# LTRE Decomposition of the Stochastic Growth Rate

## Abstract

The basic unit of comparative demography is a study that reports the value of some demographic outcome in two populations that differ in a set of vital rates. One challenge of such studies is to account for the difference in outcomes by decomposing that difference into contributions from differences in each of the parameters. It frequently happens that small differences in some parameters make large contributions to the difference in outcomes, and vice-versa.

## 9.1 Introduction

The basic unit of comparative demography is a study that reports the value of some demographic outcome in two populations that differ in a set of vital rates. One challenge of such studies is to account for the difference in outcomes by decomposing that difference into contributions from differences in each of the parameters. It frequently happens that small differences in some parameters make large contributions to the difference in outcomes, and vice-versa.

In some parts of the literature, such studies are called life table response experiment (or LTRE) analyses; versions of this analysis have appeared in Sect. 1.3.1 and Chaps. 2 , 4 , and 8 . The term was introduced by in the context of laboratory studies of the population effects of pollutants, hence the use of the word “experiment” (Caswell 1989). The conditions among which the populations are compared will be called “treatments” here, but there is no restriction to experimental manipulations.

Similar decomposition analyses have been developed independently in ecology and human demography. For example, Pollard’s (1988) study of life expectancy used methods very similar to LTRE analyses of the population growth rate. Horiuchi et al. (2008) developed a method for continuous variables that is essentially identical to that used by ecologists for regression LTRE calculations (Caswell 1996). Canudas Romo (2003) reviews the human demographic literature.

This chapter uses matrix calculus to extend LTRE analysis to stochastic models, by showing how to decompose differences in the stochastic growth rate, \(\log \lambda _s\). Because stochastic models include both environmental fluctuations and the vital rate responses to those fluctuations, their structure is richer than that of time-invariant models. Stochastic LTRE analysis thus requires a new approach to decomposing these differences. The payoffs, in terms of demographic and biological understanding, are great.

## 9.2 Decomposition with Derivatives

*(dimension*

**ξ***q*× 1), as described in Chap. 2 . Suppose that

*depends on a vector*

**ξ***of vital rates (dimension*

**θ***p*× 1), and that observations are available under two treatments, with

*is evaluated at the mean of the two parameter vectors.*

**ξ**

**ξ**^{(2)}−

**ξ**^{(1)}are contained in a matrix

**C**(dimension

*q*×

*p*) given by

**θ**^{(1)}and

**θ**^{(2)}.

**C**(

*i*,

*j*) of the contribution matrix is the contribution of the difference Δ

*θ*

_{j}to the difference in Δ

*ξ*

_{i}. The columns and rows of

**C**give

**C**is the approximation (9.3) to the treatment effect on

**ξ***θ*

_{i}or because

*ξ*does not respond much to changes in

*θ*

_{i}.

The contribution matrix **C** takes advantage of matrix calculus to provide a simple calculation for decomposition of scalar-, vector- or matrix-valued differences. Studies including more than two treatments or conditions are analyzed by defining a reference parameter vector **θ**_{r} and calculating a matrix **C**_{i} for treatment *i* in terms of the parameter difference **θ**_{i} −**θ**_{r}. The reference treatment might be the average parameter set, or the parameters for a “control” condition, etc.

## 9.3 Kitagawa and Keyfitz: Decomposition Without Derivatives

In decomposing differences in the stochastic growth rate, we encounter variables for which the derivatives in (9.3) cannot be calculated. Fortunately, an alternative method for decomposition is available that does not rely on derivatives. It was introduced by Kitagawa (1955) to explore the effects of age-specific death rates and of age distribution on crude death rates. The method was later extended by Keyfitz to decompose differences in age distributions, dependency ratios, and population growth rates into contributions from the entire mortality and fertility schedules (Keyfitz 1968, Section 7.4; Keyfitz and Caswell 2005, Section 10.1). Canudas Romo (2003) summarizes more recent extensions of the approach in demography.

