Evaluating the demographic buffering hypothesis with vital rates estimated for Weddell seals from 30 years of mark–recapture data


Correspondence author. E-mail: rotella@montana.edu


1. Life-history theory predicts that those vital rates that make larger contributions to population growth rate ought to be more strongly buffered against environmental variability than are those that are less important. Despite the importance of the theory for predicting demographic responses to changes in the environment, it is not yet known how pervasive demographic buffering is in animal populations because the validity of most existing studies has been called into question because of methodological deficiencies.

2. We tested for demographic buffering in the southern-most breeding mammal population in the world using data collected from 5558 known-age female Weddell seals over 30 years. We first estimated all vital rates simultaneously with mark–recapture analysis and then estimated process variance and covariance in those rates using a hierarchical Bayesian approach. We next calculated the population growth rate’s sensitivity to changes in each of the vital rates and tested for evidence of demographic buffering by comparing properly scaled values of sensitivity and process variance in vital rates.

3. We found evidence of positive process covariance between vital rates, which indicates that all vital rates are affected in the same direction by changes in annual environment. Despite the positive correlations, we found strong evidence that demographic buffering occurred through reductions in variation in the vital rates to which population growth rate was most sensitive. Process variation in vital rates was inversely related to sensitivity measures such that variation was greatest in breeding probabilities, intermediate for survival rates of young animals and lowest for survival rates of older animals.

4. Our work contributes to a small but growing set of studies that have used rigorous methods on long-term, detailed data to investigate demographic responses to environmental variation. The information from these studies improves our understanding of life-history evolution in stochastic environments and provides useful information for predicting population responses to future environmental change. Our results for an Antarctic apex predator also provide useful baselines from a marine ecosystem when its top- and middle-trophic levels were not substantially impacted by human activity.


Theory predicts that variation in a population’s annual growth rate (λ) will typically increase extinction probability (Lewontin & Cohen 1969). In perennial organisms, λ varies because vital rates vary and covary. However, different vital rates make different contributions to λ; and thus, similar degrees of temporal variation in different vital rates are expected to induce different consequences for variation in the annual multiplication rate of a lineage (Morris & Doak 2004). Accordingly, Pfister (1998) and Gaillard et al. (2000) developed the demographic buffering hypothesis: life histories should evolve to minimize the effects of environmental variation on fitness by favouring traits that buffer important vital rates from temporal environmental variation. In long-lived iteroparous species with delayed maturity, analyses of data from a variety of turtles, birds and ungulates have shown that changes in adult survival rate cause greater variation in λ than do changes in juvenile survival or reproduction (Heppell 1998; Pfister 1998; Gaillard, Festa-Bianchet & Yoccoz 1998; Gaillard et al. 2000; Sæther & Bakke 2000; Reid et al. 2004; Jenouvrier et al. 2005). Given this information on which vital rates have the greatest potential impact on λ, the demographic buffering hypothesis predicts that, for long-lived animals, environmental variation should have the least effect on adult survival and greater impacts on reproduction and survival of younger animals.

Recent research has, however, shown that life-history evolution in stochastic environments might be more complex than previously hypothesized (Boyce et al. 2006). Empirical results from a rigorous, 17-year study of a long-lived seabird indicated that the vital rate to which λ was most sensitive was not the vital rate with the lowest temporal process variation (Doherty et al. 2004). Several explanations have now been put forth for why demographic buffering is not always expected. Negative covariation between vital rates can, under some circumstances, actually lead to selection for variation in some parameters (Doak et al. 2005). Further, nonlinear relationships between vital rates and environmental conditions can promote demographic lability rather than buffering under certain conditions (Koons et al. 2009).

Another difficulty in evaluating the empirical support for demographic buffering hypothesis relates to approach. Doherty et al. (2004) noted three deficiencies in the methods employed for much of the work performed on the demographic buffering hypothesis: (i) estimates of vital rates might be biased because they did not account for detection probability, (ii) biological process variation and sampling variation were not distinguished and partitioned and (iii) measures of λ’s response to changes in vital rates might not have been properly scaled, especially for vital rates such as survival rates and breeding probabilities that are bounded between 0 and 1. Morris & Doak (2004) emphasized that proper analyses require estimates of vital rates whereas many studies to date, including foundational work by Pfister (1998), have not had access to the requisite data and therefore relied on less-desirable alternatives. Finally, as shown by Coulson, Gaillard & Festa-Bianchet (2005) and Doak et al. (2005), covariation between vital rates can have important implications, but it has largely been ignored in past work.

There is clearly a need for further empirical studies of the demographic buffering hypothesis. By implementing advanced estimation methods on data from long-term studies, we can improve our understanding of life-history evolution in variable environments. Such understanding has important theoretical and applied implications (Boyce et al. 2006; Tuljapurkar, Gaillard & Coulson 2009; Nevoux, Weimerskirch & Barbraud 2010). Accordingly, several recent studies have addressed the topic using long-term data and at least some of the suggestions for improved analyses (e.g. Altwegg, Schaub & Roulin 2007; Forcada, Trathan & Murphy 2008; Frederiksen et al. 2008; Karell et al. 2009). However, more such studies are needed, especially across diverse taxa and life histories (Koons et al. 2009; van de Pol et al. 2010).

Data collected to date on the Erebus Bay population of Weddell seals (Leptonychotes weddellii Lesson) provide an excellent opportunity to investigate demographic buffering in a long-lived mammal in a variable environment. This southern-most population of breeding mammal has been the subject of an on-going mark–resight research program since 1968 (Stirling 1969; Siniff et al. 1977) and contains a large number and proportion of known-age animals for which annual rates of survival and reproduction can be estimated (Hadley et al. 2006; Hadley, Rotella & Garrott 2007). Recent analyses indicate that survival rates and breeding probabilities vary not only by age and breeding state, but also among years. Recent advances in hierarchical modelling (Link & Barker 2010) now make it possible to estimate process variation and covariation among the population’s vital rates with all rates being estimated simultaneously from a multi-decade data set. Accordingly, we conducted this study to evaluate the empirical support for the demographic buffering hypothesis while employing recent suggestions for improved analyses (Link & Doherty 2002; Doherty et al. 2004; Morris & Doak 2004; Doak et al. 2005). We used rigorous estimates of all vital rates and their process variation and covariation as well as properly scaled measures of λ’s response to changes in vital rates in our evaluation.

