Estimation of population parameters from aerial counts of North American mallards: a cautionary tale.

The accuracy of population estimates strongly interferes with our ability to obtain unbiased estimates of population parameters based on analyses of time series of population fluctuations. Here we use long-term data on fluctuations in the size of Mallard populations collected as part of the May Breeding Waterfowl Survey covering a large section of North America. We assume a log-linear model of density dependence and use a hierarchical Bayesian state-space approach in which all parameters are assumed to be realizations from a common underlying distribution. Thus, parameters for different populations are not allowed to vary independently of each other. We then simulated independent time series of aerial counts, using the estimated parameters and adding various levels of observation error. These simulations showed that the estimates of stochastic population growth rate and strength of density dependence were biased even when moderate sampling errors were present. In contrast, the estimates of the environmental stochasticity and the carrying capacity were unbiased even for short time series and large observation error. Our results underline the importance of reducing the magnitude of sampling error in the design of large-scale monitoring programs of population fluctuations.     

Hierarchical Bayes estimation of hunting success rates

Although most post-season harvest surveys are conducted at the state level, the effective management of wildlife populations often requires estimates of hunting success rate, hunting pressure and harvest at the sub-area (such as management unit, regional, or county) level.Sample sizes for some sub-areas are often very small or even zero. Because of small sample sizes, estimates for small sub-areas often yield unacceptably large standard errors. In this article, a hierarchical Bayes model is used to estimate hunting success rates at the sub-area level from post-season harvest surveys. The computation is done by Gibbs sampling and adaptive rejection sampling techniques. The method is illustrated using data from the Missouri Turkey Hunting Survey 1994 Spring Season. The Bayesian estimates are close to the frequency estimates for the sub-areas with large sample sizes and more stable than the frequency estimates for those with small sample sizes. The Bayesian estimates will be more useful to wildlife biologists in estab-lishing hunting regulation on small sub-areas at no additional survey cost.     

A Bayesian approach to line-transect analysis for estimating abundance

Line-transect analysis is a widely used method of estimating plant and animal density and abundance. A Bayesian approach to a basic line-transect analysis is developed for a half-normal detection function. We extend the model of Karunamuni and Quinn [Karunamuni, R.J., Quinn II, T.J., 1995. Bayesian estimation of animal abundance for line-transect sampling. Biometrics 51, 1325–1337] by including a binomial likelihood function for the number of objects detected. The method computes a joint posterior distribution on the effective strip width and the density of objects in the sampled area. Analytical and computational methods for binned and unbinned perpendicular distance data are provided. Existing information about effective strip width and density can be brought into the analysis via prior distributions. The Bayesian approach is compared to a standard line-transect analysis using both real and simulated data. Results of the Bayesian and non-Bayesian analyses are similar when there are no prior data on effective strip width or density, but the Bayesian approach performs better when such data are available from previous or related studies. Practical methods for including prior data on effective strip width and density are suggested. A numerical example shows how the Bayesian approach can provide valid estimates when the sample size is too small for the standard approach to work reliably. The proposed Bayesian approach can form the basis for developing more advanced analyses.     

Accounting for matching uncertainty in two stage capture–recapture experiments using photographic measurements of natural marks

We propose a Bayesian hierarchical modeling approach for estimating the size of a closed population from data obtained by identifying individuals through photographs of natural markings. We assume that noisy measurements of a set of distinctive features are available for each individual present in a photographic catalogue. To estimate the population size from two catalogues obtained during two different sampling occasions, we embed the standard two-stage $M_t$ capture–recapture model for closed population into a multivariate normal data matching model that identifies the common individuals across the catalogues. In addition to estimating the population size while accounting for the matching process uncertainty, this hierarchical modelling approach allows to identify the common individuals by using the information provided by the capture–recapture model. This way, our model also represents a novel and reliable tool able to reduce the amount of effort researchers have to expend in matching individuals. We illustrate and motivate the proposed approach via a real data set of photo-identification of narwhals. Moreover, we compare our method with a set of possible alternative approaches by using both the empirical data set and a simulation study.     

