Sensitivity Analysis of Nonlinear Demographic Models
Abstract
Nonlinearities in demographic models arise due to density dependence, frequency dependence (in 2sex models), feedback through the environment or the economy, recruitment subsidy due to immigration, and from the scaling inherent in calculations of proportional population structure.
10.1 Introduction
Nonlinearities in demographic models arise due to density dependence, frequency dependence (in 2sex models), feedback through the environment or the economy, recruitment subsidy due to immigration, and from the scaling inherent in calculations of proportional population structure. This chapter presents a series of analyses particular to nonlinear models: the sensitivity and elasticity of equilibria, cycles, ratios (e.g., dependency ratios), age averages and variances, temporal averages and variances, life expectancies, and population growth rates, for both ageclassified and stageclassified models.
 Density dependence:

arises when one or more of the percapita vital rates are functions of the numbers or density of the population. Such effects have been incorporated into demographic studies of plants (e.g., Solbrig et al. 1988; Gillman et al. 1993; Silva Matos et al. 1999; Pardini et al. 2009; Shyu et al. 2013) and animals (e.g., Pennycuick 1969; CluttonBrock et al. 1997; Cushing et al. 2003; Bonenfant et al. 2009). Density dependence has been intensively studied in the laboratory (e.g., Pearl et al. 1927; Frank et al. 1957; Costantino and Desharnais 1991; Carey et al. 1995; Mueller and Joshi 2000; Cushing et al. 2003). It can arise from competition for food, space, or other resources, or from interactions (e.g., cannibalism) among individuals.
Simple density dependence is less often invoked by human demographers^{1}. Weiss and Smouse (1976) proposed a densitydependent matrix model, and Wood and Smouse (1982) applied it to the Gainj people of Papua New Guinea. Density dependence is included in epidemiological feedback models applied to a rural English population in the sixteenth and seventeenth centuries by Scott and Duncan (1998).
The Easterlin effect (1961) produces density dependence in which fertility is a function of cohort size. Analysis of the Easterlin effect has focused mostly on the possibility that it could generate cycles in births (e.g., Lee 1974, 1976; Frauenthal and Swick 1983; Wachter and Lee 1989; Chu 1998).
 Environmental (or economic) feedback.

Densitydependent models are often an attempt to sneak in, by the back door as it were, a feedback through the environment. A change in population size changes some aspect of the environment, which affects the vital rates, which in turn affect future population size. Models in which the feedback operates through resource consumption are the basis for the food chain and food web models that underlie models of global biogeochemistry (e.g.,. Hsu et al. 1977; Tilman 1982; Murdoch et al. 2003; Fennel and Neumann 2004). These models are typically unstructured, but there is a rich literature on structured models, written as partial differential equations, to incorporate physiological structure and resource feedback (de Roos and Persson 2013).
Feedback models are also invoked in human demography, with the feedback operating through the economy (Lee 1986, 1987; Chu 1998). An interesting aspect of these approaches is the possibility that, if larger populations support more robust economies, the feedback could be positive instead of negative (Lee 1986; Cohen 1995, Appendix 6). An exciting combination of ecological and economic feedback appears in the food ratio model recently proposed by Lee and Tuljapurkar (2008).
 Twosex models.

To the extent that both males and females are required for reproduction (and, in the bigger scheme of things, this is not always so), demography is nonlinear because the marriage function or mating function cannot satisfy (10.2). Nonlinear twosex models have a long tradition in human demography (see reviews in Keyfitz 1972; Pollard 1977) and have been applied in ecology (e.g., Lindström and Kokko 1998; Legendre et al. 1999; Kokko and Rankin 2006; Lenz et al. 2007; Jenouvrier et al. 2010, 2012). Their mathematical properties have been investigated by e.g, Caswell and Weeks (1986), Chung (1994) and Iannelli et al. (2005) and in a very abstract setting by Nussbaum (1988, 1989).
In their most basic form, twosex models differ from densitydependent models in that the vital rates depend only on the relative, not the absolute, abundances of stages in the population (they are sometimes called frequencydependent for this reason). This has important implications for their dynamics.
 Models for proportional population structure.

Even when the dynamics of abundance are linear, the dynamics of proportional population structure are nonlinear (e.g., Tuljapurkar 1997). This leads to some useful results on the sensitivity of the stable age or stage distribution and the reproductive value.
Linear models lead to exponential growth and convergence to a stable structure. Much of their analysis focuses on the population growth rate λ or \(r=\log \lambda \). Nonlinear models do not usually lead to exponential growth (frequencydependent twosex models are an exception). Instead, their trajectories converge to an attractor. The attractor may be an equilibrium point, a cycle, an invariant loop (yielding quasiperiodic dynamics), or a strange attractor (yielding chaotic dynamics); see Cushing (1998) or Caswell (2001, Chapter 16) for a detailed discussion.