Materials and methods

Study Area and Population

The study population occupies Erebus Bay, located in the western Ross Sea, Antarctica (−77·62° to −77·87°S, 166·3° to 167·0°E; see Cameron & Siniff 2004 for details). Each spring, c. 10 pupping colonies form along perennial cracks in the sea ice created by tidal movement of fast ice against land or glacial ice. Females generally have their first pup at about 7 years of age and have a pup in two of every 3–4 years thereafter (Hadley et al. 2006; Hadley, Rotella & Garrott 2007). Pupping occurs on the fast ice surface from late October to November (Stirling 1969). Mothers and pups are highly visible on the ice, typically close to one another and spend much of their time hauled out on the sea ice, especially in the few weeks immediately following birth. Females who have not yet had a pup (pre-breeders) or those who are skipping pupping also haul out in the study area each year and are readily visible.

Data Collection

Weddell seals have been individually marked and resighted in Erebus Bay annually since 1968 (Siniff et al. 1977). The majority of the tagging effort occurred from approximately 15 October to 15 November each year, during parturition when colonies were visited every few days to tag new-born pups. Beginning in early November, 6–8 resighting surveys were performed every 3–5 days each year. Seals could be readily approached within 0·5 m, such that observers were able to read tags on all resighted animals. Nearly all females that use the study area during the pupping season are detected at least once during annual surveys (Rotella et al. 2009). High fidelity to breeding colonies (Cameron et al. 2007) and high recapture rates (Hadley et al. 2006) allow construction of comprehensive encounter histories that include annual breeding state for each marked animal.

Data Analysis Overview

The analysis consisted of four major components: (i) estimation of vital rates with a mark–recapture analysis of data collected over 30 years, (ii) hierarchical modelling of mark–recapture estimates to decompose variances and develop estimates of mean vital rates and process variance–covariance of vital rates, (iii) matrix modelling of the vital rates to estimate population growth rate’s sensitivity to changes in vital rates and (iv) evaluation of the demographic buffering hypothesis through comparisons of process variation in vital rates and sensitivity values. Mark–recapture modelling of annual vital rates was performed with a multi-state model (Williams, Nichols & Conroy 2002), and subsequent hierarchical modelling was performed with a Bayesian approach. Our evaluation of the demographic buffering hypothesis incorporated a variety of recent suggestions for improving investigations of the hypothesis. In particular, we (i) conducted all analyses at the level of the vital rates rather than matrix elements (Morris & Doak 2004), (ii) estimated all vital rates from a single, long-term study, which allowed us to have rigorous estimates of process variance–covariance (Doak et al. 2005) and (iii) used variance-stabilized measures of sensitivity (Link & Doherty 2002).

Mark–Recapture Modelling of Vital Rates

We conducted multi-state capture–recapture analysis that (i) took advantage of Hadley et al.’s (2006) and Hadley, Rotella & Garrott’s (2007) modelling results for annual rates of apparent survival, recruitment and breeding for female seals during 1979–2003 and (ii) incorporated data from an additional 5 years. Hadley et al. (2006) provided strong evidence of annual variation in survival and recruitment rates for pre-breeding females of different ages. However, annual variation in breeding probabilities for females that had already recruited to the pup-producing portion of the population was ignored by Hadley et al. (2006) and evaluated in a subsequent analyses that used data only from recruited females (Hadley, Rotella & Garrott 2007). Thus, both previous analyses evaluated a variety of possible sources of variation in vital rates, but neither considered all of the data simultaneously. Accordingly, analyses have not yet evaluated whether vital rates for pre-breeders and recruited females might covary annually, which is a distinct possibility given that c. 62–87% of pre-breeders and recruited females are typically present in the study area during the pup-rearing season and thus experience the same environmental conditions for at least part of the year (Hadley et al. 2006; Rotella et al. 2009). In the analyses reported here, we used data from 30 years to evaluate several combinations of multi-state model structures identified by Hadley et al. (2006) as useful for pre-breeders and by Hadley, Rotella & Garrott (2007) for recruited females to investigate possible covariation in vital rates for pre-breeders and recruited females. All analyses were performed in program MARK (White & Burnham 1999).

The multi-state model included three types of parameters: apparent survival probability (φ), capture probability (p) and conditional transition probability (ψrs) between any pair of states r and s. We considered four breeding states: pre-breeder (P), first-time mother (F), experienced mother (E) and skip breeder (S). We were able to identify these states accurately for females in our study population because detection rate for mother–pup pairs is 1·0 (Hadley et al. 2006). Within each state, we used the same age and/or experience classes as those found by Hadley et al. (2006) and Hadley, Rotella & Garrott (2007) to be most parsimonious among a variety of potential classes and age-related patterns in rates for fixed-effects-only multi-state models. For females in state P, we used three age classes for φ (pups, yearlings, ≥2 years old); 4 age classes for p (1, 2, 3–6 and ≥7 years old); and 7 age classes for ψ (5, 6, …, 10, ≥11 years old). For females in states F or E, we ignored age and breeding experience when modelling φ and p (each rate was constrained to be the same for all active mothers within a year), but we allowed ψ to differ by state (but not age). For females in state S, we ignored age and breeding experience. When modelling ψ, our primary interest was in estimating breeding probabilities for females that survived from year t to + 1, i.e. given a female’s breeding state in year t, we were interested in her probability of being in a pup-producing state (F or E) in year + 1 given that she survived to year + 1. Thus, we used parameterizations that estimated the following transitions directly: (i) inline image (probability that a pre-breeder of a given age class in year t will have her first pup in year + 1; varies by year and age classes within P), (iii) inline image (probability that a female that had her first pup in year t will have another pup in year + 1; varies by year), (iii) inline image (probability that a female that produced a pup in year t and at least once before will have another pup in year + 1; varies by year) and (iv) inline image (probability that a female that had a pup prior to year t but not in year t will produce a pup in year + 1; varies by year). From any state, only two transitions were possible (e.g. for females in state P, ψPF and ψPP were possible but ψPE and ψPS were not), so the two impossible transitions were assigned probability zero in analyses [see Appendix S1 (Supporting information) for further details of capture–recapture analyses].

