Periodic Models
Abstract
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.
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.
- 1.
Seasonal variation. Plants and animals experience obvious and dramatic seasonal variation in their demographic rates. Periodic models have been used to describe this variation, with seasons variously defined in terms of monthly periods, calendar seasons, or in terms of environmental events such as rainfall or flood patterns (e.g., Smith et al. 2005).
Although annual or near-annual species are obvious candidates for periodic models, within-year time scales may also be important for long-lived species. For example, Hunter and Caswell (2005a) incorporated chick development events on a time scale of weeks into a periodic model for the sooty shearwater, which has a lifetime of decades. Similarly, Jenouvrier et al. (2010, 2014) have used periodic models to capture the timing of events in the breeding cycle within a portion of the year in the long-lived emperor penguin.
- 2.
Periodic interannual variability. Periodic models based on sequences of annual observations have been used to study effects of inter-event intervals, where events include fires, floods, ENSO events, etc.
- 3.
Harvest and management. These activities often take place at specified points within an annual or interannual cycle. Periodic models have been used to study the effects of their timing; one of the earliest periodic models being devoted to seasonal harvesting (Darwin and Williams 1964).
- 4.Conditional probabilities. Periodic matrix products appear when models are written as products of conditional probabilities. In stage-classified models, for example, a transition matrix U, is written as the product of a diagonal matrix Σ (with survival probabilities σ on the diagonal) and a matrix G of transition probabilities conditional on survival:which creates a period-2 periodic matrix product within the model.$$\displaystyle \begin{aligned} \mathbf{U} = \mathbf{G} \boldsymbol{\Sigma} {} \end{aligned} $$(8.3)
- 5.
Multistate vec-permutation models. When individuals are classified by two or more criteria (e.g., stage and location), the dynamics over the projection interval can be described in terms of the processes affecting each criterion (e.g., transitions and movement). The result is a periodic model that uses the vec-permutation matrix to generate a block-structured projection matrix over the entire interval. See Chap. 6 for analysis of such models.
- 6.
Nonlinear models. Henson and Cushing (1997) developed a model for Tribolium in an experimental system in which container size was varied periodically. Shyu et al. (2013) developed a nonlinear seasonal model of an invasive plant to account for the timing of both density effects and management actions within the year. In such models, cyclic dynamics can be produced both by the environmental periodicity and the nonlinearities (e.g., Cushing 2006).
Table of symbols in this chapter
Symbol | Meaning |
---|---|
s _{ i } | Number of stages at phase i of the cycle |
p | Period of the cycle |
q | Dimension of parameter vector θ |
r | Number of locations in spatial model |
θ _{ i } | Parameter vector evaluated at phase i |
B _{ i } | Projection matrix from phase i to phase i + 1, or in location i |
\({\mathbf {C}}_i^j\) | Ordered product B_{j}⋯B_{i} of matrices from i to j |
M _{ i } | Dispersal matrix for stage i |
A | Projection matrix over entire cycle |
A _{ i } | Projection matrix over cycle, starting at phase i |
R _{ i } | Matrix of LTRE contributions from phase i |
E _{ s, i} | s × s matrix with 1 in (i, i) position and 0 elsewhere |
I _{ s } | Identity matrix of dimension s |
\(\mathcal {D}\,( \mathbf {x} )\) | Diagonal matrix with x on the diagonal |
1 | Vector of ones |
\(\mathbb {B}\), \(\mathbb {M}\), etc. | Block-structured matrices |
∘ | Hadamard, or element-by-element product |
⊗ | Kronecker product |
8.1.1 Perturbation Analysis
In this chapter we analyze linear periodic models of the form (8.1) and the cyclic dynamics of nonlinear seasonal models with delayed density effects. We will briefly discuss the generalization of the multistate age×stage-classified models explored in Chap. 6 to an arbitrary number of classifications. We extend the LTRE decomposition analysis to the periodic case, making it possible to analyze effects of parameter changes at any point in a periodic environment.
8.2 Linear Models
Consider the basic model (8.1) with projection matrix (8.2). The period of the cycle is p. To allow for differences in the state vector at different phases within the cycle, define the number of stages at phase i as s_{i}. Thus the matrix B_{i} is of dimension s_{i+1} × s_{i}, with the subscript i interpreted mod(p) (that is, (p + 1) mod(p) = 1).
8.2.1 A Simple Harvest Model
8.3 Multistate Models
We have encountered several examples of models in which individuals are classified by two criteria (age and stage, stage and environmental state, stage and location, etc.). These multistate models can be constructed by the vec-permutation matrix approach; see Chaps. 5 and 6 or Hunter and Caswell (2005b) and Caswell et al. (2018).
