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Fishery Bulletin 118(4) 
measured in millimeters FL per year, was estimated at 2 
user-defined reference lengths, « and B (where a<f). Ref- 
erence lengths were chosen to ensure that the majority of 
lengths at initial capture (L,) fell between the 2 defined ref- 
erence lengths by taking the mean length of the 3 smallest 
and 3 largest individuals at initial capture (a and B, respec- 
tively; Dureuil and Worm, 2015) in each region-specific data 
set. These parameters have better statistical properties 
than the asymptotic length, or the theoretical maximum 
length (L.,), and the growth constant (Rk) because they are 
not highly correlated. Further, they allow easier interpre- 
tation of growth from tagging data (Francis, 1988b). The 
growth rates relate to the parameters of the von Berta- 
lanffy growth curve as follows: 
8a. — & 
ek = 14 ot a SB (2) 
a—B 
Seasonal growth is parameterized as w, the time of year 
when growth is at its maximum, and as u, with a u value 
of 0.0 indicating no seasonal growth and a wu value of 1.0 
indicating strong seasonal growth with growth likely ceas- 
ing at some point during the year: 
8 AT +(¢>-6)) 
Nia) OE 77, |e ees (3) 
8a — & a-9® 
where 6; = aa fori = 1, 2. 
Tt 
The GROTAG model is fit by minimizing the negative 
log-likelihood function (—A). Growth variability (v) is incor- 
porated into the model by the parameter wu; , the expected 
mean growth increment of the 7th individual where u; is 
normally distributed with a standard deviation (SD) of 
o;. In this study, 6, was assumed to be a function of the 
expected growth increment o,=vu;. An additional parame- 
ter p, the probability of outlier contamination, was also fit. 
For each data set, made up of i=1 to n growth increments 
where F is the range (largest and smallest) of observed 
growth increments, the following equation was used: 
Pp 
i= Byln|(.- ph +2] (4) 
~¥, (AL, — u, — m)? | (6? + s”) 
where i, = exp : 
[2n(o? + s*)]” 
The likelihood function estimates the population measure- 
ment error in AL as being normally distributed with a mean 
of m and an SD of s. The initial model estimated g,,, 8, and v 
with additional parameters (m, s, w, uw, and p) added, increas- 
ing model complexity (Table 1). Unfitted parameters were 
held at zero. Optimal model parameterization was deter- 
mined for each region by using likelihood-ratio chi-square 
tests to determine if improvement in model fit was significant 
(P<0.05). Francis (1988b) suggested that the introduction of 
Table 1 
Parameters fitted for the GROTAG models used to esti- 
mate region-specific growth rates of bonnetheads (Sphyrna 
tiburo) in the Gulf of Mexico during 1993-2006 and in the 
Atlantic Ocean off the southeastern United States during 
1998-2019: growth rate estimates at reference lengths a 
and # (g, and gz), mean (m) and standard deviation (s) of 
the measurement error, magnitude (wu) and timing (w) of 
seasonal growth, growth variability (v), and outlier con- 
tamination probability (p). 
GROTAG model Parameters estimated 
Ew 8p, s 
Su &p> S,U 
Sa» &p, $V, mM 
S02 Ep» S, UV, M, U, W, P 
Ew &p> S,p 
an additional parameter should increase the log-likelihood 
value by at least 1.92. Likelihood ratio tests were also con- 
ducted to determine significant differences in von Bertalanffy 
growth curves between regions (Kimura, 1980). 
Age-based growth model 
To allow direct comparison of growth estimates based 
on age data and those based on tag-recapture data, 
region-specific length-at-age data from Lombardi-Carlson 
et al. (2003) and Frazier et al. (2014) were remodeled (the 
authors of both publications used the Beverton and Holt, 
1957, method of modeling VBGF parameters) by using an 
alternative parameterization of the VBGF recommended 
by Francis (1988a), in which mean length (L) of fish of age 
t is determined with this equation: 
(ly L lo “ 7 ve) 
L=l, + ae 
, (5) 
where p = 04x s 
ly —la 
where X = (® + Y)/2; 
l» = mean length at age ®; 
lx = mean length at age X; and 
ly = mean length at age V. 
Values for ® and were chosen to encompass the range 
of ages represented in the published length-at-age data 
from both regions (Lombardi-Carlson et al., 2003; Frazier 
et al., 2014). The Francis (1988a) parameterization yields 
estimates of VBGF parameters that better represent the 
growth information modeled from length-at-estimated- 
age and tagging data. The growth estimates generated 
from this model allow comparison of the mean growth rate 
of fish of an estimated age with that of fish of a length 
equal to the mean length at that age (Francis, 1988a). 
