Population stage structure is fundamental to ecology, and models of this structure have proven useful in many different systems. Many ecological variables other than stage, such as habitat type, site occupancy and metapopulation status are also modelled using transitions among discrete states. Transitions among life stages can be characterised by the distribution of time spent in each stage, including the mean and variance of each stage duration and within-individual correlations among multiple stage durations. Three modelling traditions represent stage durations differently. Matrix models can be derived as a long-run approximation from any distribution of stage durations, but they are often interpreted directly as a Markov model for stage transitions. Statistical stage-duration distribution models accommodate the variation typical of cohort development data, but such realism has rarely been incorporated in population theory or statistical population models. Delay-differential equation models include lags but no variation, except in limited cases. We synthesise these models in one framework and illustrate how individual variation and correlations in development can impact population growth. Furthermore, different development models can yield the same long-term matrix transition rates but different sensitivities and elasticities. Finally, we discuss future directions for estimating realistic stage duration models from data.
Population stage structure is fundamental to ecology (Caswell 2001). For animals such as arthropods (Manly 1990), the stage is often the most important and easily observed state variable. However, stage-structured models are also used for plants (Crone et al. 2011), marine mammals (Fujiwara & Caswell 2001), terrestrial mammals (Ozgul et al. 2009), fish (Lo et al. 1995), birds (Blackwell et al. 2007), amphibians (Biek et al. 2002) and other taxa. While some organisms actually progress through discrete life stages such as insect instars or plant seeds, rosettes and flowering stages, for other organisms the stage categories can be pragmatic surrogates for size. Ecologists also apply state transition concepts to more general types of classifications, such as habitat types (Horvitz & Schemske 1995), animal location (Hunter & Caswell 2005), organism state (Brownie et al. 1993), patch occupancy and metapopulation or metacommunity status. Our focus here is on stage-structured models, but these more general state transition models involve similar issues.
A central issue for stage-structured models is that development through a life stage takes time, and the amount of time often varies between individuals. Three general classes of stage-structured models make different assumptions about stage transitions. Matrix models treat stage changes as a Markov process, in which the probability of surviving and moving from one stage to another (or not changing) depends simply on the current stage. On the other hand, in studies of stage development of cohorts or individuals, it is often obvious that the probability of transition depends on how long an individual has been developing in a stage. Such data, which are common in studies of terrestrial and aquatic arthropods, are statistically modelled using distributions of stage durations (Read & Ashford 1968; Hoeting et al. 2003). A third modelling approach is to include fixed developmental delays but ignore their variation, yielding delay-differential equations (Nisbet 1997). These models have been used for theory and have been fit to stage-structured time-series data with overlapping cohorts, such as zooplankton time series (Ohman 2012). However, distinguishing stage durations from mortality and reproduction is so problematic that researchers often must assume the stage durations are known a priori (Wood 1994). Indeed, ecologists lack flexible and widely accepted statistical methods for stage-structured time-series data without marked individuals (Twombly et al. 2007).
These three distinct domains of stage-structured modelling – matrix models, stage-duration distribution models and delay-differential equation models – represent quite different assumptions about development through stages (Fig 1), and each has limitations. For example, matrix transition probabilities can be derived under long-term density-independent population growth by assuming stage durations follow any distribution and hence are non-Markovian on the scale of individuals (Vandermeer 1975; Caswell 1983). Yet, many matrix modelling studies omit this step and instead interpret individual stage transitions as biologically realistic Markov processes. Under the latter interpretation, the distributions of stage durations are geometric (top right panels of Figs 3 and 4), implying that the most common stage duration is one time step, which is often unrealistic. This matters because the distributions of stage durations, due to individual variation in development, are known to impact population growth rate, sensitivities and elasticities (Caswell 1983; Bellows 1986; Birt et al. 2009; De Valpine 2009). Individual heterogeneity has recently been an important focus for research (Bolnick et al. 2011; González-Suárez et al. 2011; Kendall et al. 2011), but little attention has been paid to heterogeneity in stage durations. Similarly, correlations of stage durations across multiple stages and/or with other vital rates have been recognised as potentially important (Kempton 1979; Saether & Bakke 2000; De Valpine 2009) but have rarely been incorporated into theory.
Since matrix models are most common, a central question is when, and for what purposes, is it realistic to approximate real development processes with a simple transition probability? The decision to turn to matrix models for a particular application often seems to rest on two unstated rationales. The first is an assumption that simple transition probabilities are adequate for making theoretical predictions, perhaps motivated by the idea that time lags (development delays) are mostly of interest in systems with non-linear dynamics and so may be safely ignored. The second is a pragmatic statistical outlook: fractions of a population transitioning between each stage are sometimes thought to be about the most that can be extracted from many data sets. Both of these assumptions deserve to be critically examined.
