Individual Stochasticity and Implicit Age Dependence
Abstract
Demography is the study of the population consequences of the fates of individuals. As an individual organism develops through its life cycle it may increase in size, change its morphology, develop new physiological functions, exhibit new behaviors, or move to new locations.
5.1 Introduction
Demography is the study of the population consequences of the fates of individuals. As an individual organism develops through its life cycle it may increase in size, change its morphology, develop new physiological functions, exhibit new behaviors, or move to new locations. It may marry and divorce, become ill and recover, or change its employment status. It may change sex and/or change its reproductive status. These changes can be dramatic. This developmental process, and its attendant risks of death and opportunities for reproduction, determine the rates of birth and death that, in turn, determine population growth or decline.
Individuals are differentiated on the basis of age or, in general, life cycle stages. The movement of an individual through its life cycle is a random process, and although the eventual destination (death) is certain, the pathways taken to that destination are stochastic and will differ even between identical individuals; this is individual stochasticity. A stageclassified demographic model contains implicit agespecific information, which can be analyzed using Markov chain methods. The living stages in the life cycles are transient states in an absorbing Markov chain, in which death is an absorbing state.
This chapter presents Markov chain methods for computing the mean and variance of the lifetime number of visits to any transient state, the mean and variance of longevity, the net reproductive rate R_{0}, and the cohort generation time. It presents the matrix calculus methods needed to calculate the sensitivity and elasticity of all these indices to any life history parameters.
The Markov chain approach is then generalized to variable environments (deterministic environmental sequences, periodic environments, iid random environments, Markovian environments). Variable environments are analyzed using the vecpermutation method to create a model that classifies individuals jointly by the stage and environmental condition. Throughout, examples are presented using the North Atlantic right whale (Eubaleana glacialis) and an endangered prairie plant (Lomatium bradshawii) in a stochastic fire environment.
5.1.1 Age and Stage, Implicit and Explicit
The essence of demography is the connection between the fates of individual organisms and the dynamics of populations. There exist diverse mathematical frameworks in which this connection can be studied (Keyfitz 1967; Metz and Diekmann 1986; Nisbet and Gurney 1982; Caswell 1989; Tuljapurkar and Caswell 1997; Caswell et al. 1997; DeAngelis and Gross 1992; Ellner et al. 2016). Regardless of the type of equations used, demographic analysis must account for differences among individuals, and the ways in which those differences affect the vital rates.
Among the many ways that individuals may differ, age has long had a kind of conceptual priority. Age is universal in the sense that every organism becomes one minute older with the passage of one minute of time. Age is also often associated with predictable changes in the vital rates. However, in some organisms characteristics other than age provide more and better information about an individual. Ecologists recognized this long ago, and have developed demographic theory based on size, maturity, physiological condition, instar, spatial location, etc.—referred to in general as “stageclassified” demography. Human demographers, who were responsible for the classical ageclassified theory, by no means deny the importance of other properties, such as employment, parity, or health status; see Land and Rogers (1982), Goldman (1994), Robine et al. (2003), and Willekens (2014) for a sample of the kinds of issues that arise.
Even when the demographic model is entirely stageclassified, however, age is still implicitly present. Individuals in a given stage may differ in age, and individuals of a given age may be found in many different stages, but each individual still becomes one unit of age older with the passage of each unit of time. Extracting this implicit agedependent information makes it possible to calculate interesting agespecific properties, such as survivorship, longevity, life expectancy, generation time, and net reproductive rate (Cochran and Ellner 1992; Caswell 2001, 2006; Tuljapurkar and Horvitz 2006; Horvitz and Tuljapurkar 2008).^{1}
In this chapter, I show how to calculate some of these implicit agespecific properties from any stageclassified model. The trick is to formulate the life cycle as a Markov chain, and to generalize the “life” cycle to include death as a stage. Because death is permanent, it is called an absorbing state, and the theory of absorbing Markov chains provides the starting point for our analysis (Feichtinger 1971; Caswell 2001).
