Distance to stable stage distribution in plant populations and implications for near-term population projections


Correspondence author. E-mail: jwilliams@nceas.ucsb.edu


1. Matrix population models capture how variation in vital rates among life stages translates to population dynamics. Analyses of these models generally assume that populations have reached a stable stage distribution (SSD), where the proportion of individuals in each stage remains constant. However, when life stages respond differentially to environmental cues and perturbations, a population may be moved away from equilibrium. Given the multitude of stochastic processes acting in natural systems, populations may never be exactly at SSD. It is thus critical to understand how far away populations are from SSD and how distance from SSD influences near-term model projections.

2. We analysed published matrix models from 46 plant species spanning a range of life histories that reported both a current stage distribution and projection matrix. We examined the distance between observed and theoretical SSD and the associated consequences for near-term transient population dynamics for each species.

3. In the majority of studies, populations were near their expected SSD, with 80% falling within one unit of a projection distance (α0) of zero. This distribution was skewed towards positive values of α0, indicating that the majority of populations had individuals concentrated into stages with high reproductive values.

4. Half of the populations in our survey had projection distances such that transient projections of population size and growth rate were within 10% of asymptotic projections at 5 years. However, in populations where projection distance was > 2, deviations from SSD caused important (more than twofold) differences.

5. We also found that larger deviations from SSD were positively correlated with generation time and matrix size.

6.Synthesis. When some life stages within a plant population are more strongly affected by disturbances or stresses than others, the results of our literature survey suggest that equilibrium projections will tend to underestimate projections that account for the current stage distribution. Measuring the current stage distribution can help determine whether asymptotic measures of matrix model analyses are reliable, and is a crucial step to take when precise population metrics are necessary for guiding conservation and management.


Understanding the population dynamics of species is a central theme in ecology. Within species, life stages – defined by size, age or reproductive status – may respond differently to biotic or abiotic factors in ways that have important consequences for population dynamics. One way to capture how variation in vital rates among life stages affects population dynamics is to use demographic matrix models. Matrix models are widely used in basic and applied ecological and evolutionary research, in part because their general and versatile structure allows them to be applied broadly across taxa (Boyce 1992; Menges 2000; Mills & Lindberg 2002; Morris & Doak 2002; Franco & Silvertown 2004). The power of these models is that they can summarize complex demographic data into simple, straightforward descriptors for whole population dynamics, including metrics such as population growth rate (λ), net reproductive rate (R0) and measures of the relative contribution of each life stage to overall population growth (Caswell 2001; Morris & Doak 2002). However, common measures of population growth (λ1, r) require the assumption that the relative proportion of individuals in each life stage has stabilized even when the total population size is changing (Caswell 2001).

Theoretically, a population would reach an equilibrium or asymptotic state with a corresponding stable stage distribution (SSD) that is determined solely by the demographic rates in the matrix when demographic rates are constant. In reality, a population may never arrive exactly at its SSD, due to stochasticity and disturbance. Recent theoretical advances can overcome the dependence on a stable distribution by allowing population descriptors to depend on time and initial starting conditions (Fox & Gurevitch 2000; Yearsley 2004; Caswell 2007; Haridas & Tuljapurkar 2007; Koons, Holmes & Grand 2007; Townley et al. 2007; Tenhumberg, Tyre & Rebarber 2009; Wiedenmann, Fujiwara & Mangel 2009). The results from these transient analyses can differ greatly from equivalent asymptotic analyses, depending on the initial stage distribution (e.g. McMahon & Metcalf 2008; Ezard et al. 2010; Maron, Horvitz & Williams 2010). For example, using models from six vertebrate species, Koons et al. (2005) found that after 5 years, transient population growth rates could differ from asymptotic growth rates by as much as 59% and transient sensitivities by as much as 200%. In practice, the transient response that a population experiences depends on the structure of the matrix, time scale of interest and the relationship between a population’s initial and asymptotic stage distributions.