Suppose individuals classified by two criteria; e.g., stages (1, …, s) and locations (1, …, r). One might describe population dynamics in terms of stage transitions within locations, and spatial movement within stages, with the two processes acting sequentially. Thus individuals first survive and reproduce according to their stage-specific demography, and then disperse among locations, and then repeat. Let B_{i} be the s × s matrix describing transitions and reproduction within location i, and M_{j} the r × r matrix describing movement probabilities for stage j. Let \(\mathbb {B}\) and \(\mathbb {M}\) be the sr × sr block diagonal matrices with the B_{i} and the M_{j}, respectively, on the diagonal.
8.4 Nonlinear Models and Delayed Density Dependence
Anticipating the more extensive treatment in Chap. 10 , we consider the effects of nonlinearity in periodic models. You may want to return to this section after Sect. 10.7 , which analyzes periodic oscillations arising from time-invariant nonlinearities. When periodic environmental changes interact with such oscillations, the results can be complicated, and such interactions are the focus of the present section.
In a periodic nonlinear model, each of the B_{i} in (8.2) may depend on density. Especially in seasonal models, the vital rates in the matrix B_{i} may depend on densities not only at phase i, but at previous phases within the cycle as well. For example, in a study of the invasive plant garlic mustard (Alliaria petiolata) Shyu et al. (2013) found that seed production of fruiting plants in the fall reflected the density experienced by vegetative rosettes in the early spring.
8.4.1 Averages
The vector \(d \mathbb {N}/ d \boldsymbol {\theta }^{\mbox{\textsf { T}}}\) created by (8.46) contains the sensitivities of all s stages, at each of p seasons within the year, for each of the k years within the inter-annual k-cycle. If this is too much information, one can calculate the sensitivity of averages, or other linear combinations, taken in various ways.
To write these averages, let b_{m} be a m × 1 vector of weights. For a simple average of m quantities, each entry of b_{m} is 1∕m; for a weighted average, the entries of b_{m} would be non-negative numbers summing to 1. More generally, b may contain arbitrary weights, such as biomass, metabolic rate, economic value, etc. See Chaps. 7 and 10 . To calculate averages from \(\mathbb {N}\), first apply these vectors to average over rows or columns of \(\mathcal{N}\) and then apply the vec operator to express the results as averages over \(\mathbb {N}\).
Annual fixed point
Annual k-cycle
Calculation of averages of attractors of nonlinear periodic matrix population models. The upper half of the table shows averages over stages and over seasons when the dynamics are a fixed point on the inter-annual time scale, and thus a p-cycle on the seasonal time scale. The lower half of the table shows averages over all combinations of stages, seasons, and years, when the dynamics are a k-cycle on the inter-annual time scale, and thus a kp-cycle on the seasonal time scale
Average over | Formula | Vectors | Dimension |
---|---|---|---|
Stages | \(\left ( {\mathbf {I}}_p \otimes {\mathbf {b}}_s^{\mbox{\textsf { T}}} \right ) \mathbb {N} \) | 1 | p × 1 |
Seasons | \(\left ( {\mathbf {b}}_p^{\mbox{\textsf { T}}} \otimes {\mathbf {I}}_s \right ) \mathbb {N}\) | 1 | s × 1 |
Seasons and stages | \(\left ( {\mathbf {b}}_p^{\mbox{\textsf { T}}} \otimes {\mathbf {b}}_s \right ) \mathbb {N}\) | 1 | 1 × 1 |
Stages | \(\left ( {\mathbf {I}}_k \otimes {\mathbf {I}}_p \otimes {\mathbf {b}}_s^{\mbox{\textsf { T}}} \right ) \mathbb {N} \) | 1 | kp × 1 |
Seasons | \(\left ( {\mathbf {I}}_k \otimes {\mathbf {b}}_p^{\mbox{\textsf { T}}} \otimes {\mathbf {I}}_s \right ) \mathbb {N}\) | k | s × 1 |
Years | \(\left ( {\mathbf {b}}_k \otimes {\mathbf {I}}_p \otimes {\mathbf {I}}_s \right ) \mathbb {N}\) | p | s × 1 |
Seasons and years | \(\left ( {\mathbf {b}}_k^{\mbox{\textsf { T}}} \otimes {\mathbf {b}}_p^{\mbox{\textsf { T}}} \otimes {\mathbf {I}}_s \right ) \mathbb {N} \) | 1 | s × 1 |
Seasons and stages | \(\left ( {\mathbf {I}}_k \otimes {\mathbf {b}}_p^{\mbox{\textsf { T}}} \otimes {\mathbf {b}}_s^{\mbox{\textsf { T}}} \right ) \mathbb {N}\) | 1 | k × 1 |
Years and stages | \(\left ( {\mathbf {b}}_k^{\mbox{\textsf { T}}} \otimes {\mathbf {I}}_p \otimes {\mathbf {b}}_s^{\mbox{\textsf { T}}} \right ) \mathbb {N}\) | 1 | p × 1 |
Stages, seasons, years | \(\left ( {\mathbf {b}}_k^{\mbox{\textsf { T}}} \otimes {\mathbf {b}}_p^{\mbox{\textsf { T}}} \otimes {\mathbf {b}}_s^{\mbox{\textsf { T}}} \right ) \mathbb {N}\) | 1 | 1 × 1 |
8.4.2 A Nonlinear Example
At the seasonal 2-cycle (annual fixed point), increases in s_{j} or s_{a} increase density in Season 1 and reduce density in Season 2, and have little effect on the density averaged over seasons. The maximum fertility level a has little effect at either season, and the density-dependent parameter b has large negative effects throughout.
