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Fishery Bulletin 93(1), 1995 



than one, the data were discarded (<5% of all 

 otoliths). If counts differed by one, the higher value 

 was chosen (10% of otoliths). The relative frequency 

 of these discrepancies was similar for all species. 



Data analysis 



The length-at-age data were fitted to the repara- 

 meterized von Bertalanffy growth curve of Francis 

 (1988). This method has the advantage that the pa- 

 rameters estimated are independent and can be com- 

 pared directly between species and populations. 



Most previous studies of lutjanid age and growth 

 have fitted the von Bertalanffy growth equation to 

 data on length at age (e.g. Lai and Lui, 1979; 

 Manooch, 1987; Davis and West, 1992). However, the 

 estimated parameters L m , K, and t Q either do not have 

 direct biological meaning (e.g. Knight, 1968; Schnute 

 and Fournier, 1980; Ratkowsky, 1986) or are extrapo- 

 lations from the data (Ratkowsky, 1986). Francis 

 (1988) extended the equation of Schnute and 

 Fournier (1980) to derive a new set of parameters 

 L v L 2 , and L 3 (his L, I , and l w ), which correspond to 

 the length at the lower, middle, and upper limits of 

 any arbitrarily defined age range, such that: 



L t =L 1+ (L 3 -L 1 )(l-r 2{t ^ /u "^ ) )/a-r 2 ), (3) 



where r = (L 3 —L 2 )/(L 2 —L 1 ); L t is the mean length of a 

 fish at age t; and L v L 2 and L 3 are the length at the 

 lower, middle, and upper limits of two arbitrary ages 

 and w. By fitting a curve of this form, extrapola- 

 tions beyond the data are avoided, as the three fit- 

 ted parameters are chosen from within the range of 

 the data and hence can be directly compared with 

 the results of previous studies. 



In this study, we set = 1 ring and w = 6 rings for 

 each species. This equation has the advantage that 

 the age range to be examined can be chosen by the 

 investigator, rather than having to be the largest and 

 smallest age classes found, as required by the 

 Schnute and Fournier ( 1980) equation. These param- 

 eters (Lj, L 2 , and L 3 ) can also be expected to have 

 similar properties to those of Schnute and Fournier 

 (1980) and not to show the high negative correlation 

 between L x and K (Francis, 1988). 



All parameters were estimated by an iterative 

 least-squares method (SAS NLIN procedure with the 

 Marquardt option; SAS, 1989). Vaughan and 

 Kanciruk (1982) found that this procedure consis- 

 tently showed the least bias in parameter estimates, 

 converged rapidly, and provided more precise esti- 

 mates than did standard linear techniques. A mea- 



sure of goodness-of-fit was obtained by calculating 

 an r 2 value from the residual and the explained sums 

 of squares derived from the least-squares regression. 



Relationship of ring counts in whole and 

 sectioned otoliths with radiometric ages 



The estimated age from ring counts in whole and 

 sectioned otoliths used in the radiometry were com- 

 pared for all species by two methods. First, the rela- 

 tionship between radiometric age and whole and sec- 

 tioned otolith ages of the same fish were plotted. If 

 the slope of the relationship was not significantly 

 different from 1, the results of the two methods were 

 considered to be in close agreement. Second, the two 

 ageing methods were compared with the radiomet- 

 ric ages with a Wilcoxon matched-pairs ranks test 

 (Conover, 1980). The two hypotheses tested were 1) 

 that whole otolith ring counts underestimated true 

 age (radiometric age) or 2) that sectioned otolith ring 

 counts overestimated true age. 



Results 



Radiometry 



Lutjanus erythropterus— The specific activity of 226 Ra 

 and the 210 Pb/ 226 Ra activity ratio differed among the 

 three samples of L. erythropterus (Tables 1 and 2). The 

 activity ratio was highest in the cored sample (0.118 

 ± 0.031; Table 2). Ring counts in whole otoliths were 

 linearly related to otolith mass (Fig. 2A) and ring 

 count in sectioned otoliths, though the relationship 

 was significantly weaker CP<0.05). Radiometric age 

 estimates were calculated on the basis of a single 

 constant (linear) growth rate for otolith mass, which 

 removes the need to include the mass growth rela- 

 tion in the radiometric age calculation (Eq. 1). Un- 

 der the assumption of a constant growth model, ra- 

 diometric age estimates were most similar to those 

 obtained from the ring counts in whole otoliths (Table 

 2). The match was best for the cored otolith sample 

 where model assumptions are less stringent (sample 

 2673). 



Lutjanus malabaricus — The specific activity of 226 Ra 

 in L. malabaricus otoliths differed among samples 

 and among size classes (Tables 1 and 2). The 210 Pb/ 

 226 Ra activity ratios ranged from less than 0.027 to 

 0.212 and varied to a similar degree in cored and 

 whole-otolith samples (Table 2). Otolith weight was 

 linearly related to the number of rings in whole 

 otoliths (Fig. 2B), and this relationship was stron- 

 ger than that for counts from sectioned otoliths 