In this paper, we review theoretical and statistical issues for incorporating more general – and hence often more realistic – development into stage-structured population models. First, we briefly review the biological considerations of mean stage durations, individual variation in stage durations and within-individual correlations among stage durations. Second, we review how each modelling tradition handles these three aspects of stage durations. Third, we review why the approach taken for modelling stage development matters for model interpretations and predictions. To do so, we place matrix models, stage-duration distribution models and delay-differential equation models into a common framework so they can be directly compared. Using this framework, we illustrate that individual variation in development can have a major impact even on density-independent population growth rate; that different models can yield the same long-term matrix rates but have different sensitivities and elasticities to the underlying development parameters; and that correlated stage durations can also impact population growth rate. Finally, we discuss limitations and future directions for statistical methods to estimate more realistic models of stage development, by which we mean models that allow for the possibility of development delays, individual variation in development and possibly correlations among stages and/or other demographic parameters.
We use the terms ‘development time’ and ‘stage duration’ interchangeably. In the case of a single pre-adult stage, these are equivalent to ‘maturation time.’ In models with a terminal adult stage, which is often the only non-age-structured stage (Blackwell et al. 2007), adult ‘stage duration’ is equivalent to ‘time until death,’ and a non-Markov model could correspond to senescence.
Biology of Stage-Structured Demography
In this section, we briefly summarise biological considerations for theoretical or statistical modelling of mean stage durations, between-individual variance in stage durations and within-individual correlations among multiple stage durations. At the simplest level, the passage of an individual through a stage often requires sufficient time for the completion of some physiological processes such as growth, development, metamorphosis or dormancy. Some transitions are affected by stochastic events such as environmental conditions or resource flushes that may, for example, trigger seed germination or flowering from a plant rosette. Stage-structured models also need to accommodate organisms that are able to skip stages or regress to previous stages.
Intraspecific variation in stage durations is typical, regardless of whether development occurs on time scales of days or years. Life table studies of terrestrial and aquatic arthropods grown under the same conditions virtually always display variation in stage durations (Manly 1990), as do field studies of plants (Ehrlén 2000; Acker et al. 2014) and other taxa. Phenotypic plasticity in response to factors such as temperature, resource quantity and quality, density dependence, perceived mortality risk and many others can impact growth (Nylin & Gotthard 1998). Phenotypic plasticity can also lead to cohort effects, i.e. different development rates experienced by different cohorts (Beckerman et al. 2003). For modelling purposes, phenotypic variation can impact population dynamics whether the variation is genetic or environmental in origin.
For both genetic and environmental effects on stage durations, a useful distinction can be made between effects that impact a single life stage or might impact multiple life stages (Tuljapurkar et al. 2009; Zuidema et al. 2009). The latter may include genotypes for fast or slow development, maternal effects, microsite differences, birth order and size-dependent growth, among others. Such factors may correspond to the ‘fixed heterogeneity’ of Tuljapurkar et al. (2009) and to the concept of autocorrelated growth (Pfister & Stevens 2003; Fujiwara et al. 2005), which may arise from size-dependent resource acquisition (Peacor et al. 2007). On the other hand, environmental factors that change through time might impact development time of one life stage but not another. These include transient food resources, predators that must be avoided, or weather, for example. For modelling purposes, factors that impact multiple life stages can generate stage durations that are positively (or negatively) correlated within an individual, meaning that individuals who develop quickly through one stage tend to develop quickly through other stages. Of course stage durations could also be correlated with other traits such as body size, reproduction and mortality.
In summary, development through stages often takes time, varies between individuals, and may be correlated within individuals. One result is that in many cases, transition between stages will not be a Markov process. In a Markov process, the probabilities of transitioning from one stage to the next at time t depend only on the stage at time t − 1, rather than on time spent within that and/or previous stages.
Three Traditions of Stage-Structured Modelling
Next, we discuss in more detail the three stage-structured modelling traditions. Again our focus lies on treatment of stage durations, including means, variances and correlations. The three approaches represent a spectrum of how stage durations are modelled (Fig 1; top rows of Figs 3 and 4). Stage-duration distribution models and delay-differential equation models are formulated in continuous time, while matrix models are in discrete time, but this distinction is not important for our discussion. For simplicity, we will discuss these without density dependence.
Delay-differential equation models
In delay-differential equation models, the number of individuals entering one stage is the number that have survived in the previous stage for the duration of that stage. In most work, individuals born together are assumed to undergo identical development (Nisbet 1997), so stage durations introduce mean development delays but omit variances. In many cases, this implies that development times are constant, but some models incorporate phenotypic plasticity by dependence on external variables or density (McCauley et al. 1996). These models have tended to be used for theoretical work, or for empirical work where individual parameters are measured and plugged in to the model (McCauley et al. 1996). They have also been fit to time series of stage-structured data (Ellner et al. 1997; Severini et al. 2003; Wood 2010).
