149 



A new growth model for 

 red drum iSciaenops 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 

 Email address Clay Porchia'noaa gov 



Charles A. Wilson 



David L. Nieland 



Coastal Fisheries Institute 



School of the Coast and Environment 



Louisiana Stale University 



Baton Rouge, Louisiana 70803 



strong seasonal component (Goodyear-). 

 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 (/,) and the expected 

 maximum (/ ): 



dt 



Ml-L). 



(4) 



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 gi-owth 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.i|. allows 

 the rate at which an animal approach- 

 es the asymptotic length to change af- 

 ter some pivotal age, t : 



/, = 



/_(l_e-*."-'.') ift<t,, 



/, 1-e 



- *.,!(-(,! 



.f?>^, '1' 



where t = age; 



/ . = asymptotic length; 

 /?,,/;., = instantaneous growth rate 

 coefficients; and 

 f J, t., = 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: 



/, =(f)fi +bit){l-e 



-kit-t„). 



(2) 



Of course other generalizations of 

 the von Bertalanffy cui-ve may also 

 be appropriate, such as the Richards 

 (1959) equation 



/, =La-de- 



')' where 5 5^0. (3) 



tp=ik.,t.,-k,ti)Hk.2-k,), 



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 gi'owth 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 



: ^ + /i,e"^'' + k^e'^'' sin(2;r(f  



(5) 



where A, and A., are damping coeffi- 

 cients and t^ is a shifting parameter for 

 the sine wave valued between and 

 1. Substituting Equations 5 and 4 and 

 integrating with / = when ^ = ?o gives 



' Condrey, R., D. W. Beclcman 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 Agi'eement NA87- 

 WC-H-06122. Marine Fisheries Initiative 

 IMARFIN) Program. Louisiana Depart- 

 ment of Wildlife and Fisheries, Seafood 

 Division, Finfish Section. Baton Rouge, 

 Louisiana 70803-7.503. 



- 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 Lahoratorv, .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). 



