36 
Fishery Bulletin 113(1) 
600 
500 
3- 400 
LL 
E 
E 
£ 300 
CD 
C 
200 
100 
0 
i t =457(l-e<-°- 33 < t+158 ») 
Predicted, freely estimated 
* Observed 
— —Predicted, to = 0 
0 2 4 6 8 10 12 14 16 
Age (years) 
Figure 4 
Observed and predicted lengths at age, measured in fork lengths (FLs), for 
gray triggerfish (Balistes capriscus ) sampled from the southeastern Unit- 
ed States in 1990-2012. L t =length at age f; fo = theoretical age when length 
equals zero. 
we recommend estimating growth parameters with a 
fixed t o value, such as -0.5 (Burton et ah, 2012) or 
zero (Burton et al., 2014). This method has the effect 
of pulling the growth curve down to simulate smaller 
length at age for fish at the youngest ages. We re-es- 
timated growth parameters of gray triggerfish with a 
fixed value of to = 0. The value of the growth coefficient 
K increased measurably (0.75 versus 0.33, when freely 
estimated) because of the steepness of the initial part 
of the curve (Fig. 4). The theoretical maximum length 
decreased marginally, 436 mm FL versus a freely 
estimated value of 457 mm FL. This result is not unex- 
pected because L„ and K are negatively correlated with 
each other (Ricker, 1975; Xiao, 1994). The 2 growth 
curves are very similar from age 2 on, with only minor 
differences in size at age at the oldest ages. 
Our theoretical growth curve fitted the observed 
data well (Fig. 4). Growth parameters estimated in our 
study compare very closely with the study of gray trig- 
gerfish in the Gulf of Mexico (Johnson and Saloman, 
1984), and it is interesting to note that fish samples 
came primarily from fishery-dependent sources in both 
studies. In Moore’s (2001) study, samples were obtained 
from both fishery-dependent and fishery-independent 
sources and yielded markedly different growth param- 
eters between male and female gray triggerfish, but no 
differences in growth parameters estimates were found 
between the fishery-dependent and fishery-independent 
data sets. 
Natural mortality of wild populations of fishes is 
difficult to measure. A single estimate of M for the 
entire life span of a fish is unreasonable, except for 
fish that have attained a maximal size that renders 
them invulnerable to large-scale predation. The Hewitt 
and Hoenig (2005) estimate of M is based on the maxi- 
mum age a species can attain in an unfished popula- 
tion. In this sense, the point estimate of M, derived 
with Hewitt and Hoenig’s (2005) method, can serve 
as a lower boundary for the estimate of M derived for 
older ages by an age-varying method. The maximum 
observed age from our study was 15 — a result that 
compares favorably with the maximum age of 13 found 
by Escorriola (1991) and Johnson and Saloman (1984). 
We feel this age is a realistic maximum age for gray 
triggerfish because no other study has found an older 
age, including studies that might be considered more 
representative of the population at large because they 
included fishery-independent samples as well (Moore, 
2001). We think our estimates are reasonable given 
that our age-specific estimate of M= 0.33 for the older 
ages that was derived by using the method of Charnov 
et al. (2013) compares closely with the point estimate 
of M= 0.28 found with the method of Hewitt and Hoe- 
nig (2005) (Table 2). 
