The red drum (Sciaenops ocellatus) is a popular gamefish found throughout the coastal waters of the Gulf of Mexico and along the eastern seaboard as far north as Massachusetts. Juvenile red drum grow extremely rapidly, especially during the warmer months, but adults grow very little. In fact, the change in growth with age is so abrupt that the standard von Bertalanffy curve has proven inadequate— the predicted lengths of younger fish are generally too large and the predicted lengths of older fish too small (see Beckman et al., 1988; Murphy and Taylor, 1990)

Nieland, David L.

Porch, Clay E.

Wilson, Charles A.

Aquatic Commons

.149 
A new growth model for 
red drum (Sciaenops ocellatus) 
that accommodates seasonal and 
ontogenic changes in growth rates 
Clay E. Porch 
Southeast Fisheries Science Center 
75 Virginia Beach Drive 
Miami, Florida 33149-1099 
E-mail address: Clay.Porch@noaa.gov 
Charles A. Wilson 
David L. Nieland 
Coastal Fisheries Institute

School of the Coast and Environment

Louisiana State University

Baton Rouge, Louisiana 70803

strong seasonal component (Goodyear2). 
In this paper a new growth equation is 
developed that explicitly models these 
two features. The new equation provid­
ed a better fit to northern Gulf of Mex­
ico red drum age-length data than any 
of the above alternatives. 
Materials and methods 
The model 
The growth rate at any given age t is 
assumed to be in some proportion k 
to the difference between the expected 
size at that age (lt) and the expected 
maximum (l
∞
): 
dlt 
= k l
∞
− lt ).  (4)(dt 
Further, k is assumed to decline with 
age and vary with the seasons such 
that 
The red drum (Sciaenops ocellatus) 
is a popular gamefish found through­
out the coastal waters of the Gulf of 
Mexico and along the eastern seaboard 
as far north as Massachusetts. Juve­
nile red drum grow extremely rapidly, 
especially during the warmer months, 
but adults grow very little. In fact, 
the change in growth with age is so 
abrupt that the standard von Berta­
lanffy curve has proven inadequate— 
the predicted lengths of younger fish 
are generally too large and the pre­
dicted lengths of older fish too small 
(see Beckman et al., 1988; Murphy and 
Taylor, 1990). 
Two generalizations of the von Ber­
talanffy curve have been found to fit red 
drum length-at-age data better than 
the original three-parameter form. The 
first, dubbed the “double” von Ber­
talanffy growth curve (Vaughan and 
Helser, 1990; Condrey et al.1), allows 
the rate at which an animal approach­
es the asymptotic length to change af­
ter some pivotal age, tp: 
1(l
∞(1 − e− k t−t1) ) if t < tp 
2 ( 
lt = l∞(1 − e− k t−t2 ) ) if t ≥ tp (1) 
tp = (k t2 2  − k1t1 () k2 − k1), 
where t = age; 
l
∞ 
= asymptotic length; 
k1, k2 = instantaneous growth rate 
coefficients; and 
t1, t2 = age intercept parameters. 
The second, dubbed the “linear” von 
Bertalanffy curve (Hoese et al., 1991; 
Vaughan, 1996), expresses the asymp­
totic length as a linear function of age: 
(lt = (b0 + b1t) (1 − e
− k t−t0 ) ). (2) 
Of course other generalizations of 
the von Bertalanffy curve may also 
be appropriate, such as the Richards 
(1959) equation 
( 1 lt = l∞ (1 − δe− k t−t0 ) ) δ where δ ≠ 0.  (3) 
The double von Bertalanffy curve ac­
commodates the possibility that older, 
larger fish might grow more slowly in 
proportion to their length than young­
er, smaller fish. (The linear von Ber­
talanffy curve has no biological inter­
pretation.) In reality, one might expect 
the growth rate in proportion to length 
to decrease gradually with the age of 
the fish rather than at some abrupt piv­
otal point. Moreover, the growth pattern 
of juvenile red drum seems to have a 
k k0 + k1e
−λ1t + k2e
−λ2t sin(2π (t − tc )), (5)= 
where λ1 and λ2 are damping coeffi­
cients and tc is a shifting parameter for 
the sine wave valued between 0 and 
1. Substituting Equations 5 and 4 and 
integrating with l = 0 when t = t0 gives 
1 Condrey, R., D. W. Beckman and C. A. 
Wilson. 1988. Management implications 
of a new growth model for red drum.Appen­
dix D. In Louisiana red drum research, 
J. A. Shepard (ed.), 26 p. U.S. Dept. 
Commerce Cooperative Agreement NA87-
WC-H-06122. Marine Fisheries Initiative 
(MARFIN) Program. Louisiana Depart­
ment of Wildlife and Fisheries, Seafood 
Division, Finfish Section, Baton Rouge, 
Louisiana 70803-7503. 
2 Goodyear, C. P. 1996. Status of the 
red drum stocks of the Gulf of Mexico: 
report for 1996. Rep. MIA 95/96-47, 21 p. 
Miami Laboratory, Southeast Fish. Sci. 
Cent., Natl. Mar. Fish. Serv., NOAA, 75 
Virginia Beach Dr., Miami, Fl. 33149. 
Manuscript accepted 17 May 2001. 
Fish. Bull. 100:149–152 (2002). 
}+ −  
150 Fishery Bulletin 100(1) 
1 0 (lt = l∞ (1 − e
β β2 k t−t0 ) ) 
1 0 )β1 = k1 (e−λ1t − e−λ t λ1 
 e−λ2t  2π cos{2π (tc − t)} − −
 (6) 
k2 
 λ2sin{2π (tc − t)}   .β2 = 4π 2 + ( )2  e−λ t  2π cos{2π (tc − t0)  − 
λ2 
2 0  
 λ2 sin{2π (tc − t0)}   
Assuming the animal will not shrink with age, i.e., dl/dt 
≥0, implies the constraint 
k0 + k1e
−λ1t + k2e
−λ2t sin(2π (t − tc )) ≥ 0. (7) 
Equation 6 may appear formidable, but typically re­
quires only a minute or two more to enter into standard 
statistical fitting packages. It reduces to a form similar to 
the Gompertz equation when k2 = 0 and to the von Berta­
