836 



Fishery Bulletin 98(4) 



pendent factors and are useful for modeling growth at 

 age and time (e.g. from length-at-age data), incremental 

 growth at age and time increment (e.g. data on length 

 increment at age and time increment from tagging stud- 

 ies), the effects of tagging, and, if coupled with a proper 

 age- and time-dependent population dynamics model, the 

 effects on the gi-owth of animals of many population char- 

 acteristics, such as population size. 



Because of experimental constraints, such as difficul- 

 ties in taking continuous measurements (if measured at 

 all), anabolic and catabolic rates of animals are necessar- 

 ily measured either by restraining them in the laboratory 

 or in the field. Such restraints can cause stress to animals 

 and hence result in biased measurements. Experimental 

 methods should be developed to estimate the anabolic and 

 catabolic rates of animals in as natural an environment as 

 possible. 



In this study, I use an age- and time-dependent von Ber- 

 talanffy ( 19381 gj'owth equation and some of its commonly 

 used special cases for estimating the parameters from a 

 set of tagging data on times at liberty, lengths at release, 

 and lengths at recapture of a centropomid perch, barra- 

 mundi (Lates calcarifer) and provide a method for deter- 

 nnnmg the anabolic and catabolic rates of animals in their 

 natural environment. I also explain why Pauly's (1981) 

 index of growth performance is adequate and propose an 

 index of catabolic performance. 



Model 



Let 0<VVfa,?)<oo. -°°<aQ<a<oo, -<"<tQ<t<<x, denote the body 

 mass of an individual of age a at time t, with an arbitrary 

 reference age a^ and an arbitrary reference time /„. The von 

 Bertalanffy ( 1938) growth equation can be generalized as 



dW{a,t) 

 dt 



AKa,t)W{a,t)" -C(a,t)W(a,t)" , 



(1) 



where A(a,t)>0 and C(a.tl>Q = functions of age a and time 



t: and 

 B and D = positive biological constants. 



For a particular functional form o{'A(a,t) and C<a.t), Equa- 

 tion 1 can be used for estimating its parameters from data 

 on body masses of animals of different ages at different 

 times, or on two distinct body masses of the same individ- 

 ual at different times. If collected at all, such data are col- 

 lected mostly for terrestrial and occasionally for aquatic 

 animals. What is most commonly gathered for both terres- 

 trial and aquatic animals is, however, one or more linear 

 dimensions of an individual's body, such as its total length 

 at age, or two distinct measurements at different times. 

 Measurements of linear dimensions of an animal contain 

 useful information on its body mass. Indeed, it is well 

 known that body mass Wla.tl is scaled allomelricaliy to 

 body length Ua,t), i.e. 



where a and /3 = (constant) allometric parameters (Peters, 

 1983; Calder, 1984; Schmidt-Nielsen, 

 1984; Reiss, 1989). 



Substitution of Equation 2 into Equation 1 yields 



dt P 



(3) 



a"-'Cia,t}L(a,t)'' 



Thus, if a and /3 are known, as is usually assumed, para- 

 meters B and D, and those in Ala.t) and ClciJ) can be esti- 

 mated from data on length-at-age data, or on two distinct 

 lengths of the same individual at different times. 



Although too general to be solved even numerically. 

 Equations 1 and 3 are useful in formulating ideas. Now, I 

 consider a special case of Equations 1 and 3 for seasonally 

 varying A(a,?) and C(a,t), such that 



Ala.t) : 



/ 



A -I-6C0S — it -t. ) 

 i T 



In 



and 



C(a,n = A' 4- 6cos| —(/'-/ I 

 T 



where y. yb I K. T. and t^, are, respectively, the mean, ampli- 

 tude, period, and time shift for the anabolic process; A', b, 

 T, and t^ are, respectively, the mean, amplitude, period, 

 and time shift for the catabolic process. For this special 

 case. Equations 1 and 3 then become, respectively 



dWiaJ) 



dt 



,- , 1 2;r 

 A +6cos — (/-/ ) 

 T 



^wui.t)" -wui.t)" 



K 



(4) 



and 



dLiaJ) _ 1 

 dt ~~J5 



A +bcos\ — it-t ) 

 T 



K 



(5) 



Equations 4 and 5 can be solved numerically but not 

 analytically. For comparison and illustration, I now con- 

 sider five special cases (four of which are reparameteriza- 

 tions of commonly used growth equations) of Equation 5: 



If 6=0, or li A(a,t} and C(aJ) are constants. Equation 5 

 becomes 



dUaJ) 1 



-dr-~i5^'" 



" 'Uajf 



Ka"'Ua.t)" 



(6) 



W(a. t) = aLiu. tf. 



(2) 



If /3(fi-l)+l=0 (i.e. fi-l=-l//ii and /i(D-l )-(-l=l (i.e. 

 /J=l), Equation 5 becomes 



