Jackknife-corrected parametric bootstrap estimates of growth rates in bivalve mollusks using nearest living relatives

Theoretical Population Biology 90 (2013) 36–48

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Jackknife-corrected parametric bootstrap estimates of growth rates in bivalve mollusks using nearest living relatives Troy A. Dexter ∗ , Michał Kowalewski Florida Museum of Natural History, University of Florida, 1659 Museum Road, P.O. Box 117800, Gainesville, FL 32611-7800, United States

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Article history: Received 5 February 2013 Available online 23 September 2013 Keywords: Growth rates Marine bivalves Parametric bootstrap Sclerochronology von Bertalanffy

abstract Quantitative estimates of growth rates can augment ecological and paleontological applications of bodysize data. However, in contrast to body-size estimates, assessing growth rates is often time-consuming, expensive, or unattainable. Here we use an indirect approach, a jackknife-corrected parametric bootstrap, for efficient approximation of growth rates using nearest living relatives with known age–size relationships. The estimate is developed by (1) collecting a sample of published growth rates of closely related species, (2) calculating the average growth curve using those published age–size relationships, (3) resampling iteratively these empirically known growth curves to estimate the standard errors and confidence bands around the average growth curve, and (4) applying the resulting estimate of uncertainty to bracket age–size relationships of the species of interest. This approach was applied to three monophyletic families (Donacidae, Mactridae, and Semelidae) of mollusk bivalves, a group characterized by indeterministic shell growth, but widely used in ecological, paleontological, and geochemical research. The resulting indirect estimates were tested against two previously published geochemical studies and, in both cases, yielded highly congruent age estimates. In addition, a case study in applied fisheries was used to illustrate the potential of the proposed approach for augmenting aquaculture management practices. The resulting estimates of growth rates place body size data in a constrained temporal context and confidence intervals associated with resampling estimates allow for assessing the statistical uncertainty around derived temporal ranges. The indirect approach should allow for improved evaluation of diverse research questions, from sustainability of industrial shellfish harvesting to climatic interpretations of stable isotope proxies extracted from fossil skeletons. © 2013 Elsevier Inc. All rights reserved.

1. Introduction Body size is one of the most fundamental biological variables that can be estimated quantitatively for most organisms, both extant and fossil, using direct measurements. In contrast, comparably fundamental biological parameters – age of individuals and growth rates controlling age–size relationships within populations – are less trivial to estimate. Moreover, because growth rates can be affected by environmental conditions and tend to vary within and across species, determining growth rates for any given population may require new field or laboratory data, even for extant groups for which pre-existing growth rate estimates are available in the literature. Estimating age–size relationships is even more challenging for fossils even if closely related extant species exist. In this paper, we focus on three monophyletic families of bivalves (Donacidae, Mactridae, and Semelidae) to explore an indirect approach for estimating age and growth rates of individuals and populations for which direct field or laboratory data are not available.

∗

Corresponding author. E-mail addresses: [email protected], [email protected] (T.A. Dexter).

0040-5809/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.tpb.2013.09.008

In this approach, growth–size relationships are approximated indirectly, by applying a resampling approach to literature-based growth rate estimates reported for populations of nearest living relatives inhabiting comparable environments (i.e. similar temperatures, seasonality, water chemistry, nutrient levels, etc.). A two-pronged rationale motivates this study. First, estimates of growth rates (both for individuals and populations) have wide applications within biological sciences and beyond. For example, in applied and environmental contexts, from conservation biology to aquaculture industry, even an approximate knowledge of growth rates can help to assess the short-term ecological changes, recovery times from environmental disturbances and pollution, or maximum sustainable yields in food sources (Beverton and Holt, 1957; Chapman, 1961; Saucedo and Monteforte, 1997; Arneri et al., 1998; Gaspar et al., 1999; Melchor-Aragón et al., 2002; Laudien et al., 2003; De Nóbrega and Lessa, 2009). Similarly, in geochemical and paleoclimatologic studies, biomineralized skeletons (e.g., mollusk shells) are sampled along their growth axes to obtain diverse geochemical proxies of various environmental and climatic parameters (Romanek and Grossman, 1989; Ivany et al., 2000; Schöne et al., 2002; Goodwin et al., 2003; Dettman et al., 2004; Carré et al., 2005; Carroll et al., 2006; Riascos, 2006). Such geochemical data

T.A. Dexter, M. Kowalewski / Theoretical Population Biology 90 (2013) 36–48

often represent a spatial series of micro-samples collected along the growth axis of the skeleton. Consequently, an understanding of growth rates of sampled skeletons can be critical when interpreting extracted patterns (e.g., do environmental fluctuations suggested by changes in geochemical proxies record sub-seasonal, multi-seasonal, or multi-decadal records?). Another obvious application of indirect growth rate estimates involves paleontological samples, where growth estimates cannot be measured directly and often can only be inferred using other approaches such as stable isotope sclerochronology (e.g., Steuber, 1996, Kirby et al., 1998), an indirect approach requiring substantial lab work that is often beyond the scope of primary research. The second rationale for this study is the fact that whereas direct growth estimates are possible in many cases, such substantial data collecting is often not feasible. Admittedly, the indirect method proposed here is much less precise and less accurate than the direct methods applicable to biomineralizing organisms such as notching, tag/recapture, length/frequency population distributions and sclerochronological techniques (see methods in Pearson and Munro, 1991, Mitchell et al., 2000, Melchor-Aragón et al., 2002). These direct approaches are typically major undertakings that are not only time-demanding and field-intensive, but also ecologically invasive. While the approach proposed herein is not meant as a replacement to field study of growth rates, many studies require an understanding of the age of specimens based on size while the growth rates in particular do not represent the primary research target. A key advantage of the proposed approach is that for such studies an indirect estimate will suffice and can be applied when direct measurement techniques are not applicable (e.g., the fossil record). The approach requires only limited data harvesting and analytical time while providing growth rate estimates that may be adequate for many biological, paleontological, and geochemical applications. Here, we apply indirect estimates to evaluate growth rates of species from three families of marine mollusk bivalves. Indirect estimates of growth rates in bivalve mollusks are particularly feasible because there is a sizable collection of published data available on the growth rates of this taxon, due in large part to the numerous practiced methods of ascertaining age and growth rate as mentioned above. Another reason for the abundance of published bivalve growth rate data is the wide range of fields for which growth estimates are relevant, including marine biology, ecology, paleontology, geochemistry, paleoclimatology and the shellfish industry. The usefulness and veracity of resulting indirect estimates for the examined bivalve mollusks in this study is then assessed using two published reports on bivalve species for which the geochemical data were collected along the shell growth axis. These two case examples include one species with an unknown growth rate with age estimates acquired using oxygen isotope data and one species where growth rates were known and could be used to independently test the indirect estimates derived here. In addition, we illustrate briefly how indirect growth estimates can augment the fisheries research and inform harvesting and sustainability practices. 2. Indirect resampling approach for estimating growth rates 2.1. Generalized protocol We propose a four-step protocol for indirectly estimating the growth rate of a specimen (or a monospecific sample of specimens) with unknown age–size relationship using nearest living relatives for which growth rates are available: 1. empirical growth rates reported previously for the same or closely related species (preferably congeneric species) are compiled from the literature. These estimates should represent