A Markov chain is a stochastic model for the movement of a particle among a set of states (e.g., Kemeny and Snell 1976; Iosifescu 1980). The probability distribution of the next state of the particle may depend on the current state, but not on earlier states. In our context, a “particle” is an individual organism. The states correspond to the stages of the life cycle, plus death (or perhaps multiple types of death, for example deaths due to different causes). This structure is ideally suited to asking questions about individual stochasticity, because it accounts for all the possible pathways, and their probabilities, that an individual can follow through its life. I will focus on discretetime models, but much of the theory can no doubt be generalized to continuoustime models.
The use of Markov chains in demographic analysis is not new. As far as I know, Feichtinger (1971, 1973) was the first to use discretetime absorbing Markov chains in demography, paying particular attention to competing risks and multiple causes of death. At around the same time, Hoem (1969) applied continuoustime Markov chains in the analysis of insurance systems (with states such as “active,” “disabled,” and “dead”). Later, Cochran and Ellner (1992) independently proposed the use of Markov chains to generate ageclassified statistics from stageclassified models, but minimized the use of matrix notation in their presentation. Influenced by Feichtinger’s work, and relying heavily on Iosifescu’s (1980) treatment of absorbing Markov chains, I extended the calculations using matrix notation (Caswell 2001; Keyfitz and Caswell 2005), introduced sensitivity analysis (Caswell 2006), and presented results for both timeinvariant and timevarying models. At the same time, Tuljapurkar and Horvitz (2006) and Horvitz and Tuljapurkar (2008) developed the same approaches and presented a more extensive investigation of time variation.
5.1.2 Individual Stochasticity and Heterogeneity
Consider a newborn individual. As it develops through the stages of its life cycle, it may grow, shrink, mature, move, reproduce, and allocate resources among its biological processes. At each moment, it is exposed to various mortality risks. At each moment, it has some chance of reproducing. Because these processes are stochastic, the lives of any two individuals may differ. These random outcomes—this individual stochasticity—imply that the agespecific properties of an individual (say, longevity) are random variables—there is a distribution among individuals that should be characterized by its mean, moments, etc. (Caswell 2009).
It is critical to notice that the calculation of these moments explicitly assumes that every individual in a given stage experiences exactly the same rates and hazards. There is no heterogeneity among the individuals (or, at least, none that matters demographically), even though there is variation in their lifetime properties. Empirical studies of longevity or lifetime reproductive output find that the variation among individuals is usually large, but it is a mistake to jump to the conclusion that it is due to heterogeneity among individuals without first examining the variance that is inevitably created by individual stochasticity (e.g., Tuljapurkar et al. 2009; Steiner and Tuljapurkar 2012; Caswell 2011; Caswell and Kluge 2015; Caswell and Vindenes 2018; Hartemink et al. 2017; Hartemink and Caswell 2018; van Daalen and Caswell 2017).
5.1.3 Examples
The calculations will be demonstrated by means of two case studies. The first is a stageclassified model for the North Atlantic right whale (Eubaleana glacialis). Later, in Sect. 5.5.4, a stochastic matrix model for the threatened prairie plant Lomatium bradshawii will appear as part of a study of variable environments.
The North Atlantic right whale is a large, highly endangered baleen whale (Kraus and Rolland 2007). Once abundant in the north Atlantic, it was decimated by whaling, beginning as much as a thousand years ago (Reeves et al. 2007). By 1900 the eastern North Atlantic stock had been effectively eliminated, and the western North Atlantic stock hunted to near extinction. The population has recovered only slowly since receiving at least nominal protection in 1935, and now numbers only about 300 individuals. Right whales migrate along the Atlantic coast of North America, from summer feeding grounds in the Gulf of Maine and Bay of Fundy to winter calving grounds off the Southeastern U.S. They are killed by ship collisions and entanglement in fishing gear (Kraus et al. 2005), and may also be affected by pollution of coastal waters.
Individual right whales are photographically identifiable by scars and callosity patterns. Since 1980, the New England Aquarium has surveyed the population, accumulating a database of over 10,000 sightings (Crone and Kraus 1990). Treating the first year of identification of an individual as marking, and each year of resighting as a recapture, permits the use of markrecapture statistics to estimate demographic parameters of this endangered population (Caswell et al. 1999; Fujiwara and Caswell 2001, 2002; Caswell and Fujiwara 2004).