Application of adaptive cluster sampling to low-density populations of freshwater mussels

Freshwater mussels appear to be promising candidates for adaptive cluster sampling because they are benthic macroinvertebrates that cluster spatially and are frequently found at low densities. We applied adaptive cluster sampling to estimate density of freshwater mussels at 24 sites along the Cacapon River, WV, where a preliminary timed search indicated that mussels were present at low density. Adaptive cluster sampling increased yield of individual mussels and detection of uncommon species; however, it did not improve precision of density estimates. Because finding uncommon species, collecting individuals of those species, and estimating their densities are important conservation activities, additional research is warranted on application of adaptive cluster sampling to freshwater mussels. However, at this time we do not recommend routine application of adaptive cluster sampling to freshwater mussel populations. The ultimate, and currently unanswered, question is how to tell when adaptive cluster sampling should be used, i.e., when is a population sufficiently rare and clustered for adaptive cluster sampling to be efficient and practical? A cost-effective procedure needs to be developed to identify biological populations for which adaptive cluster sampling is appropriate.     

Can modeling improve estimation of desert tortoise population densities?

The federally listed desert tortoise (Gopherus agassizii) is currently monitored using distance sampling to estimate population densities. Distance sampling, as with many other techniques for estimating population density, assumes that it is possible to quantify the proportion of animals available to be counted in any census. Because desert tortoises spend much of their life in burrows, and the proportion of tortoises in burrows at any time can be extremely variable, this assumption is difficult to meet. This proportion of animals available to be counted is used as a correction factor (g0) in distance sampling and has been estimated from daily censuses of small populations of tortoises (6-12 individuals). These censuses are costly and produce imprecise estimates of go due to small sample sizes. We used data on tortoise activity from a large (N = 150) experimental population to model activity as a function of the biophysical attributes of the environment, but these models did not improve the precision of estimates from the focal populations. Thus, to evaluate how much of the variance in tortoise activity is apparently not predictable, we assessed whether activity on any particular day can predict activity on subsequent days with essentially identical environmental conditions. Tortoise activity was only weakly correlated on consecutive days, indicating that behavior was not repeatable or consistent among days with similar physical environments.     

Poor-data and data-poor species stock assessment using a Bayesian hierarchical approach

Appropriate inference for stocks or species with low-quality data (poor data) or limited data (data poor) is extremely important. Hierarchical Bayesian methods are especially applicable to small-area, small-sample-size estimation problems because they allow poor-data species to borrow strength from species with good-quality data. We used a hammerhead shark complex as an example to investigate the advantages of using hierarchical Bayesian models in assessing the status of poor-data and data-poor exploited species. The hammerhead shark complex (Sphyrna spp.) along the Atlantic and Gulf of Mexico coasts of the United States is composed of three species: the scalloped hammerhead (S. lewini), the great hammerhead (S. mokarran), and the smooth hammerhead (S. zygaena) sharks. The scalloped hammerhead comprises 70-80% of the catch and has catch and relative abundance data of good quality, whereas great and smooth hammerheads have relative abundance indices that are both limited and of low quality presumably because of low stock density and limited sampling. Four hierarchical Bayesian state-space surplus production models were developed to simulate variability in population growth rates, carrying capacity, and catchability of the species. The results from the hierarchical Bayesian models were considerably more robust than those of the nonhierarchical models. The hierarchical Bayesian approach represents an intermediate strategy between traditional models that assume different population parameters for each species and those that assume all species share identical parameters. Use of the hierarchical Bayesian approach is suggested for future hammerhead shark stock assessments and for modeling fish complexes with species-specific data, because the poor-data species can borrow strength from the species with good data, making the estimation more stable and robust.     

Estimating Population Size with Noninvasive Capture-Mark-Recapture Data

Abstract:  Estimating population size of elusive and rare species is challenging. The difficulties in catching such species has triggered the use of samples collected noninvasively, such as feces or hair, from which genetic analysis yields data similar to capture-mark-recapture (CMR) data. There are, however, two differences between classical CMR and noninvasive CMR. First, capture and recapture data are gathered over multiple sampling sessions in classical CMR, whereas in noninvasive CMR they can be obtained from a single sampling session. Second, because of genotyping errors and unlike classical CMR, there is no simple relationship between (genetic) marks and individuals in noninvasive CMR. We evaluated, through simulations, the reliability of population size estimates based on noninvasive CMR. For equal sampling efforts, we compared estimates of population size N obtained from accumulation curves, a maximum likelihood, and a Bayesian estimator. For a closed population and without sampling heterogeneity, estimates obtained from noninvasive CMR were as reliable as estimates from classical CMR. The sampling structure (single or multiple session) did not alter the results, the Bayesian estimator in the case of a single sampling session presented the best compromise between low mean squared error and a 95% confidence interval encompassing the parametric value of N in most simulations. Finally, when suitable field and lab protocols were used, genotyping errors did not substantially bias population size estimates (bias < 3.5% in all simulations). The ability to reliably estimate population size from noninvasive samples taken during a single session offers a new and useful technique for the management and conservation of elusive and rare species.     

