62 
Fishery Bulletin 107(1) 
It is assumed that the age composition of fish with ages 
t c <a<A observed in year t represents a random sample 
from a multinomial distribution and uniform selectiv- 
ity from the age of full recruitment. Thus, by ignoring 
constants, the log-likelihood, A, of the age compositions 
observed in the various years may be calculated as 
A 
A = SS"‘*.* l0g [^]’ 
t a=t c 
where n a t = the observed number of fish of age a in 
year t. 
An estimate of Z was obtained by maximizing the log- 
likelihood by using Solver in Excel (vers. 2003, Microsoft 
Corporation, Salem, MA). 
Next, the constraint that recruitment is constant in 
the analysis just described was relaxed and an estimate 
of Z was determined with relative abundance analysis 
(Deriso et al., 1985). This latter analysis, which is an 
extension of catch curve analysis, involved initially set- 
ting the relative level of recruitment, R v , for each year 
class, y, to 1. The relative levels of recruitment of the 
different year classes were then successively introduced 
as additional parameters to be estimated by the model, 
by using a stepwise forward selection algorithm (e.g., 
Sokal and Rohlf, 1995). The process was terminated 
when the introduction of R as a parameter to be es- 
timated for any further year class failed to produce a 
statistically significant improvement in the fit of the 
model to the data, as determined with the likelihood- 
ratio test (Cerrato, 1990). At each stage, the relative 
levels of recruitment for year classes not included as 
parameters to be estimated in the model continued to 
be constrained to the average level, i.e., 1. The 95% 
confidence intervals for Z for A. gouldii, when either 
constant or variable recruitment was assumed, were 
obtained from the profile likelihood for Z (Hilborn and 
Mangel, 1997). All of the above calculations for Z were 
undertaken by maximizing the above log-likelihood with 
the Solver tool in Excel. 
The point estimate and 95% confidence intervals for 
the natural mortality, M, of A. gouldii were determined 
by refitting, in SPSS, Hoenig’s (1983) regression equa- 
tion for fish to the values for mortalities and maximum 
ages for 82 unexploited or lightly exploited fish stocks 
(see Hall et al., 2004) and including the maximum re- 
corded age for A. gouldii. 
The approach of Hall et al. (2004) was used to deter- 
mine the likelihood for M, calculated by using the likeli- 
hood for Z, as derived by using the relative abundance 
analysis and assuming variable recruitment. For this 
estimation it was assumed that, for each value of Z, 
there is a uniform probability that M < Z. The resulting 
likelihood for M was then combined with the estimate 
for M derived by refitting Hoenig’s (1983) regression 
equation for fish. 
A Monte Carlo resampling approach was used to de- 
rive estimates of F for the fully recruited age classes 
of each species. Estimates of Z (from the relative abun- 
dance analysis) and M (from the method of Hall et al., 
2004) were randomly resampled, with replacement, from 
their respective probability distributions, and any pair 
of estimates for which the value for M was greater than 
Z was rejected. 10,000 sets of estimates of Z and M 
were produced, from which 10,000 estimates of F were 
determined with the equation F = Z - M. The point 
estimate of F and associated 95% confidence limits 
were taken as the median value and the 2.5 and 97.5 
percentiles of these estimates. 
The spawning stock biomass per age-0 recruit ( SSB / 
R) of A. gouldii was calculated for each sex with the 
equation 
SSB /R = Y W a P P mat a exp(-Za), 
*—‘a=t c 
where Z = total mortality; 
W a = the weight of a fish at age a; 
P sex a = the proportions of each sex at age a; and 
P ma t a ~ proportions of mature females at each 
age a. 
W a was calculated by using the von Bertalanffy growth 
curves for each sex and the relationship between total 
body weight (g) and length (mm). For males, P sexa was 
determined by using the growth curve and the logistic 
relationship describing the probability that a fish of 
a given length is a male, whereas, for females, P sexa 
was calculated as 1 minus this probability. P mat a was 
determined by using the logistic function describing 
the probability that a fish of a given length is mature, 
together with the lengths at age predicted by using 
the von Bertalanffy growth function. All males were 
assumed to be mature. Estimates of SSB/R were deter- 
mined for each of the 10,000 values generated for F by 
the resampling procedure. The point estimates and 
associated 95% confidence limits for the current level 
of SSB/R were taken as the median and 2.5 and 97.5 
percentiles of the resulting SSB/R values. The spawn- 
ing potential ratio, SPR, was calculated as the ratio of 
SSB/R at a specified level of fishing mortality to that 
for an unfished population (Goodyear, 1993). For the 
per-recruit analysis we assumed that knife-edge recruit- 
ment to the fishery occurred at 15 years, total mortality 
for fully recruited fish was constant and the maximum 
age was 100 years. 
Results 
Aging methods, length and age compositions, 
and growth pattern 
The mean monthly marginal increments on sectioned 
otoliths of A. gouldii with 2-10 zones remained rela- 
tively high (>0.39) between July and October, before 
declining sequentially to 0.36 in November and to a 
minimum of 0.26 in January and February, and then 