Hierarchical Modelling of Mark–Recapture Estimates

Having selected a multi-state model from a set of candidate models and having evaluated its fit by the data and found it satisfactory, we proceeded with inference conditional on this model. Such inference is often limited to what can be learned by interval estimation, either of model parameters or of functions of model parameters, and is based on the assumption that the sampling distribution of the maximum likelihood estimator is multivariate normal with variance matrix Σ given by the inverse of the estimated information matrix. Writing β for the vector of parameters describing the selected model and inline image for the maximum likelihood estimates (MLE), this asymptotic sampling distribution is designated as inline image, where k is the dimension of β.

The assumption of asymptotic normality of the MLE can be used for more than interval estimation: we used it for Bayesian analysis of a hierarchical extension of the multi-state model. We first describe a hierarchical model for parameters of the multi-state model using probability distributions [β|Ω], where Ω is a vector of hyperparameters. Combining the specified sampling distribution, the hierarchical structure and a prior [Ω], Bayesian analysis is based on the posterior distribution


Thus, given the sampling distribution assumption inline image, we can investigate the hierarchical structure [β|Ω] using the MLE inline image as data.

We describe the hierarchical model considered subsequently, noting for now that it involved means, precisions and correlation parameters; these were assigned diffuse normal, diffuse gamma and uniform priors, respectively.

We approximated the posterior distribution for the parameters of interest using Markov Chain Monte Carlo (MCMC) simulations in OpenBugs (Lunn et al. 2009) with four chains of length 125 000; each was started with different initial values, and values were stored after a burn-in of length 500. We calculated the Gelman–Rubin convergence statistic, as modified by Brooks & Gelman (1998) after 20 000 steps were completed for each chain. No thinning was performed.

We evaluated Monte Carlo error by calculating the standard deviation of mean values for each parameter among four independent chains, which was then used to calculate the precision of the MCMC sampling (inline image). We summarized attributes of the posterior samples using the base packages in program r version 2.11.0 (R Development Core Team 2010).

Hierarchical modelling not only allows the evaluation of process variances and covariances, but also yields improved estimation of the multi-state model parameters via shrinkage. We measured gains in precision as GP = 100% (1 − Posterior variance/Squared standard error). For example, if GP = 25%, the posterior variance is 25% smaller than the sampling variance of the MLE (squared standard error).

The top-ranked multi-state model had additive effects of year and factor level on the logit scale for survival rates φ and transition probabilities ψ. We designate the parameters estimated in program MARK by a vector β, and their MLEs are inline image.

Our hierarchical model for β begins with a latent vector


in which inline image and inline image are 29 potential year effects on survival and transition probabilities, respectively. We note that few marked animals early in the study combined with a minimum age of first reproduction of 4 years of age prevented us from obtaining MLEs of survival until 1980 and of transition probabilities until 1984. The inline image are factor-level effects on survival, and inline image are factor-level effects on transition probabilities. The five factor levels for survival were baseline survival for pups, yearlings, pre-breeders ≥2 years old, active mothers and skip breeders. The 11 factor levels for transitions were baseline probabilities of first reproduction for 8 age classes of pre-breeders (ages 4, 5, …, 10 and ≥11 years old) and baseline breeding probabilities for females in states F, E and S. We write θ ∼ N74(074Σ) and modelled it as a 74-dimension mean-zero multivariate normal random variable with covariance matrix


where bold zeros indicate matrices of zeros of specified dimension, bold Is denote identity matrices of specified dimensions and


Parameter ρ on the superdiagonal of C is the correlation between year effects on survival from year t to + 1 and breeding probabilities in year t + 1. We expected that this correlation might be positive for the following reasons. Female Weddell seals depend heavily on a capital breeding strategy and incur reproductive costs (Hadley, Rotella & Garrott 2007). Further, evidence suggests that maternal parturition mass varies with environmental conditions and is related to offspring survival (Wheatley et al. 2006; Proffitt et al. 2007a, b; Proffitt, Garrott & Rotella 2008). Thus, as has been shown in other capital-breeding seals (Boyd 2000), decisions about whether or not to produce a pup in a given year ought to be condition dependent and vary annually. Embryos are implanted in the summer (Stirling 1969), and gestation occurs throughout the winter, which might be a challenging period for survival as well as maintaining pregnancy. Thus, as discussed by Coulson, Gaillard & Festa-Bianchet (2005), survival and fertility might be elevated or depressed depending on conditions. For Weddell seals, environmental conditions during Antarctic winter months between pupping seasons t and + 1 might have similar effects on survival probability and the subsequent rate of pup production.

Survival and transition probabilities for the selected multi-state model can all be calculated from θ. For example, the 1980 survival rate for factor level one (pups) is φ1980,1, which satisfies


For consistency with the MLE, we rewrite this as


where inline image and inline image This reparameterization is necessitated by the imposition of identifiability constraints in maximum likelihood estimation: in program MARK, we set 1992 as the baseline year for survival. Similarly, we used 2007 as a baseline year for transition probabilities and defined parameters inline image and inline image. Survival estimation began in 1980 because data limitations prevented us from estimating pup survival rates for the initial cohort of pups tagged in 1979. Because of delayed maturity until at least 4 years of age, the actual transition probabilities presented here begin with estimates for inline image

The parameter vector inline image, is obtained as a linear transformation of the latent vector θ, namely β = , where


Thus, β ∼ N65(065QΣQ′). It should be noted that the Is in Q are identity matrices, 1s are vectors and 0s are vectors or matrices of zeros of required dimensions, consistent with nonzero elements of block columns. Block column four consists of × 7 matrices of zeros, corresponding to seven latent variables not needed for the calculation of β (these are inline image OpenBUGS code (available from the authors upon request) analyses inline image for a known matrix V, and β ∼ N65(065QΣQ′).