*ξ*depends on two variables, with values (

*a*,

*b*) in Treatment 1 and (

*A*,

*B*) in Treatment 2. Thus

*ξ*[

*A*,

*B*] −

*ξ*[

*a*,

*b*] into contributions from

*A*−

*a*and

*B*−

*b*, the Kitagawa-Keyfitz method proceeds by exchanging variables between the two treatments and calculating

*ξ*for all possible combinations. The effect of

*A*−

*a*, against the background of

*B*, is

*ξ*[

*A*,

*B*] −

*ξ*[

*a*,

*B*]. The effect of

*A*−

*a*, against the background of

*b*is

*ξ*[

*A*,

*b*] −

*ξ*[

*a*,

*b*]. The overall contribution of

*A*−

*a*is obtained by averaging its effect against the two backgrounds:

*B*−

*b*is

## 9.4 Stochastic Population Growth

**A**(

*t*) is generated by a realization of an ergodic stochastic environment that produces, for every environmental state, a set of vital rates that satisfy certain regularity conditions. Then, the asymptotic long-term growth rate is, with probability one,

*λ*or \(r=\log \lambda \) in stable population theory in constant environments. Cohen (1986) and Lee and Tuljapurkar (1994) have incorporated models of the form (9.12), with the addition of immigration terms, into the context of human population projections, to provide estimates of confidence intervals more rigorous than the “high, medium, low” scenarios usually reported.

In this chapter, I consider the case in which the environment is described by a finite-state Markov chain. Ecological examples include years with our without fire (Silva et al. 1991), years since fire (Caswell and Kaye 2001), years with early or late floods, or with high or low precipitation (Smith et al. 2005) and years with good or poor sea ice conditions (Hunter et al. 2010; Jenouvrier et al. 2009b). The Markovian environment case also includes the situation where the environment is modelled implicitly by selecting randomly from a set of empirically-measured matrices (e.g., Bierzychudek 1982; Cohen et al. 1983; Jenouvrier et al. 2009a). Let *u*(*t*) be the state of the environment at time *t*. The environmental dynamics are determined by the Markov chain transition matrix **P**, where \(p_{ij} =P \left [ u(t+1) = i | u(t) = j \right ]\).

*be a vector of parameters that determine the projection matrix*

**θ****A**. The vectors

**θ**_{1}, …,

**θ**_{K}correspond to environmental states 1, …,

*K*. I will write the entire set of vital rates as

**A**(

*t*) =

**A**[

*(*

**θ***t*)], and the stochastic growth rate (9.13) becomes

*(*

**θ***t*) is the parameter vector created by the environmental state

*u*(

*t*). I have written \(\log \lambda _s\) as an explicit function of

**P**and

**Θ**to emphasize that it depends on both the environment and the vital rate response.

### 9.4.1 Environment-Specific Sensitivities

**θ**_{i}in

**Θ**. These environment-specific sensitivities were given by Caswell (2005) and independently by Horvitz et al. (2005), and have been applied by Gervais et al. (2006), Aberg et al. (2009), and Svensson et al. (2009). Rewriting Tuljapurkar’s (1990) formula in matrix calculus notation yields the derivative of \(\log \lambda _s\) with respect to the vital rate vector in environment

*i*:

**w**(

*t*) and

**v**(

*t*) are the stochastic analogues of the right and left eigenvectors of a deterministic model, and

*R*

_{t}is the growth of total population size from

*t*to

*t*+ 1. See Caswell (2001, Section 14.4) for a step-by-step algorithm for the calculation.

*J*

_{t}is an indicator variable, defined as

*consist of the elements of*

**θ****A**, then

*d*vec

**A**∕

*d*

**θ**^{T}=

**I**, where

**I**is the identity matrix. If

*contains lower-level parameters, then*

**θ***d*vec

**A**∕

*d*

**θ**^{T}contains the derivatives of

**A**with respect to these parameters.

## 9.5 LTRE Decomposition Analysis for \(\log \lambda _s\)

The treatment effect on \(\log \lambda _s\) in (9.18) depends on both the differences in environmental dynamics (captured in the transition matrices **P**^{(1)} and **P**^{(2)}) and the differences in the vital rate responses (captured in the parameter arrays **Θ**^{(1)} and **Θ**^{(2)}). Because \(\log \lambda _s\) is calculated numerically from (9.15) by simulation, it cannot be differentiated^{1} with respect to **P**, so we will use the Kitagawa-Keyfitz decomposition for the environmental dynamics contribution, and environment-specific derivatives (9.16) for the vital rate response contributions

Let us consider three cases: the case where only the vital rate responses differ, the case where only the environmental dynamics differ, and finally the case where both differ.