This chapter analyzes the sensitivity and elasticity of equilibria and cycles. Because the dynamic models considered here are discrete, solutions always exist and are unique. The nature and the number of the attractors depends on the specific model. Perturbation analysis always considers perturbations of something, so the equilibria or cycles must be found before their perturbation properties can be analyzed.
10.2 DensityDependent Models
10.2.1 Linearizations Around Equilibria
If all the eigenvalues of M are less than one in magnitude, the equilibrium \(\hat {{\mathbf {x}}}\) is locally asymptotically stable. The linearization also provides valuable information about shortterm transient responses to perturbation; see Sect. 10.2.4.
10.2.2 Sensitivity of Equilibrium
The following example, applying (10.16) to a simple model, shows the basic steps and output of the analysis.
Example 1: A simple twostage model
10.2.3 Dependent Variables: Beyond \(\hat {{\mathbf {n}}}\)
 1.Weighted population density. Let c ≥ 0 be a vector of weights. Weighted population density is then N(t) = c^{T}n(t). Examples include total density (c = 1), the density of a subset of stages (c_{i} = 1 for stages to be counted; c_{i} = 0 otherwise), biomass (c_{i} is the biomass of stage i), basal area, metabolic rate, etc. The sensitivity of \(\hat N\) is$$\displaystyle \begin{aligned} {d \hat N \over d \boldsymbol{\theta}^{\mathsf{T}}} = {\mathbf{c}}^{\mathsf{T}} {d \hat{{\mathbf{n}}} \over d \boldsymbol{\theta}^{\mathsf{T}}}. {}\end{aligned} $$(10.31)
 2.Ratios, measuring the relative abundances of different stages. Letwhere a ≥ 0 and b ≥ 0 are weight vectors. Examples include the dependency ratio (in human populations, the ratio of the individuals below 15 or above 65 to those between 15 and 65; see Sect. 10.5.3), the sex ratio, and the ratio of juveniles to adults, which is used in wildlife management; see Skalski et al. (2005). Differentiating (10.32) gives$$\displaystyle \begin{aligned} R(t) = \frac{{\mathbf{a}}^{\mathsf{T}} {\mathbf{n}}(t)}{{\mathbf{b}}^{\mathsf{T}} {\mathbf{n}}(t)} {} \end{aligned} $$(10.32)$$\displaystyle \begin{aligned} {d \hat R \over d \boldsymbol{\theta}^{\mathsf{T}}} = \left( \frac{{\mathbf{b}}^{\mathsf{T}} \hat{{\mathbf{n}}} {\mathbf{a}}^{\mathsf{T}}  {\mathbf{a}}^{\mathsf{T}} \hat{{\mathbf{n}}} {\mathbf{b}}^{\mathsf{T}}}{\left( {\mathbf{b}}^{\mathsf{T}} \hat{{\mathbf{n}}} \right)^2} \right) {d \hat{{\mathbf{n}}} \over d \boldsymbol{\theta}^{\mathsf{T}}}. {} \end{aligned} $$(10.33)
 3.
Age or stage averages. These include quantities such as the mean age or size in the stable population or at equilibrium and the mean age at reproduction in the stable population. Their perturbation analysis is presented in Sect. 10.5.4.
 4.
Properties of cycles. Nonlinear models may produce population cycles. Attention may focus on the mean, the variance, or higher moments of the population vector or of some scalar measure of density, over such cycles. The sensitivity of these moments is explored in Sect. 10.7.
10.2.4 Reactivity and Transient Dynamics
The asymptotic stability of an equilibrium is determined by the eigenvalues of the Jacobian matrix M in (10.9), evaluated at that equilibrium. In the short term, however, perturbations of the population away from the equilibrium can exhibit transient dynamics that differ from their asymptotic behavior. In particular, perturbations of stable equilibria, that are destined to eventually return to the equilibrium, may move (much) farther away before that return. Neubert and Caswell (1997) introduced three indices, each calculated from M, to quantify these transient responses.^{6} The reactivity of an asymptotically stable equilibrium is the maximum, over all perturbations, of the rate at which the trajectory departs from the equilibrium. At any time following a perturbation, there is a maximum (over all perturbations) deviation from the equilibrium. This maximum is the amplification envelope. It gives an upper bound on the extent of transient amplification as a function of time. The phrase “over all perturbations” in these definitions signals that the transient amplification depends on the direction of the perturbation. The perturbation that produces the maximum amplification at any specified time is the optimal perturbation (Verdy and Caswell 2008).^{7}
10.2.5 Elasticity Analysis
Example 2: Metabolic population size in Tribolium
The sensitivity of \(\hat {{\mathbf {n}}}\) is calculated using (10.16). However, the damage caused by Tribolium as a pest of stored grain products might well depend more on metabolism than on numbers. Emekci et al. (2001) estimated the metabolic rates of larvae, pupae, and adults as 9, 1, and 4.5 μl CO_{2} h^{−1}, respectively. We define the metabolic population size as N_{m}(t) = c^{T}n(t) where \({\mathbf {c}}^{\mathsf {T}} = \left (\begin {array}{ccc} 9 & 1 & 4.5 \end {array}\right )\), and calculate the sensitivity and elasticity of \(\hat N_m\) using (10.37) and (10.31).