We used our estimates of the hyperparameters underlying φ and ψ to make predictions of probabilities φ and ψ for females in different breeding states and age classes for the sampled years (φ for 1980–2006 and ψ for 1984–2007, which yield breeding probabilities for 1985–2008) and for an as-yet-unsampled year, which are useful as values for a typical year.

Evaluating Effects of Vital Rate Changes on Population Growth Rate

We constructed an age- and breeding-state-based, post-breeding matrix model (Caswell 2001) for female seals in which age of recruitment and subsequent breeding schedules were flexible, seals could live up to 30 years of age (the maximum ever recorded in 40 years of data collection), litter size was 1 pup and sex ratio for newborn pups was 50 : 50 (Fig. S1, Supporting information). We parameterized the model with mean vital rate values estimated for an as-yet-unsampled year. Prior to using these in matrix calculations, we used methods described in Hadley et al. (2006) to adjust estimates of φ for tag loss, which is known to occur at low rates in our population, and to bias estimates low if not accounted for (Arnason & Mills 1981; Nichols et al. 1992; Bradshaw, Barker & Davis 2000). The adjustment method was based on animal age (Pistorius et al. 2000), and the probability of losing both tags given estimated probabilities of losing one tag; the method treated the probability of losing each tag as an independent event. We recognize that it has been shown for some species of pinnipeds that marker-loss rates for different tags on the same animal might not be independent (Bradshaw, Barker & Davis 2000; McMahon & White 2009). However, for the years of study reported on here, overall tag retention has been estimated to average 0·99 (range = 0·95–0·999, Cameron & Siniff 2004) such that dependencies, if present, should have resulted in little bias in our estimates and little or no effect on our estimates of process variation. The high rates of tag retention in our data relate to improvements in tag types prior to 1980 as well as the fact that we implement daily efforts during each field season to re-tag any animal with any missing or broken tags. Also, the animals are highly detectable and approachable for re-tagging. Finally, we suspect that Weddell seals might have higher rates of tag retention than that found in some other pinnipeds, because they haul out on ice and snow rather than rocky beaches. Thus, their tags are less likely to be abraded or to tangle on plants that can occur at greater abundance at lower latitudes (Bradshaw, Barker & Davis 2000). For example, Beauplet et al. (2005) estimated tag loss rates of 21% for animals crossing beaches with volcanic rocks, and Boyd et al. (1995) reported rates of 8·7% in an area with rocky beaches and abundant plants in associated shallow waters.

Using a projection interval of 1 year, we calculated asymptotic growth rate (λ1) and λ1’s sensitivity (S) to changes in vital rate means (θi) using chain-rule differentiation (inline image, Caswell 1978). Similar to Gaillard & Yoccoz (2003), we neglected stochasticity when calculating sensitivity because the life history being investigated is slow, survival rates are high, fertilities are low, coefficients of variation are low for process variation in vital rates (mean = 0·25) and population growth rates are <2. For such a scenario, stochasticity is not expected to have strong effects on mean λ1 or sensitivities (Benton, Grant & Clutton-Brock 1995); empirical support for this expectation was recently provided by Altwegg, Schaub & Roulin (2007). We calculated variance-stabilized sensitivity (VSS) of λ1 to each vital rate with the arcsine square-root formula provided by Link & Doherty (2002), i.e.


When comparing demographic probabilities, VSS provides a better measure of the effect on λ of changes in vital rates than does unscaled sensitivity because VSS operates on a scale where changes in rates are functionally independent of the magnitude of the rates (Link & Doherty 2002). The issue of scaling is important to consider as it can complicate comparisons of sensitivities. Proper scaling (more precisely, parameter transformation) allows changes in different parameters to be comparable. In some scenarios, transformations might be needed to facilitate comparisons between potential changes to survival rates (bounded between 0 and 1) and fertilities (taking on positive values that can be large). This was not the case in the work presented here as all parameters in question are rates of survival or transition. However, as explained by Link & Doherty (2002: 3301), transformations can also be important when comparing changes in different rates because ‘an increase of 10% (proportional or absolute) is one thing for a survival rate of 50%, quite a different thing for a survival rate of 90%, and an impossibility for a survival rate of 95%’. Elasticities certainly present an appealing alternative to sensitivities for evaluating the importance of changes in demographic parameters as they regard proportional changes and are unitless (Caswell 2001). However, elasticities (E) can be affected by the details of parameterization. In particular, elasticities for survival and for mortality (or 1-survival) will differ unless the survival rate equals 0·5 (Link & Doherty 2002). In the work presented here, all rates are parameterized as either the probability of surviving or of breeding, and so elasticities could have been used. We chose to use VSS so as to avoid possible problems of comparison with other studies that can be performed using different parameterizations (e.g. mortality instead of survival or probability of skipping reproduction instead of probability of reproducing).