Toward accurate and precise estimates of lion density

Reliable estimates of animal density are fundamental to understanding ecological processes and population dynamics. Furthermore, their accuracy is vital to conservation because wildlife authorities rely on estimates to make decisions. However, it is notoriously difficult to accurately estimate density for wide‐ranging carnivores that occur at low densities. In recent years, significant progress has been made in density estimation of Asian carnivores, but the methods have not been widely adapted to African carnivores, such as lions (Panthera leo). Although abundance indices for lions may produce poor inferences, they continue to be used to estimate density and inform management and policy. We used sighting data from a 3‐month survey and adapted a Bayesian spatially explicit capture‐recapture (SECR) model to estimate spatial lion density in the Maasai Mara National Reserve and surrounding conservancies in Kenya. Our unstructured spatial capture‐recapture sampling design incorporated search effort to explicitly estimate detection probability and density on a fine spatial scale, making our approach robust in the context of varying detection probabilities. Overall posterior mean lion density was estimated to be 17.08 (posterior SD 1.310) lions >1 year old/100 km², and the sex ratio was estimated at 2.2 females to 1 male. Our modeling framework and narrow posterior SD demonstrate that SECR methods can produce statistically rigorous and precise estimates of population parameters, and we argue that they should be favored over less reliable abundance indices. Furthermore, our approach is flexible enough to incorporate different data types, which enables robust population estimates over relatively short survey periods in a variety of systems. Trend analyses are essential to guide conservation decisions but are frequently based on surveys of differing reliability. We therefore call for a unified framework to assess lion numbers in key populations to improve management and policy decisions.     

Bayesian analysis of abundance for binomial sighting data with unknown number of marked individuals

The mark-resight method for estimating the size of a closed population can in many circumstances be a less expensive and less invasive alternative to traditional mark-recapture. Despite its potential advantages, one major drawback of traditional mark-resight methodology is that the number of marked individuals in the population available for resighting needs to be known exactly. In real field studies, this can be quite difficult to accomplish. Here we develop a Bayesian model for estimating abundance when sighting data are acquired from distinct sampling occasions without replacement, but the exact number of marked individuals is unknown. By first augmenting the data with some fixed number of individuals comprising a marked “super population,” the problem may then be reformulated in terms of estimating the proportion of this marked super population that was actually available for resighting. This then allows the data for the marked population available for resighting to be modeled as random realizations from a binomial logit-normal distribution. We demonstrate the use of our model to estimate the New Zealand robin (Petroica australis) population size in a region of Fiordland National Park, New Zealand. We then evaluate the performance of the proposed model relative to other estimators via a series of simulation experiments. We generally found our model to have advantages over other models when sample sizes are smaller with individually heterogeneous resighting probabilities. Due to limited budgets and the inherent variability between individuals, this is a common occurrence in mark-resight population studies. WinBUGS and R code to carry out these analyses is available from .     

Dynamics of a low‐density tiger population in Southeast Asia in the context of improved law enforcement

Recovering small populations of threatened species is an important global conservation strategy. Monitoring the anticipated recovery, however, often relies on uncertain abundance indices rather than on rigorous demographic estimates. To counter the severe threat from poaching of wild tigers (Panthera tigris), the Government of Thailand established an intensive patrolling system in 2005 to protect and recover its largest source population in Huai Kha Khaeng Wildlife Sanctuary. Concurrently, we assessed the dynamics of this tiger population over the next 8 years with rigorous photographic capture‐recapture methods. From 2006 to 2012, we sampled across 624–1026 km² with 137–200 camera traps. Cameras deployed for 21,359 trap days yielded photographic records of 90 distinct individuals. We used closed model Bayesian spatial capture‐recapture methods to estimate tiger abundances annually. Abundance estimates were integrated with likelihood‐based open model analyses to estimate rates of annual and overall rates of survival, recruitment, and changes in abundance. Estimates of demographic parameters fluctuated widely: annual density ranged from 1.25 to 2.01 tigers/100 km², abundance from 35 to 58 tigers, survival from 79.6% to 95.5%, and annual recruitment from 0 to 25 tigers. The number of distinct individuals photographed demonstrates the value of photographic capture–recapture methods for assessments of population dynamics in rare and elusive species that are identifiable from natural markings. Possibly because of poaching pressure, overall tiger densities at Huai Kha Khaeng were 82–90% lower than in ecologically comparable sites in India. However, intensified patrolling after 2006 appeared to reduce poaching and was correlated with marginal improvement in tiger survival and recruitment. Our results suggest that population recovery of low‐density tiger populations may be slower than anticipated by current global strategies aimed at doubling the number of wild tigers in a decade.     

