Age × Stage-Classified Models
Abstract
The first step in developing any kind of structured population model is choosing one or more variables in terms of which to describe the population structure. The job of these i-state variables is to encapsulate all the information about the past experience of an individual that is relevant to its future behavior (Metz and Diekmann 1986; Caswell 2001).
6.1 Introduction
The first step in developing any kind of structured population model is choosing one or more variables in terms of which to describe the population structure. The job of these i-state variables is to encapsulate all the information about the past experience of an individual that is relevant to its future behavior (Metz and Diekmann 1986; Caswell 2001). Classical demography (for both humans and for non-human animals and plants) uses age as a i-state, but other, more biologically relevant criteria (e.g., size, developmental stage, parity, physiological condition, etc.) are now widely used in ecology, with age-classified models viewed as a special case.
- 1.
Even in a stage-classified model, age still exists; every individual becomes older, by one unit of age, with the passage of each unit of time (e.g., Feichtinger 1971a; Caswell 2001, 2006, 2009; Tuljapurkar and Horvitz 2006; Horvitz and Tuljapurkar 2008). In these analyses, age dependence is implicit in the stage-classified model (see Chap. 5). Models that include both age and stage provide information that goes beyond this implicit dependence.
- 2.
If the vital rates depend on both age and stage, only a model that includes both can reveal the joint action of age-and stage-specific processes (e.g., Goodman 1969). Such models, of course, require information on the joint age-dependence and stage-dependence of the vital rates, and thus are challenging to construct. A special case that has been extensively explored is the multi-regional case, in which the stage variable describes spatial location (e.g., Rogers 1966; Lebreton 1996). Models that combine age and some measure of health or disability status are an important part of health demography (e.g., Willekens 2014; Peeters et al. 2002; Wu et al. 2006; Zhou et al. 2016).
This chapter presents a model framework in which individuals are classified by age and stage, using the vec-permutation matrix approach (so-called for the role that the vec-permutation matrix plays in rearranging age and stage categories in the population vector). This formalism was introduced by Hunter and Caswell (2005) for populations classified by stage and location, was used in Chap. 5 to classify individuals by stage and environmental state; it has also been applied to stage and infection status (Klepac and Caswell 2011), stage and age (Caswell 2012; Caswell et al. 2018), and age and frailty (Caswell 2014). Megamatrix models (e.g., Pascarella and Horvitz 1998; Horvitz and Tuljapurkar 2008) can be written using this approach, as can block-structured multiregional models (e.g., Rogers 1975; Lebreton 1996). Matrix models can describe both population dynamics and cohort dynamics. Population dynamics (population growth, age and stage structure, reproductive value) depend on both the transitions of extant individuals and the production of new individuals by reproduction. In contrast, cohort dynamics (survivorship, life expectancy, age at death, generation time) depend only on the fates of already existing individuals. This chapter describes both kinds of analysis. For a more complete review and treatment, see Caswell et al. (2018).
6.2 Model Construction
Mathematical notation used in this chapter. Dimensions are shown, where relevant, for matrices and vectors; s denotes the number of stages and ω the number of age classes
Quantity | Description | Dimension |
---|---|---|
A_{i}, F_{i}, U_{i} | Stage-classified projection, fertility, and transition matrices for age class i | s × s |
D_{U}, D_{F} | Age transition matrices for individuals already present in the population and for new individuals produced by reproduction | ω × ω |
\(\mathbb {A,F,U,D}\) | Block-diagonal matrices | sω × sω |
\(\tilde {{\mathbf {A}}}, \tilde {{\mathbf {U}}}\), etc. | Age × stage matrices constructed from block-diagonal matrices using the vec-permutation matrix | sω × sω |
K_{s,ω}, K | Vec-permutation matrix | sω × sω |
I _{ s} | Identity matrix | s × s |
1 _{ s} | Vector of ones | s × 1 |
e _{ i} | The ith unit vector, with a 1 in the ith entry and zeros elsewhere | Various |
E _{ ij} | A matrix with a 1 in the (i, j) position, and zeros elsewhere | Various |
⊗ | Kronecker product | |
∘ | Hadamard, or element-by-element, product | |
vec X | The vec operator, which stacks the columns of a m × n matrix X into a mn × 1 vector | |
\(\mathcal {D}\,({\mathbf {x}})\) | A diagonal matrix with x on the diagonal and zeros elsewhere |
The second term in (6.15), \( {\mathbf {K}}^{\mathsf {T}} \mathbb {D}_{\mathrm {F}} {\mathbf {K}} \mathbb {F} \), carries out a similar sequence of transformations for the generation of new individuals. First, newborn individuals are produced according to the block-diagonal fertility matrix \(\mathbb {F}\). The resulting vector is rearranged by the vec-permutation matrix, and then the matrix \(\mathbb {D}_{\mathrm {F}}\) places all the newborn individuals into the first age class. Finally, K^{T} rearranges the vector to the stage-within-age arrangement.