Evaluation of Demographic Buffering Hypothesis

To understand how we tested the buffering hypothesis, it is important to consider several aspects of the development of various tests that can be used. Noting the occurrence of the product inline image in Tuljapurkar’s (1982) asymptotic expansion for stochastic λ, Pfister (1998) suggested the negative association between squared sensitivity and variance. The demographic hypothesis is then that the correlation Var(θ) and S(θ) is negative. Because inline image (see Morris & Doak 2004 for details), one might equivalently express the hypothesis as a negative correlation between squared values of elasticity and coefficients of variation. In the work presented here on Weddell seals, all vital rates are probabilities measured on the unit scale. For such rates, Morris & Doak (2004) reported that a spurious correlation exists between sensitivities and vital rates because of a functional relation between the mean and the variance for such rates. Accordingly, they recommended that future investigations of demographic buffering be based on relativized variances (RV), which measure a vital rate’s level of variability relative to its maximum possible value. We followed their recommendation and calculated inline image. Noting that inline image (for rate parameters), we used the correlation between RV and VSS as the primary basis of our evaluation of the demographic buffering hypothesis as this approach maintains a strong connection to Pfister’s (1998) motivation for the demographic buffering hypothesis. To allow comparisons of results from different approaches to evaluating the buffering hypothesis, we also calculated Spearman’s rank correlations between (i) sensitivity and variance, as in the seminal work by Pfister (1998) and (ii) sensitivity and RV, as in Morris & Doak (2004). For each pairing, we predicted a negative correlation after considering estimates of process covariance between vital rates (Doak et al. 2005). We also compared our estimates of process variation in hyperparameters for survival and breeding probability and assessed how well that difference corresponded with sensitivity results for survival and breeding probabilities.


Mark–Recapture Model-Selection Results

Capture–mark–recapture data were available from 5558 females tagged as pups between 1979 and 2007 and monitored through 2008, by which time 917 had been resighted with a pup at least once. These 917 females produced 2679 pups and provided 1090 observations of skip breeders. Estimated over-dispersion was slight for these data (inline image = 1·06).

In the most-supported mark–recapture model, patterns of annual variation (on the logit scale) in φ and ψ were shared (additive structure) for pre-breeders and recruited females (effects for φ and ψ were separated from one another), and p for pre-breeders and skip breeders varied among years. A model in which patterns of annual variation were allowed to differ between pre-breeders and recruited females for φ and ψ was also well supported (ΔAICc = 0·80; Table S1, Supporting information). The two top-ranked models produced similar estimates (between-model differences in estimates: average for differences in φ = 0·002, SD of differences <0·01; average for ψ = 0·004, SD = 0·05) and had 95% confidence intervals that overlapped for all estimates. Accordingly, we used estimates from the best-supported model (Table S2, Supporting information) for variance decomposition and matrix modelling.

Hierarchical Modelling of Mark–Recapture Estimates

The Gelman–Rubin convergence statistic clearly indicated that our hierarchical model had reached convergence within a few hundred iterations. Our MCMC simulations were of sufficient length to ensure with 95% confidence that the posterior means reported were, on average, within 0·0005 (SD = 0·0003) of the true values. We summarize findings about hyperparameters by features (mean, standard deviation, etc.) of their posterior distributions.

Our analysis provides evidence that logit(ψ) was more variable than logit(φ). The average difference between σ (mean = 0·57, SD = 0·10) and σ (mean = 0·37, SD = 0·07) was 0·20 (SD = 0·11); σ was the larger of the two quantities with probability 0·97. The posterior mean of ρ the correlation between logit(φt)and logit(ψt),  was 0·34 (SD = 0·21); the posterior probability for ρ > 0 was 0·94. Because φt is the survival rate from the pupping season in year t to the pupping season in year + 1, and ψt is the probability of producing a pup in year + 1 given the breeding state in year t, the positive correlation indicates that years with higher survival rates tended to be immediately followed by pupping seasons in which the probability of producing a pup was also high.

Posterior means of inline image and inline image were typically similar to corresponding MLE, but, as expected, Bayesian estimates of year effects were shrunk towards the mean value. Accordingly, the Bayesian hierarchical analysis provided gains in precision (mean GP = 22%, SD = 19%, Table S3, Supporting information). For the 65 estimated parameters, 30 of the top 32 GP values were for parameters having to do with inline image (26 were for year effects, four were for state-specific offsets). GP was >50% for five parameters and 30–50% for another 12 parameters. In addition to improving precision, the hierarchical modelling approach also allowed us to produce parameter estimates for more years (Figs S2 and S3, Supporting information). For example, the hierarchical structure allowed us to estimate inline image based on survival of seals in other states in that year and the dependencies estimated for survival rates among seals in different states.

Posterior distributions of all vital rates for an as-yet-unsampled year (Tables 1 and S3, Fig. S4, Supporting information) reveal that φ for pups and yearlings is lower than it is for older seals. For ψ, the posterior distribution indicates that recruitment probability, which is estimated to be only 0·04 at age 5, climbs to 0·41 by age 8 and then remains at a similar level for pre-breeders >8 years old. Subsequent-year breeding probability was lower for first-time mothers (0·46) than for experienced mothers or skip breeders, for which the rate was 0·66.

Table 1.   Features of posterior distribution for Weddell seal vital rates in an as-yet-unsampled year (based on data from 1979 to 2008 from Erebus Bay, Antarctica) along with the relativized variance (RV) of each rate, λ1’s sensitivity to each of the individual vital rates, and variance-stabilized sensitivity (VSS)
Vital rateMean (SD)RVSensitivityVSS
  1. Breeding states are the following: pre-breeder (P), first-time mother (F), experienced mother (E) and skip breeder (S). Estimates of apparent survival probability (φ) were corrected for tag loss and are presented by age class for pre-breeders (pups, yearlings and ≥2 years old) but not for females that have recruited to the breeding population (states F, E or S). Estimates of ψrs represent the probability of transitioning from breeding state r in year t to a pup-producing state (F or E) in year + 1. For females in state P in year t, transition rates varied by age class in year t (5, 6, …, 10, ≥11 years old); age was ignored for females in states F, E and S in year t.

inline image0·60 (0·10)0·0410·1050·052
inline image0·51 (0·10)0·0400·1230·063
inline image0·95 (0·03)0·0140·3820·088
φor E0·89 (0·04)0·0180·3680·120
φS0·95 (0·03)0·0120·1750·040
inline image0·04 (0·03)0·0170·0090·002
inline image0·19 (0·09)0·0520·0070·003
inline image0·32 (0·12)0·0680·0050·003
inline image0·41 (0·13)0·0740·0040·002
inline image0·41 (0·14)0·0760·0020·001
inline image0·48 (0·14)0·0810·0010·001
inline image0·39 (0·14)0·0790·0020·001
ψFE0·46 (0·14)0·0750·0040·002
ψEE0·66 (0·12)0·0690·0330·016
ψSE0·66 (0·13)0·0700·0190·009