5.2 Markov Chains
5.2.1 An Absorbing Markov Chain
The right whale
5.2.2 Occupancy Times and the Fundamental Matrix
As the syllogism asserts, all men are mortal; absorbtion is certain. Our question is, how long does absorbtion take and what happens en route? From a demographic perspective, this is asking about the lifespan of an individual and the events that happen during that lifetime. The key to these questions is the fundamental matrix of the absorbing Markov chain. Consider an individual presently in transient state j. As time passes, it will visit other transient states, repeating some, skipping others, until it eventually dies. Let ν_{ij} denote the number of visits to, or the occupancy time in, transient state i that our individual, starting in transient state j, makes before being absorbed. The ν_{ij} are random variables, reflecting individual stochasticity.
The right whale
We would like to know how the entries of N vary in response to changes in the vital rates. To accomplish this, we need matrix calculus, which is the topic of the next section.
5.2.3 Sensitivity of the Fundamental Matrix
The right whale
The elasticity of n_{41} to σ_{3} (survival of mature females) is approximately 30. This implies that a 1% increase in σ_{3} would produce about a 30% increase in the expected number of reproductive events.
5.3 From Stage to Age
The fundamental matrix summarizes the agespecific information implicit in the transient matrix U, even if the model is stageclassified and age does not appear explicitly. We now extend this, to explore a series of agespecific demographic indices and their sensitivity analyses. Some are well known (R_{0}, generation time), others little explored (variance in longevity, for example). They can, however, all be easily calculated from any stageclassified model.
5.3.1 Variance in Occupancy Time
The right whale
Hint
Before looking at Appendix A.1, to derive (5.21), write N_{dg} = I ∘N, differentiate (5.18), and use the fact that \(\mbox{vec} \, ({\mathbf {A}} \circ {\mathbf {B}}) = \mathcal {D}\,(\mbox{vec} \, {\mathbf {A}}) \mbox{vec} \, {\mathbf {B}} = \mathcal {D}\,(\mbox{vec} \, {\mathbf {B}}) \mbox{vec} \, {\mathbf {A}}\).
The right whale
The elasticities of V (ν_{41}), calculated from (5.21) and (5.17), are shown in Fig. 5.2b. They are roughly proportional to the elasticities of \(E \left ( \nu _{41} \right )\); that is, the vital rates that have large effects on the expected number of reproductive events also have large effects on the variance.
5.3.2 Longevity and Life Expectancy
The sensitivity of life expectancy in ageclassified models has been studied by Pollard (1982) and Keyfitz (1971); see Keyfitz and Caswell (2005, Section 4.3), Vaupel (1986), and Vaupel and Canudas Romo (2003).
Hint
To obtain (5.25), differentiate both sides of (5.23), apply the vec operator, and use (5.16) for the derivative of N. See Appendix A.2 for the derivation.
The right whale
5.3.3 Variance in Longevity
Like the occupancy time in a transient state, longevity is a random variable, the variability of which is a measure of individual stochasticity. Individuals differ in longevity depending on the pathways taken from birth to death. This variance has been explored by human demographers, using life table methods, as one way of studying the inequality in life span generated by a given mortality schedule, and how that inequality has changed over time (e.g., Wilmoth and Horiuchi 1999; Shkolnikov et al. 2003; Edwards and Tuljapurkar 2005; Van Raalte and Caswell 2013).
Hint
To derive (5.28), differentiate (5.27) and apply the vec operator and Roth’s theorem to each term, using (5.25) for the derivative of E(η). See Sect. A.3 for details.
The right whale
The elasticities of the variance of longevity of a calf are shown in Fig. 5.3b. The variance in longevity is increased by increases in σ_{3}, less so by increases in σ_{2} and σ_{4}. The pattern of the elasticities is strikingly similar to that of the elasticities of E(η).