6.3 Sensitivity Analysis
Age-stage models pose particular challenges for perturbation analysis, because interest naturally focuses on changes in the matrices F_{i} and U_{i} (i = 1, …, ω), which are deeply embedded within \(\tilde {{\mathbf {F}}}\), \(\tilde {{\mathbf {U}}}\), and \(\tilde {{\mathbf {A}}}\).
Notice that (6.26) requires only three pieces of demographic information: the derivatives of U_{i} and F_{i} with respect to the parameters (whatever those may be in the case at hand) and the sensitivity of the dependent variable ξ (whatever that may be) to the elements of the matrix \(\tilde {{\mathbf {A}}}\) from which it is calculated. All the other pieces of (6.26) are constants. Some of these constant matrices may be large, depending on s and ω, but they are very sparse; the sparse matrix technology available in Matlab can be extremely useful in implementation. An alternative formulation of the differentials of the block matrices \(\mathbb {U}\) and \(\mathbb {F}\) is given in Caswell and van Daalen (2016).
6.4 Examples
Here we consider two examples of the sensitivity analysis of age-stage model to extract age-classified information from a stage-classified model. The first example will derive the sensitivity of the population growth rate λ, obtaining the sensitivity of λ to both age- and stage-specific survival, permitting examination of how selection pressures on senescence-inducing traits would vary from stage to stage. The second example is an analysis of the joint distribution of age and stage at death.
These examples are based on a stage-classified model (Parker 2000) for Scotch broom (Cytisus scoparius). Scotch broom is a large (up to 4 m tall) leguminous shrub, introduced into North America from Europe in the late nineteenth century. It is an invasive plant, considered a pest in the northwestern parts of North America. Stage-classified demographic models have been used to evaluate potential management policies for the plant (Parker 2000) and to investigate its potential for spatial spread (Neubert and Parker 2004).
The model contains seven stages (stage 1 = seeds, 2 = seedlings, 3 = juveniles, 4 = small adults, 5 = medium adults, 6 = large adults, 7 = extra-large adults), and parameters were estimated at a number of locations in Washington State. As is typical with many perennial plant species, survival is low for seeds and seedlings, but increases dramatically in larger stages. Parker’s study presented estimated projection matrices for plants at the edge, at intermediate locations, and at the center of an invading stand. Plants near the center experience more crowding, with resulting reduced rates of survival, growth, and fertility.
6.4.1 Population Growth Rate and Selection Gradients
Results
The matrix U is obtained from A by setting all elements in the first row, except for a_{11}, to zero. The matrix F is a 7 × 7 matrix with the elements of row 1, columns 2–7 of A in the corresponding positions, and zeros elsewhere. The maximum age was set to ω = 30. The aging matrices D_{U} and D_{F} are given by (6.2) and (6.3) with ω = 30. Because the vital rates do not depend on age, the dominant eigenvalues of A and \(\tilde {{\mathbf {A}}}\) should be identical, and they are; λ = 1.268.
It is now known that this pattern is widespread in plant populations. It appears in all eight of the Scotch broom populations studied by Parker (2000), and in almost all of 36 species of plants examined by Caswell and Salguero-Gómez (2013). It has important implications for the evolution of senescence. Hamilton (1966) showed that the selection gradient on age-specific mortality is always decreases with age, and argued that this implied that selection would always lead to senescence. Incorporating stage-dependence as well as age-dependence of the vital rates means that, over some range of ages, the selection gradient increases (contra-senescent selection in the terminology of Caswell and Salguero-Gómez 2013). Thus conclusions that follow from the general decline in selection gradients with age may not apply to traits that affect age-specific survival differentially depending on developmental stage. Traits that affect survival in adult stages should postpone senescence for at least some time.
