838 



Fishery Bulletin 98(4) 



a -ag> t-tg) for a group of individuals. It is, however, more 

 convenient to use only one segment in a single analysis. 

 Indeed, although growth parameters can be estimated by 

 use of either segment of any of Equations 8, 9, 11, and 12, it 

 is easier to use the segment for a - a,, < t-t^, by letting time t 

 start before the animals whose giowth is to be modeled are 

 born, unless time is allowed to take negative values. Use of 

 the other segment, i.e. that for a - Qq > t-tg, gives identical 

 results, but it requires first calculating Llt^^ + a -t, t^). 



The amount of data required to estimate parameters 

 in a growth model is a function of the generality of that 

 model: the more general a model is, the more data it 

 usually requires. Age- and time-dependent growth models 

 generally require knowledge of two ages Qq and a, time t, 

 and two .sizes LiUf,, t - a -\- a^) and L(a,t) if a - «„ < '~'o' "'" 

 knowledge of two times /q and t, age a. and two sizes Lit^^ 

 + a - 1. 1„) and L(a,t) if a - Cq - '"V However, use of Equa- 

 tions 4, 5, 8, and 11 only requires knowledge of the differ- 

 ence between two ages a - a^, time t, and two sizes Lia^, t - 

 a + Qij) and L(a,t); or of the difference between two times 

 t - tg, time /, and two sizes L(/„ + a - f , ?„) and Ua.t). By con- 

 trast, use of Equations 9 and 12 only requires knowledge of 

 the difference between two ages, a - Oq, and two sizes, Lia^, 

 t-a+ a„) and L(a,t), or of the difference between two times 

 t - tg, and two sizes L{tg + a - t,tg) and Ua.t). 



Interestingly, a reparameterization of Equation 9 has 

 been widely used to model tagging data (Xiao, 1999), 

 where a^ or t^ is interpreted as time at release, o or ? as 

 time at recapture, a - a„ or f - t^ as time at liberty, Uaf,, 

 t - a + a,,) or LU^ + a - t, t^) as size at release, and Ua.t) 

 as size at recapture. It has also been used extensively to 

 model size-at-age data (obtained, say, by aging animals by 

 reading marks in scales and otoliths) (e.g. Moulton et al., 

 1992 ), where Oq or ?„ is interpreted as age at birth, a or t as 

 age, Lia^^, t - a + Oq) or L(/,, + a - t. t^) as size at birth, and 

 L(a,t) as size at age. However, it is rare to know two ages 

 and the corresponding sizes of an animal; what are com- 

 monly measured are one age and its corresponding size. 

 Consequently, it is common practice to fit Equation 9 to 

 such size-at-age data to estimate age at birth a„ or t^, as 

 well as the growth parameters, thereby implicitly assum- 

 ing, for all animals concerned, that the size at birth Lia,^, t 

 - a + Qq) or Utg + a -t,tQ) is zero and that the age at birth 

 a, I or ^11 is the same. Exactly the same argument applies to 

 Equation 12. 



The barramundi L. calcarifer is a protandrous fish found 

 in estuaries and other coastal areas of the Indo-West 

 Pacific (Cxriffin, 1987). Between August 1977 and June 

 1980, 4933 barramundi with a body total length range of 

 about 10-100 cm were captured by a combination of lure 

 fishing, tidal trap, seine and gill net. Fish were measured 

 to the nearest cm, tagged with Floy FT-2 dart tags for 

 fish >35 cm and FD-67 anchor tags for fish <35 cm, and 

 released in rivers flowing into the Van Diemen Gulf and 

 the Gulf of Carpentaria of northern Australia (Davis and 

 Reid, 1982). Of those tagged, 312 of a total length of 23-92 

 cm with a mean of 60 ±13 (moan +SE) cm were recaptured, 

 but only 308 were used in the analysis below due to incom- 

 plete recapture information. The time at liberty ranged 

 from zero to 932 d, with a mean of 219 ±21 1 d, and the 



length increment from -21 to 35 cm, with a mean of 6 ±8 

 cm. Negative increments in length are often observed in 

 a tagging experiment, because tagged animals can shrink 

 in size immediately after tagging, or because of recording 

 errors at both release and recapture. The estimates of allo- 

 metric parameters for barramundi, used in the present 

 paper, were those obtained by Reynolds (1978): a = 1.06 x 

 lO-'^ kg X cm-/J and /3 = 3.02. 



Let Oq or t^ denote time at release, a or f time at recap- 

 ture, a - flg or f - ?,| time at liberty, Lia^, t - a + f/f,) or Litg 

 + a - t, tg) the length of a fish at release, and Lla.t) its 

 length at recapture. Equation 6 and the segments of Equa- 

 tions 8, 9, 11, and 12 for a -af^< t-t^, were fitted to the tag- 

 ging data, by using the nonlinear least squares method, 

 under the assumptions that T=365.25 d, time started (i.e. 

 time t=0) on 1 January 1960 (see Xiao 119991 for its sig- 

 nificance), and errors in L(a.t) follow independent normal 

 distributions, with a mean of L(a,t) and a constant vari- 

 ance of O' (Table 1). In these calculations. Equation 6 

 was numerically solved as an initial value problem with 

 Lia. t)\i ^ I = L(?Q + a - t. t^) for a - Qq > t-t^^ using the 

 fourth order Runge-Kutta method (Beyer, 1978). A likeli- 

 hood ratio test suggests that Equation 8 is significantly 

 different from Equation 9 (n3,„=48.6892, P<0.0001); and 

 Equation 11 is significantly different from Equation 12 

 <^2,.M4=45.3460, P<0.0001). Thus, Equations 8 and 11, and 

 their associated estimates of parameters seem adequate 

 for describing the tagging data. Selection among Equa- 

 tions 6, 8, and 11 is difficult because little is known of the 

 underlying mechanisms of the growth process. 



Discussion 



Fitting of the original von Bertalanffy growth model (Eq. 

 6) to the tagging data for barramundi suggests that its 

 anabolic rate changes proportionally with the 5=0.872 1 

 power of its body mass and that its catabolic rate changes 

 proportionally (D = l. 03.567=1) with its body mass (i.e. at a 

 1:1 ratio). Such an estimate of B is 9.0125';^ higher than 

 that (S=4/5) obtained for many fish under laboratory con- 

 ditions (Pauly, 1981). More data are needed to examine the 

 generality of this finding. By contrast, little information 

 is available on the value of D. Nonetheless, it is interest- 

 ing that the catabolic rate of barramundi increases pro- 

 portionally with its body mass; a I'/f increase in body mass 

 corresponds to about a V'i increase in catabolic rate. Con- 

 sequently, it is unnecessary to use the analytically solv- 

 able special cases of the original von Bertalanffy growth 

 equation in data analysis, unless their assumptions are 

 validated. 



Alternatively, like many tropical and subtropical spe- 

 cies of fish lAppeldoorn, 1987; Pauly et al.. 1992), barra- 

 mundi may be interpreted as exhibiting a strong seasonal 

 growth. For both models (Eqs. 8 and 11), its growth rate 

 reaches its maximum on 3 or 4 March (i.e. at the start of 

 autumn), slows down to zero on 17 July (i.e. in the middle 

 of winter), reaches its minimum on 2 or 3 September (i.e. 

 at the start of spring), returns to zero on 19 or 20 October 

 (i.e. in the middle of spring), and comes back to its maxi- 