A population is expected to behave more similarly to its asymptotic dynamics if it is ‘close’ to its theoretical asymptotic stage distribution (Bierzychudek 1999). Yet, forces that could move populations away from asymptotic stage distributions by affecting particular life stages are common, including stochastic events such as disturbance and weather, as well as disease, herbivory, harvest and temporal changes in environmental conditions. At the same time, it is unknown whether differences from SSD are infrequent occurrences or typical patterns, and thus it is unknown how often the assumption of populations being in their SSD is violated. We conducted an analysis of published demographic models of plant populations for which both the projection matrix and current stage distribution were reported to determine: (i) how similar populations are to their theoretical stable distributions, (ii) how the observed divergence from stable distribution affects near-term projections of population size and growth rate, and (iii) whether there are characteristics of populations based on life history or model attributes that relate to the distance between current and asymptotic stage distributions.

Materials and methods

Literature search

We used Biological Abstracts (ISI Web of Knowledge; Thomson Scientific, New York, NY, USA) to search the literature (1969–2005) for papers with population projection matrices and current stage distributions that we could extract directly or reconstruct with the available data. Specifically, we searched the 448 studies relating to population biology that were returned from the search: TS = (matrix) AND (projection OR demog*) NOT (neuro* OR medicine OR protein). From these studies, we were able to extract irreducible matrices for 46 plant species from studies that reported an associated observed stage distribution (see Appendix S1 in Supporting Information); only three studies of animals met our selection criteria.

One of our objectives was to describe, in general, how near or far species are to/from their SSD, and so in our analyses, one matrix represented each species in each study. If matrices were given for several populations in the same type of habitat or in qualitatively different habitats with the same likelihood of being near SSD, we chose one matrix at random. If the study reported matrices from multiple years, we also chose one matrix at random. If the study reported matrices from multiple habitats with different likelihoods of being near SSD, we used either the control matrix or the matrix representing the conditions that had persisted for the longest time to gain a conservative estimate of distance from SSD. We used the given stage distribution (n0) that corresponded with the starting year and location of the chosen projection matrix, rescaled to sum to 1 (see Appendix S1 for study details and the matrices and n0’s used in the analyses).

Measures of distance from stable stage distribution

A number of different metrics have been used to assess the difference between an observed stage distribution and the distribution that would be expected at equilibrium. We compared three of the more commonly used measures to determine which would be most informative for understanding near-term population projections: Keyfitz’s delta (Keyfitz 1968), Cohen’s cumulative distance (Caswell 2001), and projection distance (Haridas & Tuljapurkar 2007).

Keyfitz’s delta is a measure of the distance between any two probability vectors (Keyfitz 1968): Keyfitz’s

image(eqn 1)

where n0,i is the observed proportion of individuals in stage i and wi is the proportion expected at SSD. This measure reflects the proportion of individuals in the vector that describes the observed stage distribution (n0) that are in a stage class different from that expected from the vector describing the population at SSD (w). It therefore provides a ‘snapshot’ measurement of the distance that a population is from SSD at the time the population vector was measured. The range of the index is 0 (no difference) to 1.

Cohen’s cumulative distance (D1) measures the difference between observed and expected vectors along the matrix path that the population would take to reach the expected population vector. It is a function of both the observed stage distribution (n0) and the structure of the matrix (A). We used the analytical solution for Cohen’s D1 (Cohen 1979):

image(eqn 2)

where I is an identity matrix of the same size as the transition matrix A, w is the dominant right eigenvector of A rescaled so that its elements sum to 1, v is the dominant left eigenvector of A (v is rescaled so that v′w = 1; H. Caswell, pers. comm.), and λ1 is the dominant eigenvalue of A.

Projection distance describes the proportion of an observed vector that parallels the expected vector, taking into account both the differences between observed and expected proportions and the reproductive value of a stage (Haridas & Tuljapurkar 2007). It is measured as: α0 = α(0)−1 = vn0−1, using the same notation as in eqn 2. When the population is in its SSD, the projection distance will be 0 (α0 = v′n0−1 = v′w−1 = 0). Negative values imply that the population is more concentrated into stages with low reproductive values compared to SSD, and positive values imply that the population is concentrated into stages with high reproductive values.

Measures of transient dynamics

When a population is in its SSD, the dominant eigenvalue (λ1) describes the long-term population growth rate, but for populations perturbed away from their SSD, all of the matrix eigenvalues are important for describing the near-term population growth rate. Even when the environment is constant following a perturbation, at each time step the transition probabilities in the matrix will redistribute individuals to different stage classes and the population size can fluctuate. We summarized the near-term or transient dynamics for each population using estimates of the transient population growth rate (λ(t)) and population size (N(t)), because we were interested in the changes in population size that might be observed over the time period of a standard ecological study or management application (similar to Koons et al. 2005).