When the stages are weighted by their metabolic rate, the elasticity of \(\hat N_m\) to fecundity is positive, but the elasticities to all the other parameters (cannibalism rates and mortalities) are negative. The positive effects of c_{pa} and μ_{a} on \(\hat {{\mathbf {n}}}\) disappear when the stages are weighted according to metabolism. \(~~\blacksquare \)
10.2.6 ContinuousTime Models
10.3 Environmental Feedback Models
Environmental (or economic) feedback models write the vital rates as functions of some environmental variable, which in turn depends on population density. Feedback models may be static or dynamic. In static feedback models, the environment depends only on current conditions, with no inherent dynamics of its own. In dynamic feedback models, the environment can have dynamics as complicated as those of the population (e.g., if the environmental variable was the abundance of a prey species, affecting the dynamics of a predator species). The sensitivity analysis of dynamic feedback models is given in Sect. 10.8.
10.4 Subsidized Populations and Competition for Space
A subsidized population is one in which new individuals are recruited from elsewhere rather than (or in addition to) being generated by local reproduction. Subsidy is important in many plant and animal populations, especially of benthic marine invertebrates and fish. Many of these species produce planktonic larvae that may disperse very long distances (Scheltema 1971) before they settle and become sessile for the rest of their lives. Thus a significant part—maybe even all—of the recruitment at any location is independent of local fertility (e.g., Almany et al. 2007). Subsidized models have been used to analyze conservation programs in which captivereared animals are released into a wild or reestablished population (Sarrazin and Legendre 2000). They have been applied to the demography of human organizations; e.g., schools, businesses, learned societies (Gani 1963; Pollard 1968; Bartholomew 1982). They are also the basis of cohortcomponent population projections that include immigration.
In the simplest subsidized models, both local demography and recruitment are densityindependent. Alternatively, recruitment may depend on some resource (e.g., space) whose availability depends on the local population, or the local demography after settlement may be densitydependent. All three cases can lead to equilibrium populations.
10.4.1 DensityIndependent Subsidized Populations
Example 3: The Australian Academy of Sciences
Most human organizations are subsidized; recruits (new students in a school, new employees in a company) come from outside, not from local reproduction. In an early example of a subsidized population model, Pollard (1968) analyzed the age structure of the Australian Academy of Sciences, recruitment to which takes place by election.^{10} The Academy had been founded in 1954, and between 1955 and 1963 had elected about 6 new Fellows each year, with an age distribution (Pollard 1968, Table 2) given by
Age  Percent 

30–34  0.0 
35–39  12.2 
40–44  24.5 
45–49  26.5 
50–54  20.4 
55–59  4.1 
60–64  10.2 
65–69  2.0 
As parameters, consider the agespecific mortality rates \(\mu _i =  \log P_i\), and define the parameter vector \(\boldsymbol {\theta } = \left (\begin {array}{ccc} \mu _1 & \mu _2 & \ldots \end {array}\right )^{\mathsf {T}}\). Equation (10.58) then gives the sensitivity of the equilibrium population to changes in agespecific mortality. The sensitivity of the total size of the Academy, \(\hat N = {\mathbf {1}}^{\mathsf {T}} \hat {{\mathbf {n}}}\), calculated using (10.31), is shown in Fig. 10.2b. It shows that increases in mortality reduce \(\hat N\) (not surprising), with the greatest effect coming from changes in mortality at ages 48–58.
Because learned societies are often concerned with their age distributions, Pollard (1968) examined the proportion of members over age 70. At equilibrium, this proportion is \(\hat R = 0.26\). The sensitivity \(d \hat R/d \boldsymbol {\theta }^{\mathsf {T}}\), calculated using (10.33), is shown in Fig. 10.2c. Increases in mortality before age 48 would increase the proportion of members over 70, while increases in mortality after age 48 would decrease it.^{11}\(~~\blacksquare \)
10.4.2 Linear Subsidized Models with Competition for Space
Recruitment in subsidized populations may be limited by the availability of a resource. Roughgarden et al. (1985; see also Pascual and Caswell 1991) presented a model for a population of sessile organisms, such as barnacles, in which recruitment is limited by available space. Barnacles^{12} produce larvae that disperse in the plankton for several weeks before settling onto a rock surface or other suitable substrate, after which they no longer move.
Example 4: Intertidal barnacles
Gaines and Roughgarden (1985) modelled a population of the barnacle Balanus glandula in central California. In one site (denoted KLM in their paper), they reported ageindependent survival with a probability of P_{i} = 0.985 per week, i = 1, …, 52. The growth in basal area of an individual barnacle could be described by g_{x} = π(ρx)^{2}, where x is age in weeks and ρ is the radial growth rate (ρ = 0.0041 cm/wk). The mean settlement rate was ϕ = 0.107. The matrix B contains survival probabilities P_{i} on the subdiagonal, terms of the form − ϕg_{i} in the first row, and zeros elsewhere.