5.3.4 Cohort Generation Time
Generation time measures the typical age at which offspring are produced, or the age at which the typical offspring is produced. It appears in the IUCN criteria for classifying threatened species (IUCN Species Survival Commission 2001) as well as in various evolutionary considerations. There are several definitions of generation time (Coale 1972); here we will examine the cohort generation time, defined as the mean age of production of offspring in a cohort of newborn individuals. From the definition it is clear why calculation of generation time is a problem in stageclassified models, in which the age of parents does not appear. Moreover, in stageclassified models, individuals may be born into several stages (e.g., cleisthogamous vs. chasmogamous seeds; LeCorff and Horvitz 2005), each with a different subsequent pattern of development, survival, and fertility. There could be a different generation time for each type of offspring, and if individuals may produce more than one type of offspring, the average age at which they are produced could differ from one kind of offspring to another.
Hint
To derive (5.34), it helps to note that, for any vector z, one can write \(\mathcal {D}\,({\mathbf {z}}) = {\mathbf {I}} \circ {\mathbf {z}} {\mathbf {1}}^{\mathsf {T}}\). Apply this to X, differentiate all the terms in μ^{(j)}, and apply the vec operator. With any luck, you will come out to this answer. See Sect. A.5.1 for derivation.
The right whale
5.4 The Net Reproductive Rate
 C_{1} :

R_{0} measures the expected lifetime production of offspring.
 C_{2} :

R_{0} measures the rate of increase per generation (in contrast to the rate of increase per unit of time, which is given by λ or r).
 C_{3} :

R_{0} is an indicator function for population persistence. If R_{0} > 1 then an individual will, on average, produce more than enough offspring to replace itself, the next generation will be larger than the present generation, and the population will grow. If R_{0} < 1, each generation is smaller than the one before, and the population will decline to extinction.
In stageclassified models, however, the calculation of R_{0} must account for the multiple pathways that an individual may follow through the life cycle, and the production of multiple kinds of offspring along each of these pathways. Rogers (1974; see also Lebreton 1996) considered R_{0} in the context of an ageclassified population distributed across a set of spatial regions. However, these calculations assume that agespecific survival and fertility schedules are available for each region. A more general solution was provided by Cushing and Zhou (1994) for stageclassified populations with no agespecific information. Their analysis produces an index that satisfies as many as possible of the conditions C_{1}, C_{2}, and C_{3}. de CaminoBeck and Lewis (2007, 2008) have derived graphtheoretic ways to calculate R_{0}.
The relation between lifetime offspring production and R_{0} (condition C_{1}) is more complicated when the life cycle contains multiple types of offspring. If only a single type of offspring is produced (call it stage 1), then F will have nonzero entries only in its first row, and FN will be upper triangular, with its dominant eigenvalue appearing in the (1, 1) position. i.e., the sum of the fertilities of each stage weighted by the expected time spent in that stage. This is precisely the expected lifetime offspring production, so for the case of a single type of offspring, the CushingZhou R_{0} also satisfies C_{1}.
However, if the life cycle contains multiple types of offspring (say stages 1, …, h), the upper left h × h corner of FN will contain the expected lifetime production of offspring of types 1, …, h by individuals starting life as types 1, …, h. Since such a life cycle contains more than one kind of expected lifetime production of offspring, R_{0} cannot satisfy C_{1} in the sense of being the expected lifetime reproduction. Instead, R_{0} is calculated from all these expectations (as the dominant eigenvalue of this h × h submatrix). It determines pergeneration growth and population persistence as a function of the expected lifetime production of all types of offspring in a way that satisfies C_{2} and C_{3}.
The right whale
5.4.1 Net Reproductive Rate in Periodic Environments
Periodic timevarying models (Caswell 2001, Chapter 13) are an interesting special case of the multiple offspring type problem. In a periodic model, apparently identical offspring (e.g., seeds) produced at different phases of the cycle (e.g., seasons) are, in effect, of different types of. To the extent that they face different environments, they will differ in their expected offspring production, and R_{0} will differ depending on the phase of the cycle in which it is calculated.