We estimated population size at time t under transient dynamics (N(t)) by iteratively multiplying observed stage distributions (starting with n0) by the matrix A for t time steps. We then compared this estimate with the population size estimate at time t based on an asymptotic growth rate assumption: inline image, where N(0) is the sum of n0, or 1 for all populations due to the rescaling described above. We calculated transient lambda iteratively as λ(t) = N(t)/N(t−1) following Koons et al. (2005). To facilitate comparisons among species, we measured the transient response for each population in two ways: first, as the proportional difference (Δλ(t)) between λ(t) and λ1 at = 1, 5, and time-averaged across = 1–5 as ((λ(t)–λ1)/λ1); second, as the proportional difference ΔN(t) between transient and asymptotic population size as inline image.

These measures are similar to measures of amplification and attenuation (Neubert & Caswell 1997; Townley et al. 2007; Townley & Hodgson 2008) in that they measure the relative differences between transient and asymptotic dynamics. However, amplification and attenuation refer to maximum differences across all possible initial stage distributions, whereas we were interested in the effect of the specific observed stage distributions. The behaviour at = 1 that we describe above is analogous to reactivity, which is defined as the maximum amplification across all initial vectors at = 1 (Neubert & Caswell 1997; Townley et al. 2007; Townley & Hodgson 2008), and the behaviour at = 5 accounts for longer-lasting transient dynamics. The time at which a population reaches its maximum response following a disturbance will depend on the species and the disturbance; however, in a review of matrix models for plant populations, Stott et al. (2010) found very strong correlations between measures of amplification over time. This result suggests that although we may lose some detail in simplifying our measures of the transient response, we are unlikely to be missing key dynamics.

Assessing predictors of deviations from SSD

One way to characterize the life history of organisms across a wide variety of taxa is to calculate the generation time of each species (Gaillard et al. 2005). Generation time measures the time required for a population at its SSD to grow by the net reproductive rate R0, and is represented in its simplest form by = ln(R0)/ln(λ) (Coale 1972; Caswell 2001). Calculating R0 requires separating survivorship and growth parameters from fecundity terms and can be complicated by the presence of multiple newborn stages (Cochran & Ellner 1992). We separated each matrix into its fecundity (F) and growth and survival (P) matrix components based on the life cycle descriptions in the text of each paper. We then used P and F to calculate the stage-specific effective net reproductive rate, R0(j), standardized in terms of newborn equivalents (Cochran & Ellner 1992):

image(eqn 3)

Here, γi is the fecundity of an individual in stage i in terms of newborn equivalents, found by weighting the original fecundity (F) by the reproductive value of each newborn stage compared to a reference newborn stage. To standardize across species, we used the average reproductive value of newborn stages, weighted by the distribution of individuals in newborn stages at SSD, as the reference newborn stage. Finally, we found the average generation time of the population by weighting R0(j) by the distribution of newborns at SSD:

image(eqn 4)

Of the 46 populations used in our analyses, we were able to calculate generation time for 39; the remaining seven did not contain enough information to separate the matrix into two parts.

We tested whether deviations from SSD were influenced by generation time, matrix size or λ1 using multiple linear regression. Matrix size and generation time were log transformed to meet model assumptions. All matrix simulations were carried out in matlab version 7.10 (MathWorks 2010) and all statistical analyses in R version 2.12.1 (R Development Core Team 2010).


Of the 46 studies reporting both current stage distributions and population projection matrices, a wide variety of plant life histories were represented including herbs, shrubs, and trees with different life spans, as well as cacti, lianas and a bryophyte and brown alga (see Appendix S1). The sample of species included both common (32) and rare (7) species, some of which are of conservation concern due to rarity or harvest. In our conservative estimate of deviation of observed stage distribution from the theoretical SSD, we found that the majority of the populations were near their SSD, as assessed by all three measures of distance (Fig. 1). Specifically, over half the populations had fewer than 20% of their individuals in classes other than expected at SSD (Keyfitz’s Δ, Fig. 1a) or within one unit of Cohen’s D1 (Fig. 1b), and 80% of populations were within one unit of a projection distance (α0) of 0 (Fig. 1c). The distribution of projection distances was skewed such that the majority of populations had positive projection distances, indicating that they had individuals concentrated into stages with high reproductive values.