### 9.5.1 Case 1: Vital Rates Differ, Environments Identical

**P**is identical in the two sites, but the vital rates differ. The stochastic growth rates are

**Θ**

^{(1)}and

**Θ**

^{(2)}. The

*i*th term of the summation in (9.21) is the contribution of differences in the

*i*th environment. These can be written as the elements of a contribution matrix (dimension 1 ×

*p*)

### 9.5.2 Case 2: Vital Rates Identical, Environments Differ

**P**

^{(1)}and

**P**

^{(2)}) but not the vital rate responses. For example, a comparison of population growth before and after implementing a fire control strategy that changes the frequency of fire, but has no effect on how the vital rates respond to fire. The stochastic growth rates are

**P**

^{(1)}and

**P**

^{(2)}may differ in their long-term frequencies of environmental states. Those long-term frequencies are given by the stationary distributions, i.e., the right eigenvector

*corresponding to the dominant eigenvalue of*

**π****P**(which always equals 1), scaled so that

*sums to 1. The same frequency of environmental states, however, can be obtained from processes with different autocorrelation patterns, from negative autocorrelation (where states tend to alternate) to positive autocorrelation (characterized by long runs of the same state; see Caswell and Kaye (2001, Fig. 2) for an example). So,*

**π****P**

^{(1)}and

**P**

^{(2)}may differ in their stationary distributions, autocorrelation patterns, or both. To separate the contributions from these, using the Kitagawa-Keyfitz decomposition, we construct a Markov chain with the same stationary distribution

*as*

**π****P**, but in which successive environmental states are independent, and hence there is no autocorrelation. This chain has the transition matrix

**1**is a vector of ones. Because the next state is independent of the previous state, and the same matrix is applied at each time, this process is called “independent and identically distributed,” and abbreviated “iid.”