6.4.2 Distributions of Age and Stage at Death
The pattern of longevity within a population is captured by the probability distribution of the age at death, one of the standard results of age-classified life table analysis. The moments of the age at death and their sensitivity can also be calculated directly from stage-classified models using Markov chain methods (Feichtinger 1971b; Caswell 2001, 2006, 2009; Tuljapurkar and Horvitz 2006; Horvitz and Tuljapurkar 2008); see Chaps. 4 and 5. Here we can go beyond that and get the full joint distribution of stage and age at death, along with the marginal distributions of age at death and stage at death, implied by an age × stage classified model.
6.4.2.1 Perturbation Analysis
Derivation
Results
6.5 Discussion
Models in which individuals are classified by both age and stage extend demographic analyses in several directions. They permit biodemographic analyses of aging to take advantage of the many stage-classified demographic analyses accumulated by ecologists (Salguero-Gómez et al. 2015, 2016). They also permit human demographers to take account of factors other than age in determining mortality, longevity, fertility, and population dynamics.
Age- and stage-specific demographic processes are often combined in demography using multistate life table methods (e.g., Rogers 1975; Willekens 2002, 2014). These are usually focused on cohort dynamics and associated survival statistics (but see Rogers 1975, Chap. 5 for an explicit consideration of population projection). Multistate life table models are written as continuous-parameter, discrete-state Markov chains, where the parameter represents age and the states represent stages. In order to solve the resulting equations, the dynamics must be approximated over a (usually short) finite age interval; this would correspond to the sequence of matrices A_{i} in the model here. The age × stage-classified model described by \(\tilde {{\mathbf {A}}}\) is a way to solve the discretized equations in a single step, and makes possible a variety of analyses that are difficult or impossible in the usual life table formulation. Further investigation of the relation between continuous multistate life table methods and age × stage-classified models will be interesting.
These analyses blur the distinction (Chap. 5) between implicit and explicit age dependence. If the A_{i} are truly identical, by definition only implicit age dependence is revealed. But the structure of the age × stage model separates all of the age-dependent A_{i}, and thus is ready to include any degree of explicit joint dependence of the vital rates on age and stage.
Given sufficient longitudinal data on both age and stage, it is possible to estimate the stage-specific matrices A_{i} as explicit functions of age; see Peeters et al. (2002) for an example of a study of human heart disease, and Lebreton et al. (2009) for a review of methods used in multistate capture-mark-recapture analysis in ecology. Needless to say, the data requirements for a full age × stage parameterization are challenging. I suspect that the development of estimation methods at intermediate levels of detail will be an important step.
6.5.1 Reducibility and Ergodicity
The properties of \(\tilde {{\mathbf {A}}}\) raise an important theoretical and technical issue regarding population growth, fitness, and selection gradients. The use of λ as a measure of fitness is usually justified by the strong ergodic theorem (Cohen 1979, Caswell 2001, Section 4.5.2), which guarantees the eventual convergence to the stable population structure and growth at a rate given by the dominant eigenvalue λ. A sufficient condition for this convergence is that the projection matrix be irreducible; i.e., that there exist a pathway connecting any two stages. Stott et al. (2010) surveyed published population projection matrices and found that reducible matrices were not uncommon, and explored the implications for ergodicity. Reducible matrices are not as bad as some people think, but it is important to understand their implications, especially for age × stage models.
General results about the irreducibility of block-structured matrices are difficult; see Csetenyi and Logofet (1989), Logofet (1993, Chap. 3), and Logofet and Belova (2007) for some important graph-theoretical results. However, the age × stage matrices developed here are unusual among population models in that they are (almost) always reducible, because they contain categories to which there are no possible pathways. This arises because age 1 individuals are produced only by reproduction. Hence there can never be age 1 individuals in any stage that is not produced by reproduction. For example, Scotch broom reproduces only by seeds, so age 1 seeds appear in the model. However, the matrix \(\tilde {{\mathbf {A}}}\) also contains entries corresponding to age 1 seedlings, age 1 juveniles, age 1 adults, etc. These do not exist, and because there are no pathways to these stages from any other stages, the matrix \(\tilde {{\mathbf {A}}}\) is reducible.