Cushing and Ackleh (2012) returned to this issue. They argue that the standard approach for studying dynamics of periodic models is to study the “periodic composite map”, which is the map for the entire cycle composed of the product of the phasespecific matrices, as in (5.41), which projects over the entire cycle, rather than from one season to the next. They separate transitions and reproduction as in Eqs. (5.43) and (5.44), and prove that R_{0} calculated in this way satisfies C_{1} (with a different lifetime reproductive output for each starting season) and C_{3} (so that the values of R_{0} in each season agree in their determination of positive or negative growth). Cushing and Ackleh (2012) also explore the net reproductive rate in nonlinear models, in which R_{0} calculated at zero density determines whether the extinction equilibrium is stable.
In the end, it is valuable to have two different ways of calculating R_{0}, but it highlights the need to carefully specify which properties one wants the index to have.
5.4.2 Sensitivity of the Net Reproductive Rate
Hint
To derive (5.50), write R_{0} = ρ[FN] and write dR_{0} in terms of the right and left eigenvectors of FN and the differential of FN. Then expand d(FN) = (dF)N + Fd(N) and apply the vec operator and the chain rule.
The right whale
5.4.3 Invasion Exponents, Selection Gradients, and R_{0}
Selection on life history traits can be studied in terms of the invasion exponent, which measures the rate at which a mutation, introduced at low densities, will increase in the environment created by a resident phenotype (Metz et al. 1992; Ferriére and Gatto 1993); for a recent introduction see Otto and Day (2007). The selection gradient on a trait is the derivative of the invasion exponent with respect to the value of the trait. If the derivative is positive, selection favors an increase in the trait, and viceversa. The invasion exponent in a densityindependent model is given by \(\log \lambda \). In a densitydependent model, the invasion exponent is given by the growth rate at equilibrium, \(\lambda [\hat {{\mathbf {n}}}]\). The net reproductive rate R_{0} is not, strictly speaking, an invasion exponent, but because it measures expected lifetime reproduction, it is attractive as a measure of fitness (see, e.g., the discussion in Kozlowski 1999). Using R_{0} as a measure of fitness will lead to erroneous conclusions unless the selection gradients, measured in terms of λ and of R_{0}, give the same answers, i.e., unless \(d R_0 / d \theta \propto d \log \lambda / d \theta \).
The right whale
5.4.4 Beyond R_{0}: Individual Stochasticity in Lifetime Reproduction
Variation among individuals is fundamental to population biology. As argued here, two sources of variation must be distinguished: heterogeneity and individual stochasticity Heterogeneity refers to genuine differences among individuals, because of which the individuals experience different vital rates. Individual stochasticity refers to the apparent differences that result from the random outcome of identical vital rates, applied to identical individuals. We have seen above that individual stochasticity is always present. That is particularly true of lifetime reproductive output (LRO). The net reproductive rate is the expectation of LRO, but what can we say about the variance among individuals.
Empirical measurement shows that LRO is usually highly variable among individuals and positively skewed. Typically, a few individuals produce many offspring while most produce few, or none at all (CluttonBrock 1988; Newton 1989). If this variance reflected heterogeneity among individual properties, and if the heterogeneity had a genetic basis, the variance would provide material for natural selection (the “opportunity for selection” of Crow 1958). Population and quantitative genetics are replete with methods to measure such genetic variation; e.g., Lande and Arnold (1983) and Endler (1986).
However, variance among individuals in LRO is not evidence of heterogeneity, genetic or otherwise; some is due to individual stochasticity. Only after evaluating the extent of individual stochasticity can data on LRO be interpreted as evidence for heterogeneity (Caswell 2011; Tuljapurkar et al. 2009; Steiner et al. 2010; Steiner and Tuljapurkar 2012). Caswell (2011) developed a method to calculate the mean, variance, and higher moments of lifetime reproductive output for any age or stageclassified life cycle, using Markov chains with rewards; see van Daalen and Caswell (2015, 2017) for full details. In these models^{4} the movement of the individual through its life cycle is described by an absorbing Markov chain; mortality appears as transitions to an absorbing (dead) state. At each step, the individual accumulates a “reward.” In our context, the reward is the production of offspring. The reproductive reward is a random variable with a specified set of moments. The reward accumulated by the inevitable death of the individual is its LRO. Although every individual experiences the same vital rates—there is no heterogeneity—each individual may experience a different life and thus a different lifetime reproductive output.