Evaluating Effects of Vital Rate Changes on Population Growth Rate

We estimated λ1 as 0·98 when the mean predicted value of each vital rate in an as-yet-unsampled year (Table 1) was used to parameterize our matrix model. Based on estimates of λ1 obtained using each set of vital rates in the posterior distribution for an as-yet-unsampled year, it was clear that estimated process variation in vital rates can change population growth from strongly declining to rapidly increasing (mean = 0·99; SD = 0·06; 0·025, 0·5 and 0·975; quantiles = 0·87, 0·99, and 1·08, respectively). Despite such variation, the geometric mean of λ1 was only 0·002 lower than the arithmetic mean.

Our estimates of vital-rate variation, RV, sensitivities, and VSS are useful for evaluating the demographic buffering hypothesis because each varied substantially across the vital rates (Table 1). For example, vital-rate-specific values for RV ranged from 0·012 to 0·81, and those for VSS ranged from <0·001 to 0·120. Because correlations between all pairs of vital rates were positive and because each vital rate represented either the probability of survival or of producing a pup, process variance in any vital rate is expected to have negative effects on λ1.

Regardless of whether we used sensitivity or VSS to evaluate effects of changes in vital rates on population growth, changes in φ were always predicted to have greater effects than changes in ψ. However, the details of rankings for sensitivity to changes in φ for different classes of females did vary among sensitivity metrics and scalings (Table 1). Variance-stabilized metrics consistently ranked φpup and φyearling higher than did unscaled sensitivity, kept φmother as one of the two most important vital rates and lessened the importance of inline image and φskip. Among breeding probabilities, changes in ψEE and ψSE are expected to have the largest impact on fitness.

Evaluation of Demographic Buffering Hypothesis

In support of the buffering hypothesis, we found, as reported earlier, greater temporal variation in breeding probability (mean σ = 0·57, SD = 0·10) than in survival rate (mean σ = 0·37, SD = 0·07), whereas λ1 was more sensitive to changes in survival rates than to changes in breeding probabilities. Results of correlation analyses were also in keeping with the prediction that vital rate variation would be lower for those vital rates that had the greatest effects on fitness. We found evidence of a strong negative correlation between (i) sensitivity and variance (Spearman’s ρ = −0·82, < 0·001, one-tailed test of no negative correlation), (ii) sensitivity and RV (Spearman’s ρ = −0·89, < 0·001, one-tailed test of no negative correlation) and (iii) RV and VSS (Spearman’s ρ = −0·78, < 0·001, one-tailed test of no negative correlation).


Our evaluation of multi-state model structures for pre-breeders and recruited females advanced previous work by Hadley et al. (2006), Hadley, Rotella & Garrott (2007), which explored variation in vital rates for either pre-breeders or recruits but not both simultaneously. The top-ranked model indicated that females of different ages and breeding states shared similar patterns of annual variation in φ (additive year effects on the logit scale) and also in ψ (a separate set of additive year effects on the logit scale independent from those for φ). Subsequent hierarchical modelling of mark–recapture estimates indicated that survival and breeding probabilities were positively correlated across years such that there was a tendency for a given year to be good or bad for all vital rates. This is an interesting result because young pre-breeders are thought to emigrate temporarily from breeding colonies until they are near the typical ages of first reproduction (age 7–8), whereas females that have recruited to the breeding population are likely to be present in the breeding colonies even in years when they skip pup production (Testa & Siniff 1987; Hadley, Rotella & Garrott 2007; Rotella et al. 2009). Thus, despite the fact that females of different ages and breeding states appear to occupy different locations, at least during the breeding season, their vital rates tend to follow similar patterns. This could be interpreted as evidence that environmental conditions at broader, rather than finer, spatial scales in the Ross Sea are important drivers of population dynamics for the study population. However, knowledge of the region’s food web and its connections with broad-scale drivers such as the Southern Oscillation and features such as sea-ice extent is incomplete (Smith, Ainley & Cattaneo-Vietti 2007) and must be improved to allow an understanding of what underlies the observed patterns in seal vital rates.

Regardless of the ultimate drivers of environmental conditions, changes in food conditions seem a possible proximate cause of annual variation in demographic performance for Weddell seals. Weddell seals are capital breeders that rely heavily on stored reserves during lactation. For animals in our study population, average parturition mass of mothers and weaning masses of pups vary strongly among years and are related to broad-scale oceanographic variables (Proffitt et al. 2007a, b). Further, weaning mass is positively related to maternal parturition mass (Wheatley et al. 2006) and to the probability of surviving to age three (Proffitt, Garrott & Rotella 2008). However, it is not yet known whether an adult female’s body mass is also related to her survival rate and subsequent breeding probability in Weddell seals. In the few studies of large mammals that investigated such questions for adults, the results are mixed (Festa-Bianchet, Gaillard & Jorgenson 1998; Gaillard et al. 2000) such that further work is needed to understand what underlies positive correlations.

Positive correlations in annual vital rates at the population level have been reported for many species and are more common than negative correlations (Clutton-Brock 1988). For example, Jenkins, Watson & Miller (1963) reported that years with good breeding success for red grouse (Lagopus lagopus scoticus Lath) also tended to be years with higher adult survival. Similarly, Nur & Sydeman (1999) reported that survival and breeding probability were positively correlated in Brandt’s cormorants (Phalacrocorax penicillatus Brandt). Positive correlations between survival and breeding strengthen selection on life histories (Orzack & Tuljapurkar 1989; Benton & Grant 1996, 1999), increase variability in annual population growth rates and eliminate any buffering against environmental variability that would occur with negative correlations between rates (Jongejans et al. 2010). Given the importance of correlations in vital rates (van Tienderen 2000; Coulson, Gaillard & Festa-Bianchet 2005; Doak et al. 2005), it is important to estimate them for a greater variety of species using appropriate methods that account for sampling variance (Link & Nichols 1994).