Multimethod,multistate Bayesian hierarchical modeling approach for use in regional monitoring of wolves

In many cases, the first step in large‐carnivore management is to obtain objective, reliable, and cost‐effective estimates of population parameters through procedures that are reproducible over time. However, monitoring predators over large areas is difficult, and the data have a high level of uncertainty. We devised a practical multimethod and multistate modeling approach based on Bayesian hierarchical‐site‐occupancy models that combined multiple survey methods to estimate different population states for use in monitoring large predators at a regional scale. We used wolves (Canis lupus) as our model species and generated reliable estimates of the number of sites with wolf reproduction (presence of pups). We used 2 wolf data sets from Spain (Western Galicia in 2013 and Asturias in 2004) to test the approach. Based on howling surveys, the naïve estimation (i.e., estimate based only on observations) of the number of sites with reproduction was 9 and 25 sites in Western Galicia and Asturias, respectively. Our model showed 33.4 (SD 9.6) and 34.4 (3.9) sites with wolf reproduction, respectively. The number of occupied sites with wolf reproduction was 0.67 (SD 0.19) and 0.76 (0.11), respectively. This approach can be used to design more cost‐effective monitoring programs (i.e., to define the sampling effort needed per site). Our approach should inspire well‐coordinated surveys across multiple administrative borders and populations and lead to improved decision making for management of large carnivores on a landscape level. The use of this Bayesian framework provides a simple way to visualize the degree of uncertainty around population‐parameter estimates and thus provides managers and stakeholders an intuitive approach to interpreting monitoring results. Our approach can be widely applied to large spatial scales in wildlife monitoring where detection probabilities differ between population states and where several methods are being used to estimate different population parameters.     

Minimum area requirements for an at‐risk butterfly based on movement and demography

Determining the minimum area required to sustain populations has a long history in theoretical and conservation biology. Correlative approaches are often used to estimate minimum area requirements (MARs) based on relationships between area and the population size required for persistence or between species’ traits and distribution patterns across landscapes. Mechanistic approaches to estimating MAR facilitate prediction across space and time but are few. We used a mechanistic MAR model to determine the critical minimum patch size (CMP) for the Baltimore checkerspot butterfly (Euphydryas phaeton), a locally abundant species in decline along its southern range, and sister to several federally listed species. Our CMP is based on principles of diffusion, where individuals in smaller patches encounter edges and leave with higher probability than those in larger patches, potentially before reproducing. We estimated a CMP for the Baltimore checkerspot of 0.7–1.5 ha, in accordance with trait‐based MAR estimates. The diffusion rate on which we based this CMP was broadly similar when estimated at the landscape scale (comparing flight path vs. capture‐mark‐recapture data), and the estimated population growth rate was consistent with observed site trends. Our mechanistic approach to estimating MAR is appropriate for species whose movement follows a correlated random walk and may be useful where landscape‐scale distributions are difficult to assess, but demographic and movement data are obtainable from a single site or the literature. Just as simple estimates of lambda are often used to assess population viability, the principles of diffusion and CMP could provide a starting place for estimating MAR for conservation.     