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comparable environmental settings to those from which the evaluated sample was collected. A maximally exhaustive literature search of all relevant taxa should be conducted (in most cases the number of reported growth curves will be relatively limited and will need to be restricted further to studies reporting the usable growth rate metrics from relevant environmental settings). The resulting dataset provides a sample of empirical growth curves for the nearest living relatives of the species that is being estimated. 2. an average growth curve of the nearest living relatives is computed using some measure of central tendency applied to empirical growth–size data. The growth rate metric used most commonly across the compiled studies should be targeted to maximize the number of usable growth curves and derive maximally robust growth rate estimates. In a case example used below, a parametric approximation (3-parameter von Bertalanffy growth functions) was used to summarize growth curves and compute average curves. 3. a resampling strategy is applied to estimate the standard errors and confidence bands around the average growth curve obtained in Step 2. A parametric bootstrap (based on resampling of von Bertalanffy parameters in this study) is used to estimate an expected sampling distribution of growth curves. In addition, a ‘‘leave-one-out’’ (jackknife-style) correction is applied to the growth curves. By iteratively removing individual populations and comparing the remaining growth curves, this jackknifestyle resampling method provides larger error rate estimates. The jackknife correction, by evaluating sequentially each species/population with known growth rates as if it were unknown, generates more conservative and realistic estimates of error when estimating growth rates for an unknown specimen. 4. the jackknife-corrected bootstrap estimates are then applied to the sample of interest to convert size data into age data. We recommend that 2.5th and 97.5th percentiles of jackknifecorrected sampling distributions be applied to estimate growth rate uncertainties for the specimen(s) of interest. The standard error of the estimate would also be an applicable (if less conservative) strategy for assessing errors around the growth rate estimates. The proposed four-step protocol assumes that the empirically estimated growth rates of nearest living relatives and the unknown growth rates of the species of interest came from the same underlying population of possible growth rates. In practice, the generalized protocol outlined above will vary in detail when applied to specific datasets. In particular, decisions have to be made regarding: (1) the method of measuring growth rates; (2) an appropriate estimate of central tendency; and (3) practical details of resampling procedure. Below, we discuss those decisions in the context of mollusk bivalves characterized by accretionary skeletons and indeterministic growth. Following the analysis, post-hoc evaluations of resulting indirect estimates are recommended, including sensitivity analyses and cross-testing against empirical case studies (example of such post-hoc evaluations are illustrated later in this report). 2.2. Measuring growth rates Whereas direct empirical curves can be generated in many cases to describe the relation between age and size of specimens, researchers frequently use parametric approximations (or growth functions) that provide a continuous representation of growth rates and can be summarized in terms of a few parameters. These parametric measures of growth rates are often reported in individual studies and facilitate comparisons across studies (especially when considering that empirical methods of deriving growth rates vary across studies even for closely related taxa). Also, the parametric functions provide an attractive system for developing statistical estimates of growth rates, as outlined in our 4-step protocol above.

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T.A. Dexter, M. Kowalewski / Theoretical Population Biology 90 (2013) 36–48

Numerous procedures have been developed to determine a specimen’s age at a given body size, including tag/recapture, length/frequency population distributions, and skeletal component measurements (e.g., growth banding of mollusk shells or inner ear fish otoliths). The tag/recapture approach measures the rate of growth directly against empirically established population age data. Length/frequency distributions are collected periodically through the year and shifts in modes observed in size–frequency distributions through time are then used to estimate rates of growth. Skeletal components often develop in pulses that leave behind traces (banding) that can then be correlated to seasonal fluctuations. Data collected on the age/body size from these procedures can then be fit to a given growth rate model to parameterize the species general growth rate. Numerous models have been developed to determine the relationships between ontogenetic age and size of organisms (Ricker, 1979). These models represent a wide range of mathematical functions, from linear to logistic, applied to fit various age–size relationships observed in macroscopic life. They include among others the isometric growth model of Huxley and Teissier (1936) and the asymptotic growth model of Von Bertalanffy (1938, 1957). For most organisms, a maximum biological limit on size imposes a boundary condition that can often be approximated efficiently using logistic models, such as those of Blumberg (1968), Gompertz (1825), Richards (1959), or Von Bertalanffy (1938; see also Ricker, 1975, Schnute, 1981, Tsoularis, 2001). These models are often manipulated or redesigned to fit a particular organism or to dynamically model time for seasonal or periodic growth; a growth pattern common to many organisms (Pitcher and Macdonald, 1973; Cloern and Nichols, 1978; Fontoura and Agostinho, 1996). The von Bertalanffy growth function (further referred to as VBF) was selected in this study as a means of relating age and size of targeted organisms (i.e., bivalve mollusks), though other growth models would also be applicable. This model was selected for multiple reasons. First, it has been used widely by mollusk researchers (i.e., a wealth of growth rate data reported in terms of VBF parameters is available in the literature). This practical criterion is particularly important: the efficacy of the indirect estimation proposed here and the statistical quality of resulting estimates improves with the availability of applicable literature datasets for closely related sets of species. Second, the model is easy to implement into resampling algorithms because it represents a 3-parameter function; note here that a variety of programs are available to fit collected growth rate data to the VBF model (as well as other models) and then calculate these defining parameters. Third, and most important, VBF provides an excellent approximation of empirical growth curves for bivalve mollusks. As organisms with indeterminate growth, bivalve mollusks have a relatively rapid rate of shell accretion early in their ontogeny, followed by a tapering off of accretion later in ontogeny as an increasingly larger volume of shell material is required for an equivalent change in length. This pattern of growth fits the asymptotic VBF model as rate of growth slows while moving toward the parameterized maximum length. Admittedly, there are multiple external sources of variation that can alter the rate of growth between populations within a species or individuals in a population. Ambient temperature, environmentally triggered shifts in onset of sexual maturation, parasitism, and other extrinsic factors may affect growth rates and make a simple mathematical function, such as VBF, an unrealistic proxy. Furthermore, the VBF is calculated from age data that often do not include the earliest stages of an organism’s ontogeny. Missing stages of an organism’s ontogeny, such as the early post-larval stages in the case of bivalves, will be less constrained and subject to greater error. However, numerous empirical studies demonstrating satisfactory performance of VBF and averaging across multiple samples, on which the approach proposed here relies, support viability of the estimates. The von Bertalanffy growth rates are modeled using three variables; L∞ , K , and t0 , where L∞ is the asymptotic maximum

length (in mm), K is the growth constant (per year), and t0 is the age at zero length (in years). The variable t0 does not necessarily need to be zero and may account for post-displacement of development or the lack of mineralized shell material during early larval stages. These three variables allow one to determine the expected length at a given time (Lt ): Lt = L∞ (1 − e−K (t −t0 ) ). To calculate these von Bertalanffy growth rate parameters, many authors use ELEFAN (electronic length frequency analysis; Pauly and David, 1981) of the FiSAT II software package (www.fao. org/fishery/topic/16072/) which is particularly appropriate since length frequency data is a method commonly used to estimate rate of growth. A few authors have used different software packages R such as SAS⃝ (www.sas.com), MULTIFAN-CL (www.multifan-cl. R org), or SYSTAT⃝ (www.systat.com), or developed their own algorithms to conduct the nonlinear regressions of their data in order to calculate VBF variables or other growth model parameters (Ansell and Lagardère, 1980; Cranfield et al., 1996; Urban, 1998; Gaspar et al., 1999; Laudien et al., 2003; Riascos, 2006). For this project, time was converted from years to months to improve resolution as the asymptotic growth of the bivalve species targeted here achieve maximum length after only a few years. For animals with greater longevity, it would be appropriate to maintain calculated ages in terms of years. The previous formula was also reformulated to estimate the expected age (t) in years as a function of a given length (Lt ):

 t =

ln(1 − (Lt /L∞ ))

−K



+ t0 .

