RALSTON and WILLIAMS: AGEING OF TROPICAL FISHES 



dian = 14, range = 5-22) of increments and measur- 

 ing the axial length of the short segment in which 

 they occurred. In addition, the curvilinear distance 

 between the midpoint of each segment and the 

 otolith focus was measured along the focus to 

 postrostral growth axis. Up to 12 readings were 

 made from each preparation. The focus was defined 

 to be the most posterior of what typically were 

 several primordia (e.g., Radtke 1987). 



The data were summarized by computing the ratio 

 of segment length in micrometers to the included 

 number of increments at each specific site examined, 

 providing an estimate of the average increment 

 width at some measured distance from the otolith 

 focus. Under the assumption that one increment 

 forms each day, these data can be used to estimate 

 the instantaneous growth rate of the otolith (Ralston 

 and Miyamoto 1981, 1983; Ralston 1985). 



To estimate age. a simple form of numerical in- 

 tegration was employed. Starting at the focus, the 

 data were subdivided into 500 jjm intervals of otolith 

 length. For each interval, the arithmetic mean 

 growth rate of the otolith was calculated based upon 

 the number of readings falling therein. This aver- 

 age growth rate was then divided into 500 fxm to 

 estimate the number of days needed to complete 

 growth through the intervals, which were sequen- 

 tially accumulated away from the focus, and finally 

 divided by 365.25 to convert age estimates to years. 

 The size of the otolith upon completion of growth 

 through each interval was used to predict the cor- 

 responding FL of the fish after the natural loga- 

 rithm of FL was regressed on the logarithm of total 

 otolith length. These data (age [in years] and FL 

 [mm]) were then fitted to the von Bertalanffy 

 growth equation (Ricker 1979) by using a nonlinear 

 regression routine (SAS Institute Inc. 1979, NLIN 

 procedure). 



Monte Carlo simulation techniques (Naylor et al. 

 1966) were applied to this analytical procedure to 

 evaluate the accuracy (i.e., bias) of the estimator and 

 to study the precision of parameter estimates. The 

 structure of the simulation model was such that von 

 Bertalanffy growth was assumed by stipulating a 

 decreasing linear relationship between somatic 

 growth rate and length, i.e., d(FL)ldt = A'(L„ - 

 FL). Likewise, the relationship between otolith 

 length (OL) and FL was assumed to be governed 

 by the power function, so that FL = oOL''. Otolith 

 growth rate, rf(OL)/rff, was then obtained by form- 

 ing the ratio of d{FL)ldt and rf(FL)/rf(OL). All 

 parameters in the model were otherwise set equal 

 to the estimates obtained from the otolith study. 

 and the specific probability distributions invoked 



were similar to those encountered with the actual 

 data. 



Length-Frequency Analysis 



As an independent means of verifying results ob- 

 tained through the study of otoliths, the regression 

 method of Wetherall et al. (1987) was used to esti- 

 mate specific growth and mortality parameters 

 characterizing the study population. The analysis 

 was based on the combined length-frequency distri- 

 bution (FL rounded to the nearest 10 mm) of all gin- 

 dai sampled (see Ralston [in press a] for a discus- 

 sion of the effects of pooling length data taken at 

 different times throughout the year). 



Initially, this method requires determination of the 

 least FL at which fish are fully represented in the 

 catch ((c.min)- ^or this purpose, the first size class 

 larger than the mode was assumed to be the small- 

 est length category fully sampled (see, for example, 

 Ricker 1975). Moreover, for this and any larger cut- 

 off value (^f,,), we were able to compute the mean 

 size of fully vulnerable fish in the catch (f ,), i.e., 

 those fish greater than (',,,. As I,, was successive- 

 ly advanced through the fully vulnerable size range, 

 the mean and variance in size of larger fish were 

 recalculated at each step, and a series of ordered 

 pairs was developed. The actual estimation proce- 

 dure involved regressing values of I , against suc- 

 cessive values of f^ ,. The inverse of the standard 

 error of (^ was used as a statistical weight for 

 each point, leading to the best linear unbiased esti- 

 mates of the slope (6) and intercept (C). With the 

 resulting regression statistics, the formulae pro- 

 vided in Wetherall et al. (1987) were used to obtain 

 point estimates of the ratio of total instantaneous 

 mortality rate to the von Bertalanffy growth coef- 

 ficient (ZIK) and the von Bertalanffy asymptotic size 

 parameter (LJ). In particular, they showed that 

 ZIK = 6I{1 - 6) and L„ = ^/(l - 6). Likewise, 

 error estimates for these statistics were calcuated 

 as well. 



RESULTS 



Age Estimation from Increment 

 Microstructure 



In all, 440 otoliths were extracted, and of these, 

 94 were sectioned and examined for daily incre- 

 ments. As expected, there is a clear statistical basis 

 for predicting FL from OL (Fig. I). The regression 

 equation relating these variables is highly signifi- 

 cant (P < 0.0001) and is given by 