At the 4-point seasonal cycle (2-cycle on the annual time scale), the patterns are more complicated. We describe them in terms of the kp = 4 seasons in the cycle. The maximum fertility a has little effect at any point. The survival probabilities s_{j} and s_{a} have effects that are opposite in sign: an increase in s_{j} increases the density in seasons 1 and 4, and reduces it in seasons 2 and 3. An increase in s_{a} has the opposite effect. Averaged over years, both s_{a} and s_{j} increase density in season 1 and reduce it in season 2, thus increasing the amplitude of the oscillation. Averaged over seasons, s_{a} and s_{j} have opposite effects. When averaged over stages, seasons, and years, the effects of s_{a} cancel each other out, and only s_{j} and b have appreciable effects.
Even in this simple example, it is clear that parameter changes can have effects that differ among seasons and years. A set of Matlab scripts to carry out these calculations appears in an online supplement to Caswell and Shyu (2012).
8.5 LTRE Decomposition Analysis
The LTRE decomposition analysis introduced in Sects. 2.9 and 4.5 can be extended to obtain the contributions, to any given outcome, of differences in parameters at each phase of the cycle.
The contribution matrix (8.71) requires \(d \boldsymbol {\xi } / d \boldsymbol {\theta }_k^{\mbox{\textsf { T}}}\), that is, the derivative of ξ to the parameter at phase k of the cycle. In the linear model (8.2), this is given by the kth term in the summation in (8.11). In the case of the nonlinear model (8.36), the derivative is obtained from Eq. (8.46) by setting all blocks of \(\mathbb {D}\), except those corresponding to phase k, to zero.
8.6 Discussion
The distinguishing feature of periodic models is that the dynamics over a projection interval are given by a periodic product of matrices. The periodic product may reflect the existence of multiple timescales (e.g., seasonal and annual), or the operation of multiple processes (e.g., demography and harvest), or express conditional probabilities, or arise from classifying individuals by multiple criteria. The sensitivity analysis of periodic models must account for the chain of causation (Fig. 8.1) from demographic parameters at each phase in the cycle to the corresponding projection matrices, and thence to the periodic matrix product over the whole cycle, and finally to demographic outcome ξ. Matrix calculus makes this easy to do, starting with a simple chain rule expression (see Eq. (8.5)) and then using an appropriate version of (8.9) to calculate the derivative dvec A∕dθ^{T}.
Footnotes
- 1.Although we will not address it in this chapter, the model (8.1) can be written in a way that explicitly defines the starting phase in the cycle. As written, A in (8.2) projects from phase 1 to phase 1; if desired we could write this as A_{1} and define matrices$$\displaystyle \begin{aligned} \begin{array}{rcl} {\mathbf{A}}_2 &\displaystyle =&\displaystyle {\mathbf{B}}_1 {\mathbf{B}}_p \cdots {\mathbf{B}}_2 \\ &\displaystyle \vdots&\displaystyle \\ {\mathbf{A}}_p &\displaystyle =&\displaystyle {\mathbf{B}}_{p-1} \cdots {\mathbf{B}}_1 {\mathbf{B}}_p \end{array} \end{aligned} $$
The A_{i} are obtained by cyclic permutations of the sequence {B_{p}, …, B_{1}}; each of these projects from a different phase in the cycle. Some demographic properties (e.g., the population growth rate λ) are invariant with respect to such permutations; others (e.g., the eigenvectors) are not (Caswell 2001). In this chapter, we will start with phase 1 and refer to A rather than A_{1}.
- 2.
Alternatively, let μ_{i} be the mortality due to harvest experienced by an individual in stage i. Then \(\mathbf {H} = \exp \left [ - \mathcal {D}\,(\boldsymbol {\mu }) \right ]\). Harvest imposes an additional, additive hazard on top of the natural mortality contained in B.
- 3.
The extension to more than two conditions is easy; see Caswell (2001).
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