In the work reported here, we estimated process variation and covariation in survival and breeding probabilities with a hierarchical model. For our species’ life history, wherein litter size is fixed at one pup, the results of our hierarchical modelling provided insights into all vital rates simultaneously. Until recently, it was difficult to estimate process variation and covariation in vital rates; accordingly, most past reports of correlations must be considered with caution until re-analysis employing recent developments in hierarchical models can be performed.

In this study, we used a two-phase analysis in which we first produced standard mark–recapture estimates of vital rates and then analysed those results with a Bayesian hierarchical analysis. One might ask what is gained (or lost) by such hierarchical analysis of MLE relative to analysis based on posterior distributions from the original data X. Given that the maximum likelihood estimate is unique, it is a function of a sufficient statistic. Thus, it can be shown that, given the sampling distribution inline image is multivariate normal (an asymptotic result typically taken for granted), the posterior distribution inline image is the same as the posterior distribution [βψ|X]. Hierarchical analysis based on MLE can be performed even if the original data are not available, might sometimes result in computational efficiencies and provides a means of estimating process variation from existing data for diverse species.

As has been shown in a number of other long-lived species (Brault & Caswell 1993; Caswell, Fujiwara & Brault 1999; Sæther & Bakke 2000; Gaillard & Yoccoz 2003; Oli & Dobson 2003), λ1 was most sensitive to changes in survival of prime-age animals. Population growth rate was less sensitive to changes in breeding probabilities, which is also in keeping with findings for other long-lived species with flexible breeding schedules (Jenouvrier et al. 2005; Forcada, Trathan & Murphy 2008). We found strong evidence of buffering in Weddell seal vital rates. In accordance with the demographic buffering hypothesis, our estimates of process variation in vital rates were inversely related to sensitivity. Specifically, variation was greatest in breeding probabilities, intermediate for survival rates of young animals and lowest for survival rates of older animals. These results expand on work performed by Forcada, Trathan & Murphy (2008) that used published estimates of vital rates for the Erebus Bay Weddell seal population and showed evidence of buffering but that was not able to incorporate estimates of process variance and covariance.

Our results suggest that adult survival rate is canalized to buffer fitness from environmental variation (Gaillard & Yoccoz 2003). However, female Weddell seals do incur reproductive costs to survival rate (Hadley, Rotella & Garrott 2007). We estimated that survival rate for mothers averaged 0·89, whereas the mean for skip breeders and older pre-breeders was 0·98. Thus, Weddell seals are not able to buffer their mean survival rate from being lower when they produce a pup. It seems likely that they are unable to avoid such costs because of their heavy reliance on a capital breeding strategy whereby they undergo extreme reductions in body mass during pup rearing (Wheatley et al. 2006). However, it does appear that they use a flexible breeding tactic to minimize variation in survival rate and thereby achieve some buffering. Wheatley et al. (2006) also found that females adjust their maternal expenditure during lactation in a manner consistent with demographic buffering. Thus, it appears that female Weddell seals can use reproductive flexibility at different stages in the reproductive cycle that are in keeping with the buffering hypothesis.

Although we found evidence for demographic buffering, several aspects of the work merit discussion as there is room for improvement in future analyses. First, our models assumed that several vital rates were constant across a variety of ages (e.g. inline image, inline image, inline image, inline image). Our age classes were those identified by Hadley et al. (2006), Hadley, Rotella & Garrott (2007) in investigations that considered models with more complex age structure and that allowed for possible senescent declines in each of the vital rates. However, modelling of Weddell seal vital rates performed to date has not considered individual heterogeneity in individual fitness components that might reasonably be expected to occur in life-history traits (Vaupel, Manton & Stallard 1979) and that, if present, could mask senescence in vital rates (Cam et al. 2002). When senescence is not properly accounted for, estimates of demographic responses to perturbations can be biased (Festa-Bianchet, Gaillard & Côté 2003). Recent advances in mark–recapture analyses now make it possible to evaluate individual heterogeneity in vital rates (e.g. King et al. 2009; Schofield, Barker & MacKenzie 2009; Gimenez & Choquet 2010; Link & Barker 2010). Further, it seems possible that senescence can occur in Weddell seal vital rates given the widespread evidence for actuarial senescence (increase in mortality with age) reported across diverse vertebrate species (Ricklefs 2010), and a recent report of physiological senescence in muscles of Weddell seals 17 years of age and older that could impair survival and/or reproduction (Hindle et al. 2009). Accordingly, we intend to use hierarchical models that include individual random effects to re-evaluate possible senescence in vital rates in future analyses. If senescence occurs in the vital rates, we suspect that our current analyses might overestimate temporal process variance in rates for prime-age females that were estimated based on pooled data from prime-age females and older females, i.e. females that might be less able to resist harsh environmental conditions. If true, our results regarding demographic buffering might be conservative, and actual buffering of the rates for prime-age females might be even stronger than reported here.

An additional issue worthy of investigation is possible serial autocorrelations in Weddell seal vital rates. If environmental conditions are auto-correlated, between-year correlations in vital rates can have sizeable effects on population growth rates (Tuljapurkar & Haridas 2006), which could influence the optimal life-history tactics in a stochastic environment. Further, the effects of between-year correlations in vital rates can have positive or negative effects on stochastic growth rate depending on circumstances (Tuljapurkar, Gaillard & Coulson 2009). Temporal correlations in vital rates can come from lagged responses, and it is certainly possible to envision how lagged responses in reproductive costs might occur in a capital breeder (Tuljapurkar, Gaillard & Coulson 2009). In future analyses, we plan to investigate possible serial correlations in vital rates and to incorporate findings on between-year correlations in further analyses of demographic buffering.