Figure 1.

 Distributions of distances between current and asymptotic theoretical stage distribution for 46 published matrix models for unique plant species as measured by (a) Keyfitz’s Δ, (b) Cohen’s D1 and (c) projection distance (α0). Note that the breaks on the x-axes in panels b and c are not evenly spaced to show more detail at small distances.

Populations that had larger deviations from SSD also had increased transient responses as measured by the proportional differences between transient and asymptotic projections of population size (ΔN(t)) and transient and asymptotic calculations of population growth rate (Δλ(t)) (Table 2, Fig. 2). The relationships between measures of distance between current and stable stage distributions and the transient responses were continuous and best described by projection distance (Table 1). For populations with projection distances ± 0.5 units of 0, transient projections of population size and λ were within 29.5 ± 5.1% of asymptotic projections at = 1 (± 1 SE, see Table 2 for regression coefficients), a measure analogous to reactivity (Fig. 2d). The proportional difference in N(t) increased over time, such that with projections of 5 years, populations with projection distances within 0.5 units of 0 had transient projections of population size that were within 48.7 ± 8.8% of asymptotic projections (Fig. 2b), and the populations furthest from SSD (α0 > 5) overestimated population size by 4–16 times (Fig. 2a). In contrast to projections of N(t), the proportional difference between transient and asymptotic λ became smaller over time. In fact, for projection distances within 0.5 units of 0, transient λ at = 5 was, on average, within 0.26 ± 0.21% of asymptotic λ (Fig. 2d), and even for populations furthest from SSD (α0 > 8), transient λ was on average within 6.5 ± 7.3% of asymptotic λ (Fig. 2c). When the population growth rate was averaged across a transient period of 5 years (see triangle symbols in Fig. 2c,d), transient λ was very similar to asymptotic for α0 < ± 0.5 (within 4.1 ± 0.41%). In populations with an excess of individuals with low reproductive values (α< 0), transient population sizes and growth rates were smaller than projected by their asymptotic equivalents, while the opposite was true for populations with individuals of high reproductive value.

Table 2.   Coefficients and standard errors (SE) for linear models describing the relationship between projection distance (α0) and measures of the magnitude of the transient response, measured as the proportional difference in N or λ at = 1 and = 5 between transient and asymptotic projections. Models were run separately for each measure of transient response
Measure of transient responseInterceptα0Model fit statistics
Δλ(average across 5 years)0.002130.009360.07770.00415351.4<0.0001
Figure 2.

 Effect of distance between current and theoretical asymptotic stage distribution (measured by projection distance, α0) on near-term population dynamics. The proportional differences between transient and asymptotic projections for population size (ΔN(t)) (a, b) and population growth rate (Δλ(t)) (c, d) for simulations starting at the current or theoretical stage distribution. Panels b and d are insets of panels a and b to illustrate the relationships when projection distance is small. Open squares and dashed lines denote proportional differences derived from projections at = 1, closed squares and solid lines at = 5, and grey triangles in panels c and d denote the time-averaged transient population growth rate (λ) after 5 years. Horizontal dashed lines in panels c and d indicate bounds where transient projections fall within 10% of asymptotic projections.

Table 1.   Adjusted R-squares for linear models describing the relationship between a measure of magnitude of the transient responses and one of three metrics of distance between the current and predicted asymptotic stage distributions (projection distance (α0), Cohen’s D1 and Keyfitz’s Δ). The magnitude of transient responses was measured as the proportional difference in N or λ at = 1 and = 5 between transient and asymptotic projections. See Materials and methods for more details
Measure of transient responseα0log(D1)Keyfitz’s Δ
Δλ(average across 5 years)0.8860.3200.119

To investigate the characteristics of a population that corresponded to greater deviations from theoretical SSD, we used projection distance as the response variable since that was the best indicator of the magnitude and direction of the transient response (Table 1). While projection distance increased with increasing generation time and matrix size, the trend was statistically significant only for matrix size ([log(generation time)]: F1,35 = 3.03, = 0.091, coeff ± SE = 0.59 ± 0.34; [log(matrix size)]: F1,35 = 7.04, P = 0.011, coeff ± SE = 2.49 ± 0.94). The equilibrium rate at which a population was growing or shrinking as measured by λ1 had no effect on projection distance (F1,35 = 1.93, = 0.173).