**P**is

*C*(

**R**), is obtained by subtraction;

### 9.5.3 Case 3: Vital Rates and Environments Differ

**P**

^{(1)}and

**P**

^{(2)}) and the vital rate responses (

**Θ**

^{(1)}and

**Θ**

^{(2)}). The stochastic growth rates are

*C*(

**Q**)), in the autocorrelation pattern (

*C*(

**R**)), and in the vital rates in each environmental state (

*C*(

**θ**_{1}), …,

*C*(

**θ**_{K})). The decomposition analysis proceeds in three steps.

- 1.Write the contributions of the environmental differences using the Kitagawa-Keyfitz method$$\displaystyle \begin{aligned} \begin{array}{rcl} C(\mathbf{P})&\displaystyle =&\displaystyle \frac{1}{2} \left( \rule{0in}{3ex} \log \lambda_s \left[ {\mathbf{P}}^{(2)}, \boldsymbol{\Theta}^{(2)} \right] - \log \lambda_s \left[ \mathbf{ P}^{(1)}, \boldsymbol{\Theta}^{(2)} \right] \right. \\ {} &\displaystyle &\displaystyle + \left. \log \lambda_s \left[ {\mathbf{P}}^{(2)}, \boldsymbol{\Theta}^{(1)} \right] - \log \lambda_s \left[ {\mathbf{P}}^{(1)}, \boldsymbol{\Theta}^{(1)} \right] \rule{0in}{3ex} \right) {} \end{array} \end{aligned} $$(9.31)$$\displaystyle \begin{aligned} \begin{array}{rcl} C(\mathbf{Q}) &\displaystyle =&\displaystyle \frac{1}{2} \left( \rule{0in}{3ex} \log \lambda_s \left[ {\mathbf{Q}}^{(2)}, \boldsymbol{\Theta}^{(2)} \right] - \log \lambda_s \left[ \mathbf{ Q}^{(1)}, \boldsymbol{\Theta}^{(2)} \right] \right. \\ {} &\displaystyle &\displaystyle + \left. \log \lambda_s \left[ {\mathbf{Q}}^{(2)}, \boldsymbol{\Theta}^{(1)} \right] - \log \lambda_s \left[ {\mathbf{Q}}^{(1)}, \boldsymbol{\Theta}^{(1)} \right] \rule{0in}{3ex} \right) {} \end{array} \end{aligned} $$(9.32)Each of$$\displaystyle \begin{aligned} \begin{array}{rcl} C(\mathbf{R}) &\displaystyle =&\displaystyle C(\mathbf{P}) - C(\mathbf{Q}) {} \end{array} \end{aligned} $$(9.33)
*C*(**P**),*C*(**Q**), and*C*(**R**) is a scalar. - 2.Write the contributions of the vital rate differences using the Kitagawa-Keyfitz method$$\displaystyle \begin{aligned} \begin{array}{rcl} C(\boldsymbol{\Theta}) &\displaystyle =&\displaystyle \frac{1}{2} \left\{ \rule{0in}{3ex} \log \lambda_s \left[ {\mathbf{P}}^{(2)}, \boldsymbol{\Theta}^{(2)} \right] - \log \lambda_s \left[ {\mathbf{P}}^{(2)}, \boldsymbol{\Theta}^{(1)} \right] \right\} \\ {} &\displaystyle +&\displaystyle \frac{1}{2} \left\{ \rule{0in}{3ex} \log \lambda_s \left[ {\mathbf{P}}^{(1)}, \boldsymbol{\Theta}^{(2)} \right] - \log \lambda_s \left[ {\mathbf{P}}^{(1)}, \boldsymbol{\Theta}^{(1)} \right] \right\} {} \end{array} \end{aligned} $$(9.34)
*C*(**Θ**) is a scalar, summing the effects of differences in all of the parameter responses at all states of the environment. It is decomposed further in the next step: - 3.Use the environment-specific derivatives of \(\log \lambda _s\) to decompose each term in (9.34) into contributions from the vital rates in each environment, using (9.22)for$$\displaystyle \begin{aligned} \begin{array}{rcl} \mathbf{C} \left( \boldsymbol{\theta}_i \right) &\displaystyle =&\displaystyle \frac{1}{2} \left( \left. {\partial {\log \lambda_s \left[ {\mathbf{P}}^{(2)}, \bar{\boldsymbol{\Theta}} \right]}\over \partial {\boldsymbol{\theta}^{\mbox{\textsf{ T}}}}} \right|{}_{u=i} \right) \mathcal{D}\, \left( \boldsymbol{\theta}_i^{(2)} - \boldsymbol{\theta}_i^{(1)} \right) \\ &\displaystyle +&\displaystyle \frac{1}{2} \left( \left. {\partial \log \lambda_s \left[ {\mathbf{P}}^{(1)}, \bar{\boldsymbol{\Theta}} \right] \over \partial \boldsymbol{\theta}^{\mbox{\textsf{ T}}}} \right|{}_{u=i} \right) \mathcal{D}\, \left( \boldsymbol{\theta}_i^{(2)} - \boldsymbol{\theta}_i^{(1)} \right) \quad i=1,\ldots,K\\ {} \end{array} \end{aligned} $$(9.35)
*i*= 1, …,*K*, with the derivatives evaluated at \(\bar {\boldsymbol {\Theta }}\), the mean of the vital rates in the two treatments being compared. The matrix**C**(**θ**_{i}) is (1 ×*p*) vector, whose entries give the contributions to the differences in \(\log \lambda _s\) from each of the vital rates in environment*i*.The total contribution of the parameter differences given in (9.34) is$$\displaystyle \begin{aligned} C(\boldsymbol{\Theta}) = \sum_{i=1}^K \mathbf{C} \left( \boldsymbol{\theta}_i \right) \, {\mathbf{1}}_p. \end{aligned} $$(9.36)

## 9.6 An Example: Fire and an Endangered Plant

I know of no comparative studies of stochastic population growth that include differences in both the environmental dynamics and the vital rate responses, so here is an artificial example, based on a model for an endangered plant, *Lomatium bradshawii*, in a stochastic fire environment (Caswell and Kaye 2001). *L. bradshawii* (Apiaceae) is a polycarpic herbaceous perennial plant. It exists in only a few isolated populations in prairies of Oregon and Washington. These habitats were, until recent times, subject to natural and anthropogenic fires, to which *L. bradshawii* seems to have adapted. Fires increase plant size and seedling recruitment, but the effect fades within a few years. Populations in recently burned areas have higher growth rates and lower probabilities of extinction than unburned populations. For more information, see Pendergrass et al. (1999), Caswell and Kaye (2001), and Kaye et al. (2001).

