1. Stable isotopes are increasingly used in ecology to investigate ontogenetic shifts in foraging habitat (via δ13C) and in trophic level (via δ15N). These shifts are in essence an individual-level phenomenon, requiring repeated measures throughout the life of individuals, i.e. longitudinal data. Longitudinal data require in turn specifying an appropriate covariance structure. Here we present a hierarchical model to jointly investigate individual ontogenetic shifts in δ13C and δ15N values.
2. In a Bayesian framework, we used a Cholesky decomposition for estimating a moderately-sized covariance matrix, thereby directly estimating correlations between parameters describing time-series of isotopic measurements. We offer guidelines on how to select the covariance structure.
3. The approach is illustrated with a hierarchical change-point (or broken stick) model applied to a data set collected on Southern Elephant Seals, Mirounga leonina. Ontogenetic shifts in foraging habitat, following a juvenile and variable stage, were detected and interpreted as fidelity to a foraging strategy; while ontogenetic shifts in trophic level were more likely the result of complete independence from maternal resources followed by a gradual increase in trophic level as seals aged.
4. Specifying both an appropriate covariance and mean structure enabled us to draw strong inferences on the ecology of an elusive marine predator, and has wide applicability for isotopic ecology provided repeated isotopic measurements are available.
A large number of studies looking at ontogenic shifts concerns species with ‘cryptic lifestages’, in particular marine organisms such as turtles (Reich, Bjorndal & Bolten 2007), fish (Post 2003; Estrada et al. 2006) or marine mammals (Hobson & Sease 1998; Mendes et al. 2007; Drago et al. 2009; Newsome et al. 2009). In some studies, repeated isotopic measurements were available for the same individual using so-called archive tissues, because they are metabolically inert after synthesis, such as vertebrae (Estrada et al. 2006), or teeth (Hobson & Sease 1998; Mendes et al. 2007; Newsome et al. 2009). These studies addressed the estimation of a change-point in the time-series of isotopic measurements, yet they typically pooled data from all individuals to infer a population-level change-point, or ontogenetic shift. For example, Newsome et al. (2009) fitted a four parameters logistic model to estimate a change in dentin δ15N of Californian Killer Whales (Orcinus orca) after weaning. The model is fit at the population level, that is assuming all individuals experienced an ontogenetic shifts at the same age, despite apparent individual heterogeneity in the raw plot (their Fig. 2a). Ignoring individual heterogeneity when it is in fact present may hinder our ability to draw accurate inferences (Cooch, Cam & Link 2002; Petrovskii, Mashanova & Jansen 2011). In addition, the change-point is often treated as known even when it was first estimated from the same data: no confidence interval for the change-point is usually reported, and all subsequent inferences are conditional on the point estimate for the change-point.
Stable isotopes in ecology of wild animals are often hailed as a powerful technique. Yet, inferences are typically drawn from statistical analyses that tend to (1) emphasize testing over estimation and goodness-of-fit (Graham 2001; Martìnez Abraìn 2010); and (2) focus on the mean response at the expense of variability (but see Hénaux et al. 2011). In the case of detecting an ontogenic shift, the problem is clearly one of estimation: when does an organism change its habitat use or trophic level? Further questions may arise as to what are the ecological, life-history and ultimately population consequences of such an individual change (Werner & Gilliam 1984; Graham et al. 2007). This paper thus deals with the problem of estimating individual ontogenic shifts with longitudinal isotopic data, that is repeated measurements of δ13C and δ15N on the same organism throughout its life. We present a Bayesian change-point model to estimate jointly individual ontogenic shifts in δ13C and δ15N. Our aim is to bring forward to a larger audience the vast literature on change-point models (Hall et al. 2000; Beckage et al. 2007; Ghosh & Vaida 2007; Muniz-Terrera, den Hout & Matthews 2011), and how to fit them using the BUGS language (Lunn et al. 2000).
