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Fishery Bulletin 97(3), 1999 



its length at recapture, and y,,,^,, its maximum length. 

 The segments of Equations 6.1, 10.1, 14.1; Equations 

 6.2 (p=l), 10.2 (p=l), 14.2 (p=l); and Equations 6.3, 

 10.3, 14.3, all for a-0||<^ were fitted into the tagging 

 data, by using the nonlinear least squares method, 

 under the assumptions that T'=365.25 d, time started 

 (i.e. time /=0) on 1 January 1960 (see below for its 

 significance), and errors my(a,t) follow independent 

 normal distributions, with a mean of y(a,t) and a 

 constant variance of o^ (Table 3). A likelihood ratio 

 test suggests that Equation 6.1 is significantly dif- 

 ferent from Equation 6.2 (F^ 304=48.6892, P<0.0001 ) 

 or from Equation 6.3 iF.,.^^^^=4.l238, P=O.Oni)\ Equa- 

 tion 10.1 (p=l) is significantly different from Equa- 

 tion 10.2 (p=l) (i^2 304=45-3460, P<0.0001) or from 

 Equation 10.3 (p=i) (F., ,,,,^=3.3241, P=0.0373); and 

 Equation 14.1 is significantly different from Equa- 

 tion 14.2 (F,,.jr,4==46.8516, P<0.0001) or from Equa- 

 tion 14.3 (^.,".^^,4=3. 5345, P=0. 0304). Thus, Equations 

 6.2, 6.3; Equations 10.2 (p=l), 10.3 (p=l); and Equa- 

 tions 14.2, 14.3, and their associated estimates of pa- 

 rameters seem adequate for describing the tagging 

 data. Selection between equations 6.2 and 6.3, between 

 Equations 10.2 and 10.3, and between Equations 14.2 

 and 14.3 by developing more general models of K(a.t) 

 was not successful because of a lack of data. 



Are equations 6.1, 6.2, 6.3; 10.1, 10.2, 10.3; 

 14.1, 14.2, 14.3 independent ofthe start of time? 



An age- and time-dependent growth model is useful, 

 if and only if it is independent ofthe start of time or 

 if it is time-homogeneous. The reason for this is that 



start of time is unknown. For Equations 6.0, 10.0 

 and 14.0 to be useful, 



J K{s + a-t,s)ds 



if a -Qq < / or 



K(s + a -f,s)ds 



if a-a,j>^ must be independent ofthe start of time t. 

 Obviously, Equations 6.1, 6.2, 6.3; 10.1, 10.2, 10.3; 



14.1, 14.2, and 14.3 all are independent ofthe start 

 of time, where time / appears as time differences t- 

 t,, or t-t^. 







However, interesting differences exist among them. 

 Equations 6.1, 6.3, 10.1, 10.3, 14.1, and 14.3 apply 

 on any time scales, without any adjustment of esti- 

 mates of their parameters in subsequent applications 

 because they depend on time difference t-tf^ or age 

 difference a-a^^ only. By contrast. Equations 6.2, 10.2 

 and 14.2 and estimates of their parameters must be 

 properly adjusted for this purpose. Specifically, the 

 estimate of parameter t^ in Equations 6.2, 10.2, and 

 14.2 must be correctly adjusted before their subse- 

 quent applications. To make such an adjustment, 

 suppose that all growth parameters are estimated 

 from tagging data by using one segment of Equation 

 6.2, 10.2, or 14.2 on one time scale (regression time 

 scale. Fig. 1), with time t, parameter / (estimated), 

 and a reference time t^, (known). Now. Equation 6.2, 



10.2, or 14.2 is to be applied in a future fish stock 

 assessment on another time scale (application time 

 scale. Fig. 1), with time t', parameter t ' (unknown. 