Estimating Population Size of Elusive Animals with DNA from Hunter-Collected Feces: Four Methods for Brown Bears

Abstract:  Noninvasive genetic methods can be used to estimate animal abundances and offer several advantages over conventional methods. Few attempts have been made, however, to evaluate the accuracy and precision of the estimates. We compared four methods of estimating population size based on fecal sampling. Two methods used rarefaction indices and two were based on capture-mark-recapture (CMR) estimators, one combining genetic and field data. Volunteer hunters and others collected 1904 fecal samples over 2 consecutive years in a large area containing a well-studied population of brown bears ( Ursus arctos ). On our 49,000-km² study area in south-central Sweden, population size estimates ranged from 378 to 572 bears in 2001 and 273 to 433 bears in 2002, depending on the method of estimation used. The estimates from the best model in the program MARK appeared to be the most accurate, based on the minimum population size estimate from radio-marked bears in a subsection of our sampling area. In addition, MARK models included heterogeneity and temporal variation in detection probabilities, which appeared to be present in our samples. All methods, though, incorrectly suggested a biased sex ratio, probably because of sex differences in detection probabilities and low overall detection probabilities. The population size of elusive animals can be estimated reliably over large areas with noninvasive genetic methods, but we stress the importance of an adequate and well-distributed sampling effort. In cases of biased sampling, calibration with independent estimates may be necessary. We recommend that this noninvasive genetic approach, using the MARK models, be used in the future in areas where sufficient numbers of volunteers can be mobilized.     

Bayesian predictive inference for the proportion of eroded land in a small area in Iowa

There is an increasing interest in the quality of soil, especially for small geographical areas. We present a method to estimate the percent of the area in a county or hydrological basin that is eroded. There are sample data (for several counties in eastern Iowa) from the National Resources Inventory and population data on land use, land capability class, rainfall and slope length and steepness. Using the Gibbs sampler we perform Bayesian predictive inference to obtain estimates for the non-sampled units. These estimates, together with the sample data, provide an estimate of the proportion of the total area that is eroded. We assess the quality of fit of our model using two cross-validation exercises and graphical methods.     

Bayesian estimation of age-specific bird nest survival rates with categorical covariates

The populations of many North American landbirds are showing signs of declining. Gathering information on breeding productivity allows critical assessment of population performance and helps identify good habitat-management practices. He (Biometrics (2003) 59 962–973) proposed a Bayesian model to estimate the age-specific nest survival rates. The model allows irregular visiting schedule under the assumption that the observed nests have homogeneous nest survival. Because nest survival studies are often conducted in different sites and time periods, it is not realistic to assume homogeneous nest survival. In this paper, we extend He’s model by incorporating these factors as categorical covariates. The simulation results show that the Bayesian hierarchical model can produce satisfactory estimates on nest survival and capture different factor effects. Finally the model is applied to a Missouri red-winged blackbird data set.     

Assessing tiger population dynamics using photographic capture-recapture sampling

Although wide-ranging, elusive, large carnivore species, such as the tiger, are of scientific and conservation interest, rigorous inferences about their population dynamics are scarce because of methodological problems of sampling populations at the required spatial and temporal scales. We report the application of a rigorous, noninvasive method for assessing tiger population dynamics to test model-based predictions about population viability. We obtained photographic capture histories for 74 individual tigers during a nine-year study involving 5725 trap-nights of effort. These data were modeled under a likelihood-based, "robust design" capture-recapture analytic framework. We explicitly modeled and estimated ecological parameters such as time-specific abundance, density, survival, recruitment, temporary emigration, and transience, using models that incorporated effects of factors such as individual heterogeneity, trap-response, and time on probabilities of photo-capturing tigers. The model estimated a random temporary emigration parameter of gamma" = gamma' = 0.10 +/- 0.069 (values are estimated mean +/- SE). When scaled to an annual basis, tiger survival rates were estimated at S = 0.77 +/- 0.051, and the estimated probability that a newly caught animal was a transient was tau = 0.18 +/- 0.11. During the period when the sampled area was of constant size, the estimated population size N(t) varied from 17 +/- 1.7 to 31 +/- 2.1 tigers, with a geometric mean rate of annual population change estimated as lambda = 1.03 +/- 0.020, representing a 3% annual increase. The estimated recruitment of new animals, B(t), varied from 0 +/- 3.0 to 14 +/- 2.9 tigers. Population density estimates, D, ranged from 7.33 +/- 0.8 tigers/100 km2 to 21.73 +/- 1.7 tigers/100 km2 during the study. Thus, despite substantial annual losses and temporal variation in recruitment, the tiger density remained at relatively high levels in Nagarahole. Our results are consistent with the hypothesis that protected wild tiger populations can remain healthy despite heavy mortalities because of their inherently high reproductive potential. The ability to model the entire photographic capture history data set and incorporate reduced-parameter models led to estimates of mean annual population change that were sufficiently precise to be useful. This efficient, noninvasive sampling approach can be used to rigorously investigate the population dynamics of tigers and other elusive, rare, wide-ranging animal species in which individuals can be identified from photographs or other means.     