Change-point, or broken-stick, models aim at finding an abrupt rupture in a time-series. The time-series is assumed to be the juxtaposition of piece-wise linear homogeneous segments, each segment separated from the next by a change-point. Such models have been used in epidemiology to infer the onset of cognitive decline (Hall et al. 2000; Muniz-Terrera, den Hout & Matthews 2011), of prostate cancer (Bellera et al. 2008) or of HIV immunologic response decline (Ghosh & Vaida 2007). In ecology, Beckage et al. (2007) used a change-point model to study allometric relationships between tree height and tree diameter or to study seedling recruitment with respect to canopy cover along a transect; while Da-Silva et al. (2008) studied post-reproductive survival in a partially semelparous marsupial. These models are very flexible as they allow specifying different probability distributions to describe different parts of a time series. Change-point models thus seem appropriate to describe ontogenetic shifts (e.g. Post 2003), but are not prescriptive. Other models (e.g. Newsome et al. 2009) may prove useful when investigating ontogenic shifts. We will illustrate our methodology with an example using data on Southern Elephant Seals, Mirounga leonina.
Southern Elephant Seal example
Southern Elephant Seals are marine carnivores with a very elusive lifestyle since they can spend more than 80% of their lifetime at sea (McIntyre et al. 2010). Where they are foraging remained a mystery until the advent of miniaturized electronic tags (Biuw et al. 2007). Seals from îles Kerguelen (49°30’S, 69°30’E) in the Southern Indian Ocean show a dual foraging strategy: animals forage either in Antarctic waters or in Polar Frontal waters (Bailleul et al. 2010). Across the Southern Ocean, δ13C decreases with increasing latitude (Bentaleb et al. 1998; Trull & Armand 2001). Carbon stable isotopes can thus help identify the foraging areas of marine predators: a relative difference of ≈2‰ is expected between the two strategies (Bentaleb et al. 1998). Processes underlying carbon isotopic fractionation in marine foodwebs are briefly reviewed in MacKenzie et al. (2011).
With Southern Elephant Seals, we were interested in answering the following questions:
Throughout we assume the data are composed of N measurements of δ13C and δ15N on m individuals. For the jth individual, there are nj measurement, such that . These measurement are collected along a biologically-meaningful ordered scale such as age (or size). This scale is assumed continuous for convenience. We also posit that a piecewise linear, or broken-stick model, provides an adequate description of the data, although this may be relaxed to consider non-linear functions as well. With the broken-stick model, we denote by () the age of the jth individual when an ontogenetic shift in foraging habitat (trophic level) occurs.
The time-series of isotopic measurements for the jth individuals is then modelled as: for i ∈ [1:nj]
and σδ13C is the residual standard deviation, which is allowed to be different before and after the ontogenetic shift. A logarithmic transformation is used to guarantee positive values for all or . We implicitly assume that only the consumer, not its prey, can experience an isotopic shift, but the model cannot be used to distinguish between these two alternatives (Matthews & Mazunder 2004).
The individual coefficients ak ∈ [1:4],j are assumed to be exchangeable and drawn from a multivariate normal distribution of vector mean αk ∈ [1:4] and covariance matrix of dimension 4:
This formulation allows to directly estimate correlations between parameter of interest via the covariance matrix. For example, one could be interested to assess whether an ontogenetic shift occurs later or earlier depending on the steepness of the slope a2,j. The interpretation of such correlations would depend on the biology of the studied organism.
The same broken-stick model can be applied to δ15N: this model then calls for the estimation of two independent covariance matrices each of dimension 4: one for δ13C and one for δ15N (hereafter referred to as 2 × 4 × 4). An obvious question is whether ontogenetic shifts in δ13C and δ15N are simultaneous or correlated. Answering this question requires the estimation of covariance matrix V of dimension 8, as represented on Fig. 1 (this model is referred to as 8 × 8 hereafter).