10.4.3 DensityDependent Subsidized Models
10.5 Stable Structure and Reproductive Value
The linear model n(t + 1) = An(t) will, if A is primitive, converge to a stable age or stage distribution. But while the dynamics of the population vector n(t) are linear, the dynamics of the proportional population structure are nonlinear (Tuljapurkar 1997). We can take advantage of this to analyze the sensitivity of proportional structures by writing them as equilibria of nonlinear maps.
10.5.1 Stable Structure
The sensitivity of the stable stage distribution has been approached as an eigenvector perturbation problem (e.g., Caswell 1982, 2001; Kirkland and Neumann 1994), but those calculations are complicated. Analysis of the equilibrium of the nonlinear model (10.69) is much easier.
Example 5: A human age distribution
10.5.2 Reproductive Value
10.5.3 Sensitivity of the Dependency Ratio
This result can be generalized in several ways. The analysis may be performed separately for the dependent young and the dependent old, by suitable modification of a and b. These two components are likely to be influenced by different demographic factors and can respond to perturbations in opposite directions. The 01 vectors a and b may be replaced by vectors of weights; e.g., agespecific consumption and agespecific income (FürnkranzPrskawetz and Sambt 2014). For an example applied to a population projection for Spain, see Caswell and Sanchez Gassen (2015). The analysis also applies to stageclassified models, provided that dependent and independent stages can be identified. It also applies to nonlinear models, with the stable stage distribution \(\hat {{\mathbf {p}}}\) replaced by the equilibrium population \(\hat {{\mathbf {n}}}\) in (10.81). It can be extended to transient dynamics, where the age distribution, and thus the dependency ratio, varies over time (Caswell 2007), as is the case in population projections (Caswell and Sanchez Gassen 2015). Finally, the sensitivity (10.81) makes it possible to carry out LTRE analyses to decompose differences in dependency ratios into components due to differences in each of the vital rates (see Chaps. 2 , 8 , and 9).
Example 5: (cont’d) Dependency ratios in human populations
Breaking D into its young and old components helps to explain these differences. In both countries, there is a crossover in survival effects. Increases in survival at early ages increase D_{y} and reduce D_{o}. At later ages, increases in survival reduce D_{y} and increase D_{o}. Increases in fertility increase D_{y} and reduce D_{o}. In the U.S. population, both these effects are large, with the negative effect on D_{o} larger than the positive effect on D_{y}. In the Kuwaiti population, the positive effect on D_{y} is much greater than the negative effect on D_{o}. \(~~\blacksquare \)
10.5.4 Sensitivity of Mean Age and Related Quantities
From an age distribution \(\hat {{\mathbf {p}}}\), it is possible to compute the mean age of any agespecific property (e.g., production of children, collection of retirement benefits, exposure to toxic chemicals); see Chu (1998, p. 26) for general discussions. The most familiar of these is the mean age of reproduction, which is one measure of generation time (Coale 1972).
Example 5: (cont’d) Mean age of reproduction
Increases in fertility reduce \(\bar a_{{\mathbf {f}}}\) if they happen before age 25 and increase \(\bar a_{{\mathbf {f}}}\) if they happen after age 25. These sensitivities are quite large, although this is somewhat irrelevant since the largest sensitivities are for ages at which fertility is zero and unlikely to be modified. \(~~\blacksquare \)
10.5.5 Sensitivity of Variance in Age
10.6 FrequencyDependent TwoSex Models
Because of the homogeneity of A[θ, n], frequencydependent models do not converge to an equilibrium density \(\hat {{\mathbf {n}}}\). Instead, there may exist^{15} a stable equilibrium proportional structure \(\hat {{\mathbf {p}}}\) to which the population will converge, at which point it grows exponentially at a rate λ given by the dominant eigenvalue of \({\mathbf {A}}[\boldsymbol {\theta }, \hat {{\mathbf {p}}}]\). Thus sensitivity analysis of twosex models must include both the population structure and the population growth rate.
Note that matrix models that include Mendelian genetics are also homogeneous of degree zero, but it is confusing to call them frequencydependent, because doing so creates confusion with the genetic phenomenon of frequencydependent fitness, which is a different thing altogether (de Vries and Caswell 2018).
10.6.1 Sensitivity of the Population Structure
10.6.2 Population Growth Rate in TwoSex Models
Because a population with the equilibrium structure grows exponentially, I once suggested treating \({\mathbf {A}}[\boldsymbol {\theta }, \hat {{\mathbf {p}}}]\) as a constant matrix and applying eigenvalue sensitivity analysis to it, in order to examine life history evolution in 2sex models (Caswell 2001, p. 577). This was incorrect, because it ignored the effect of parameter changes on A through their effects on the equilibrium \(\hat {{\mathbf {p}}}\). A correct calculation obtains the sensitivity of λ including effects of parameters on both A and \(\hat {{\mathbf {p}}}\).