A stochastic demographic model for *L. bradshawii* was developed by Caswell and Kaye (2001), based on data from an experimental burning study. Individuals were classified into six stages based on size and reproductive status: yearlings, small and large vegetative plants, and small, medium, and large reproductive plants. The environment was classified into four states defined by the time since the most recent fire: the year of a fire and 1, 2, and 3+ years post-fire, and vital rates were estimated in each of these environmental states. The matrices are given in Caswell and Kaye (2001).

*L. bradshawii*, with RP superior to FB. Population growth rates were generally higher at RP than at FP (Table 9.1), and the stochastic growth rate was higher in RP than FB at any fire frequency. The critical fire frequency required to maintain

*L. bradshawii*populations was about 0.8–0.9 at FB, but only 0.4–0.5 at RP. The causes of the differences between the sites are not known (Pendergrass et al. 1999).

The population growth rate *λ* calculated from the environment-specific matrices **A**[**θ**_{i}] for *L. bradshawii*. (From Caswell and Kaye 2001)

Fisher Butte | Rose Prairie | |
---|---|---|

Years post-fire | | |

0 | 1.020 | 1.155 |

1 | 0.984 | 1.118 |

2 | 0.662 | 0.483 |

≥3 | 0.869 | 0.906 |

### 9.6.1 The Stochastic Fire Environment

*f*be the long-term frequency of fire, and

*ρ*the temporal autocorrelation coefficient of the fire process (the magnitude of

*ρ*determines the rate of decay of correlation as time increases, the sign of

*ρ*determines whether the correlation is of one sign, or oscillates). In the two-state fire model, the probability of fire in year

*t*+ 1 if there was no fire in year

*t*is

*q*=

*f*(1 −

*ρ*). The probability of a fire if there was a fire in year

*t*is

*p*=

*q*+

*ρ*(see Caswell 2001, Section 14.1). The resulting transition matrix for the four environmental states is

*ρ*< 0,

*f*must satisfy

*ρ*= 0, the environmental process given by (9.38) is not iid.

### 9.6.2 LTRE Analysis

There is no information on differences in fire dynamics at the two sites, so Caswell and Kaye (2001) studied the response of \(\log \lambda _s\) to the frequency and autocorrelation of fires. Here, we use stochastic LTRE analysis to decompose the differences in \(\log \lambda _s\) in three hypothetical scenarios of environmental differences. I will use the matrix entries as the vital rates * θ*, there being no natural lower-level parameterization in this model. Matlab code for the calculations is available as an appendix to Caswell (2010).

The stochastic growth rate \(\log \lambda _s\) increases with fire frequency for both species. The RP site has a growth advantage, with \(\log \lambda _s^{(RP)} > \log \lambda _s^{(FB)}\) at all fire frequencies. The RP advantage, measured by \(\log \lambda _s^{(RP)} - \log \lambda _s^{(RP)}\) increases from ≈0.02 when *f* = 0 to ≈0.13 when *f* = 1.

### Differences in vital rates and environmental transitions (Case 3)

To decompose the treatment effect \(\log \lambda _s^{(RP)} - \log \lambda _s^{(FB)}\), we construct the Markov chain transition matrices from (9.38), and calculate the stationary distributions **π**^{(RP)} and **π**^{(FB)} as eigenvectors of **P**. For each site, we generate the iid transition matrix **Q** from (9.25), and compute the contributions *C*(**P**) from (9.31), *C*(**Q**) from (9.32), and *C*(**R**) from (9.33). Then we compute the environment-specific sensitivities of \(\log \lambda _s\) from (9.16), for both **P**^{(RP)} and **P**^{(FB)}, and use these to calculate the contributions *C*(**θ**_{i}) of the vital rates in each environmental state, using (9.35). Finally, we sum the *C*(**θ**_{i}) to obtain the integrated effect of all vital rate differences in each environment.