lanffy equation when k1 = k2 = 0. 
Fitting the model to data 
Equations 1, 2, 3, and 6 were fitted to observations of 
length-at-age from red drum collected in the northern Gulf 
of Mexico between September 1985 and October 1998 (see 
Beckman et al., 1988, or Wilson et al.3 for further details 
regarding the data collection and aging procedures). The 
fitting was accomplished by ordinary least squares by 
using a Nelder-Mead simplex search4 and, as a check, proc 
NLIN of SAS (1990). The least-squares solution is equiva­
lent to the maximum likelihood solution when the distri­
bution of length at age is normal with constant variance, 
which seems to be approximately true of this particular 
data set (Porch, unpubl. data). 
Akaike’s (1973) information criterion (AIC) was used to 
rank the growth models in terms of their ability to provide 
statistically parsimonious explanations of the data. The 
formula for the AIC may be written 
AIC = −2log (L) + 2 p, 
where L is the likelihood function and p is the number 
of parameters (see Buckland et al., 1997). In this case, 
–2log(L) is equal to the residual sums of squares. 
3 Wilson, C. A., D. L. Nieland and A. L. Stanley. 1993. Varia­
tion of year-class strength and annual reproductive output of 
red drum Sciaenops ocellatus and black drum Pogonias cromis 
from the northern Gulf of Mexico. Final Report 1991–1992, 
31 p. U.S. Dept. Commerce Cooperative Agreement NA90AA-
H-MF724. Marine Fisheries Initiative (MARFIN) Program. 
Coastal Fisheries Institute. Louisiana State University, Baton 
Rouge, La 70803-7503. 
4 Shaw, D. E., R. W. M. Wedderburn, and A. Miller. 1991. A 
Program for function minimization using the simplex method. 
CSIRO, Division of Mathematics and Statistics, P.O. Box 218, 
Lindfield, N.S.W. 2070, Australia. 
Table 1 
Akaike’s information criteria (AIC) quantifying the fit of 
the various growth models to the red drum length-at-age 
data. Smaller AIC values indicate statistically better fits. 
Number Negative 
of log- AIC 
Model likelihood (–25,000) 
von Bertalanffy 3 16045.9 7098 
Richards 14169.8 3348 
linear von Bertalanffy 4 12883.8 776 
double von Bertalanffy 5 12876.6 763 
damped (Eq. 6, k2=0) 12651.3 313 
seasonal + damped 
(Eq. 6) 8 12584.9 186 
parameters 
4 
5 
The parameter estimates for the Richards equation 
tended to be unstable unless good initial estimates were 
provided. This was accomplished by conducting the esti­
mation in two stages. In the first stage the exponent δ was 
fixed to 1 and the other parameters were estimated, reduc­
ing the Richards equation to the von Bertalanffy form. In 
the second stage the initial guesses were set equal to the 
final estimates from the first stage and then all four pa­
rameters were estimated simultaneously. 
Results and discussion 
All five alternative growth models fitted the data signif­
icantly better than the von Bertalanffy equation accord­
ing to the AIC statistic (Table 1). The Richards equation, 
however, did not fit the data nearly as well as the other 
alternative formulations and suffered from well-known 
instability problems (Ratkowsky, 1983), therefore it can 
probably be dropped from any future consideration with 
respect to red drum. The double von Bertalanffy curve 
fitted the data better than the linear von Bertalanffy 
curve, but the comparison is rendered moot by the perfor­
mance of the new model. The five-parameter version with­
out seasonal oscillations (Eq. 6 with k2=0) fitted the data 
significantly better than either. The eight-parameter ver­
sion with seasonal oscillations fitted the data significantly 
better still (see Fig. 1). 
The estimated seasonal component to the growth rate 
was fairly substantial initially, having an amplitude at 
age 0 of 0.301 (k2) and a peak in June, but declined rap­
idly with age (Fig. 2). It is possible that an even stronger 
seasonal signal would have been estimated if age-0 fish, 
which exhibit the strongest seasonal pattern (Goodyear4), 
had been adequately represented in the sample. 
Some of the parameter estimates were highly correlat­
ed, as is the case in most growth studies. In particular, 
the correlations between the estimates for the growth rate 
and asymptotic length coefficients were typically above 
0.8. However, the asymptotic variance-covariance matrix 
NOTE Porch et al.: A growth model for Sciaenops ocellatus 151 
Table 2 
Parameter estimates and corresponding coefficients of vari­
ation (CV) derived from the asymptotic covariance matrix. 
Model arameter Estimate CV (%) 
von Bertalanffy l
∞
 37.7 1 
k 0.323 1 
t0 –0.646 
linear von Bertalanffy b0 32.5 1 
b1 0.284 1 
k 0.590 9 
t0 0.126 1 
double von Bertalanffy l
∞
 40.0 1 
k1 0.412 1 
t1 0.0530 23 
k2 0.114 2 
t2 –8.41 
damped (Eq. 6, k2=0) l∞ 44.1 1 
k0 0.0416 7 
t0 0.362 3 
k1 0.667 2 
λ1 0.464 1 
seasonal and damped l
∞
 43.4 1 
k0 0.0475 5 
t0 0.443 3 
k1 0.695 2 
λ1 0.476 1 
k2 0.301 16 
λ2 0.344 22 
tc 0.439 3 
To
ta
l l
en
gt
h 
(in
ch
es
) 
Age 
50 
40 
30 
20 
10 
0 
35 
30 
25 
20 
15 
10 
5 
0 5 10 15 20 25 30 35 40 
0 1 5 
Figure 1 
Fit of the proposed seasonal growth model to red 
drum length at age data: (A) the fit for all ages; 
(B) the fit for the younger ages. 
Figure 2 
Growth rate coefficient from seasonal model as a 
function of age. 
G
ro
w
th
 ra
te
 c
oe
ffi
 cie
nt
 
