216 



Fishery Bulletin 100(2) 



crepancy between the counts recorded by the independent 

 reader and the senior author was five. After consultation, 

 it was agreed that, in many of the cases where the counts 

 differed by one opaque zone, the independent reader had 

 failed to discern the outermost opaque zone at the periph- 

 ery of the otohths. Moreover, the extent of any discrepancy 

 between the counts of the independent reader and the se- 

 nior autiior declined if the independent reader continued to 

 recount the number of opaque zones on the otoliths. 



Validation that the opaque zones in the otoliths of G. he- 

 braiciim are formed annually was carried out by analyzing 

 the trends exhibited throughout the year by the marginal 

 increments on whole otoliths, when only one opaque zone 

 was present, and on sectioned otoliths when two or more 

 opaque zones were present. For this purpose, the marginal 

 increment on each otolith, i.e. the distance between the out- 

 er edge of the single or outermost opaque zone and the 

 edge of the otolith, was expressed either as a proportion of 

 the distance between the primordium and the outer edge 

 of the opaque zone, when only one opaque zone was pres- 

 ent, or as a proportion of the distance between the outer 

 edges of the two outermost opaque zones, when two or more 

 opaque zones were present. Each of the above requisite dis- 

 tances was measured perpendicular to the opaque zone(s) 

 and without knowledge of the date of capture of the fish 

 and was recorded to the nearest 0.01 mm by using Optimas 

 5. The values for the marginal increments were separated 

 into groups according to the number of opaque zones on the 

 otoliths, i.e. 1, 2-5, 6-8, 9-11 etc., after which the values 

 for each of those groups in each corresponding month of the 

 year between May 1996 and April 1998 were pooled. 



Von Bertalanffy growth equations 



The time when spawning peaked was estimated from the 

 trends exhibited throughout the year by gonadosomatic 

 indices, gonadal maturity stages, and pattern of oocyte 

 development. Tliis time was considered to coiTespond to the 

 birth date of G. hehraicum and could thus be used, in combi- 

 nation with the number of opaque zones on the otolith and 

 the time when the annulus becomes delineated on the oto- 

 lith, to determine the age of individual fish on their date of 

 capture. Because the sex offish <150 mm could not be deter- 

 mined, the lengths-at-age of these small fish were randomly 

 allocated in equal numbers to the data sets for female and 

 male fish used for constructing the gi-owth curves. 



Assumptions are made concerning the distribution of er- 

 rors when fitting von Bertalanffy growth cui-ves to length- 

 at-age data. Kimura (1980) discussed the implications of 

 the following three assumptions, namely that 1) the indi- 

 vidual lengths-at-age have a constant variance, 2) the mean 

 lengths-at-age have a constant variance and 3) the vari- 

 ance of the lengths-at-age is dependent on age. The assump- 

 tion most frequently adopted in growth studies is that the 

 individual lengths-at-age have a constant variance. As dis- 

 cussed by Kimura (1980), different assumptions regarding 

 the error variance require modifications to the objective 

 function to ensure that the parameters are estimated ac- 

 curately and that any comparisons between gi-owth cui-ves, 

 that are based on the likelihood ratio, are appropriate. 



A von Bertalanffy growth equation was fitted to the 

 lengths-at-age of female and male fish with the traditional 

 assumptions that the lengths-at-age are normally distrib- 

 uted around the values predicted from the growth equa- 

 tion and that the variance of this distribution is constant 

 for each sex over all ages. However, visual examination of 

 the residuals for each curve suggested that it was not ap- 

 propriate to make the latter assumption. Further study 

 showed that the variance of the residuals is approximately 

 proportional to the age of the fish, as above in assump- 

 tion 3 of Kimura ( 1980). Thus, the von Bertalanffy growth 

 equation was fitted to the length-at-age data for each sex 

 by using the assumption that the residuals were normally 

 distributed, where the variance of this distribution was 

 proportional to age but dependent on sex. That is, 



L, =L.{l-exp[-/?(^, -^„)]} 

 and Li = L, 4- f,, 



where, for each sex, L^ = the observed length at age; 

 L = the estimated length-at-age; 

 t^ = the age; and 



fj= the error associated with the 7th 

 fish. 



For the growth curves for females and males, L .^ is the 

 mean asymptotic length predicted by the equation, k is 

 the growth coefficient, and t^^ is the hypothetical age at 

 which fish would have zero length if growth had followed 

 that predicted by the equation. The errors are assumed to 

 be normally distributed, such that f~NiO, cj^), where c,. 

 is the constant of proportionality between the variance of 

 the residuals and age for fish of sex .s. The growth equa- 

 tions were fitted to the observed length-at-age data for 

 both sexes by maximizing the log-likelihood of the data. 

 The log-likelihood for the combination of male and female 

 fish. A, may be written as 



A: 



\m-^ ^f' 



where A^ = the log-likelihood associated with females or 

 males and may be calculated as 



A, = -^log(2ff)-- > log(cJ,) > \ — '- ^— k 



and n^ = the number offish of that sex in the length-at- 

 age data. 



The maximum likelihood estimate off,, for each sex is given 

 by 



The SOLVER routine in Microsoft EXCEL (Microsoft 

 Corp., 2000) was used to estimate the parameters that 



