150 



Fishery Bulletin 100(1) 



^1 



Po 



fc 



4;r-+(A,)' 



2KCOs{2!t{t^.'t)}-' 



A.,sin{2nit^ -t)] 

 '2;rcos{2;r(/, -^o)}-' 

 A._,sin{2;r(*,, -<„)} 



(6) 



Assuming the animal will not shrink with age, i.e., dl I dt 

 >0, implies the constraint 



/fo + V"^''+ V~ 



'sin(2;r(?-r))>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 k., = and to the von Berta- 

 lanffy equation when ^j = ^2=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.'^ for further details 

 regarding the data collection and aging procedures). The 

 fitting was accomplished by ordinary least squares by 

 using a Nelder-Mead simplex search^ 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 gi-owth models in terms of their ability to provide 

 statistically parsimonious explanations of the data. The 

 formula for the AIC may be written 



A/C=-21og(L) + 2p, 



where L is the likelihood function and p is the number 

 of parameters (see Buckland et al., 1997). In this case, 

 -21og(L) is equal to the residual sums of squares. 



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 Agi-eement NA90AA- 

 H-MF724. Marine Fisheries Initiative (MARFINi Program. 

 Coastal Fisheries Institute. Louisiana State University, Baton 

 Rouge, La 70803-7503. 



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, 

 Lmdfield, N.S.W. 2070. Australia. 



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 5 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 cui-ve 

 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 /?.,=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 of 0.301 (k.-,'> 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 (Goodyear*), 

 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 