The formulas require that the maximum length (L∞ ) be greater than any given length (Lt ) of a specimen under consideration. This requirement may not always be met when multiple species are analyzed simultaneously because some of those species may have maximum lengths that are shorter than the length of specimens that are being estimated using the indirect approach. By using this particular parametric approach, those species must be removed from the group of all curves as it is impossible to calculate a negative natural logarithm. This may not be the case using metrics from alternative parametric growth models and would not pose a problem when applied to direct measures of size and age (raw data). The reliability of our parametric estimates provided by the variant of the 3-parameter VBF function, in the specific context of the data presented in this study, is further evaluated below. 2.3. Estimating central tendency for sets of growth rates When estimating an average curve from a set of growth functions, two approaches can be readily employed: (1) in the parametric approach, an average VBF function can be computed by averaging the three individual VBF parameters (or alternative model parameters) across all included species (i.e., L∞ , K , and t0 are estimated as arithmetic means of L∞ , K , and t0 of the nearest living relative species used in a given model); (2) in the incremental approach, an average growth curve can be calculated incrementally for each size class, with average body length calculated at discrete time steps. In the case of bivalves, these species-level curves can be plotted in terms of shell length in mm vs. time expressed in months (Fig. 1(A)–(C)). The average curve can then be computed by averaging the shell length values across species at monthly increments for a relevant time span of growth (36 months in the case of data analyzed here). For both of these approaches, averaging can be performed by using various descriptors of central tendency. Here, initially, we use two different common metrics: the arithmetic mean and the median, and apply each metric to both the parametric and incremental estimates for four proxies total (Fig. 1(D)–(F)).

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Fig. 1. Empirical literature derived growth rate curves (A–C) and average curves calculated using the von Bertalanffy growth functions (D–F) for three families of bivalve species. Four different ways were used to calculate the average growth curve for a given family. Mean/month and median/month incremental estimates were calculated by determining the expected length of each of the growth curves for each month and taking the mean and median length of those curves. Mean/formula and median/formula were calculated by taking the mean and median of the variables (L∞ , K , and t0 ) for all of the von Bertalanffy growth functions within that family and creating a new, averaged VBF formula. (A) and (D) Donacidae, (B) and (E) Semelidae, and (C) and (F) Mactridae.

For the three families of bivalve mollusks analyzed in this study, these four proxies of central tendencies provided remarkably consistent results (Fig. 1(D)–(F)) suggesting that specific datasets analyzed here are reasonably insensitive to the way in which the average curve is estimated. However, the parametric method is far more efficient to apply: it simply involves computing arithmetic mean (or median) for each of the three VBF parameters. The simplicity of this parametric approach is particularly attractive when implementing iterative resampling techniques. Note that, the application of the incremental approach in resampling simulations would require modeling average growth curves by incremental averaging at each discrete (monthly) time steps, adding

computational time, discretizing rate estimates, and introducing additional error by back estimating the VBF parameters from the generated curves. Though more complicated, the incremental method can be applied to models that lack a simple set of descriptive parameters or can be applied in situations where growth rates are given as a direct measure of body size to age (raw data) rather than being fit to a model. Because the two approaches yielded very similar results for all three bivalve groups in this study (for both arithmetic mean and the median) only the first approach (based on direct averaging of the VBF parameters) was used in the simulations (Fig. 1). Also, because the arithmetic mean and the median yield consistent estimates of the average VBF curves (Fig. 1(D)–(F)),

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and because the arithmetic mean tends to behave more smoothly in resampling simulations (e.g., Kowalewski and Novack-Gottshall, 2010), we employ the arithmetic mean in all subsequent analyses. 2.4. Resampling procedure Two resampling strategies have been employed. First, a parametric bootstrap protocol (resampling with replacement of empirical VBF parameters obtained from the literature; see above) was employed to estimate the mean growth curve and standard errors and confidence bands for each of the three groups of bivalves. As in the case of misclassification rates in discriminant analysis, these errors are apparent because all growth curves are included in the analysis. To address this problem, a second approach, a jackknifecorrected parametric bootstrap, was implemented by sequentially removing each species curve from the dataset and conducting a parametric bootstrap on the remaining species. At each iteration, the offset between the mean growth curve and the growth curve for the removed species was then used to compute the error. The offsets were then compiled for all sequentially removed species to derive a sampling distribution from which standard errors and confidence bands can be derived. Note that when the mean is calculated using a group of five species for example and one of the extreme growth curves is removed as the ‘‘unknown species’’, the mean is no longer pulled toward that extreme curve and offsets (errors) inevitably increase (when the removed curve is close to the mean, the increase is less notable, but still present). The jackknifecorrected offsets represent a more conservative estimate of standard errors, and thus, jackknife-corrected confidence bands are expected to be wider than those estimated using the parametric bootstrap alone. Whereas the standard errors should differ between the two resampling protocols, the estimates of the arithmetic mean of the sampling distributions of growth curves should converge for the two approaches, with differences reflecting imprecision controlled by number of iterations. However, minor inaccuracies may result in jackknife-corrected means when the distribution of growth curves is strongly skewed. An accelerated bias correction may be recommended in such cases (this correction has not been applied here because the magnitude of this bias is trivial in the specific case of our data). 3. Material and methods A literature search was conducted to compile the growth rate estimates of marine bivalves for sets of closely related species. Three monophyletic families of bivalves (Donacidae, Mactridae, and Semelidae) were selected from within the order Veneroida in the subclass Heterodonta (Bouchet et al., 2010). Veneroids are a common, diverse, and widespread group within the bivalve clade. The three families selected for this study represent distinct morphologies within this order, making them a reasonably representative sample for bivalves. These three families were targeted because the published data for each of these groups included estimates for multiple species from similar environmental settings (Table 1). Some of the compiled studies included multiple sites, sampled and analyzed separately for the same species, thus providing multiple rate estimates. In some of the studies, different methods were used to analyze the age or the same age data were fit to different growth models. In those cases, authors reported VBF growth rate estimates derived by averaging across these different methods. In this analysis, we utilized average VBF growth rate rather than all the separate rates derived using different methods. For each site, the following variables were recorded: species name, site latitude and longitude, water depth, and the three reported VBF parameters. Average water temperature at the given depth of each site was acquired independently from the National