Stagespecific reproductive output is specified by a set of reward matrices R_{k}. The (i, j) element of R_{k} is the kth moment of the reproductive output associated with the transition from stage j to stage i. Given the reward matrices, the Markov chain transition matrix P, and the reasonable assumption that the dead do not reproduce, all the moments of LRO can be calculated (van Daalen and Caswell 2017).
One of the most significant findings of this line of research has been that, in many cases, individual stochasticity can account for most or all of the observed phenotypic variance in LRO (Steiner and Tuljapurkar 2012; van Daalen and Caswell 2017). It appears that the contribution of stochasticity to variance in lifetime reproductive output has been underappreciated.
5.5 Variable and Stochastic Environments

Deterministic aperiodic environments. These usually appear as specific historical sequences; e.g., the specific sequence of vital rates exhibited by the right whale between 1980 and 1998 (Caswell 2006). That sequence is fixed, and is neither random nor periodic.

Periodic environments. A periodic model may describe seasonal variation within a year, or may approximate interannual variability in events such as floods, fires, or hurricanes.

Stochastic iid environments. In such environments, successive states are drawn independently from a fixed probability distribution; hence the identifier iid, short for “independent and identically distributed.”

Markovian stochastic environments. In a Markovian environment the probability distribution of the next environmental state may depend on the current state. This permits study of the effects of environmental autocorrelation. Markovian environments include periodic and iid environments as special cases.
When studying variable environments, it is important to distinguish period and cohort calculations. Period calculations are based on the vital rates in a given year. They describe the results of the hypothetical situation where the conditions of year t are maintained indefinitely, and compare those to the results for conditions in year t + 1, etc. Period calculations are a way to summarize the effects of changing environment. But an individual born in year t does not live its life under the conditions of year t. It spends its first year of life under the conditions in year t, its second year under the conditions of year t + 1, and so on. Results calculated in this way are called cohort calculations, because they describe a cohort born in year t and living through the environmental sequence starting then. Periodspecific calculations are easy; simply apply the timeinvariant calculation to the vital rates of each year and tabulate the results. Cohort calculations, however, must account for all the possible environmental sequences through which a cohort may pass. Caswell (2006) and Tuljapurkar and Horvitz (2006) independently introduced two different, complementary approaches to doing so. I will present the former approach here.
5.5.1 A Model for Variable Environments
Tuljapurkar and Horvitz (2006), whose paper I highly recommend, work directly from (5.60) to develop the means and variances of N, η, and survivorship, in periodic, iid, and Markovian environments. Here, we consider an approach in which an individual is jointly classified by stage and environment, using the vecpermutation model developed by Hunter and Caswell (2005b).
The first block of entries gives stage 1 individuals in environments 1 through q. The second block gives stage 2 individuals in environments 1 through q, and so on.
Matrices of similar form, but not using this formalism, were introduced by Horvitz to study populations in habitat patches where the habitat patches change state over time, for example in recovering from disturbance (Horvitz and Schemske 1986; Pascarella and Horvitz 1998). Horvitz introduced the term “megamatrix” to describe these models. A megamatrix, in the sense of Horvitz, is a special case of (5.70) when the population is classified by stages within environmental states, the demographic matrices are applied first, and the environmental transition matrices D_{i} are identical for all stages, as is the case in (5.62).