Age 
0.8 
0.6 
0.4 
0.2 
0 
0 2 3 4 5 8 9 10 
suggests that the parameters for all of the models were 
estimated fairly precisely (Table 2). 
The new model, either with or without the seasonal 
component, has both practical and theoretical advantages 
over the four other models examined in this study. By vir­
tue of its greater flexibility, it was able to fit the red drum 
data significantly better than the linear and double von 
Bertalanffy curves (its nearest competitors). Moreover, the 
Richards and linear von Bertalanffy curves are theoret­
ically disadvantaged because their parameters have no 
physical interpretation. The double von Bertalanffy curve, 
although it has a physical interpretation, suffers because 
it allows only a single discontinuous change in the growth 
rate at one age rather than a continuous change through 
time. For these reasons, the new model should be more 
widely applicable than the others, particularly for species 
that change habitat preferences with age or are subject to 
strong seasonal environmental fluctuations. 
Acknowledgments 
We thank C. Legault, G. Scott, S. Turner, D. Vaughan, and 
an anonymous reviewer for their helpful comments. 
Literature cited 
Akaike, H. 
1973. Information theory and an extension of the maximum 
likelihood principle. In Second international symposium 
A 
B 
P
3 
3 
4 3 2 
1 7 6 
152 Fishery Bulletin 100(1) 
on information theory (B. N. Petrov and F. Csaki, eds.), 
p. 267–281. Akademiai Kiado, Budapest. 
Beckman, D. W., C. A. Wilson, and A. L. Stanley. 
1988. 	Age and growth of red drum, Sciaenops ocellatus 
from offshore waters of the Gulf of Mexico. Fish. Bull. 87: 
17–28. 
Buckland, S. T., K. P. Burnham, and N. H. Augustin. 
1997. Model selection: an integral part of inference. Bio­
metrics 53:603–618. 
Hoese, H. D., D. W. Beckman, R. H. Blanchet, D. Drullinger, and 
D. L. Nieland. 
1991. A biological and fisheries profile of Louisiana red drum 
Sciaenops ocellatus. Fishery management plan series, 
number 4, part 1. Louisiana Department of Wildlife and 
Fisheries, Baton Rouge, LA, 93 p. 
Murphy, M. D., and R. G. Taylor. 
1990. Reproduction,growth,and mortality of red drum, Sciae­
nops ocellatus, in Florida waters. Fish. Bull. 88:531–542. 
Ratkowsky, D. A. 
1983. Nonlinear regression modeling. Marcel Dekker, New 
York, NY, 276 p. 
Richards, F. J. 
1959. A flexible growth function for empirical use. J. Exp. 
Botany 10:290–300. 
SAS. 
1990. SAS/STAT user’s guide, vol. 2, version 6, 4th ed. SAS 
Institute Inc., Cary, NC, 1686 p. 
Vaughan, D. S. 
1996. Status of the red drum stock on the Atlantic Coast: 
stock assessment report for 1995. U.S. Dep. Commer., 
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A new growth model for red drum (Sciaenops ocellatus) that accommodates seasonal and ontogenic changes in growth rates

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A new growth model for red drum (Sciaenops ocellatus) that accommodates seasonal and ontogenic changes in growth rates

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