Oceanic and Atmospheric Administration’s National Oceanographic Data Center 2005 World Ocean Atlas (http://www.nodc. noaa.gov/OC5/WOA05/pr_woa05.html). Because a single species often had separate growth rates reported for multiple populations, the number of VBF growth estimates compiled here exceeds the total number of species included in the dataset. Because numerous studies reported growth rates for Donax, all species used here for the Donacidae belong to this single genus. For the other two groups, species from multiple genera were combined. The three groups are likely to be meaningful phylogenetically because their monophyly is well supported by molecular data (Taylor et al., 2007). For Donacidae, our data include 31 reported growth rates for the following species of Donax: D. cuneatus, D. denticulatus, D. dentifer, D. faba, D. hanleyanus, D. incarnates, D. obesulus, D. serra, D. striatus, D. trunculus and D. vittatus. For Mactridae, two genera (Mactra and Mesodesma), which are closely related (Taylor et al., 2007), were included producing 8 growth rates for the following species: Mactra discors, Mactra murchisoni, Mesodesma donacium, and Mesodesma mactroides. For Semelidae, three monophyletic (Taylor et al., 2007) genera (Abra, Gari, and Semele) yielded 11 growth rates for Abra alba, Gari solida, and Semele solida. Gari belongs to the family Psammobiidae, which is closely related to Semelidae (Taylor et al., 2007). Each of the groups in the dataset contains species growth rates collected from localities of similar environments. All populations were collected from tropical to subtropical water masses in sandy beach environments. Specimens of Donacidae were collected primarily from the tropics, whereas populations of Mactridae and Semelidae were collected primarily from subtropical waters. A few of the included populations may have experienced some seasonal temperature changes as they are on the edge of subtropical waters, such as a population of Donax striatus and D. trunculus from the northern Atlantic Ocean on the coast of France or Mesodesma mactroides from the southern Atlantic Ocean on the coast of Argentina. The average ocean temperatures and the associated standard deviations for the populations of Donacidae, Mactridae, and Semelidae are 21 ± 6 °C, 18 ± 4 °C, and 15 ± 2 °C, respectively. Populations used to calculate growth rates for the Donacidae and Mactridae groups were collected from shallow marine to intertidal beach sands. Populations of Semelidae were collected from the ocean floor at depths of 7 m or greater, with an average depth of 12 m. For each of the three targeted bivalve groups, the parametric bootstrap and jackknife-corrected parametric bootstrap statistics were derived by randomly reassigning the values of the VBF growth rate parameters and then recomputing age estimates using a size range from 5 to 20 mm at 0.5 mm increments. The jackknifecorrection was implemented by sequentially removing each of the published VBF rates from the total group and computing errors for the removed growth curve using simulation outcomes derived for the remaining species. For each dataset at each size increment, resampling simulations were carried out independently for parametric bootstrap and jackknife-corrected parametric bootstrap. The iteration outputs were binned by one month increments. Pilot runs indicated that standard error estimates stabilized and distribution means became reasonably convergent at around 500 iterations. In final analyses, 1000 iterations were used for each simulation. 4. Results There were 31 species-level growth curves for the Donacidae dataset, 11 for the Semelidae dataset, and 8 for the Mactridae dataset, respectively. All of the growth curves for the three families were plotted in millimeters using monthly increments from 0 to 36 months (Fig. 1). For the family Donacidae most of the curves follow

b

21.47 22.87 33.02 31.47 30.02 29.30 23.40 26.30 28.60 46.63 26.15 44.00 28.50 33.00 28.82 30.94 46.00 82.00 82.00 38.00 25.10 20.20 35.99 47.30 48.90 52.80 41.80 35.55 39.70 47.56 35.90 31.28 89.60 86.10 106.30 86.00 101.60 77.50 68.00 88.00 110.00 83.00 75.47 70.42 77.73 79.13 78.00 85.30 100.60 67.00

Linf (mm) 0.67 0.06 0.04 1.48 1.79 0.62 2.12 1.76 0.84 0.30 0.10 0.47 0.90 0.80 0.09 0.16 1.00 0.27 0.28 0.68 1.16 0.29 0.96 0.58 0.38 0.55 0.71 0.79 0.77 0.30 1.01 1.06 0.31 0.35 0.19 0.70 0.49 0.67 0.41 0.57 0.38 0.82 0.90 0.54 0.42 0.47 0.30 0.19 0.13 0.77

K (yr−1 )

−0.25 0.00 0.00 0.50 0.20 0.00 0.00 0.00 0.00 0.69 0.00 0.00 0.00 0.00 0.00 0.04 0.00 0.00 0.00 0.71 −0.03 0.30 −0.70 0.52 0.29 0.52 0.35 0.00 0.00 0.00 0.00 0.00 0.35 0.00 0.00 0.00 0.65 0.00 0.20 0.10 0.00 0.00 −0.04 −0.36 −0.51 −0.45 0.37 0.00 0.00 0.00

T (°C) 14.0 28.2 27.4 27.5 26.1 25.7 25.7 25.7 25.7 28.4 27.3 15.2 22.9 17.8 21.1 28.6 27.6 15.9 14.9 12.6 27.3 27.2 17.1 18.3 12.6 15.2 17.3 15.4 15.4 19.1 15.4 15.4 13.1 13.1 13.1 17.2 16.6 16.6 15.0 15.0 27.6 17.8 17.8 15.3 14.3 14.3 13.1 13.1 13.1 17.2

t0 (yr−1 ) 39 <1 <1 <1 <1 15 15 15 15 <1 <1 <1 <1 <1 <1 <1 <1 2 <1 <1 <1 <1 <1 6 <1 <1 2.5 <1 <1 <1 <1 <1 5 5 5 10 20 20 7 7 <1 <1 <1 <1 <1 <1 5 5 5 10

Depth (m)

Location Atlantic Ocean, France Indian Ocean, India Indian Ocean, India Atlantic Ocean, Venezuela Caribbean Sea, Venezuela Pacific Ocean, Columbia Pacific Ocean, Columbia Pacific Ocean, Columbia Pacific Ocean, Columbia Pacific Ocean, Costa Rica Indian Ocean, India Pacific Ocean, Argentina Atlantic Ocean, Brazil Atlantic Ocean, Uruguay Atlantic Ocean, Brazil Indian Ocean, India Pacific Ocean, Peru Atlantic Ocean, Namibia Atlantic Ocean, South Africa Atlantic Ocean, France Atlantic Ocean, Brazil Caribbean Sea, Venezuela Mediterranean, France Atlantic Ocean, Portugal Atlantic Ocean, France Atlantic Ocean, Spain Mediterranean Sea, Spain Atlantic Ocean, France Atlantic Ocean, France Adriatic Sea, Italy Atlantic Ocean, France Atlantic Ocean, France Pacific Ocean, Chile Pacific Ocean, Chile Pacific Ocean, Chile Pacific Ocean, Peru Pacific Ocean, Peru Pacific Ocean, Peru Pacific Ocean, New Zealand Pacific Ocean, New Zealand Pacific Ocean, Peru Pacific Ocean, Urugauy Pacific Ocean, Urugauy Pacific Ocean, Argentina Pacific Ocean, Argentina Pacific Ocean, Argentina Pacific Ocean, Chile Pacific Ocean, Chile Pacific Ocean, Chile Pacific Ocean, Peru