Specifying the covariance structure of a model has generally received less attention than specifying its mean response, but the problem is no less relevant (Pourahmadi 2010). Estimating a covariance matrix of size >2 is a challenge: in addition to the usual restriction to lie between −1 and 1, correlations are jointly constrained. For example, with a 3 × 3 covariance matrix, ρ1,2 and ρ1,3 can take any value between −1 and 1, but ρ2,3 must then conform to the following constraints for the matrix to be positive-definite and invertible (Budden et al. 2007):
Estimating a matrix such as represented in Fig. 1 presents some additional challenges since some elements are constrained to be 0. We opted for a Cholesky decomposition of V into a diagonal matrix Γ and a lower triangular matrix L with 1 s on the diagonal:
In a Bayesian framework, priors are specified on each parameter. We used weakly-informative priors: for parameters on the same scale as the data (α1, α2 and α4) we used normal priors with a large variance. For the parameter governing the distribution of ages at ontogenetic shifts, a logarithmic transformation in eqn 1 guarantees positive values for all or . For the parameter α3, we used a Student-t prior [with location, scale and degrees of freedom set to 0, 10 and 7 respectively (Gelman et al. 2008)]. For modelling V, we used the priors similar to those of Chen & Dunson (2003): independent half-normal priors of mean 0 and standard deviation 1·5 for the elements, γp ∈ [1:8], of the diagonal matrix Γ, and independent normal priors of mean 0 and standard deviation 0·5 for the elements, λp ∈ [2:8],q<p, of L. A prior covariance matrix of dimension 4 (8) with such a specification is depicted on Fig. S1 (Fig. S3). This prior gives reasonable values (i.e. between 0 and 10) for the variances of the ai,j and is somewhat conservative: most of the probability mass is put on values <5. This prior thus reflects skepticism for large differences between individuals. Uniform priors were put on the residual standard deviations (Gelman 2006).
With hierarchical models, model selection is a challenge and several methods have been suggested, such as Deviance Information Criterion (DIC) (Spiegelhalter et al. 2002; Barnett et al. 2010); but there is currently no consensus (Jordan 2011). We choose to avoid using the DIC because of drawbacks such as lack of invariance to reparametrization [Spiegelhalter et al. (2002) and the following discussion]. In fact, DIC was computed but yielded non-sensical results for the estimated number of parameters when the Cholesky decomposition was used (see Table S2). To select an appropriate model, we focused on Posterior Predictive Checks (Gelman, Meng & Stern 1996; Berkhof, van Mechelen & Gelman 2003) wherein each fitted model is used to predict (hypothetical) repetitions of the data set. From this hypothetical dataset, we compared an observed summary statistic (Tobs) to its predicted values (Trep) and computed a Pvalue:
A Pvalue close to 0·5 flags a good fit (Trep≈Tobs), while an extreme Pvalue (0 or 1) betrays a major model misfit. We chose the range of observed isotopic values as discrepancy statistics to assess model fit. The rational for choosing the range as a test statistic was: if a change-point is necessary to describe the time-series of isotopic measurement, the range of predicted value is likely to be underestimated when fitting a model with no change-point. The tip of the broken stick will be missed by a simple linear regression, hence an underestimation of the range. Posterior Predictive Checks can be used to test whether a broken-stick model is justified or to select a covariance structure. For example, we can compare the covariance structure depicted in Fig. 1 with a simpler structure where the matrix is block diagonal with no correlation between δ13C and δ15N (i.e. ρ1,5 = ρ2,6 = ρ3,7 = ρ4,8 = 0 in Fig. 1).
and are the means of the observed and fitted values respectively, while the numerator in eqn 5 is the sum of squared-residuals εi for the jth individual.
Southern Elephant Seal data
Teeth were collected from elephant seals that died of natural causes on îles Kerguelen. Canines grow continuously throughout the whole life without closing of the pulp cavity, allowing for age determination (Laws 1952, 1993). Canines from 47 males and 20 females were analyzed and sampled for isotopic analysis. Eighteen teeth were sampled on animals that died before a population crash in the 1970 s, while the remaining 49 were sampled in the 2000 s, after the population had stabilized (Authier, Delord & Guinet 2011).