Lifting a veil on diversity: a Bayesian approach to fitting relative-abundance models.

Bayesian methods incorporate prior knowledge into a statistical analysis. This prior knowledge is usually restricted to assumptions regarding the form of probability distributions of the parameters of interest, leaving their values to be determined mainly through the data. Here we show how a Bayesian approach can be applied to the problem of drawing inference regarding species abundance distributions and comparing diversity indices between sites. The classic log series and the lognormal models of relative- abundance distribution are apparently quite different in form. The first is a sampling distribution while the other is a model of abundance of the underlying population. Bayesian methods help unite these two models in a common framework. Markov chain Monte Carlo simulation can be used to fit both distributions as small hierarchical models with shared common assumptions. Sampling error can be assumed to follow a Poisson distribution. Species not found in a sample, but suspected to be present in the region or community of interest, can be given zero abundance. This not only simplifies the process of model fitting, but also provides a convenient way of calculating confidence intervals for diversity indices. The method is especially useful when a comparison of species diversity between sites with different sample sizes is the key motivation behind the research. We illustrate the potential of the approach using data on fruit-feeding butterflies in southern Mexico. We conclude that, once all assumptions have been made transparent, a single data set may provide support for the belief that diversity is negatively affected by anthropogenic forest disturbance. Bayesian methods help to apply theory regarding the distribution of abundance in ecological communities to applied conservation.     

Inference about density and temporary emigration in unmarked populations

Few species are distributed uniformly in space, and populations of mobile organisms are rarely closed with respect to movement, yet many models of density rely upon these assumptions. We present a hierarchical model allowing inference about the density of unmarked populations subject to temporary emigration and imperfect detection. The model can be fit to data collected using a variety of standard survey methods such as repeated point counts in which removal sampling, double-observer sampling, or distance sampling is used during each count. Simulation studies demonstrated that parameter estimators are unbiased when temporary emigration is either "completely random" or is determined by the size and location of home ranges relative to survey points. We also applied the model to repeated removal sampling data collected on Chestnut-sided Warblers (Dendroica pensylvancia) in the White Mountain National Forest, U.S.A. The density estimate from our model, 1.09 birds/ha, was similar to an estimate of 1.11 birds/ha produced by an intensive spot-mapping effort. Our model is also applicable when processes other than temporary emigration affect the probability of being available for detection, such as in studies using cue counts. Functions to implement the model have been added to the R package unmarked.     

Estimating age from recapture data: integrating incremental growth measures with ancillary data to infer age-at-length

Estimating the age of individuals in wild populations can be of fundamental importance for answering ecological questions, modeling population demographics, and managing exploited or threatened species. Significant effort has been devoted to determining age through the use of growth annuli, secondary physical characteristics related to age, and growth models. Many species, however, either do not exhibit physical characteristics useful for independent age validation or are too rare to justify sacrificing a large number of individuals to establish the relationship between size and age. Length-at-age models are well represented in the fisheries and other wildlife management literature. Many of these models overlook variation in growth rates of individuals and consider growth parameters as population parameters. More recent models have taken advantage of hierarchical structuring of parameters and Bayesian inference methods to allow for variation among individuals as functions of environmental covariates or individual-specific random effects. Here, we describe hierarchical models in which growth curves vary as individual-specific stochastic processes, and we show how these models can be fit using capture-recapture data for animals of unknown age along with data for animals of known age. We combine these independent data sources in a Bayesian analysis, distinguishing natural variation (among and within individuals) from measurement error. We illustrate using data for African dwarf crocodiles, comparing von Bertalanffy and logistic growth models. The analysis provides the means of predicting crocodile age, given a single measurement of head length. The von Bertalanffy was much better supported than the logistic growth model and predicted that dwarf crocodiles grow from 19.4 cm total length at birth to 32.9 cm in the first year and 45.3 cm by the end of their second year. Based on the minimum size of females observed with hatchlings, reproductive maturity was estimated to be at nine years. These size benchmarks are believed to represent thresholds for important demographic parameters; improved estimates of age, therefore, will increase the precision of population projection models. The modeling approach that we present can be applied to other species and offers significant advantages when multiple sources of data are available and traditional aging techniques are not practical.