The obvious limitation of delay-differential equation models is that they omit variation in development. To overcome this, researchers have replaced one fixed-delay stage with a series of intermediate virtual stages, each with matrix-like transition rates rather than fixed delays, to compose a biological stage (Lewis 1977; Plant & Wilson 1986). This is sometimes called a ‘time-distributed delay’ model. Theoretical work has shown that individual heterogeneity in development can dramatically impact non-linear dynamics (Blythe et al. 1984; Wearing et al. 2004). However, the distributions of stage durations created by virtual stages are limited to independent gamma distributions. Hence, this approach is mathematically appealing but cannot handle correlated stage durations or non-gamma distributions.
Stage-duration distribution models
Stage-duration distribution models arose to accommodate the obvious heterogeneity in development seen from monitoring cohorts through time (sometimes called stage-frequency data, Manly 1990). Such data comprise the abundance of each stage at a series of sampling occasions, starting from a cohort of new individuals such as grasshopper eggs hatching in the spring or replicated cohorts in a laboratory. With such data in hand, a model with fixed delays cannot even be considered because multiple stages are present at many sampling occasions. Such heterogeneity is also evident in studies that track individuals. In practice, many researchers estimate separate distributions of cumulative time spent to each stage transition for statistical comparisons, rather than trying to disentangle multiple stage durations to obtain a population model (Murtaugh et al. 2012; Ohman 2012). In a landmark paper, Read & Ashford (1968) proposed estimating a model with a distribution of time spent in each stage. Subsequent researchers built upon this idea with different distributions, estimation methods and handling of mortality (Bellows & Birley 1981; Manly 1990; Hoeting et al. 2003). These methods lack full generality (see section on statistical methods below), but they point towards an essential ingredient for realistic population dynamics: flexible assumptions about distributions of stage durations.
A major challenge in this field is that the models quickly lead away from analytical methods towards computational ones. The distributions for stage durations draw on the field of statistical ‘survival’ analysis, including gamma, Weibull, log-normal, logistic and others. However, in general, the distribution of the sum of two durations (e.g. the net time spent in any two consecutive stages) does not follow the same distribution family as those of the individual stage durations. There are a few exceptions such as independent gamma distributions with the same shape parameter, which have been exploited for some estimation methods described below, but these are limited in generality.
A second modelling challenge is that multivariate versions of these distributions, which would be necessary to accommodate correlated stage durations, are not easy to formulate or estimate (Hougaard 2000). Two general approaches are copula models, which are discussed more below, and mixture models. Regardless of how stage durations are modelled, they imply that stage alone is not an adequate state variable for individuals, but rather that age within stage, individual quality, and/or other variables are important. It is notable that the most realistic of the three modelling traditions arose from dealing with data, but the realism of its models have not been incorporated much into the more theoretical modelling traditions.
Stage-structured matrix models can be interpreted in two ways. In the first interpretation, the matrix serves as an approximation of a more realistic model with stage-duration distributions (Caswell 1983; Crouse et al. 1987; Lo et al. 1995; Caswell 2001 pp. 159–166; Birt et al. 2009; De Valpine 2009). For any distributions of stage durations, if the population is growing at its long-term growth rate, there is not only a stable distribution of stages but also a stable distribution of ages within each stage. As a result, a constant fraction of each stage matures at each time step, and these fractions are the transition probabilities of the matrix model (compare top right and top middle panels of Fig 1, which yield the same total fraction of a stage maturing). Thus, the matrix model approximation is exact when the population is at its stable stage distribution. However, the fraction maturing does not correspond to a constant transition probability for each individual in the stage. Rather, it corresponds to the fraction of the individuals sufficiently developed within the stage that they are ready to mature. Hence, the transition probability, sometimes decomposed as a survival probability times a ‘growth rate’ (e.g. Morris & Doak 2002), is a derived parameter rather than a fundamental vital rate. In other words, any population that is at its stable stage distribution can be accurately represented by an equivalent matrix model, derived from its actual stage-duration distributions. The first interpretation of a matrix model, then, is to assume that transition probabilities are approximately accurate if the population deviates only mildly from its stable stage distribution (Vandermeer 1975).
The other way to interpret matrix models is as a directly realistic description of transition probabilities based upon the Markov assumption that transition probabilities depend only on the current stage. In this view, all individuals in a stage have the same transition probabilities in a time step. In other words, the transition probabilities are interpreted as valid for each individual at each time step, rather than as the fraction of a stage class that has developed sufficiently to mature. This implies that, if the transition probability is constant, the distribution of time spent in each stage – for individuals who survive the stage – is geometric. A geometric distribution puts the highest probability on spending one time step in a stage, with decreasing probabilities of more time steps (top right panels of Figs 3 and 4). In many (but not all) cases, this is unlikely to be realistic. Because matrix models are the most widely used tool for stage-structured modelling, we discuss their uses and interpretations more extensively.