Each tooth was cut longitudinally and observed under diffused light to count growth layers. The alternate pattern of two opaque and two translucent growth layers corresponds to the annual biological cycle of Southern Elephant Seals (Laws 1952). Translucent bands are enriched in vitamin D and synthesized when seals are ashore to breed and to moult, while opaque ones are synthesized when at sea (Wilske & Arnbom 1996). Within a year, a Southern Elephant Seal comes onshore to breed, returns to the sea, then comes onshore to moult and forages once more at sea before the next breeding season. Thus each growth layer was assumed to correspond to one forth of a year (Martin et al. in press). Each growth layer was sampled for 1 mg of bulk dentin using a MicromillTM sampler (ISEM, Université de Montpellier 2). Organic matter δ13C and δ15N signatures of the bulk dentine were measured with an elemental analyzer (EA-IRMS, Euro-Vector EA 3000, Euro-Vector, Redavalle (PV), Italy) coupled to a continuous flow mass spectrometer (Optima-Micromass) at the Université de Montpellier 2.
As a recent study raised concerns about non-linear offsets of organic %C, %N and C/N after acid treatment (Brodie et al. 2011), we forwent any acid (or demineralization) treatment prior to isotopic measurement. As a result, the measured δ13C is a mixture of organic carbon with a small amount of inorganic carbon. To test the impact of the inorganic fraction, Martin et al. (in press) compared acid-treated and untreated samples but found no differences (±0·02‰). Schulting et al. (2008) found similar C/N ratios between bulk dentin and collagen, with a lower carbon and nitrogen contents in bulk dentin most likely due to the mineral fraction. Here we assumed that the impact of the mineral fraction is negligible. If not, relative trends (see Results) should be unaffected under the assumption of a systematic bias.
Stable isotopic signatures are presented in the usual δ notation (in ‰) relative to Pee Dee Belemnite and atmospheric N2 for δ13C and δ15N respectively. Typical precisions for isotopic measurement were 0·20 ‰ for both carbon and nitrogen. We used C/N ratio thresholds of bone and tooth collagen (2·9 to 3·6) as criteria for the identification of diagenetic alteration (Ambrose 1990); assuming that total dentin, whose organic phase is mainly collagen and water (Moyes & Doidge 1984), has the same C/N ratio than bone and tooth collagen. One thousand five hundred and ninety samples were analyzed, but 176 were discarded because of anomalous C/N ratios, yielding a final sample size of 1414 (1115 from males and 299 from females) analyses. The first δ15N value of each time-series was also removed as it is clearly a reflection of maternal diet (Hobson & Sease 1998; Martin et al. in press). Summary statistics of the data are available in Table S1 and depicted in Fig. S2. Females were under-represented in this data set, and samples collected from dead females on beaches were biased toward young females. Thus time-series of isotopic measurement were usually shorter for females (Table S1).
To answer questions about any differences between males and females, or between animals living before and after the population crash, we modified the hierarchical change-point model defined by eqn 1 by specifying that the vector of means (αk ∈ [1:4]) depended on the sex of seals and whether seals lived before or after the population crash:
All models were fitted with winBUGS (Spiegelhalter et al. 2003) called from R (R Development Core Team 2009) with the package R2WinBUGS (Sturtz, Ligges & Gelman 2005). We used normal priors for regression parameter on the natural scale and Student priors with 7 degrees of freedom (Gelman et al. 2008) for regression parameters on the log scale. Three chains were initialized with overdispersed starting values. After appropriate burn-in (200 000 iterations) and thinning of the chains (1 value every 200 iterations stored), convergence was assessed using the Gelman-Rubin convergence diagnostic (Cowles & Carlin 1996) with the coda package (Plummer et al. 2008). Posterior mean (or median when posterior distributions were asymmetric) with 95% highest probability density intervals are reported as 2·5%Mean97·5% following Louis & Zeger (2009). Inferences are based on a posterior sample of 3000 iterations. Annotated BUGS code is available in the Appendix S1, along with an R script and a simulated data set.
Model selection and fit
A hierarchical change-point model provided an adequate fit to the elephant seal data (Figs 2 and 3). Ontogenetic shifts in δ13C and δ15N values were generally supported, except for short time-series and a few individuals. The broken-stick model provided a better fit than a null model with no change-point. The model with the most complex covariance structure (8 × 8 model) was not supported (Table 1). Besides, the estimated correlations δ13C and δ15N were small, with a posterior mean of ≈0·1 in absolute magnitude (Fig. 1). Results from the hierarchical model with no correlation between δ13C and δ15N are thus reported, although results from the other hierarchical model were very similar. There was no statistical support for distinguishing between sexes or between individuals sampled before or after the population crash (Figs S4 and S5): the posterior distribution of regression coefficients for both factors was as diffuse as that of its prior and included 0.