**A**[

**θ**_{1}] and

**A**[

**θ**_{2}] (the year of a fire and the year immediately following a fire). The difference in the long-term frequency of environmental states, and the differences in autocorrelation patterns, make relatively little contribution.

The accuracy of the approximations involved in the LTRE analysis is good. The sum of the contributions in Fig. 9.2 is 0.1192, while the actual difference in \(\log \lambda _s\) is 0.1219 (an accuracy of 98%).

*f*= 0.1 (well below the critical threshold for persistence), but a fire management program increased the fire frequency in the FP site to

*f*= 0.9. Now \(\log \lambda _s^{(FB)} > \log \lambda _s^{(RP)}\), despite the general advantage in vital rates of RP over FB in most environmental states. Figure 9.3 presents the contributions to \(\Delta \log \lambda _s\) from differences in fire frequency, autocorrelation, and vital rates, and shows how the contributions of the vital rate differences are, in this case, overwhelmed by the RP disadvantage due to the stationary distribution of the environment.

The sum of the contributions in Fig. 9.3 is − 0.1326, while the actual difference in \(\log \lambda _s\) is − 0.1395 (an accuracy of 95%, even with a very large difference in growth rate).

## 9.7 Discussion

This application of matrix calculus provides a general framework for decomposition analysis of the stochastic growth rate in Markovian environments. It is a direct generalization of the familiar LTRE approaches for time-invariant and periodic models, but combined with the powerful Kitagawa-Keyfitz decomposition. Comparative studies of the stochastic growth rate require additional data on the stochastic dynamics of the environment, beyond that needed for time-invariant models (Fig. 9.1). Many stochastic studies present conditional results; for example, the study of *L. bradshawii* provides \(\log \lambda _s\) as a function of *f*, *ρ*, and **Θ**, but does not estimate the value of \(\log \lambda _s\) actually exhibited in either of the two sites. To do so would require long-term data on the stochastic environment, which is hard to come by. However, such information may possibly be extracted from historical data (e.g., Smith et al. 2005; Lawler et al. 2009), or projected from climate models (Hunter et al. 2010; Jenouvrier et al. 2009b).

The methods presented here are not limited to Markovian environments in which the environmental states have an interpretation (years since fire, flood conditions, etc.). They can also be used when matrices are randomly selected from a series collected over time (e.g., the early study of Bierzychudek (1982) based on two yearly matrices, or the study by Jenouvrier et al. (2009b) based on 44 years of matrices for emperor penguins). Although such models are indeed Markov chains, if years are simply a random sample of environmental variation, then it is of little interest to know the contribution of vital rate differences in, say, 1988 compared to 1989 or 1987. In these models, the mean and variance of the vital rates may be of more interest. Davison et al. (2010), drawing on the stochastic elasticity results of Tuljapurkar et al. (2003), have presented an approach to LTRE analysis in terms of the contributions of differences in the mean and the variance of the vital rates. That method nicely complements the approach presented here.

In the analysis of *Lomatium bradshawii*, even large differences in environmental autocorrelation made small contributions to treatment effects on \(\log \lambda _s\). This is not surprising, given the generally small impact of changes in autocorrelation on the stochastic growth rate in this model (Caswell and Kaye 2001). It is, however, not guaranteed. Given the proper interaction between environmental states and the stage structure, autocorrelation can have dramatic impacts on the growth rate (Caswell 2001, Example 14.1). How often this happens in nature will only be revealed by further comparative studies.

Changing focus from plants in a fluctuating fire environment to human populations projected in response to stochastic fluctuations in mortality and fertility (e.g., Tuljapurkar 1992; Lee and Tuljapurkar 1994), there are possibilities for applying this approach to population projections. However, such attempts will be challenging because the stochastic environments are not stationary, and the interest is not in asymptotic stochastic growth, but in short term transient dynamics. A combination of the transient analyses in Chap. 7 with the decomposition approach here might yield interesting results.

## Footnotes

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