The Perron-Frobenius theorem guarantees that a reducible non-negative matrix will have a real, non-negative, dominant eigenvalue that is at least as large as any of the others. However, the asymptotic population growth rate and structure may depend on initial conditions (Caswell 2001, Section 4.5.4) This means that one must ascertain that the eigenvalues and eigenvectors under analysis correspond to initial conditions of interest.
Appendix A shows that a necessary and sufficient condition for population growth to be described by the dominant eigenvalue λ of \(\tilde {{\mathbf {A}}}\), regardless of the (non-negative and non-zero) initial population vector, is that the left eigenvector v be strictly positive, and that this corresponds to a particular block-triangular form of \(\tilde {{\mathbf {A}}}\). This provides a simple check for the ergodicity of population growth, and justifies the use of λ as a population growth rate and measure of fitness.
Primitivity may be difficult to evaluate for an age × stage matrix (but see Logofet 1993) but as with any projection matrix model, the long-term average growth rate of a primitive matrix is still given by the dominant real eigenvalue.
The matrix \(\tilde {{\mathbf {A}}}\) for Scotch broom in (6.34) is reducible, as shown by calculating \(\left ( {\mathbf {I}}_{s \omega } + \tilde {{\mathbf {A}}} \right )^{s \omega }\) and finding that this matrix contains zeros (Caswell 2001). However, the left eigenvector v is strictly positive, so we know that the population eventually grows at the rate λ regardless of initial conditions.
6.5.2 A Protocol for Age × Stage-Classified Models
- 1.
Choose a question, and a corresponding demographic outcome. Are you interested in population dynamics (growth, structure, transients)? Or in cohort dynamics (survival, longevity)? Or in some combination of the two?
- 2.
Obtain the stage-classified projection matrices A_{i} for ages i = 1, …, ω.
- 3.
Decompose A_{i} = U_{i} + F_{i}.
- 4.
Construct the block-diagonal matrices \(\mathbb {A}\), \(\mathbb {F}\), \(\mathbb {U}\), and \(\mathbb {D}\), according to Eqs. (6.5)–(6.10).
- 5.
Construct the age × stage matrices \(\tilde {{\mathbf {A}}}\), \(\tilde {{\mathbf {F}}}\), \(\tilde {{\mathbf {U}}}\) using (6.16) and, if appropriate for the question at hand, also \(\tilde {{\mathbf {M}}}\) and \(\tilde {{\mathbf {P}}}\) using (6.35) and (6.36).
- 6.
Analyze the model, e.g., by computing eigenvalues, eigenvectors, the fundamental matrix, etc., as appropriate. If necessary, check for reducibility and ergodicity using the methods in Sect. 6.5.1.
- 7.For sensitivity analysis,
- (a)
choose a dependent variable ξ and a vector of parameters θ,
- (b)
compute the sensitivity matrix \(d \boldsymbol {\xi } / d \mbox{vec} \,^{\mathsf {T}} \tilde {{\mathbf {A}}}\),
- (c)compute the matrices:$$\displaystyle \begin{aligned} {d \mbox{vec} \, {\mathbf{A}}_i \over d \boldsymbol{\theta}^{\mathsf{T}}}, ~~ {d \mbox{vec} \, {\mathbf{U}}_i \over d \boldsymbol{\theta}^{\mathsf{T}}}, ~~\mbox{and} ~~ {d \mbox{vec} \, {\mathbf{F}}_i \over d \boldsymbol{\theta}^{\mathsf{T}}}\end{aligned} $$
- (d)
compute dξ∕dθ^{T} according to (6.18).
- (a)
The explicit connection between matrix population models and absorbing Markov chain theory makes it possible to analyze both population dynamics and cohort dynamics in a unified framework (cf. Feichtinger 1971a; Caswell 2001, 2006, 2009). Cohort dynamics are, in essence, the demography of individuals. It may seem paradoxical to speak of the demography of individuals, but that is what it is, because the statistical properties of a cohort (e.g., average lifespan) are probabilistic properties of an individual (e.g., life expectancy). Demography in general, and matrix population models in particular, provides the link between the individual and the population.
Footnotes
- 1.
By assuming that F does not depend on σ, I am in effect choosing a pre-breeding census and excluding neonatal mortality from σ.
- 2.
This is the matrix for the Discovery Park population, 1993–1994, edge conditions; taken from the Appendix of Parker (2000).
- 3.
- 4.
This holds provided that A is diagonalizable, which is a generic property for linear operators (Hirsch and Smale 1974, p. 157).
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