The direct interpretation of transition probabilities as valid at the individual level is used in many applications of stage-structured matrix models. For example, a common view of the fundamental demographic parameters that enter matrix cells is that they include reproductive rates, survival probabilities and ‘growth’ rates, with the latter two combining to form transition probabilities. One implication is that the transition probabilities are assumed to be accurate even when the population is far from its stable stage distribution, either to model transient dynamics (Fox & Gurevitch 2000; Haridas & Tuljapurkar 2007; Ezard et al. 2010; Stott et al. 2011) or evolution (Koons et al. 2008; Barfield et al. 2011). Another implication is that sensitivities and elasticities are calculated for the ‘growth’ rates (De Kroon et al. 2000) rather than for underlying development parameters from which they are derived. Similar issues arise for models of habitat transitions (Pascarella & Horvitz 1998; Horvitz et al. 2005) and animal movement (Hunter & Caswell 2005; González-Suárez & Gerber 2008). For example, Markov movement probabilities between sites will be realistic at the individual level only if time spent at a site follows a geometric distribution.
Another way in which stage transition probabilities are interpreted directly is in estimation of how environmental stochasticity enters stage-structured dynamics. When data are available for multiple years, a common procedure is to estimate a matrix model for each year and assume these represent interannual variation in demographic rates due to environmental stochasticity (Fieberg & Ellner 2001; Morris & Doak 2002). However, if the matrix is really only valid at the stable stage distribution (first interpretation), this procedure has the potential to confound deviations from the stable stage distribution with environmental stochasticity (Vandermeer 1978). For example, if an above-average cohort follows a below-average cohort and some members of both cohorts are in the same stage, then that stage will deviate from the stable age within-stage distribution. What appears to be a year of high or low stage transition probabilities could simply be a year with more or fewer individuals within the stage that are ready to mature. This is not an easy problem to address, although some researchers have used additional past stage information as an explanatory variable to determine non-Markovian dynamics or have added stages to the model (Horvitz & Schemske 1995; Ehrlén 2000; Gremer et al. 2012). Models with autocorrelated vital rates (Fieberg & Ellner 2001; Tuljapurkar & Haridas 2006) could be interpreted as an indirect way to address this issue.
A final way in which transition rates are interpreted as valid at the individual level is in the calculation of ages from stages (Cochran & Ellner 1992; Boucher 1997). If the matrix model is instead an approximation derived from realistic stage-duration distributions, it would be impossible to determine age from stage based on the matrix alone (see example 2 below).
We do not mean to imply that these examples of direct interpretations of stage-structured matrix models are necessarily invalid but rather that they must be interpreted with caution. Matrix models have allowed tremendous insights into demography and evolution. Moreover, some (but not all) of these examples could be reframed using the first interpretation of matrix models as stable stage distribution approximations. This would then lead to important questions such as how elasticity to a transition probability can be decomposed into elasticities of the parameters of the underlying distributions of stage durations, which we illustrate below. In other cases, the state transition process may in fact be approximately Markovian, but it is common to assume so rather than check whether this assumption holds.
The question around which all of the above issues revolve is whether stage is an adequate individual state variable. Pfister & Stevens (2003) found with simulations that stage-structured matrix models may not be accurate when more realistic individual dynamics are really occurring. Simple matrices can be inaccurate for transient dynamics when compared to data (Bierzychudek 1999; Tenhumberg et al. 2009). Some studies have estimated individual heterogeneity in matrix probabilities from capture–recapture data using latent states for individual quality (Knape et al. 2011). Adding age to size-structured models is supported in studies of grasses and trees (Zuidema et al. 2009; Chu & Adler 2014). When multiple state variables can be compared, it is recommended to test for significant relationships with vital rates before using them in a matrix model (Caswell 2001; Morris & Doak 2002). Nonetheless, obtaining a significant relationship does not guarantee that all the heterogeneity within stages has been accounted for.
There have been several extensions of matrix models to incorporate both stage and age-within stage to allow flexible assumptions about variation in development times (Schaalje & van der Vaart 1989). Approaches include expanding the matrix with categories representing both age and age within stage (Plant & Wilson 1986; Zuidema et al. 2009), deriving an age-structured matrix from stage-structured development (Woodward 1982), and setting up different age-structured matrix pathways representing different stage durations (Acker et al. 2014). Matrix models for stage and total age (Caswell 2012; Chu & Adler 2014) could accommodate limited types of heterogeneity in development summarised below.
Why Realistic Stage Durations Matter
In this section, we illustrate the impacts of delays, individual heterogeneity and stage correlations on stage-structured ecology and evolution. To do so, we contrast the three modelling approaches described above by putting them in a common framework, using the gamma distribution for stage durations (De Valpine 2009). Different values of the coefficient of variation in the gamma distribution represent the three different approaches. We separate the roles of development and survival by formulating stage-duration distributions for individuals that survive and then imposing constant stage-specific survival rates. The gamma distribution is for continuous time durations, but we round times up to the next integer to obtain a distribution for discrete stage durations.