Latitude/longitude 45° 35′ N, 1° 15′ W 9° 17′ N, 79° 05′ W 16° 98′ N, 73° 17′ E 10° 41′ N, 63° 46′ W 11° 00′ N, 64° 00′ W 4° 05′ N, 77° 16′ W 4° 05′ N, 77° 16′ W 4° 05′ N, 77° 16′ W 4° 05′ N, 77° 16′ W 8° 56′ N, 83° 37′ W 9° 15′ N, 79° 08′ E 37° 19′ S, 57° 00′ W 23° 03′ S, 43° 34′ W 33° 40′ S, 53° 20′ W 29° 38′ S, 49° 56′ W 12° 27′ N, 74° 48′ E 12° 24′ N, 76° 46′ W 22° 47′ S, 14° 33′ E 32° 19′ S, 18° 21′ E 48° 05′ N, 4° 19′ W 3° 42′ S, 38° 27′ W 10° 46′ N, 68° 19′ W 43° 51′ N, 5° 23′ E 36° 57′ N, 7° 56′ W 48° 05′ N, 4° 19′ W 43° 43′ N, 7° 42′ W 39° 10′ N, 14° 20′ W 45° 50′ N, 1° 15′ W 45° 50′ N, 1° 15′ W 41° 55′ N, 15° 27′ E 45° 50′ N, 1° 15′ W 45° 50′ N, 1° 15′ W 36° 32′ S, 73° 57′ W 36° 32′ S, 73° 57′ W 36° 32′ S, 73° 57′ W 14° 15′ S, 76° 10′ W 14° 15′ S, 76° 14′ W 14° 15′ S, 76° 14′ W 41° 26′ S, 174° 07′ E 41° 26′ S, 174° 07′ E 12° 24′ N, 76° 46′ W 33° 40′ S, 53° 20′ W 33° 40′ S, 53° 20′ W 39° 00′ S, 61° 15′ W 40° 36′ S, 62° 11′ W 40° 36′ S, 62° 11′ W 36° 32′ S, 73° 57′ W 36° 32′ S, 73° 57′ W 36° 32′ S, 73° 57′ W 14° 15′ S, 76° 10′ W

von Bertalanffy growth paramters calculated from length frequency distributions by Herrmann et al. (2009). Gari belongs to the family Psammobiidae, which is closely related to Semelidae. Gari is included with Semelidae in all analyses presented here.

Semelidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Donacidae Psammobiidaeb Psammobiidaeb Psammobiidaeb Psammobiidaeb Psammobiidaeb Psammobiidaeb Mactridae Mactridae Mactridae Mactridae Mactridae Mactridae Mactridae Mactridae Semelidae Semelidae Semelidae Semelidae

Abra alba Donax cuneatus Donax cuneatus Donax denticulatus Donax denticulatus Donax dentifer Donax dentifer Donax dentifer Donax dentifer Donax dentifer Donax faba Donax hanleyanus Donax hanleyanus Donax hanleyanus Donax hanleyanus Donax incarnatus Donax obesulus Donax serra Donax serra Donax striatus Donax striatus Donax striatus Donax trunculus Donax trunculus Donax trunculus Donax trunculus Donax trunculus Donax trunculus Donax trunculus Donax trunculus Donax vittatus Donax vittatus Gari solida Gari solida Gari solida Gari solida Gari solida Gari solida Mactra discors Mactra murchisoni Mesodesma donacium Mesodesma mactroides Mesodesma mactroides Mesodesma mactroides Mesodesma mactroides Mesodesma mactroides Semele solida Semele solida Semele solida Semele solida

a

Family

Species Bachelet and Cornet (1981) Nayar (1955)a Talikhedkar et al. (1976)a García et al. (2003) Marcano et al. (2003) Riascos and Urban (2002) Riascos (2006) Riascos (2006) Riascos (2006) Palacios et al. (1983) Alagarswami (1966)a Herrmann et al. (2009) Cardoso and Veloso (2003) Defeo (1996) Gil and Thomé (2000) Thippeswamy and Joseph (1991) Arntz et al. (1987) Laudien et al. (2003) de Villiers (1975) ´ Guillou and Le Moal (1980) Rocha-Barreira et al. (2002) McLachlan et al. (1996) Bodoy (1982) Gaspar et al. (1999) ´ Guillou and Le Moal (1980) Mazé and Laborda (1988) Ramón et al. (1995) Ansell and Lagardère (1980) Ansell and Lagardère (1980) Zeichen et al. (2002) Ansell and Lagardère (1980) Ansell and Lagardère (1980) Urban and Campos (1994) Urban and Campos (1994) Urban and Campos (1994) Urban (1991) Urban (1998) Urban (1998) Cranfield et al. (1996) Cranfield et al. (1996) Arntz et al. (1987) Defeo et al. (1992a) Defeo et al. (1992b) Fiori and Morsán (2004) Fiori and Morsán (2004) Fiori and Morsán (2004) Urban and Campos (1994) Urban and Campos (1994) Urban and Campos (1994) Urban (1991)

Reference

Table 1 Species selected for this study and their associated phylogenetic affiliations. Table includes the three von Bertalanffy growth rate parameters for each species (L∞ is the asymptotic maximum length, K is the growth constant, and t0 is the age at zero length), as well as site location, depth, and average annual water temperature at the each locality.

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Fig. 2. Average age values and their confidence bands. The dark squares are the means calculated from the VBF formulae for each family. The open circles are the means calculated from the iterations of the parametric bootstrap statistic. The gray triangles are the average of the jackknife-corrected mean ages. The dashed lines are the 2.5% and 97.5% percentiles calculated using the parametric bootstrap statistic. The solid lines are 2.5% and 97.5% percentiles calculated using the jackknifecorrected parametric bootstrap. (A) Donacidae; (B) Semelidae; and (C) Mactridae.

similar growth trajectories (Fig. 1(A)). There were a few shallow curves; D. cuneatus, D. faba, D. incarnatus, and D. striatus grew at a much slower pace and attained smaller maximum body size when comparing with other Donax populations included here (Fig. 1(A)). At 24 months of age, the populations of Donacidae ranged from 3 to 35 mm in terms of calculated VBF body size. Semelidae displayed interspecific variation in growth rates comparable to that observed for Donacidae (Fig. 1(B)). At 24 months of age, populations of Semelidae ranged from 17 to 65 mm in body size. In contrast, at 24 months of age, the populations of Mactridae species ranged from 35 to 67 mm in body size displaying the narrowest range of growth rates (Fig. 1(C)). However, this is not surprising given the notably lower number of growth curves compared with the two other datasets. There is little difference across the four metrics of central tendency (see above) that were used to estimate the average growth curve (Fig. 1(D)–(F)). Thus, all four metrics are applicable for the datasets used here. However, for reasons explained above in the material and methods section, the VBF formula-based metric is used to compute the average growth curves (estimated using an arithmetic mean) in all simulations presented below.