5.5.2 The Fundamental Matrix
Notation alert
Superscript notation for timevarying models. The tilde indicates quantities calculated from the complete transient matrix \(\widetilde {{\mathbf {U}}}\) in (5.70). Occupancy and times to absorbtion depend on the initial and final demographic and environmental states. The superscripts (‡, §, ♡) indicate choices of summing and averaging over the environmental states. The superscripts are shown here for the fundamental matrix \(\widetilde {{\mathbf {N}}}\)
Symbol  Definition  Description  Equation 

\(E \left ( \nu _{ij,\epsilon }\epsilon _0 \right )\)  Expected visits to state i in environment 𝜖, starting from state j in environment 𝜖_{0}  (5.72)  
\(E \left ( \nu _{ij}\epsilon _0 \right ) \)  Expected visits to state i, summed over environments, starting from state j in environment 𝜖_{0}  (5.73)  
Rearrangement of the rows and columns of \(\widetilde {{\mathbf {N}}}^\ddag \)  (5.74)  
\(E \left ( \nu _{ij,\epsilon } \right )\)  Expected visits to state i and environmental state 𝜖, averaged over initial environmental states  (5.75)  
Rearrangement of the rows and columns of \(\widetilde {{\mathbf {N}}}^\S \)  (5.76)  
\(E \left ( \nu _{ij} \right ) \)  Expected visits to state i summed over environments, starting from state j and averaged over initial environmental states  (5.77) 
5.5.3 Longevity in a Variable Environment
5.5.3.1 Variance in Longevity
The choice of the mixing distribution π is important. HernandezSuarez et al. (2012) present an alternative where π is the stationary distribution of births across environments, rather than the distribution of environments itself.
5.5.4 A TimeVarying Example: Lomatium bradshawii
Lomatium bradshawii is an endangered herbaceous perennial plant, found 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. Fallseason fires increase plant size and seedling recruitment, but the effect fades within a few years. Populations in burned areas have higher growth rates and lower probabilities of extinction than unburned populations (Caswell and Kaye 2001).
A stochastic demographic model for L. bradshawii was developed by Caswell and Kaye (2001), Kaye et al. (2001), and Kaye and Pyke (2003) based on data from an experimental study using controlled burning. 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 fire history: the year of a fire and 1, 2, and 3+ years postfire. Projection matrices were estimated in each environment; the example here is based on one of the two sites (Rose Prairie) in the original study. The matrices are given in Caswell and Kaye (2001).
These patterns in the mean and variance of longevity (Fig. 5.7) depend on the stochastic properties of the environment—in this case, the frequency f and autocorrelation ρ of fires. Even with an environmental model this simple, the effects of f and ρ can be complicated. I know of no previous attempts to examine their effects on longevity. To do so, I calculated life expectancy with f = 0.5 for autocorrelation − 1 < ρ < 1, and with ρ = 0 for fire frequency 0 < f < 1.
The autocorrelation of fires has little effect on the life expectancy of seedlings, but a larger effect on that of large plants. For the latter, life expectancy is maximized as ρ →−1 (alternating fire and nonfire years) or as ρ → 1 (long periods of fires alternating with long periods without fire). The standard deviation of longevity also shows a strong Ushaped response to ρ for all stages. The generality of this pattern is unknown.
5.6 The Importance of Individual Stochasticity
The concept of individual stochasticity strikes to the heart of one of the most fundamental problems in population biology: the sources of variability among individuals. Heterogeneity—genuine differences among individuals—translates into differences in the age or stagespecific vital rates to which they are subject. Heterogeneity may arise from genetics, from physiological effects, from health conditions, or from unknown causes (“frailty,” “quality”). Stochasticity results from the random outcomes of probabilistic processes. Markov chains naturally treat individual trajectories (i.e., individual lives) as realizations of an underlying stochastic process, and so much of this chapter has been focused on the analysis of individual stochasticity. The distinction is particularly important in evolutionary demography, where variance in lifetime reproductive output is routinely treated as variance in fitness, or a component of fitness. See Sect. 5.4.4 for some recent work on this problem.
Individual stochasticity is an important component of demography, for both human and nonhuman populations. It complements environmental stochasticity (externally imposed random changes in vital rates) and demographic stochasticity (randomness in the growth of populations due to stochastic survival and reproduction) (Caswell and Vindenes 2018). Individual stochasticity reflects randomness in the pathways that individuals take through the life cycle. It expresses itself in interindividual variation in occupancy times, longevity, lifetime reproductive output, and other outcomes. The availability of methods based on Markov chains promises to change the way population biologists approach the analysis of variance among individuals (Caswell 2011; Tuljapurkar et al. 2009; Steiner and Tuljapurkar 2012; van Daalen and Caswell 2015; van Daalen and Caswell 2017).