Note that λ is the invasion exponent for this model, and thus the sensitivity of λ to a parameter gives the selection gradient on that parameter. Tuljapurkar et al. (2007) used this fact to explore the effect of male fertility patterns on the evolution of aging; the sensitivity (10.97) could be used to generalize such results. Recent work by Shyu has coupled these calculations to the methods of adaptive dynamics to examine the evolution of sex ratios (Shyu and Caswell 2016a,b).
Although twosex models are an important case of homogeneous models, they are not the only case. Keyfitz’s (1972) interpretation of the Easterlin hypothesis describes fertility as dependent on only the relative, not absolute, size of a cohort. A model based on this premise would be frequencydependent (homogeneous) and would lead to an exponentially growing population to which (10.97) would be applicable.
Example 6: A twosex model for passerine birds
Elasticity of \(\hat {{\mathbf {p}}}\) to parameters in twosex model for passerine birds, under two mortality scenarios. When male mortality is greater than female mortality, males are rarer than females and fertility at equilibrium is limited by the mating function. When male mortality is less than female mortality, females are rare and fertility is not affected by the mating function
Males rare  

Stage  σ _{0}  ρ  σ _{1}  σ _{2}  σ _{3}  σ _{4}  ϕ _{1}  ϕ _{2} 
\(\hat p_1\)  0.455  0.453  0.226  0.229  0.000  0.000  0.266  0.189 
\(\hat p_2\)  0.890  1.799  0.774  0.783  0.398  0.268  0.521  0.369 
\(\hat p_3\)  0.455  1.547  0.226  0.229  0.000  0.000  0.266  0.189 
\(\hat p_4\)  0.664  0.428  0.226  0.229  0.669  0.450  0.389  0.275 
Females rare  
Stage  σ _{0}  ρ  σ _{1}  σ _{2}  σ _{3}  σ _{4}  ϕ _{1}  ϕ _{2} 
\(\hat p_1\)  0.455  1.547  0.000  0.000  0.226  0.229  0.320  0.135 
\(\hat p_2\)  0.664  0.428  0.669  0.450  0.226  0.229  0.467  0.197 
\(\hat p_3\)  0.455  0.453  0.000  0.000  0.226  0.229  0.320  0.135 
\(\hat p_4\)  0.890  1.799  0.398  0.268  0.774  0.783  0.627  0.264 
The elasticity of λ to parameters in the twosex model for passerine birds, under two mortality scenarios. The correct calculation is based on (10.97). The naive calculation incorrectly treats \({\mathbf {A}}[\hat {{\mathbf {p}}}, \boldsymbol {\theta }]\) as a fixed matrix, ignoring the effect of parameters on the equilibrium population structure \(\hat {{\mathbf {p}}}\)
Males rare  Females rare  

Correct  Naive  Correct  Naive  
σ _{0}  0.669  0.545  0.669  0.669 
ρ  0.669  0.545  0.669  0.669 
σ _{1}  0  0.226  0.198  0.198 
σ _{2}  0  0.229  0.133  0.133 
σ _{3}  0.198  0  0  0 
σ _{4}  0.133  0  0  0 
ϕ _{1}  0.392  0.319  0.471  0.471 
ϕ _{2}  0.277  0.226  0.198  0.198 
Sometimes the correct calculations lead to apparent paradoxes. Jenouvrier et al. (2010) developed a twosex model for the Emperor penguin. It was a periodic model, with phases defined by events within the breeding cycle (cf. Chap. 8 ), and included a mating function applied to adults at the breeding colony. Because Emperor penguins breed, and share parental care, in the midst of the Antarctic winter,^{17} they must be strictly monogamous, and hence Jenouvrier used the minimum as a mating function.
Analysis of the equilibrium growth rate revealed that the sensitivity of λ to adult female survival was negative. This is impossible in a linear model, but happens in this frequencydependent model because increasing adult female survival increases the proportion of females (already greater than the proportion of males) and thus decreases mating probability. The negative effect of reduced mating overwhelms the positive effect of improved adult survival; the net result is a reduction in population growth rate; see Jenouvrier et al. (2010) for details. \(~~\blacksquare \)
10.6.3 The Birth MatrixMating Rule Model
 1.
A birth matrix whose entries give the expected number of male and female offspring produced by a mating of a male of age (or stage) i and a female of age j.
 2.
A mating rule function that gives the number of matings u_{ij} between males of age (or stage) i and females of age j.
 3.
A set of sexspecific mortality schedules, which project surviving individuals to the next age class, or, in our generalization, include other stagespecific life cycle transitions.
10.7 Sensitivity of Population Cycles
Equilibria are not the only attractors relevant in nature (e.g., CluttonBrock et al. 1997) or the laboratory (Cushing et al. 2003). Cycles, invariant loops, and strange attractors also occur, and are sensitive to changes in parameters. This section examines the sensitivity of cycles.
10.7.1 Sensitivity of the Population Vector
10.7.2 Sensitivity of Weighted Densities and Time Averages
 Weighted densities.