We illustrate three main points. First, different distributions with the same average stage durations yield different population growth rate and long-run matrix transition fractions. Second, different stage-duration distributions can yield the same equivalent matrix but different sensitivities and elasticities, especially with respect to development rate. Third, correlations among stage durations affect population growth rates and long-run transition rates of the equivalent matrices.
Common framework of the gamma distribution
This section provides mathematical details that may be skipped by readers interested only in the results. The first step is to place matrix models into our common framework by showing that the distribution of stage durations in a matrix model is a special case of the rounded gamma distribution. In the direct interpretation of a matrix model, in the absence of mortality, the probability of spending k time steps in a stage (k > 0), given growth probability of p at each step, is p(1 − p)k−1. This has mean 1/p and variance (1 − p)/p2. This geometric distribution for integer stage durations is equivalent to a (continuous) exponential distribution rounded up to the next integer. An exponential distribution for non-integer k has probability density function λe−λk, which has mean 1/λ and variance 1/λ2. If one rounds up k to the next integer, the probability of integer k is the area under the distribution from k − 1 to k, which is If λ= −log(1 − p), this distribution matches the geometric distribution of stage durations created by a matrix model in the absence of mortality.
Next, we can generalise using the gamma distribution, which includes the exponential as a special case. The gamma probability density function for k is
where a is the shape and ρ is the rate. Biologically, ρ can be interpreted as a rate for processes such as resource acquisition or metabolism that drive development. This interpretation comes from noting that ρ scales the k axis, so if ρ doubles, the corresponding development times k would be halved. Another useful interpretation is that the coefficient of variation (CV, defined as standard deviation/mean) for a gamma is . When the shape (a) is 1, the gamma is an exponential (with ρ = λ). Therefore, the distribution of stage durations in matrix models corresponds to a gamma with CV = 1 and can be compared to models with more (CV > 1) or less (CV < 1) variation in stage duration. (Note that the CVs of the rounded gamma distributions are different from the CVs of the continuous gamma distributions).
Since CV = 1 represents substantial variation, we focus on comparisons to CV < 1. In the extreme case of CV near 0, we get a model that is approximately equivalent to a delay-differential equation model with no variation. We are considering delay-differential equations with exact integer delays so they correspond to the time steps of matrix models. Intermediate values of 0 < CV < 1 correspond to the types of stage-duration distributions typically estimated from stage-frequency cohort data.
Next, we consider how to relate a model with any stage-duration distributions to an equivalent matrix model at stable stage distribution. We use a simple three-stage model for exploration (the ‘standard size-classified model’ of Caswell 2001, p. 59, with only adult reproduction):
Here, pi is the maturation fraction (or ‘growth rate’) for stage i (1 or 2), si is the survival rate for stage i (1 or 2), sA is the adult survival rate and F is adult fecundity. Given assumptions of a stage-duration distribution for stage i, one can compute the fraction of the stage maturing in each time step when the population is at its stable stage distribution. This is the value of pi that makes the matrix A equivalent to the stage-duration distribution model at stable stage distribution. Conversely, one can numerically determine combinations of the shape and rate parameters that, at stable stage distribution, yield a chosen matrix equivalent. De Valpine (2009) provided tools for accomplishing these steps, as well as calculating sensitivities and elasticities. The approach uses the framework of integral projection models (Ellner & Rees 2006) with an internal development state instead of size as a state variable for each stage.
To model correlations among stage durations, we use Gaussian copulas (Fig 2). Copula models provide a way to formulate multivariate versions of univariate distributions. For non-normal distributions such as the gamma, there is typically no natural multivariate extension of the univariate version. To create a multivariate gamma using a Gaussian copula, one matches quantiles in each dimension to those of a multivariate normal. For example, in Fig 2, 50 simulated values from a multivariate normal with correlation 0.7 are transformed into 50 correlated gamma variables. For each simulation, the marginal cumulative density value is calculated for each dimension. Then the gamma values are determined by using the same cumulative density values and the corresponding quantiles from the gamma distributions. In this way, we can maintain the same marginal distributions of times spent in each stage but vary the strength of their correlation.
An alternative way to make correlated distributions from gammas would be to include a shared random effect in each gamma, as is common in ‘shared frailty’ models in survival analysis. However, in that approach the marginal distributions are no longer gammas, and there is no way to vary the correlation without also changing the marginal distributions, so one could not isolate the role of correlation.
In the following examples, we use the tools of De Valpine (2009) with rounded-up gamma distributions representing a spectrum of three cases of individual heterogeneity in development: a nearly delay-differential equation model (low variation, CV = 0.1), a substantial but realistic amount of variation (CV = 0.5) and a literal interpretation of a matrix model (CV = 1, yielding geometric stage-duration distributions). For other parameters, we use p1 = p2 = 1/3, so that the mean time spent in each stage for the geometric case (literal matrix interpretation) is 3. In some cases, we use low immature survival, s1 = s2 = 0.6, and in others high immature survival, s1 = s2 = 0.9. We use sA= 0.5 and F = 8. Using this framework, we illustrate three essential points about the roles of development and heterogeneity on demography.
