Zug et al Age and growth of Hawaiian Chelonia 



119 



witli S(^Ls <60 cm and resorption core diameters <19.0 

 mm. Selecting only small turtles with minimum core di- 

 ameters reduces the frequency of the narrower periosteal 

 layers found in the outer margin of the humerus in larger 

 turtles; hence, it reduces the possibility of overestimating 

 the number of layers in the resorption core. The resulting 

 correction factor was used to estimate the number of re- 

 sorbed periosteal layers. 



A second method, spline integration (SI), is introduced 

 here. The SI method uses a scatterplot smoothing spline 

 (Hardle, 1990; Hastie and Tibshirani, 1990) to model the 

 relationship between the aging rate and humerus diam- 

 eter. Once the aging function is estimated, a turtle's age 

 is estimated by integrating the spline over the total diam- 

 eter of the turtle's humerus section. The method of model- 

 ing increment width patterns in hard parts and of estimat- 

 ing age by the integration of the resulting aging function 

 was first formalized by Ralston and Miyamoto ( 1983) for a 

 Hawaiian snapper and first applied to seaturtles by Zug et 

 al. ( 1995). In those applications, the aging rate was a para- 

 metric function of size. In our analysis, we modeled the ag- 

 ing rate nonparametrically by fitting a smoothing spline 

 to pairs of obsei-\-ations of growth-layer width and humer- 

 us diameter The SI approach uses the same source of data 

 as the CF method but without selection. Lines of arrested 

 growth (LAG) delimit each observable growth layer Incre- 

 ment width is measured as the difference between the hu- 

 merus diameters at the outer LAG and the inner LAG. As- 

 suming an increment represents one year of growth, each 

 increment width measurement provides a measure of the 

 humerus growth rate (nini/yr) and its reciprocal, a mea- 

 sure of the aging rate (yr/mm) at the obsein-'ed humerus di- 

 ameter (the mean diameter of the pair of LAGs). The skel- 

 etochronological sample yielded 269 such observations of 

 aging rate and humerus diameter. The aging rates were 

 grouped in 1-mm intei-vals of humerus diameter and aver- 

 aged. A cubic smoothing spline was fitted to the mean ag- 

 ing rates by usnig S-PLUS (MathSoft, Inc., 1999). The age 

 (yr) of each turtle was estimated by integrating the aging 

 spline from its origin to the observed outside diameter of 

 the humerus section. 



To assess the effect of estimation method on age esti- 

 mates, the data were divided into 10-cm SCL groups. With- 

 in each gi'oup a Student's t statistic was used to test the 

 hypothesis that the two methods give equal age estimates. 



Nonparametric growth models were estimated based 

 on the CF- and Sl-derived age estimates and associated 

 carapace lengths, by using the same S-PLUS procedure 

 employed for the Sl-method aging spline. The validity of 

 the growth models was judged qualitatively by comparing 

 growth predicted by the models with obsei"ved giowth in 

 a sample of 171 Hawaiian gi-een turtles tagged and recap- 

 tured in waters around Molokai (Balazs et al, 1999). 



To assess uncertainty in the Sl-based growth curve, the 

 269 pairs of aging rate and mean humerus diameter data 

 were resampled 100 times, and the SI procedure applied 

 to each bootstrap replicate data set. The 100 aging curves 

 derived in this manner generated a bootstrap distribution 

 of estimated age for each turtle. Nonparametric growth 

 curves were then fitted to each derived data set, produc- 



ing bootstrap distributions of predicted mean length at 

 age. Empirical confidence intervals for the predicted mean 

 length at age were approximated by using percentiles of 

 the latter bootstrap distributions. 



A linear regression of S('L on outside humerus diam- 

 eter was estimated for the 104 sample turtles. The slope 

 of the linear predictor was applied to the 269 humeral in- 

 crements to estimate a corresponding set of carapace in- 

 crements, presumed to represent annual growth. These 

 growth rate estimates were summarized in box plots over 

 10-cm intervals of SCL. Mean growth rate as a function of 

 estimated age was also estimated by computing finite dif- 

 ferences of the Sl-based growth model. 



Results 



Patterns in humerus growth and aging 



Carapace length has a strong linear association with 

 humerus diameter (7=0.643 + 2.326A: (where y=SCL cm; 

 A''=humerus diameter mm), r- =0.98, P<0. 001, « = 104 includ- 

 ing the hatchlingl. Humerus growth-increment width, on 

 the other hand, is nonlinearly associated with humerus 

 diameter at the point of growth (Fig. 2 ). Specifically, growth 

 increments tend to be larger when the turtles are smaller 

 (i.e. at smaller humerus diameters) and decline as the tur- 

 tles grow. Variation in humerus increment width (growth 

 rate) shows the same pattern. The estimated aging rate, 

 as the reciprocal of growth rate, increases as the turtles 

 grow. The aging rate does not increase uniformly (Fig. 3). 

 Rather, it increases gradually in small turtles, plateaus 

 over a broad range of length for mid-size turtles, increases 

 abruptly as turtles approach maturity, and maintains an 

 increased rate as the mature turtles grow. 



Age and growth-rate estimates 



In the CF-method analysis, the correction factor, C, was 

 estimated as 1.14 yr/mm. The resulting age estimates 

 range from 4.1 to 34.6 yr (/;=70; excluduig the hatchling 

 with age zero). The smallest turtle in the sample had the 

 lowest age estimate and the two largest turtles, the high- 

 est estimates. Only 68% could be aged by the CF method. 

 Skeletochronology requires a pattern of distinct layering 

 within the bony element examined. Such patterns are 

 most evident in the smaller, presumably younger, individ- 

 uals, and the frequency of individuals with distinct perios- 

 teal layers decreases as body size increases. In selecting 

 specimens for the CF analysis, growth layers were suffi- 

 ciently distinct to estimate the number of resorbed layers, 

 and hence the age, in decreasingly fewer turtles: 899c of 

 turtles in the 30-69 cm SCL group, 729;^ in the 70-79 cm 

 group, 18% in the 80-89 cm group, and 29% in the >89 

 cm gi'oup were used in the CF analysis. Of the individ- 

 uals for which we were unable to obtain an estimate of 

 resorbed layers, a nearly equal number (48%) had fibro- 

 papillomas. The prevalence of tumors for the CF-aged sub- 

 sample (31%) was somewhat less than in the total sample 

 (37%). Importantly, the tumor prevalence in the subsam- 