As exemplified by a number of recent papers on stochastic demography (e.g. Boyce et al. 2006, Engen et al. 2009; Barbraud et al. 2011; Sim et al. 2011), an improved understanding of demographic responses to environmental variation is important for both basic and applied reasons. Knowledge of vital rate distributions across a range of environmental conditions for species of interest provide valuable information on how species have coped with environmental variation in the past and might respond to possible future scenarios regardless of whether that variation is directional (e.g. global warming), periodic (e.g. El Niño Southern Oscillation) or non-periodic and non-directional (Berteaux & Stenseth 2006). However, it is difficult at present to reach broad conclusions about demographic buffering in animal populations because of the dearth of detailed studies and because of analytical issues in many existing studies.

Among published analyses, a variety of approaches have been used for estimating process variance and covariance and for scaling variances and sensitivities, and the approach used can influence results (Link & Doherty 2002; Doherty et al. 2004; Morris & Doak 2004; Doak et al. 2005). In many instances, process variation has been ignored or strong simplifying assumptions have been made, and scaling issues have not been considered. In other cases, despite having a long-term study with careful work on covariation based on detailed data, it has still not been possible to estimate process covariation for all vital rates (Reid et al. 2004). To allow for more detailed analyses to be conducted in the future, Morris & Doak (2004) made a plea for biologists to publish vital rate estimates. It would also be useful to publish details of the underlying estimates of model parameters (e.g. logit-scale coefficients), including estimates of variances and covariances, as we have shown here how these can be used in hierarchical modelling of process variation. If that happens, new versions of previous meta-analyses of demographic buffering (e.g. Gaillard, Festa-Bianchet & Yoccoz 1998; Pfister 1998; Sæther & Bakke 2000; Gaillard & Yoccoz 2003) can be performed that incorporate new developments in analysis methods.

Despite the continued need for more studies across a greater diversity of taxa and life histories (Koons et al. 2009; van de Pol et al. 2010), there is growing, though not universal, support for the demographic buffering hypothesis. A number of recent studies with careful analyses of excellent demographic data report that the vital rates to which population growth rate was most sensitive also tended to be those with lower variation (e.g. Reid et al. 2004; Jenouvrier et al. 2005; Forcada, Trathan & Murphy 2008; van de Pol et al. 2010). For a long-lived seabird, however, results were equivocal (Doherty et al. 2004). Further, for the Antarctic fur seal (Arctocephalus gazella Peters), buffering was in evidence during the earliest decade of the study but lost in recent years when the population was in decline (Forcada, Trathan & Murphy 2008). This last result emphasizes the importance of considering population status and ecological context when interpreting the results of studies of demographic buffering.

The data reported on here for Weddell seals were collected in a marine ecosystem during a period when its top- and middle-trophic levels were not substantially impacted by human activity (Smith, Ainley & Cattaneo-Vietti 2007) and when the population of seals was stable (Rotella et al. 2009). Accordingly, the vital rate estimates reported here provide useful baselines that can be compared with what is found under contrasting circumstances for Weddell seals or other closely related species in the Southern Ocean. Estimates of adult survival rates and breeding probabilities for Weddell seals in eastern Antarctica are available for 26 recent years (Lake et al. 2008). Similar to what we report here, Lake et al. (2008) found that breeding females had consistently high annual survival rates but variable breeding probabilities and used a flexible breeding strategy to avoid reproductive costs to survival. Results from studies of other pinnipeds in the Southern Ocean indicate that there might be some interspecific consistency in this pattern.

Studies of subantarctic fur seals (Arctocephalus tropicalis Gray) in the southern Indian Ocean found that the proportion of pups surviving from birth to their first return to their birth island ranged widely among cohorts (27–75%), whereas survival rates for adult females were high and consistent among years (Beauplet et al. 2005, 2006). Relevant results for Antarctic fur seals also show evidence that demographic buffering can occur (Forcada, Trathan & Murphy 2008). Southern elephant seals (Mirounga leonina Linnaeus) have been studied in several locations with different population trajectories, food resources and predator communities. As in Weddell seals, elephant seals had high temporal variation in fertility (Bradshaw et al. 2002) and survival through the first year of life (McMahon & Burton 2005). Within a location, survival rate was lowest in the first year of life and then higher and relatively stable across ages later in life (Pistorius, Bester & Kirkman 1999). Further, juvenile survival rate and age at primiparity varied by location, whereas survival rates for older animals were more consistent (McMahon, Burton & Bester 2003). Similar to what we report here for Weddell seals, population growth rate for elephant seals was more sensitive to changes in survival rates than to changes in fertility, and survival rates for immature seals were important to growth rate and variable among years (McMahon et al. 2005). It appears that in both species, future research such as that performed by de Little et al. (2007) on extrinsic and intrinsic drivers of survival of young seals will be provide useful insights into stochastic demography in these species. Although not all of studies of Southern Ocean pinnipeds provide all of the elements necessary for a full evaluation of demographic buffering, they do provide compelling examples of how studies of vital rates can play a crucial role in understanding how populations will respond to future environmental change. They also play an important role in developing hypotheses that can be evaluated with fuller analyses.


We thank the many individuals who have worked on projects associated with the Erebus Bay Weddell seal population since the 1960s. We thank J. D. Nichols for helpful suggestions during analysis, and we thank D. B. Siniff for discussions that improved this manuscript. We are grateful to J.-M. Gaillard, J. D. Nichols and two anonymous reviewers for their useful comments on earlier drafts of the manuscript. The project was supported by the National Science Foundation, Division of Polar Programs (grant no. DEB-0635739 to R. A. Garrott, J. J. Rotella, and D. B. Siniff) and prior NSF grants to R. A. Garrott, J. J. Rotella, D. B. Siniff and J. W. Testa. Logistical support for fieldwork in Antarctica was provided by Raytheon Polar Services Company, Antarctic Support Associates, the United States Navy and Air Force, and Petroleum Helicopters Incorporated. Animal handling protocol was approved by Montana State University’s Animal Care and Use Committee (Protocol #41-05).