The mean growth curves calculated for the actual datasets, parametric bootstrap samples, and jackknife-corrected parametric bootstrap samples all yielded highly consistent estimates (Fig. 2). Jackknife-derived means appear to slightly overestimate the bootstrap means (Fig. 2) because all three sets of distributions are right skewed (i.e., the effect of outliers on estimates of mean are more pronounced when n − 1 curves are resampled). However, this offset is trivial in magnitude and irrelevant for the accuracy of estimates reported at the monthly resolution. When comparing the 2.5% and 97.5% confidence limits for the parametric bootstrap versus the jackknife-corrected parametric bootstrap, the jackknife-corrected parametric bootstrap confidence bands are wider, as expected for this more conservative metric (Fig. 2). The 97.5% confidence bands are more distinctly separated between the parametric bootstrap analysis and the jackknife-corrected parametric bootstrap analysis than are the 2.5% confidence bands (Fig. 2). In fact, the minimum age limits overlap for the Donacidae and Mactridae datasets (Fig. 2(A), (C)). In all cases, confidences widen as body size increases. For each of the three datasets, the distribution of resampled ages for a given body size class, calculated using the jackknifecorrected parametric bootstrap, centers around the mean growth curve, with the mode (note that the darkest boxes represent the highest frequencies of curves for a given size class on Fig. 3) typically coincident with the mean curve from the actual data (Fig. 3). The frequencies of resampled ages decrease away from the curve in both directions (Fig. 3). The shapes of sampling distributions vary across datasets reflecting differences in distribution of actual empirical growth curves in those datasets. Note, for example, that for Donacidae, for all size classes, the distributions are pronouncedly right skewed (a wide range of ages exceeding the mean curve and relatively narrow range of ages below the mean curve) (Fig. 3). In contrast, the distributions appear much more symmetrical for the other two datasets (Fig. 3). The resampled curves vary over a considerable range of ages within a given size class. For example, at shell length of 20 mm, the possible shell ages range between 6 and 30 months for Donacidae, 4 and 15 months for Semelidae, and 1 and 11 months for Mactridae, respectively (Fig. 3). However, this is the complete range of all possible ages for each iterative step, and not the 95% confidence bands presented in Fig. 2, which are notably narrower. Indeed, the frequency of curves suggesting very young or very old ages for any given size class is very low (white boxes), as most of the resampled growth curves cluster around the mean curve (dark gray and black boxes) (Fig. 3). 5. Sensitivity tests 5.1. Phylogenetic distance Because the number of growth rates reported in the literature is necessarily limited for conspecific populations, an exhaustive search of all appropriate populations from environmentally constrained settings was conducted up to family level. The inclusion of growth curves across multiple confamilial genera allowed for increasing the sample size, but may have introduced substantial phylogenetic variability. To assess the impact of phylogeny on growth rate estimates, absolute differences in length at a given age were compared for all possible pairs of populations at four taxonomic levels: within species, within genus, within family, and across families. These four levels served here as an ordinal proxy for the phylogenetic distance. Note that lower-level pairwise comparisons were excluded from higher-level comparisons (e.g., within-family comparisons excluded all within-species and within-genus comparisons within a given family). Comparisons were carried out at monthly growth increments (from 6 to 24 months) by computing

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Fig. 4. Difference in absolute median deviations in growth rates based on pairwise comparisons between populations compared at four taxonomic levels (used as proxies for phylogenetic distance). The four levels included in pairwise comparisons are within species (n = 81), within genera (n = 419), within families (n = 45), and between families (n = 676). See text for additional explanation.

in a given analysis may depend on the desired precision of the indirect growth estimates and is thus study specific. 5.2. Sample size and outlier effects

Fig. 3. Frequency plots for the jackknife corrected parametric bootstrap. These graphs show the age frequency distributions at a given length derived using jackknife-corrected parametric bootstrap. (A) Donacidae; (B) Semelidae; and (C) Mactridae. The solid white line shows the average curve for the specific family based on the mean of the VBF variables for all the formulae within the group. In all cases, the calculated mean curve follows the modal age class of the bootstrap distribution of possible growth curves.

absolute pairwise differences between all pairs of populations at a given level (Fig. 4). For each of the four comparison levels, median values were used to summarize an average pairwise absolute difference between populations for each monthly age class (e.g., median of 10 mm for within-species comparisons at 6 months would indicate that an average pair of conspecific populations differed by 10 mm for 6 month old specimens). For conspecific and congeneric populations, median pairwise differences were comparably small (∼7 mm) and age invariant (Fig. 4). In contrast, comparisons within families (intergeneric) and across families (i.e., within the order Veneroida to which all three families belong) yielded substantial median absolute differences in length and the observed differences increased progressively with age (Fig. 4). This outcome demonstrates a substantial influence of phylogeny on variation in growth rates (although environmental differences, especially water temperature, may have also partly contributed to difference across families) and suggests that it is desirable to assemble datasets that are limited to the most closely related species (ideally, congeneric or conspecific populations). However, the highest acceptable taxonomic rank to be included

Because the number of known growth curves is limited within each of the three datasets used here, it is useful to assess if the resulting estimates are stable as a function of sample size (i.e., the number of known growth curves). To examine this issue each of the three datasets was subjected to a rarefaction-style analysis. For each family, an age estimate at a given shell length was recorded for the whole dataset (n = all available growth curves). The shell length of 20 mm was used here because for larger/older shells the variability tends to be higher yielding more conservative estimates of errors (the simulations were also conducted for smaller shell lengths with similar results to those presented below). Subsequently, subsets of k curves were iteratively resampled without replacement while successively decreasing the number of retained curves from k = n − 1 down to k = 1 curves. A total of 1000 iterations was carried out at each k. It should be noted here that for values of k approaching n or 1 (especially k = n − 1 and k = 1) the number of possible combinations is much less than 1000 (e.g., for n = 30 and k = 29 there are 30 [30!/29!1!] combinations, but for k = 15 there are 155,117,520 [30!/15!15!] possible combinations). In most cases, 1000 iterations represent a Monte Carlo approximation of a systematic/exhaustive randomization, which would be feasible only for a few values of k. The absolute difference between the actual age estimate for the total data and the age estimate produced for a subsample of k curves was then computed for each iteration and the median absolute difference and the upper confidence limit (here 75th percentile was used) were calculated at a given k. In evaluating the sample size effects here, we use an offset of 3 months as the maximum acceptable absolute deviation, which may be deemed sufficient if the goal of a study is to distinguish between the sub-annual and multi-annual growth patterns. A 75th percentile rather than the median is then used to evaluate a sample size at which estimates stabilize within a 3-month error envelope. For Donacidae, the simulation (Fig. 5(A)) highlights the impact of a few outlier curves which induced huge errors when a small number of curves were resampled. However, once more than 10 curves were included in the analyses, the median and 75th percentile both drop below the 3 month cutoff value. Once >20 growth curves were retained, the estimates of median and 75th percentiles stabilized (the sharp fluctuations for k approaching n