Table 1. Posterior predictive checks. The statistic considered is the range of isotopic values and the reported Pvalues are the probability that the predicted range exceeds the observed one. The percentage of individuals with a 0·1 <Pvalue <0·9 is reported for both carbon and nitrogen isotopic time-series. Broken-stick models decreased the proportion of individuals with extreme Pvalues: a broken-stick model was appropriate for most individuals. There was however little support for an increase in covariance complexity: overall changes in δ13C were not correlated with changes in δ15N
8 × 8
2 × 4 × 4
Results for the selected hierarchical change point model are summarized in Tables 2 and 3. The residual variances for both isotopes were larger before the ontogenetic shift (Table 2). We found individual heterogeneity in all four parameters ak ∈ [1:4]: all variance components were well estimated (Table 3, Fig. S3). The estimated age at ontogenetic shift was larger for δ13C values (3·2 years) than for δ15N values (1·9 years, Table 2). This difference was statistically significant at the 5% level. δ13C values at ontogenetic shifts were more variable than δ15N values, but the variability in age at ontogenetic shift was similar for the two elements (Table 3). There is a sign reversal in slopes before and after the ontogenetic shift in both carbon and nitrogen isotopes (Table 2): the slope was positive and then negative for δ13C and the opposite for δ15N. Slopes were more variable before than after the ontogenetic shift for both δ13C and δ15N values (Table 3). There was respectively a small and no correlation between slopes before and after the change-point in δ13C and δ15N values (Fig. 1).
Table 2. Estimated marginals from a broken-stick model fit to the Southern Elephant Seal data. σε,1 and σε,2 are respectively the residual standard deviations before and after the shift; α1 and Kδ the isotopic value and age at the shift respectively, and α2 and α4 the slopes before and after the shift respectively
‰ per year
‰ per year
Table 3. Estimated individual-level variances in all four parameters governing the broken-stick model fit the Southern Elephant Seal data. Medians are reported instead of means because some posterior distributions were slightly asymmetric
Value at shift
Age at shift
Southern Elephant Seal foraging ecology
Despite the on-going ‘biologging’ revolution, some questions are still not easily addressed with miniaturized tags (Hebblewhite & Haydon 2010). Equipping a large sample of individuals with data recorders that may be lost is prohibitively expensive. In contrast, stable isotopes are inexpensive and no longer studied in ecology as a complementary ‘side-kick’ to biologging, but in their own right (Newsome, Martinìnez del Rio Bearhop & Phillips 2007; Wolf, Carleton & Martìnez del Rio 2009). With the example of Southern Elephant Seal, a species with a cryptic life-style, we analyzed stable isotope data with a hierarchical change-point model to draw inferences on the ontogeny of foraging. We used repeated measurements of dentin δ13C and δ15N values over the whole life of individuals to estimate ontogenetic change-points in both foraging habitats and in trophic level; and found individual variability in both the trajectory and timing of shifts.
A hierarchical modelling approach answered all five questions we asked. After a juvenile stage characterized by a large residual variance, Southern Elephant Seals became faithfull to a foraging strategy. Inferences drawn from longitudinal isotopic data are in agreement with those of ‘biologging’ studies (Bradshaw et al. 2004), but the former involved a larger sample over a longer time-period than the latter. This commitment to a foraging strategy occurred at an early age, on average at about 3 years, but there was substantial individual heterogeneity (Table 3, Figs S6 and S7). An ontogenetic shift in δ15N was also detected, but this shift occurred earlier (around 1·9 year-old on average).
