times s} \\ I_{s} \\ 0_{s false(normalω - i false) \times s} \end{matrix}) {boldQ}_{i} = (\begin{matrix} 0_{s \times false(i - 1 false) s} & I_{s} & 0_{s \times false(normalω - i false) s} \end{matrix}) .

Then

\frac{d vec U}{d vec {boldU}_{i}} = \frac{d vec F}{d vec {boldF}_{i}} = {boldQ}_{i}^{⊤} \otimes {boldP}_{i} .

Again, note that this term is independent of the dependent variable and the parameters being perturbed.…”

Section: Sensitivity Analysis Of Age × Stage‐classified Modelsmentioning

confidence: 97%

“…A new way to obtain these derivatives is based on Eqs. 14 and 15 of Caswell and van Daalen (2016). Define the matrices P i and Q i , of dimension xs Â s and s Â xs, respectively, Vol.…”

Section: Termmentioning

confidence: 99%

Age × stage‐classified demographic analysis: a comprehensive approach

Caswell

Vries

Hartemink

et al. 2018

Ecological Monographs

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“…We want to express dvec B and dM in terms of the derivatives of their diagonal entries B i and M j . This can be done using equations (14) and (15) of Caswell and van Daalen (2016). Define the matrices P i and Q i , of dimension rs × s and s × rs, respectively,…”

Section: Multistate Modelsmentioning

confidence: 99%

Periodic Models

Caswell¹

2019

Sensitivity Analysis: Matrix Methods in Demography and Ecology

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Periodic Models 8.1 Introduction Periodic matrix models are often used to study cyclical temporal variation (seasonal or interannual), sometimes as a (perhaps crude) approximation to stochastic models. However, formally periodic models also appear when multiple processes (e.g., demography and dispersal) operate within a single projection interval. The models take the form of periodic matrix products. A familiar example is when population projection over an annual interval is described as a product of seasonal operators. The perturbation analysis of periodic models (Caswell and Trevisan 1994; Lesnoff et al. 2003; Caswell and Shyu 2012) must specify both the vital rates affected by the perturbation and the timing of the perturbation within the cycle. This chapter presents a general approach to the perturbation analysis of both linear and nonlinear periodic models. The results consist of a series of analyses of some of the most commonly encountered periodic models. If the environment is time-invariant on the scale of a chosen projection interval (e.g., from year to year), the result is a periodic matrix population model in which the seasonal product repeats itself. Such a model can be written as n(t + 1) = B p • • • B 2 B 1 n(t) (8.1) Here, B i is the matrix at phase i of the cycle and p is the period. The period is the number of phases in the cycle; i.e., the number of matrices in the periodic matrix product in (8.1). Neither the identities nor the number of stages need be the same from one phase to the next, so the matrices B i may be rectangular rather than square.

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“…(37) Caswell and Van Daalen (2016). This permits expressing all the effects of θ on the transition matrix P in terms of effects on U (see Appendix A.2).…”

Section: Sensitivity Analysis Of Lromentioning

confidence: 99%

“…where (Caswell and van Daalen 2016). In many applications, only a single absorbing state (death) exists, in which case α = 1 and Eqs.…”

Section: A2 Proof Of Theorem 2: Sensitivity Analysis Of Moments Of Lromentioning

confidence: 99%

Lifetime reproductive output: individual stochasticity, variance, and sensitivity analysis

2017

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Lifetime reproductive output (LRO) determines per-generation growth rates, establishes criteria for population growth or decline, and is an important component of fitness. Empirical measurements of LRO reveal high variance among individuals. This variance may result from genuine heterogeneity in individual properties, or from individual stochasticity, the outcome of probabilistic demographic events during the life cycle. To evaluate the extent of individual stochasticity requires the calculation of the statistics of LRO from a demographic model. Mean LRO is routinely calculated (as the net reproductive rate), but the calculation of variances has only recently received attention. Here, we present a complete, exact, analytical, closed-form solution for all the moments of LRO, for age-and stage-classified populations. Previous studies have relied on simulation, iterative solutions, or closed-form analytical solutions that capture only part of the sources of variance. We also present the sensitivity and elasticity of all of the statistics of LRO to parameters defining survival, stage transitions, and (st)agespecific fertility. Selection can operate on variance in LRO only if the variance results from genetic heterogeneity. The potential opportunity for selection is quantified by Crow's index I, the ratio of the variance to the square of the mean. But variance due to individual stochasticity is only an apparent opportunity for selection. In a comparison of a range of age-classified models for human populations, we find that proportional increases in mortality have very small effects on the mean and variance of LRO, but large positive effects on I. Proportional increases in fertility increase both the mean and variance of LRO, but reduce I. For a sizeclassified tree population, the elasticity of both mean and variance of LRO to stage-specific mortality are negative; the elasticities to stage-specific fertility are positive.

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A Note on the vec Operator Applied to Unbalanced Block-Structured Matrices

Cited by 8 publications

References 8 publications

Age × stage‐classified demographic analysis: a comprehensive approach

Age × stage‐classified demographic analysis: a comprehensive approach

Periodic Models

Lifetime reproductive output: individual stochasticity, variance, and sensitivity analysis

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