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reflects volatility of median values when only few curves are being removed). Thus, Donacidae dataset appears adequate for providing growth rate estimates, despite the presence of outliers. For the two other datasets, the errors are much lower and less than 10 curves suffice to provide error estimates within 3 month error band: 6 curves for Semelidae (Fig. 5(B)) and 3 curves for Mactridae (Fig. 5(C)). However, these low error rates reflect absence of any notable outlier curves, which may be due to small sizes of those datasets. Thus, even though the error bands are much narrower for those two datasets, they yield estimates that should be considered more suspect than those provided by the Donacidae dataset. 6. Practical feasibility tests 6.1. Cross-evaluation against the previously published sclerochronological studies To assess the value of the indirect estimates of growth in mollusk bivalve species derived here, we selected two studies in which oxygen isotope ratios were analyzed for relevant bivalves with body size estimates (the distance from the umbo) recorded (Krantz et al., 1987; Etayo-Cadavid, 2010) (Fig. 6). These two studies were the first two reports with appropriate types of geochemical, environmental, and taxonomic data that we had encountered while conducting a literature search for oxygen isotopes and bivalves. Thus, they are not the best-case scenarios selected from among a variety of examples, but rather the first two cases that had the appropriate valve length and isotope data. The negative oxygen isotope values are re-plotted here inversely so that peaks indicate warm seasons in the growth history of a given bivalve specimen (Fig. 6). The specimen from EtayoCadavid (2010) was Donax obesulus and the specimen from Krantz et al. (1987) was Spisula solidissima. We evaluate these two specimens using our collected data for Donacidae and Mactridae, respectively. These two examples represent samples that came from environmental and climatic settings comparable to those represented by the datasets above. The work from Etayo-Cadavid (2010) was conducted on a sub-fossil specimen collected from Salaverry, Peru at 8°14′ S, 78°16′ W with average annual water temperatures of 19.8 °C. The average temperature of the specimens from the modified Donacidae dataset is 21 °C (although the paleotemperature at the time of life of this sub-fossil specimen cannot be directly determined, it is not expected to have been too far off from modern tropical waters and likely occurred within the range of temperatures represented by growth curves included in our modified Donacidae dataset). The specimen utilized in Krantz et al. (1987) was collected live from a site off of Virginia Beach, Virginia, USA at 36°57′ N, 75° 39′ W from a depth of 14 m. This site should experience a similar seasonality in ocean temperatures as the latitudinally comparable localities of the Mactridae dataset. The average annual temperature at 14 m at this locality is 17.2 °C. The average temperature for our Mactridae dataset is 18 °C. Example 1 (Testing Estimates Against an Independently Derived Growth Rate). The δ 18 O values in Fig. 6(A) were collected from a valve of Donax obesulus in Etayo-Cadavid (2010). For this valve, a published growth rate for the species Donax obesulus is available. The growth rate of Donax obesulus, albeit for a different population (from Arntz et al., 1987), was in our dataset, but it was excluded for this evaluation to make the test more realistic. One growth curve (Donax striatus from McLachlan et al., 1996) was also removed from the data because its maximum length (Linf ) was smaller than the maximum length of the specimen under consideration (see section on Measuring Growth Rates). This left a total of 30 growth curves used to calculate Donacidae to determine an expected age and its uncertainty for the Donax obesulus specimen analyzed by EtayoCadavid (2010) (Fig. 6(A)–(C)). The specimen of Donax obesulus was micro-sampled for δ 18 O through ontogeny using the distance from umbo (Etayo-Cadavid,

Fig. 5. Sensitivity analysis assessing errors in median age estimates as a function of sample size (i.e., the number of growth curves retained in the analysis). Estimated specimen age (in months) at valve length of 20 mm was used here. Each data point represents an absolute deviation in median age between a given iteration and the actual age estimate for the complete dataset. The black line marks the median (50th percentile) of absolute deviations at a given sample size (k). The gray line marks the 75th percentile of absolute deviations at a given sample size (k). The dashed line represents the maximum acceptable deviation assumed here (3 months). Y -axis is log-transformed for all plots. The analyses were carried out separately for each dataset: (A) Donacidae; (B) Semelidae; and (C) Mactridae. See text for additional explanation.

2010). A published growth rate from a different population of Donax obesulus calibrated with maximum size data from the specimen’s population was used to calculate age (Etayo-Cadavid, 2010). The time interval represented by the δ 18 O samples on this specimen approximates 12 months of shell growth (Etayo-Cadavid, 2010). Our mean estimate based on jackknife-corrected parametric bootstrap is 15.1 months (Fig. 6(C)) with 95% confidence limits of 9.2 and 23.3 months, respectively (Fig. 6(C)). Thus, the approach yields a mean age estimate within 4 months of the actual estimate used by Etayo-Cadavid’s (2010). If the growth rate of

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Fig. 6. Conversion of valve length to time from two published micro-sampled specimens using the resampling approach proposed here based on closely related species. (A) and (D) show the δ 18 O values from samples in published literature collected along the valve with the length as a measure of distance from the umbo. (A–C) are the δ 18 O values collected from a valve of Donax obesulus (shell #368497) in Etayo-Cadavid (2010). (D–F) are the δ 18 O values collected from a valve of Spisula solidissima (shell #DS41) in Krantz et al. (1987). (B) and (E) show the mean age estimates (solid line) and 95% confidence limits for each δ 18 O data point using the parametric bootstrap method with the datasets of Donacidae and Mactridae, respectively. (C) and (F) show the mean age estimates (solid line) and 95% confidence limits for each δ 18 O data point from the jackknife-corrected parametric bootstrap method. The dashed error bars in (B–C) and (E–F) show the lower (2.5%) and upper (97.5%) confidence limits around the age estimate for each δ 18 O data point.

the studied species was unknown, and only the indirect estimate was applied, the oxygen data would be interpreted concordantly, as representing seasonal fluctuations rather than multi-annual patterns. In addition, the indirect estimates allow for plotting confidence intervals around each isotope data point to express uncertainty regarding its temporal location (Fig. 6(C)). Example 2 (Testing Estimates for Specimens with Unknown Growth Rates). The δ 18 O values in Fig. 6(D) were collected from a valve of Spisula solidissima in Krantz et al. (1987). Phylogenetic analyses (Taylor et al., 2007) indicate that Spisula solidissima is a member of the Mactridae, and our estimate from the jackknife-corrected parametric bootstrap for Mactridae may thus be applicable to this species. The more seasonal, temperate waters from which this specimen was collected distinguish this sample from the populations used in our dataset. However, the mean water temperature is comparable to the mean temperature estimated for our dataset. No growth rates for this species were available in the literature, so the age range was determined by Krantz et al. (1987) using solely the oxygen stable isotope data collected along the valve’s growth axis. The specimen of Spisula solidissima was collected live in August, 1982, and sampled from the middle portion of the shell to the ventral edge (most recent area of growth) (Krantz et al., 1987). The authors interpret the three peaks in δ 18 O as records of three seasonal warming periods (Krantz et al., 1987). Two years are represented by complete pulses of warming and cooling in the δ 18 O