In the models without stage correlations, the state variables are stage and age within stage. In the models with stage correlations, the state variables are stage, age within stage and previous stage durations. In principle either of these could be modelled with a large matrix of transition probabilities, but especially in the latter case the number of state categories can become quite large, so we use the computational methods of De Valpine (2009) as an alternative.
Different distributions with the same average stage durations yield different population growth rate and matrix equivalents
First, we hold the average durations of the first two stages constant at 3 and vary the CV of the corresponding gamma distributions (Fig 3). By doing so we are varying the equivalent matrix. Population growth rate (r) shifts from −3% per year at CV = 0.1 to +11% per year at CV = 1. This has been demonstrated by Caswell (1983), Bellows (1986) and Birt et al. (2009), among others. This occurs because fast-developing individuals increase r more than slow-developing individuals decrease r. In addition to reaching a reproductive state more quickly, fast-developing individuals experience the survival rate of each stage for shorter durations. Furthermore, sensitivity and elasticity of some parameters (black bars) change as CV increases and differs from those determined from the equivalent matrix at stable stage distribution (grey bars).
Different stage-duration distributions can yield the same matrix equivalent but different sensitivities and elasticities
Next, we hold the matrix maturation rates constant and search for different stage-duration distributions that yield the same matrix when the population is at its stable stage distribution (Fig 4). Again we vary the CV for the first two stages, and for each CV value we solve for the mean stage durations (via the resource acquisition rate, ρ) that yield the same equivalent matrix. Mean durations of 2.2 for CV = 0.1, of 2.4 for CV = 0.5, or of 3.0 for CV = 1 all give the same matrix. Since the two stages have identical matrix rates, we get the same mean stage duration for each. However, these models do not all have the same implications for demography, management, or evolution. The elasticities for the resource acquisition rate, ρ, change two-fold over the range of CVs considered, and those for fecundity also change substantially. Moreover, when compared to the values that would be interpreted from a standard matrix analysis (grey bars), there are qualitative differences in which traits seem most important. When elasticities are used for management considerations, this implies that the relative importance of management actions impacting different life stages depends on the details of the stage-duration distributions. Similarly, when sensitivities are used for evolutionary considerations (Barfield et al. 2011), this means that the selection gradient also depends on the stage-duration distributions.
Correlations among stage durations affect population growth rates and matrix equivalents
Next, we vary the correlation between stages while holding the stage-duration distributions constant. When the first two-stage durations are correlated, the effect of CV on population growth rate is stronger than when they are independent (Fig 5, left panel). We can further consider the case that adult longevity is correlated with the first two-stage durations. We make this correlation negative, so that slow development is associated with short adult longevity. In this case, the impact of CV on population growth rate is even stronger (Fig 5, right panel).
Dealing with Data: Estimating Models with Variable Stage Durations
If the mean, variance and correlations of stage durations are important, how can we estimate models that take such realism into account? We will address this question for different types of data – on marked or unmarked individuals, with distinct cohorts or not – and highlight important open problems.
Statistical models for marked individuals
When data are collected on the stage category of marked individuals through time, there is potential for estimating more detailed models more accurately than when only groups can be counted without individual identities. Interestingly, plant and animal ecologists often employ different methods for individual data. This has occurred not just because animal mark-recapture models must accommodate imperfect detection but also because of different ecological considerations. Plant ecologists have often directly estimated stage transitions from the proportion of individuals that change stages between samples, effectively ignoring individual identity. However, use of individual history and generalised linear mixed models (Ehrlén 2000; Gremer et al. 2012; Chu & Adler 2014) are examples of more detailed models allowing individual random effects and/or non-Markovian dynamics. In cases with size data, researchers are moving away from artificial size classes towards continuously size-structured integral projection models (Ellner & Rees 2006).
Animal ecologists must often accommodate imperfect detection and hence embed models of demographic transitions in capture–recapture models, which are usually age structured. More general multistratum capture–recapture methods (Brownie et al. 1993) allow animal states, including stages, to be modelled. These methods are often applied to estimate movement between sites, in which case state corresponds to site, or physiological states, such as breeders and non-breeders. In most analyses, transitions between states are modelled with Markov chains. They therefore suffer the same limitations in their treatment of delays and varying durations as previously discussed for matrix models. However, exceptions to the Markov assumption have been considered by letting the probability of state transitions depend on multiple preceding states (Brownie et al. 1993; Pradel 2005).