Across the range of populations surveyed, the majority of plant species had current stage distributions that closely resembled those predicted at equilibrium. This result is surprising in that environmental perturbations, both biotic and abiotic, are common in nature, and these perturbations can lead to observed stage distributions that differ substantially from asymptotic predictions (e.g. Horvitz & Schemske 1995; Bierzychudek 1999; Clutton-Brock & Coulson 2002; Ujvari et al. 2010). Our results are in line with findings from the only two previous studies that have compared observed to asymptotic stage distributions. Similar to results from the 13 studies surveyed by Ramula & Lehtilä (2005), we also found that observed distributions were on average 25% different (using Keyfitz’s Δ) from asymptotic predictions. In contrast, Bierzychudek (1999) reported ‘large, significant differences’ in 10 out of 17 studies, although distances were not quantified. Our study builds on these results by providing a systematic survey of available studies of plants and a more thorough description of the variation in observed distance from asymptotic stability. Furthermore, our study can also serve as a guide for how distance from stable stage distribution (SSD) will translate into the strength of the transient response. The implication from our results that most species are near their SSD is important because it suggests that relying on the equilibrium assumption to calculate population metrics may be valid much of the time.

Of the surveyed populations that were farthest from their predicted SSDs, most had positive projection distances, and the distribution was skewed in the positive direction even for populations near α0 = 0 (Fig. 1c), suggesting that these populations had an excess of individuals with high reproductive values. This result could be an artefact of study designs in which study sites are chosen due to the presence of large, visible individuals or older, known populations. Conversely, the positive projection distances could be viewed as a lack of individuals with low reproductive values, which at least for plants, tend to be small individuals (Franco & Silvertown 1996). Such circumstances are likely for species that have episodic recruitment. When short sampling periods exclude episodic recruitment events but count larger individuals from previous recruitment events, the resulting population models are likely to produce SSDs that are different from current observations. Detecting new recruits can be difficult, because for many plant species, this life stage can be cryptic and difficult to measure. More research is needed on how influential variable estimates of recruitment are on population dynamics, whether those are due to lack of detection or to true episodic events. Regardless of the mechanism leading to the propensity for populations to be biased toward stages with high reproductive values (Fig. 1c), this bias in the species we sampled suggests that in the short term, these populations should tend to amplify the near-term population growth rate as compared to the asymptotic growth rate.

Current theoretical work on transient dynamics tends to focus on the largest transient responses possible, regardless of the initial distributions required to achieve these responses (Fox & Gurevitch 2000; Yearsley 2004; Koons et al. 2005; Caswell 2007; Haridas & Tuljapurkar 2007; Townley et al. 2007). Essentially, these results focus on the transient potential of a demographic matrix model, whereas actual transient responses will depend both on the model and on other dynamics that may be outside the scope of the model, including the current stage distribution of a population. The results from our study affirm that deviations from SSD can have important effects on near-term projections of population size and growth rate (Fig. 2). Although many populations were close to their stable distributions and would not react dramatically to the observed stage structure, we also observed deviances from stable that produced substantial transient responses, with the largest deviance leading to a nearly 16-fold difference between transient and asymptotic projections in population size after 5 years, and seven studies with greater than twofold differences.

Given the body of theoretical work on the transient potential of populations, it would be useful to be able to gauge when populations or species are likely to have larger projection distances. Species with longer generation times are known to have greater population momentum, which is a persistent effect of a transient response (Koons et al. 2005; Koons, Rockwell & Grand 2006). We found a trend for generally increasing projection distances for populations with longer generation times, supporting the idea that the effects of a disturbance on the stage distribution of a population will remain longer in populations of species that live longer (see also Doak & Morris 1999). Work on insect populations also suggests that species with later ages of first reproduction will return to a stable distribution more slowly (Taylor 1979). A second factor that may influence projection distance is the choices of matrix size and complexity, because projection distance and other measures of distance are calculated based on the design of a particular matrix. Several studies have shown that the transient potential of a projection matrix model may be affected by model design (e.g. Doak & Morris 1999; Tenhumberg, Tyre & Rebarber 2009; Stott et al. 2010), and our result of larger distances being associated with larger matrices supports this previous work. Although it can be challenging to separate the influence of a species’ life history from choices of model complexity on near-term population dynamics (Stott et al. 2010), researchers should be aware of both influences.