data (peaks and troughs) with half a year as seen by the partial peak of warming at the distal edge of growth representing the summer in which this specimen was collected (Fig. 6(D)). Thus, the isotope data alone suggest that the sampled shell interval represents 2 and a half years of growth (∼30 months). The jackknife-corrected parametric bootstrap (Fig. 6(F)) yielded an estimate of 30.7 months of growth, which is nearly identical with isotopic estimates provided by Krantz et al. (1987). The 95% confidence interval for the jackknife-corrected parametric bootstrap predicts the shortest and longest growth spans of 12.3 and 58.7 months, respectively. The isotope-constrained estimates of Krantz et al. (1987) are well within the 95% confidence interval from the parametric bootstrap and jackknife-corrected parametric bootstrap. Although the actual age of this specimen at the time of collection is unknown, the resampling approach produces confidence intervals around age estimates at each sampling point along the growth axis of the specimen. Thus, for example, the age of the specimen at the point of the ontogenetically oldest sample (farthest from the umbo) is 40.1 months with jackknife-corrected 95% confidence limits of 26.0 and 65.1 months, respectively (Fig. 6(F)). Similarly, the earliest isotope data-point for Spisula solidissima (closest to the umbo) had an estimated age of 9.4 months (with confidence intervals between 6.3 and 13.7 months) for the jackknife-corrected parametric bootstrap (Fig. 6(F)). One can thus predict that had the entire specimen been sampled for δ 18 O, a fourth peak of seasonal warming would be present.

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6.2. Practical application to applied fisheries research Beyond basic scientific pursuits, the proposed methods may be pertinent to applied research in fisheries and other societally relevant management problems. For example, Murray-Jones and Steffe (2000) focused on resource management of the species Donax deltoides. Donax deltoides has a number of uses including as a commercial food resource, with the recommended smallest catch-size for the local fisheries industry of 30 mm (valve length) (Murray-Jones and Steffe, 2000). Because D. deltoides was not included in our data, we can use the indirect estimates derived for all species included in the Donacidae dataset with a maximum length of 30 mm or more. Approximate mean sea surface temperature for this study area is 22 °C, which closely approximates the average temperature of 21 °C for our Donax dataset. The indirect approach estimates suggest that 28 months of growth are required (with a 95% confidence interval of 20–40 months) from the time of spawning to a length of 30 mm in order to replenish those individuals harvested. This estimate augments the study of Murray-Jones and Steffe (2000) by providing a specific time frame for sustainable harvesting. In practical examples of this type, it may be instructive to examine the effect of minor temperature differences on the growth rate estimates. Here, the temperature at sites from which the compiled Donacidae populations were sampled was evaluated for correlation with valve length for 28 month old specimens. The resulting reduced major axis regression model suggests that the difference between populations of Donax growing at 21 °C versus 22 °C was 1.1 mm, indicating that the effect of minor differences in temperature on growth rates is unlikely to have been substantial, although such subtle environmental differences may be beyond detection limits of this analysis. 7. Discussion and conclusions The jackknife-corrected parametric bootstrap approach offers a non-invasive, literature-based strategy that may be applicable to bivalve mollusks (and likely can be expanded to other groups of organisms). The examples above show that published data may be often adequate for creating datasets of living relatives while controlling for environmental and climatic settings. These datasets of living relatives appear sufficient to derive useful (timeconstraining) resampling models for estimating the expected growth rates for populations of bivalve mollusks with unknown growth rates. The feasibility tests conducted above demonstrate that resampling models based on the nearest living relatives can provide the accurate age and growth rate estimates. Moreover, by combining the geochemical data with independent growth rate estimates, a more detailed history of the specimen can be acquired, including estimates of age uncertainty. Also, as illustrated above for shellfish fisheries, the proposed approach may augment the research in applied sciences. Whereas the approach to develop indirect growth rate estimates may be versatile topically and taxonomically, obvious caveats apply. First, the approach is unlikely to work if the analyzed specimens came from climatic and environmental setting notably different from that of the reference dataset. Bivalves in particular demonstrate changes in rates of shell growth based on external water temperatures; accreting valve material most rapidly during warm seasons and ceasing shell accretion in cold seasons (Schöne et al., 2003). In our feasibility tests, the published isotope data involved specimens that were collected from comparable ocean temperature settings to those represented by our dataset. Although a seasonal pattern was present in one of the specimens while primarily absent from the specimens used to create the

dataset, similar temperature regimes appear to have been sufficient to obtain accurate age estimates (Fig. 6(D)–(F)). If the climatic/environmental setting of specimens is poorly understood (as may be the case for some fossil taxa), the approach may yield unrealistic growth estimates. Selecting specimens above the congeneric level further increases error in growth rate estimates (Fig. 4). Second, the precision of estimates is affected by specimen age. The confidence bands widen notably with size/age of specimens (Figs. 2 and 3). Thus, the quality of estimates will tend to decrease with increase in specimen age. As specimens get older, age estimates become increasingly uncertain. This can be seen in our feasibility tests where the upper 95% confidence limit increasingly diverges from the average age estimate as the size and age of specimens increase. This pattern is particularly pronounced in the case of the jackknife-corrected parametric bootstrap (Fig. 6(B)–(C), (E)–(F)). However, there are numerous applications for which the age estimates at young specimen ages are important. For example, in fisheries and aquaculture, it is important to know how quickly specimens will reach production value size. The higher uncertainty for estimates at older ages has little impact when attempting to gauge how quickly a species will reach a minimum acceptable harvestable size. Regardless of the caveats above, the examples above illustrate that this approach (efficient, inexpensive, and non-invasive) can be applicable in practice to extant organisms. Moreover, it can also be applied to the fossil record to study extinct populations and extinct species that belong to extant clades. And even when other indirect proxies of age are utilized (e.g., stable isotopes), the indirect estimates based on nearest relatives allow for a secondary independent means of testing age interpretations. Often growth records can represent anything from sub-seasonal to multi-annual fluctuations. By constraining the growth rates using the approach proposed here, interpretative utility of growth records can be thus enhanced. We are not proposing this method as a replacement for field collected growth rates, which is certainly the best method for understanding the structure of a population. Our approach provides a means of quantifying an age range when the field collection of growth rates is not readily attainable or when an understanding of actual growth rates is a means and not the goal of a particular research project. Because this approach should be suitable for bracketing growth rates and longevity of individual organisms for a wide range of taxa, it should be potentially applicable broadly in both academic and industrial research settings, including ecology, paleontology, geochemistry, and fisheries industries. Acknowledgments This is a University of Florida contribution to Paleobiology #653. This study is an outgrowth of projects supported by grants from the National Science Foundation [EAR-0125149 (Virginia Tech, George Washington University, and University of Delaware), OCE0602375 (Virginia Tech, Savannah River Ecology Laboratory of the University of Georgia, and Northern Arizona University)]. The support provided by Jon L. and Beverly A. Thompson Endowment Fund (University of Florida) is gratefully acknowledged. Additional support was also provided by the Department of Geosciences at Virginia Tech and by F. Dexter. The authors would like to thank J. Huntley, D. Kaufman, R. Krause, C. Romanek, M. Simões, S. Barbour Wood, and Y. Yanes for research collaborations that motivated this study. The authors would like to thank N. Chassot for the help with data collection and J. D. Schiffbauer for commenting on early drafts of the paper. S. Tuljapurkar and one anonymous reviewer provided numerous useful suggestions that greatly improved the clarity and quality of this report.

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