The ontogenetic shifts we identified can be the result of several processes, such as complete independence from maternal resources acquired before weaning (Hobson & Sease 1998; Polischuk, Hobson & Ramsay 2001) or a shift in foraging habitat (interfrontal vs. Antarctic waters) and trophic level (Bailleul et al. 2010). If the estimated shift solely resulted from a decay of maternal resources, we would not expect a difference in residual variances before and after a shift. In the case of Southern Elephant Seals, not only residual variances, but also slope variances were larger before the shift (Tables 2 and 3). This pattern may be interpreted as an individual switching from a very variable state to a more stable one, or in other words for carbon isotopes, in seals becoming faithfull to a foraging strategy. The posterior mean for the slope after the ontogenetic shift was negative, which we interpreted as individuals foraging in Antarctic waters. These seals have to haul out on îles Kerguelen for reproduction and moulting, and they are very likely to feed on the way (Thums, Bradshaw & Hindell 2011), thus diluting a ‘pure’ Antarctic signature for δ13C. Hence a negative slope, as the Antarctic signal becomes preponderant over the years. The estimated individual variability showed that some slopes after the shift were null or slightly positive, which can be a reflection of seals foraging always in the same water mass, for example, in pelagic waters of the Polar Front (Bailleul et al. 2010). Finally, a few individuals had a large positive slope before the shift and a shift late in life. The large positive slope before the shift may be a reflection of seals foraging on the Kerguelen Plateau (Bailleul et al. 2010), which has an enriched δ13C signature compared to pelagic water masses (Cherel & Hobson 2007); before switching to an alternative strategy.
Concerning trophic level (δ15N), the shift occurred on average earlier than for the δ13C data (Table 2). Slopes before the shift were negative, yet they reversed sign after. Their magnitude also halved before and after the shift, with very few individual variability left after the shift (Table 3). This pattern suggested the shift in δ15N values to mostly reflect the gradual decay of maternal influence on δ15N (Hobson & Sease 1998), followed by a gradual elevation in the trophic web as seals grew in size. Growth is indeterminate in these seals: they keep growing until their death although growth is very slow in adults (McLaren 1993). This continuous growth means that older seals can physically catch bigger preys, which may explain why we observed a gradual elevation in trophic levels. Additionally, the large energy stores males must accumulate before the breeding season may also drive a shift toward large and energetically profitable preys. Residual variances were also larger before than after the shift but the decrease was not as dramatic as for δ13C values (Table 2). Thus this shift may mostly reflect complete independence from maternal inputs.
This pattern of an elevation in trophic level with age (Fig. 2) does not conflict with blood isotopic data for males, but was not expected for females: in a previous study, Bailleul et al. (2010) collected blood samples on juvenile males and on adult females. This study evidenced an elevation in δ15N with increasing snout-to-tail length (a proxy for age) only in juvenile males. This discrepancy probably results from the imbalance of the female data compared to males: few time-series for females spanned more than 4 years (Table S1, Figs S6 and S7). The limited number time-series spanning more than 4 years means that the male pattern largely dominates the population-level pattern in our hierarchical model. Thus blood isotopic data is more reliable to infer the female pattern (Bailleul et al. 2010), although the dentin isotopic analysis suggested that a few females too underwent an elevation in trophic position as they aged (i.e. individuals with increasing slope after the ontogenetic shift; Fig. 2, Figs S6 and S7).
The explicit modelling of correlations between parameters governing a broken-stick model for both δ13C and δ15N values allowed us to investigate whether ontogenetic shifts in foraging habitat and trophic level were concomitant. There was a very small positive correlation between the ages at shift. The explicit incorporation of this correlation into the model did not substantially improve its predictive ability for δ13C or for δ15N values (Table 1). There seemed to be such a large variability in individual trajectories of foraging strategy and trophic level in this population that there is no meaningfull ’average’δ13C profile associated with an ‘average’δ15N profile.
Finally, the hierarchical modelling approach enabled us to assess whether there were differences between sexes and between seals living before and after a population crash. The data at hand suggested none (Figs S4 and S5), but the Bayesian framework is explicit about inferences being drawn conditional on the observed data. Thus, failure to detect any differences in this peculiar data set may stem for the imbalance between males and females (respectively 70% vs. 30% of seals), and between animals living before and after the population crash (respectively 28% vs. 72% of seals).