As transient dynamics have the potential to be strong when populations are moved away from their SSD following a disturbance, we were particularly interested in understanding how important we should expect transient responses to be when a recent disturbance was not obvious. Thus, we present a conservative distribution of projection distances based on populations of species in the least-disturbed habitat that was sampled when multiple populations were studied. For a very small subset of species, which included matrices from both disturbed and undisturbed habitats (= 10), we found no difference in projection distance between treatments (J. L. Williams and M. M. Ellis, unpublished data), suggesting that disturbances might not always lead to deviations from asymptotic predictions of SSD. Although our sample of populations was not exhaustive, it included a range of plant life histories and was not biased toward plant species that would be expected to have high or low population growth rates due to management goals. One limitation of this study is that we randomly chose one time interval for which to assess projection distance, and yet it is likely that environmental stochasticity is important for population dynamics for many of the surveyed species. Although the distribution of distances from SSD when calculated from a mean matrix and mean vector was very similar (J.L. Williams, M.M. Ellis, M.C. Bricker, J.F. Brodie and E.W. Parsons, unpublished data), it is currently unknown how much either current stage distributions or projection distances vary across years within a species or population.

The literature on transient dynamics is full of metrics for measuring distance to SSD and transient response that are subtly different and idiosyncratic. However, given that important deviations from SSD may occur in populations without obvious disturbances, it is important to determine which measure of distance best explains the transient response of a population. We found that the degree to which measures of the transient response over- or underestimate near-term growth compared to using equilibrium projections is very tightly linked to projection distance (Fig. 2), much more so than to any of the other measures of distance. This tight correlation can be explained in part because projection distance takes into account the reproductive value of each stage. The asymptotic population growth rate will overestimate near-term population growth rate for populations with individuals concentrated into stages with low reproductive value and underestimate it when individuals are concentrated into stages with high reproductive value. Thus, estimating the starting stage distribution and then calculating projection distance will be crucial for projects that estimate population growth rate or project population size. Doing so will allow the importance of transient dynamics in population projections to be assessed using the guidelines from the analyses presented here (Fig. 2, Table 2). For example, in conservation of rare species, overestimates of the near-term population growth rate using asymptotic values of lambda could lead to an overly optimistic outlook of population viability.

In conclusion, current theoretical research on transient population dynamics demonstrates that estimates of population metrics can be quite inaccurate when populations are far from their predicted asymptotic stage distributions (Fox & Gurevitch 2000; Caswell 2007; Haridas & Tuljapurkar 2007). Although the precise meaning of ‘far’ is not typically defined, half of the populations in our survey had projection distances within the bounds such that transient projections of population size and growth rate were within 10% of asymptotic projections at 5 years. This means that although it is useful to predict how population dynamics will respond to extreme deviations in stage distribution, asymptotic metrics may be adequate much of the time when other assumptions of projection matrix models, such as lack of density dependence and constant vital rates through time, are met. Measuring the current stage distribution can inform whether asymptotic measures of population growth will be reliable, and we argue that this is an important step to take in studies where precise population metrics are necessary.


L.S. Mills and M. Kauffman provided important input on the initial framework and analysis for this project, and E. Kerr assisted with data base management. T. Coulson, A. de Roos, G. Fox, D. Koons, B. Kendall, J. Maron, W.F. Morris, C. Pfister, B. Sandercock and G. White provided valuable feedback on earlier versions of this manuscript. Support to J.L.W. was provided by the National Center for Ecological Analysis and Synthesis, a Center funded by NSF (Grant #EF-0553768), the University of California, Santa Barbara, and the State of California; to M.M.E by an NSF Graduate Research Fellowship and a grant from the U.S. Environmental Protection Agency’s Science to Achieve Results (STAR) program; and to J.F.B. by a David H. Smith Conservation Research Fellowship through the Society for Conservation Biology and the Cedar Tree Foundation.