 Let c be a vector of weights, and let \(\hat N_i = {\mathbf {c}}^{\mathsf {T}} \hat {{\mathbf {n}}}_i\) be the (scalar) weighted density at the ith point on the cycle. Then writeThe vector \(\hat {{\mathbf {n}}}\) can be calculated from \(\mathbb {N}\) as$$\displaystyle \begin{aligned} \hat{{\mathbf{n}}} = \left(\begin{array}{c} \hat N_1 \\ \vdots\\ \hat N_k \end{array}\right) \end{aligned} $$(10.124)$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{{\mathbf{n}}} &=& \left(\begin{array}{ccc} {\mathbf{c}}^{\mathsf{T}} \hat{{\mathbf{n}}}_1 & \cdots & {\mathbf{c}}^{\mathsf{T}} \hat{{\mathbf{n}}}_k \end{array}\right)^{\mathsf{T}} \\ &=& \mbox{vec} \, \left( {\mathbf{c}}^{\mathsf{T}} {\mathbf{G}} \right) \\ &=& \left( {\mathbf{I}}_k \otimes {\mathbf{c}}^{\mathsf{T}} \right) \mbox{vec} \, {\mathbf{G}} \\ &=& \left( {\mathbf{I}}_k \otimes {\mathbf{c}}^{\mathsf{T}} \right) \mathbb{N} \qquad \mbox{dimension}=k \times 1. {} \end{array} \end{aligned} $$(10.125)
 Timeaveraged population vector.
 Let b be a probability vector (b_{i} ≥ 0, 1^{T}b = 1) and define the timeaveraged population vector asThen$$\displaystyle \begin{aligned} \bar{{\mathbf{n}}} = \sum_{i=1}^k b_i \hat{{\mathbf{n}}}_i. \end{aligned} $$(10.126)$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar{{\mathbf{n}}} &\displaystyle =&\displaystyle {\mathbf{G}} {\mathbf{b}} \\ &\displaystyle =&\displaystyle \left( {\mathbf{b}}^{\mathsf{T}} \otimes {\mathbf{I}}_s \right) \mbox{vec} \, {\mathbf{G}} \\ &\displaystyle =&\displaystyle \left( {\mathbf{b}}^{\mathsf{T}} \otimes {\mathbf{I}}_s \right) \mathbb{N} \qquad \mbox{dimension}=s \times 1 {} \end{array} \end{aligned} $$(10.127)
 Timeaveraged weighted density.
 Taking the time average of the \(\hat N_i\) gives$$\displaystyle \begin{aligned} \begin{array}{rcl} \bar N &\displaystyle =&\displaystyle \sum_i b_i {\mathbf{c}}^{\mathsf{T}} \hat{{\mathbf{n}}}_i \\ &\displaystyle =&\displaystyle {\mathbf{c}}^{\mathsf{T}} {\mathbf{G}} {\mathbf{b}} \\ &\displaystyle =&\displaystyle \left( {\mathbf{b}}^{\mathsf{T}} \otimes {\mathbf{c}}^{\mathsf{T}} \right) \mathbb{N} {} \end{array} \end{aligned} $$(10.128)
Example 7 A 2cycle in the Tribolium model
First, the elasticities of the \(\hat {{\mathbf {n}}}_i\) differ from stage to stage and from one point on the cycle to another (Fig. 10.8a). Increases in fecundity, for example, increase the density of larvae and reduce the density of pupae in \(\hat {{\mathbf {n}}}_1\), but have the opposite effects in \(\hat {{\mathbf {n}}}_2\). The elasticities to b, c_{ea}, and c_{el} are much larger than those to the other parameters (cf. the elasticities of the equilibrium \(\hat n\) in Fig. 10.1).
The elasticities of total population are similar at the two points in the cycle (Fig. 10.8b), except that larval mortality μ_{l} has a large negative effect on \(\hat N_2\), but only a small effect on \(\hat N_1\). The elasticities of total respiration \(\hat R_i\), however, are different at the two points in the cycle (Fig. 10.8c).
The elasticities of the timeaveraged population vector \(\bar {{\mathbf {n}}}\) (Fig. 10.8d) are similar to those of the equilibrium vector in Fig. 10.1 (although they need not be). This pattern is not predictable from the patterns of the elasticities of the population vectors \(\hat {{\mathbf {n}}}_1\) and \(\hat {{\mathbf {n}}}_2\) (Fig. 10.8a).
Finally, the elasticities of the time averages, \(\bar N\) and \(\bar R\), of the weighted densities are similar to each other and to the elasticities of the timeaveraged population \(\bar {{\mathbf {n}}}\).