5.7 Discussion
Taking advantage of the Markov chain formulation of the life cycle opens up a wealth of demographic information. The ageclassified information extracted from a stageclassified model can form a valuable component of behavioral studies, especially if the model (like the right whale example) includes reproductive behavior as part of the life cycle structure. Longevity provides a powerful way to compare mortality schedules among species, populations, or environmental conditions, but it has been inaccessible to stageclassified analysis prior to the development of Markov chain methods. The generation time characterizes an important population time scale, with implications in conservation (IUCN Species Survival Commission 2001), but there has been no way to compute it from stageclassified models.
Stageclassified life cycles may have consequences that are not yet appreciated, but must be considered when interpreting the results. For example, any stageclassified model eventually leads to an ageindependent mortality rate (Horvitz and Tuljapurkar 2008), and so is of limited use in the study of senescence. This fact has consequences for life expectancy and variance in longevity that are not well understood (at least by me). For the right whale, expected longevity at birth is 32 years with a standard deviation of 34 years. It is unlikely that there are appreciable numbers of whales alive at even one standard deviation above this mean. The high survival probability and the assumption of ageindependence lead to the high standard deviation. Those of us who work with stageclassified models are accustomed to this, but discount its importance because it (often) has little effect on λ. It will be important to determine the stochastic consequences of simplifying assumptions in the life cycle graph.
This chapter does not begin to exhaust the information that can be extracted from the Markov chain formulation of a stageclassified model. Three examples of particular interest are the occupancy of sets of states, the problem of competing risks, and the calculation of passage times. It is often of interest to calculate the statistics of occupancy of sets of states (e.g., all reproductive classes, or all stages in some particular health condition). We have seen how to calculate the moments of the occupancy time of single states. The mean occupancy time of a set of states is the sum of the mean occupancy times of each state, but that is not true for the variance or higher moments. Roth and Caswell (2018) derived a general expression for all the statistics, and the complete distribution, of occupancy time for any set of states. If more than one absorbing state exists (e.g., death at different stages, or from different causes), then the risks of absorbtion compete, because an individual can only be absorbed (i.e., die) once. It is possible to calculate the probability of absorbtion in each state, and to explore the effects of changing one risk on the probability of experiencing another (Caswell and Ouellette 2018). Passage times refer to the time required to get from one stage to another in the life cycle. An important passage time is the birth interval: the time from one birth to the next. This can only be calculated for individuals that do reproduce a second time (otherwise the interval is infinite), and so it requires developing a chain that is conditional on successfully reaching the reproductive state (Caswell 2001). In species that produce only one or a few offspring, reproduction cannot be adjusted in response to the environment by changing offspring number, and so changes in the birth interval are particularly important in such species.
Footnotes
 1.
 2.
Note that n_{43} = n_{44} = n_{45} = n_{55} and n_{53} = n_{54}. This seems to be due to the fact, specific to these data, that the survival probability of stages 3 and 5 is indistinguishable from 1.0, and influences the results below.
 3.It might be easier to apply the CushingZhou theorem directly to \(\tilde {{\mathbf {A}}}\) and write$$\displaystyle \begin{aligned} R_0 = \rho \left( \tilde{{\mathbf{F}}} \left( {\mathbf{I}}  \tilde{{\mathbf{U}}} \right)^{1} \right)\end{aligned} $$(5.48)
but Bacaër does not do this.
 4.
 5.
Note that (5.68) computes the expected population at t + 1 from the expected population at t. It might be tempting to do this with the projection matrix \(\mathbb {A}\) and use the eigenvalues of \(\tilde {{\mathbf {A}}}\) to calculate the stochastic population growth rate. However, this would give the growth rate of the mean population, but not the stochastic growth rate (which is always less than or equal to the growth rate of the mean population). For calculations such as moments of longevity, which are explicitly properties of the expected population, the difference does not arise.
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