## Introduction

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.