The sensitivity analysis of cycles thus depends very much on the dependent variables of interest. The matrix \(d \mathbb {N}/d \boldsymbol {\theta }^{\mathsf {T}}\) (Fig. 10.8a) contains 36 pieces of information: the effects of 6 parameters on 3 stages at 2 points in the cycle. A focus on weighted density reduces this to 12 (Fig. 10.8b,c), but the results may depend very much on the particular weighting vector chosen. A focus on time averages reduces the information from 36 to 18 numbers (Fig. 10.8d), and the response of the timeaveraged weighted densities finally are described by just 6 numbers. The good news is that Eqs. (10.121), (10.125), (10.127), and (10.128) make it easy to compute all these sensitivities. \(~~\blacksquare \)
10.7.3 Sensitivity of Temporal Variance in Density
10.7.4 Periodic Dynamics in Periodic Environments
Periodic environments (e.g., seasons within a year) are described by periodic products of matrices. If the environmental cycle contains p phases, then matrices A_{1}, …A_{p} describe the dynamics at each phase, and the periodic product A_{p}⋯A_{1} projects the population over an entire cycle. Nonlinear periodic models permit the A_{i} to depend on the population vector at any point in the cycle, including delayed dependence (e.g., the reproductive success of an individual plant in the fall may depend on the density it experienced in the spring). A fixed point on the interannual time scale is a pcycle on the seasonal time scale. A kcycle on the interannual scale corresponds to a kpcycle on the seasonal time scale. The sensitivity analysis of these models is given by Caswell and Shyu (2012) and presented here in Chap. 8. For an application to the dynamics of an invasive plant population, see Shyu et al. (2013).
10.8 Dynamic Environmental Feedback Models
10.9 StageStructured Epidemics
The transmission of infectious diseases is a source of nonlinearity because the rate of transmission depends on the abundance of infected and noninfected individuals. When demographic structure is added to the picture, the models can become complicated because the transmission process, the recovery process, and the consequences of infection may all vary among age classes or stages.
Klepac and Caswell (2011) developed a general framework for stageclassified epidemics, using the vecpermutation formulation (e.g., Chaps. 5 and 6). Individuals were jointly classified by stage and infection category, and nonlinearity was introduced by the disease transmission process. Klepac and Caswell (2011) calculated sensitivities and elasticities of equilibria and cycles of the stage × infection distribution and, of stagespecific prevalence, to parameters specifying demographic, infection, and recovery processes.
Coupling demography and epidemiology requires attention to time scales. Suppose that the demographic processes operate on one time scale: say, years. For some diseases, the infection/recovery process might operate on a much longer time scale (decades). Or the disease might play out on a much shorter time scale (weeks). When the disease time scale is shorter than the demographic time scale, the matrices in Klepac’s model that define disease transmission operate many times within a single year; the result is a periodic model on the infection time scale. See Klepac and Caswell (2011) for details.
10.10 Moments of Longevity in Nonlinear Models
This approach can be used to generalize the results for higher moments of longevity (Chaps. 4, 5, and 11) to the nonlinear case.
10.11 Summary
Model  Sensitivity of …  Equation[s]  

Densitydependent  …equilibrium  \(\displaystyle {d \hat {{\mathbf {n}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.16) 
\({\mathbf {n}}(t+1) = {\mathbf {A}} \left [ \boldsymbol {\theta }, {\mathbf {n}}(t) \right ]{\mathbf {n}}(t)\)  …cycle  \(\displaystyle {d \mathbb {N} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.121) 
…weighted density, time average  
…temporal variance  \(\displaystyle {d V (\hat N ) \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.137)  
…life expectancy  \(\displaystyle {d \eta _1 \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.155)  
Environmental feedback  
n(t + 1) = A[θ, n(t), g(t)] n(t)  
g(t) = g[θ, n(t)]  … equilibrium (static)  \(\displaystyle {d \hat {{\mathbf {n}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\), \(\displaystyle {d \hat {{\mathbf {g}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  
g(t + 1) = B[θ, n(t), g(t)] g(t)  …equilibrium (dynamic)  \(\displaystyle {d \mathbb {N} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.152) 
Frequencydependent (twosex)  …equilibrium structure  \(\displaystyle {d \hat {{\mathbf {p}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.114) 
\({\mathbf {n}}(t+1) = {\mathbf {A}} \left [ \boldsymbol {\theta }, {\mathbf {n}}(t) \right ]{\mathbf {n}}(t)\)  …population growth rate  \(\displaystyle {d \lambda \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.97) 
A[θ, n] homogeneous of degree zero in n  
Subsidized (linear or nonlinear)  …equilibrium population  
\({\mathbf {n}}(t+1) = {\mathbf {A}} \left [ \boldsymbol {\theta }, {\mathbf {n}}(t) \right ]{\mathbf {n}}(t) + {\mathbf {b}}[\boldsymbol {\theta }, {\mathbf {n}}]\)  
Proportional structure  …age or stage distribution  \(\displaystyle {d \hat {{\mathbf {p}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.73) 
\(\displaystyle {\mathbf {p}}(t+1) = \frac {{\mathbf {A}}[\boldsymbol {\theta }] {\mathbf {p}}(t)}{{\mathbf {1}}^{\mathsf {T}} {\mathbf {A}}[\boldsymbol {\theta }] {\mathbf {p}}(t)}\)  …reproductive value  \(\displaystyle {d \hat {{\mathbf {v}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.82) 
…dependency ratio  \(\displaystyle {d D \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.81)  
…mean age of reproduction  \(\displaystyle {d \bar a_{{\mathbf {f}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.83)  
…variance in age of reproduction  \(\displaystyle {d V_{{\mathbf {f}}} \over d \boldsymbol {\theta }^{\mathsf {T}}}\)  (10.89) 
 1.