A promising direction for individual data is to estimate distributions of the time spent in each stage by drawing on the extensive field of ‘survival’ analysis (Lawless 2003). This would represent taking the core idea of stage-duration distribution models for unmarked cohorts and applying it with more informative individual data. In the parlance of survival analysis, individual data are often interval censored, meaning that exact times of stage transitions are not observed. Instead, the stage of individuals at chosen times is recorded, and any transition is known to have occurred only within an interval between observations (Scranton et al. 2013). When individuals enter a study at unknown age, the data would be right censored. In such a model, the time an individual has been in a stage could represent an individual state in the multistratum capture–recapture framework. Correlations within individuals (or subpopulations) could be incorporated either as random effects or via copula models. The combination of censored data, multiple stages, correlations and possibly imperfect detection would require non-trivial adaptation of existing survival analysis methods. Such data could also potentially allow separation of environmental stochasticity and cohort effects. Finally, such models could be compared to models that simply supplement stage with age (Chu & Adler 2014). In summary, there is a large subfield of statistics for estimating distribution models of time durations that could be tapped more thoroughly for estimating models of stage or state durations in many areas of ecology.
Statistical models for data without marked individuals
When individuals are not identified at repeated samples, data may be collected as a stage-structured time series. This is done by following a population and recording a count or estimate of the number of individuals in each stage at each sample time. In some studies, individuals can be identified as representing a cohort, meaning a group of individuals entering the population at the same or similar times. When cohorts are not clearly identifiable, the data have non-distinct, overlapping generations, representing a much more challenging statistical problem. An intermediate case is when batches of individuals are marked at one or more times so that their fates may be distinguished as a cohort within a more complex population (Viallefont et al. 2012). For many studies, the purpose of estimating stage durations is to disentangle them from mortality, so it is important to consider both goals together.
Statistical models for unmarked cohort data
When unmarked data track a cohort, the stage-duration distribution models introduced above can be used. When there is variation in the time of entry into the cohort, Kiritani-Nakasuji-Manly (KNM) methods (Manly 1990) assume that there is no additional variation in stage durations. Despite some obvious limitations, this approach is pragmatic and has continued to gain attention (Manly 1997; Yamamura 1998; Aubry et al. 2010). Read & Ashford (1968) and Ashford et al. (1970) introduced models with a sequence of stage-duration distributions, but they assumed the shape parameter of a gamma distribution for each stage to be the same for estimation convenience. Subsequent extensions continued to require arbitrary assumptions for estimation (Kempton 1979; Bellows & Birley 1981; Braner & Hairston 1989; Klein Breteler et al. 1994; Hoeting et al. 2003). Some of these methods allowed stage-specific mortality while others did not. Yet another approach is to model an internal developmental state undergoing a stochastic process. When the state advances past a threshold, a stage maturation has occurred (Dennis et al. 1986).
A distinction that has rarely been discussed is between cohort data where samples are from the same group of individuals at each time or from independent groups. Data on the same group would come, for example, from laboratory cohorts. Data from independent groups come from two different situations: from random field samples where different individuals known to be from the same cohort are counted at each time (Manly 1990); or from different laboratory replicates that are destructively counted at each time (Jellison et al. 1995). When the same individuals in a group are counted at each time, the deviations from model predictions will be correlated across observations. For example, if a cohort happens to have some fast growers, they will appear as fast growers throughout the time series. Virtually none of the above methods are designed for dependent data (but see Gouno et al. 2011, Knape et al. 2014).
A promising direction for estimating models from unmarked cohort data would be to use modern computational approaches to relax the arbitrary assumptions and estimation methods that have been used. The primary difficulty is that the likelihood can be difficult to calculate. For independent samples, if stages durations are uncorrelated, it involves convolution of multiple distributions. If stage durations are correlated, the problem of computing the probability of being in a particular stage at a particular time depends on previous transition times (copula model) or on a shared random effect (mixture model). And if the data follow the same individuals, the likelihood is even more intractable. In that case, the model would need to be similar to a state-space model and explicitly integrate over the unobserved state information, even if there is no observation error. Or, viewed another way, the classification of individuals by stage as a surrogate for an unobservable internal development state represents a kind of ‘observation error’ in the sense of incomplete information.
In summary, although stage-duration distribution models provide the most realistic treatment of development, we still lack a general framework with computational methods for estimation of stage durations and stage-specific mortality (Hoeting et al. 2003; Murtaugh et al. 2012). It is important to note that many studies of unmarked cohort data do not even attempt to estimate each stage-duration distribution. Rather, they estimate separate distributions for time spent in the first stage, in the first two stages, in the first three stages, and so on. Mean or median stage durations for stage n may be estimated as the differences between times spent in the first n − 1 and n stages (Klein Breteler et al. 1994; Kimmerer & Gould 2010). This may be fine for practical questions such as testing of experimental treatment effects, but it skips the problem of separating stage-duration distributions from stage-specific mortalities to gain a complete demographic picture.