We believe that the piecewise linear formulation of our change-point model is biologically sound for this species since the change-points reflect life-history events such as complete independence from maternal resources or commitment to a foraging strategy. This assumed model suggested gradual changes after a shift (non-null slopes), which we deemed to be reasonable with longitudinal isotopic data. The interpretation of isotopic data in ecology crucially depends on the rate of tissue turn-over/synthesis, and the accuracy (not the precision) of isotopic data can be quite crude depending on the sampled tissue. Turn-over rates may be very short for some tissues (for example blood plasma), but one order of magnitude larger for others (for example claws) (Carleton et al. 2008). These rates also scale with body mass (Carleton & Martìnez del Rio 2005), which may allow to use experimentally-estimated rates from one species on similar-sized species. However, this is still somewhat of a blackbox for wild animals (Wolf, Carleton & Martìnez del Rio 2009).
Assumptions are unavoidable, but the Bayesian framework is very flexible, allowing to fit models to peculiar data sets rather than ‘adjusting the data to fit the model’. The broken-stick model we assumed reasonable for Southern Elephant Seal need not be so for other species. With little modification in the prior specification of the covariance matrix, non-linear functional responses such as a logistic curve, which also has four parameters, can be easily fitted. However, a logistic curve carries also assumptions such as symmetry and asymptotic isotopic values at the end of the time scale. Finally, the broken-stick model was useful for estimating individual shifts for Southern Elephant Seals, but it did not accommodate cyclic-patterns discernible during the first years in some individuals (Fig. S6). The broken-stick model lumped these cycles into a residual variance which was larger in early life compared to late life.
Carbon and nitrogen stable isotope analyses are a powerful technique to peek into the ecology of cryptic species: even a cursory glance at the plethora of studies using this technique cannot fail to notice how often ‘stable isotopes revealed’ biological surprises. The technique is hailed as powerful, which it is even more so conditional on using statistical analyses specifically designed to investigate a particular question (see for example Hénaux et al. 2011). Here, we presented a hierarchical model to investigate individual patterns of ontogenetic shifts in foraging habitat and trophic level (Werner & Gilliam 1984). The most important aspect of the model is not the specification of the mean response, which can readily be modified to conform to the biology of the studied species, but of the covariance structure. The methodology we outlined can be useful for drawing inferences at the individual level (Cooch, Cam & Link 2002; Semmens et al. 2009). Bayesian methods allow to fit with relative ease sophisticated models, and thereby to accommodate the (usually complex) structure of ecological data (Ellison 2004; Clark 2005). This move towards Bayesian methods is not confined to ecology (O'Hara et al. 2008; Link & Barker 2010) or even the biological sciences (Treier & Jackman 2008; Wainer 2010). Rather, it stems for a growing realization that uncertainties need to be quantified and to flow freely across different levels of an analysis to avoid overconfident claims. As more data become available, more complex models can also be fit to refine our knowledge (Gelman & Shalizi 2010). The modelling approach outlined here can be further extended to incorporate, for example, a survival analysis (Guo & Carlin 2004; Vonesh, Greene & Schluchter 2006; Horrocks & van Den Heuvel 2009) of Southern Elephant Seals to assess the life-history consequences of a foraging strategy; thereby harnessing the power of stable isotope analyses.
We would like to thank all volunteers who helped collecting teeth from dead animals found on îles Kerguelen. We thank the Museum National d'Histoire Naturelle (Paris) for kindly providing teeth collected before the population crash. We thank Hubert Vonhof and Els Ufkes for discussing the results. We are also indebted to Christophe Barbraud, Emmanuelle Cam, Luca Börger, two anonymous reviewers and the associate editor for helpful and constructive comments that greatly improved the manuscript. This study is part of a national research program (no. 109, H. Weimerskirch and the observatory Mammifères Explorateurs du Milieu Océanique, MEMO SOERE CTD 02) supported by the French Polar Institute (Institut Paul Emile Victor, IPEV). The Territoire des Terres Australes et Antarctiques Françaises (TAAF), the TOTAL Foundation and ANR-VMC 07 IPSOS-SEAL program contributed to this study. The ethics committee of the French Polar Institute (IPEV) approved this study. All animals in this study were cared for in accordance with its guidelines. This is Publication ISE-M no. ISEM 2011–127.