Write the model, specifying the dependence of the vital rates on θ and n.
 2.
Write a matrix expression for the demographic outcome of interest (e.g., the equilibrium population).
 3.
Differentiate this expression.
 4.
Use the vec operator and Roth’s theorem to obtain an expression that involves only the differentials of vectors.
 5.
Use the chain rule for total differentials to expand the operators (e.g., dvecA) that are functions of both θ and n, as in (10.14).
 6.
Use the first identification theorem and the chain rule to extend the results to sensitivities of any desired dependent variable with respect to any set of parameters
Footnotes
 1.
Lee (1987) reviewed the situation and said “…we might say that human demography is all about Leslie matrices and the determinants of unconstrained growth in linear models, whereas animal population studies are all about Malthusian equilibrium through density dependence in nonlinear models …”. He admits that this is an exaggeration, and there clearly are nonlinear concerns in human demography (Bonneuil 1994), but a nonexhaustive survey finds no mention of density dependence in several contemporary human demography texts (e.g., Hinde 1998; Preston et al. 2001; Keyfitz and Caswell 2005).
 2.It is possible to generalize to continuoustime models, that would be written$$\displaystyle \begin{aligned} {d \mathbf{n} \over d t} = \mathbf{A}[\boldsymbol{\theta}, \mathbf{n}(t)] \; \mathbf{n}(t) \end{aligned}$$
for some appropriately defined matrix function A; see Verdy and Caswell (2008). Such models are less often used, but see Shyu and Caswell (2016a, 2018) for a twosex model example.
 3.
The explicit dependence on θ and n(t) will be neglected when it is obvious from the context.
 4.
 5.It is reassuring to check that the dimensions of all these quantities are compatible:$$\displaystyle \begin{aligned} \underbrace{{d \hat{\mathbf{n}} \over d \boldsymbol{\theta}^{\mathsf{T}}}}_{s \times p} = \underbrace{\left( \hat{\mathbf{n}}^{\mathsf{T}} \otimes {\mathbf{I}}_s \right)}_{s \times s^2} \left( \underbrace{{\partial \mbox{vec} \, \mathbf{A} \over \partial \boldsymbol{\theta}^{\mathsf{T}}}}_{s^2 \times p} + \underbrace{{\partial \mbox{vec} \, \mathbf{A} \over \partial {\mathbf{n}}^{\mathsf{T}}}}_{s^2 \times s} \underbrace{{\partial \hat{\mathbf{n}} \over \partial \boldsymbol{\theta}^{\mathsf{T}}}}_{s \times p} \right)+ \underbrace{\mathbf{A}}_{s \times s} \underbrace{{d \hat{\mathbf{n}} \over d \boldsymbol{\theta}^{\mathsf{T}}}}_{s \times p}. \end{aligned}$$
 6.
 7.
 8.
The same model could describe harvest if b ≤ 0 (e.g., Hauser et al. 2006). This form of harvest produces unstable equilibria, and is not considered further here.
 9.
If λ > 1, the population grows exponentially and the subsidy eventually becomes negligible. The equilibrium in this case is nonpositive (and hence biologically irrelevant) and unstable. If λ = 1 then the population would remain constant in the absence of subsidy; any nonzero subsidy will then lead to unbounded population growth.
 10.
Pollard’s paper is remarkable for its treatment of both deterministic and stochastic models, but here I consider only the deterministic case.
 11.
It is possible to calculate the average age of the Academy, and its sensitivity, using results to be introduced in Sect. 10.5.4. The response is very similar to that of the proportion over age 70.
 12.
The temptation to draw analogies between barnacles and the members of learned academies is almost irresistible.
 13.
Because B contains negative elements, its dominant eigenvalue may be complex or negative, leading to oscillatory approach to the equilibrium.
 14.
Or, equivalently, reductions in mortality. For these parameter values, the sensitivity to mortality is approximately the sensitivity to survival with the opposite sign.
 15.
 16.
In a survey of the literature, adult mortality for female passerines exceeded that for males in 21 out of 28 cases (Promislow et al. 1992). Birds differ from mammals in this respect.
 17.
Dramatically portrayed in the movie, March of the Penguins.
 18.
Early writers even interpreted the simple logistic equation as an interplay between a biotic potential for exponential growth and an environmental resistance due to lack of resources or interaction with predators (e.g., Chapman 1931). Incorporating a fully dynamic feedback greatly expands the range of phenomena that can be explained (see de Roos and Persson (2013) for an extensive development of this approach).
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