Statistical models for non-cohort data without marked individuals
Early methods for unmarked non-cohort data of marine copepods used simple graphical estimation of stage durations when distinct ‘pulses’ (cohorts) were clearly detectable, leading to debates about what really is estimable from such data (Hairston & Twombly 1985). More general approaches included fitting matrix models in a variety of ways: with the Kalman Filter or extended Kalman Filter to incorporate both sampling and process variation (Ennola et al. 1998); with Bayesian methods (Gross et al. 2002; David et al. 2010); with explanatory variables for matrix parameters (Benton et al. 2004; Twombly et al. 2007); with a model of internal development rate (Gilioli & Pasquali 2007); with uneven sampling intervals (De Valpine & Rosenheim 2008); and with both ages and stages (Moe et al. 2005). However, these methods often rely on regularly spaced observation intervals and on the realism of the matrix model, which may be questionable if variation in the data arises from variation in environmental conditions.
A major new approach came from the work of Wood (1994; see also Nelson et al. 2004). He sought to estimate temporal patterns of fecundity and mortality that explain a time series by smoothing the age-time surface with constraints to avoid negative mortality. The unfortunate limitation of this approach is that it requires fixed stage durations that are known a priori. Thus, rather than estimating stage durations, it addresses the related problem of estimating stage-specific mortality and fecundity from noisy field data. The premise, based on Wood & Nisbet (1991), was that stage durations and mortality are not simultaneously identifiable (estimable) from time-series data. This approach has been applied to field data (Ohman & Hirche 2001), but whether estimating mortality from zooplankton data is realistically possible continues to be a fundamental debate (Ohman 2012).
The lack of general methods for statistical modelling of non-cohort stage-structured time series represents a major gap. Modelling such data realistically is much harder than dealing with distinct cohorts. In the extreme case of a population growing at its stable stage distribution, clearly not all parameters are identifiable (example 2 above, showing that multiple models can give the same long-term matrix transition rates). In contrast, for populations that are far from a stable stage distribution, in principle the data contain information similar to that from distinct cohorts. In the spectrum between these two extremes, however, some cohorts may be apparent in a time series but they may not be cleanly separable. In summary, this represents an important open problem in statistical time-series modelling for ecology.
Many stage-structured models in ecology lack flexible treatment of stage durations. We have drawn upon several distinct traditions of theoretical and statistical modelling to bring together ideas on this problem. By considering the cases of fixed delays with no variation in stage duration (delay-differential model) and a geometric distribution of stage durations (matrix model) as two ends of a spectrum, we have highlighted the fact that assumptions about stage durations matter to the interpretations and predictions made with stage-structured models. These issues are often overlooked, especially in situations where transition probabilities may not be valid as a Markov process at the individual level. As noted in the introduction, similar issues could be considered for other state transition models in ecology. The role of demographic heterogeneity in density-dependent models also remains little explored (Stover et al. 2012).
Statistical modelling of cohort data led previous investigators to propose stage-duration distributions as realistic models for variation in individual development. We have argued that this type of formulation makes sense as a way to improve the realism of some theoretical and statistical population models. Where matrix models and delay-differential equation models represent a stage transition or duration by one parameter, stage-duration distribution models use two or more parameters. Including correlations in stage durations requires additional parameters. By correctly specifying the variation and correlations in stage durations, one can more realistically determine population growth rates, elasticities and sensitivities. These models would also differ in prediction of transient dynamics, which we have not explored.
Many types of data, especially with marked individuals, contain information on both the mean and variance of individual stage durations. However, statistical methods are often limited in using that information. For data from marked individuals, methods are needed to estimate multivariate stage-duration distributions and to disentangle observed stage transitions from environmental stochasticity and cohort effects. A large field of statistical methods exists for data on time durations, but such methods have rarely been applied to ecological state durations. Generalising current stage-duration distribution models to include stage skipping and regression could be very useful, especially for plant models. For data from unmarked individuals in distinct cohorts, methods are needed to provide greater flexibility in assumptions about stage-duration distributions and mortality as well as to account for the non-independence created by counting the same individuals repeatedly. For data from unmarked individuals in overlapping cohorts, greater insight is needed into what can realistically be estimated and how to do so. We hope to have called attention to these important problems in statistical ecology.
Despite their usefulness and popularity, stage-structured matrix models are limited by their assumption that either a population is at its stable stage distribution or that Markov stage transitions are valid at the individual level. Such models have provided many important insights, but they also have many recognised limitations, among which we have focused on non-Markovian stage transitions on an individual level. To address this issue in the future, we suggest that in some cases more can be done with the same data by estimating distributions of time spent in each stage, similar to the way integral projection models have allowed more thorough use of size-structured data than was previously possible.
We thank E. Crone, B. Kendall, S. Tuljapurkar, three anonymous reviewers and the Associate Editor for comments. This work was partially funded by NSF grant DEB-1021553.
P deV drafted the manuscript and conducted the modelling. KS, JK, KR and NJM contributed content for different sections and edited drafts.