RALSTON and WILLIAMS: AGEING OF TROPICAL FISHES 



ments beyond 7,500 /^m (see above). Still, we must 

 assume that otolith growth rates calculated from 

 regions where daily increments are visible are other- 

 wise no different from regions where they are not. 



Numerical integration of otolith growth rates pro- 

 vided a series of ordered pairs of age and otolith 

 length. Otolith lengths were then converted to FL 

 through regression analysis. The only FL data in- 

 cluded in the nonlinear von Bertalanffy regression 

 (Fig. 4), however, were based on otolith lengths in 

 excess of 3,000 fxm. Note that the excluded data (in- 

 tervals 1-6) represent the first year's growth, i.e., 

 the early life history. Although the von Bertalanffy 

 growth equation has historically been the model of 

 choice in stock-assessment applications, including 

 especially the Beverton and Holt (1957) dynamic 

 pool model, it provides a poor description of growth 

 during the early life history. Inflected growth 

 typically characterizes this stage, which is better fit 

 with a Gompertz-type curve (Zweifel and Lasker 

 1976). By excluding ages <0.8 years from the von 

 Bertalanffy regression analysis, we constrain the 

 data used to estimate the model to the domain over 

 which meaningful predictions are made. Moreover, 

 predictions of FL based on otolith length are also 

 obtained from regression analysis (Fig. 1). Because 

 the smallest otolith used in developing the regres- 

 sion equation was 5,043 ^im (see Figure 1), applica- 

 tion of the equation to predict the FL of a fish whose 

 otolith is less than this size represents an unneces- 

 sary extrapolation of the fitted model. 



One of the side effects of deleting points from the 

 early life history is to diminish the importance of 

 weighting. Note that the statistical weights of the 

 data used in the regression (Table 2) are very similar 

 (coefficient of variation = 0.78%). Thus, although 

 it may be desirable from a theoretical perspective, 

 weighting had a negligible effect on the parameter 

 estimates. 



One of the principal advantages recommending 

 this approach is an increase in efficiency and objec- 

 tivity relative to studies that obtain complete counts 

 of daily growth increments (Uchiyama and Struh- 

 saker 1981; Brouard et al. 1984; Radtke 1987). 

 Because all increments need not be visually con- 

 spicuous for a particular preparation to provide 

 useful information, as is true of studies relying on 

 whole counts, the observer can utilize only those por- 

 tions of the otolith where the microstructure is clear- 

 ly expressed. Enumeration of ill-defined increments 

 in poorly developed regions of the otolith is avoided. 

 This feature also makes it possible to automate the 

 procedure (Casselman 1983; McGowen et al. 1987) 

 and ultimately to realize the goal of standardizing 



age determinations (Boehlert and Yoklavich 1984; 

 Boehlert 1985). 



Powerful statistical tests of growth heterogeneity 

 also are possible with the acquisition of increment 

 width data (Table 1, Fig. 3). Evaluation of statistical 

 differences in populations with respect to the param- 

 eters of the von Bertalanffy growth equation is 

 cumbersome at best (Gallucci and Quinn 1979; Ber- 

 nard 1981; Kappenman 1981). Analysis of covari- 

 ance of increment width data provides a convenient 

 and widely available means of testing for growth 

 heterogeneity among any statistical populations of 

 interest. 



One of the principal disadvantages of the method 

 outlined here is that growth variation among indivi- 

 duals within the sampled population is lost through 

 averaging of the data. The final growth curve given 

 in Figure 4 describes the mean growth of the sam- 

 pled population of gindai. Of course, length varia- 

 tion at age is extremely important, and its descrip- 

 tion is required for application of the more powerful 

 and realistic stock-assessment models, especially in 

 cohort or virtual population analysis (Ricker 1975). 

 Nonetheless, given the difficult conditions surround- 

 ing assessment work in tropical environments 

 (Gulland 1982), the application of yield/recruit 

 models is a significant step forward (Munro 1982; 

 Pauly 1982). In conjunction with the analysis of 

 length-frequency distributions, the method proposed 

 here is well suited to help meet that need. 



ACKNOWLEDGMENTS 



This work is the result of the Resource Assess- 

 ment Investigation of the Mariana Archipelago at 

 the Southwest Fisheries Center Honolulu Labora- 

 tory, National Marine Fisheries Service, NOAA. 

 Numerous people provided substantive comments 

 as the research progressed and as the paper was 

 variously revised. Some of these suggestions were 

 invaluable to the development of the ideas presented 

 here, and we gratefully acknowledge the efforts of 

 those who spent some time thinking about them. 



LITERATURE CITED 



Allen, G. R. 



1985. FAO species catalogue. Snappers of the world. An an- 

 notated and illustrated catalogue of lutjanid species known 

 to date. FAO Fish. Synop. (125) 6:1-208. 

 Beamish, R. J., and G. A. McFarlane. 



1983. The forgotten requirement for age validation in fish- 

 eries biology. Trans. Am. Fish. Soc. 112:735-743. 

 1987. Current trends in age determination methodology. In 

 R. C. Summerfelt and G. E. Hall (editors). The age and 



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