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Fishery Bulletin 97(4), 1999 



Table 1 



Summary of the different cases of Schnute's (1981) size-at-age growth model that was fitted to age estimates from different 

 calcified structures and Francis's ( 1995) mark-recapture analogue of Schnute's growth model that was fitted to the tagging data. 



Schnute's ( 1981 ) size at age model 



Francis's (1995) mark-recapture analogue of 

 Schnute's (1981) growth model 



Case 1: a;tO, b?tO 

 Case 2: a?:0, b^O 

 Case 3: a?iO, b^iO 

 Case 4: sl=^Q, b^tO 

 Meaning of terms 



Y(t)-- 



1 rt~"""'l' 



Yit) = yjoxp 



1-e' 





1-e" 



Y{t)-- 



y? + (^^yf)^^^ 



y(<) = ,v,exp 



log(V2 /.Vi) 



t-T, 



y(n = fish size IFL) at age (, Tj and r, = lower 

 and upper ages offish respectively where Tg 

 > fi-.v, and v., = mean sizes at ages r, and Tg, 

 respectively, a and b describe the shape of 

 the cui-ve. 



AY = -Y,+[Y,''e-''+ca-e-°^)\'' 



Ay = -y, -I- y;""""^ ' exp[c( 1- e""" )] 

 1 



AY = -Y, +[Y,'' +iX\ - y''^)AtY 



Ay = -y, + y,(A,/3',)^ 



Ay = mean growth 

 y, = size at marking 

 y'l and y., = lower and upper sizes of fish, 



respectively, where y, < v., 

 g, andg., = mean annual growth for fishes of 

 sizes y, and v., respectively. 

 6 = describes curvature in model, 

 ^1 = yi + gi and A, = Vo + g-,^ 



a = In 



An 



h b 



y-i - yi 



if6;tOor 



ln(y2 /yj) 



ln(A, /A,) 



if 6 = 0, 



c= r- rV^ . iffe;^Oor 



ln(y)ln(A,)-ln(v,)ln(A„) ._, „ ,, , 



c = = '—'' ^- 11 o = 0, A? = 1 



ln(A,y„)-ln(A., /y,) 



To determine the timing of zone formation, the 

 edges of the various structures were examined. The 

 growth of the structure, subsequent to the most re- 

 cent zone, was estimated as a percentage (20, 40, 60, 

 and 80%) of the previously completed zone. It was 

 also noted whether the zone was considered to be on 

 the edge of the structure. Only fish with 2-4 gi-owth 

 zones were used. Fish were examined individually 

 by structure in a random order with no knowledge of 

 date of collection. 



Estimation of growth models from calcified structures 



Growth models using age estipiates from different 

 calcified structures were derived by using procedures 

 outlined in Schnute ( 1981 ). Schnute's model relates size 

 (FL) to age by several parameters, including two that 

 describe the shape of the curve (a and 6; Table 1 1. These 



latter parameters combine to describe a range of com- 

 mon growth curves, including the von Bertalanffy 

 (a>0, 6=1), Richards (a>0, 6<0), logistic (o>0, b=-l) 

 and Gompertz (a>0, 6=0) (Table 1; Schnute, 1981). 

 The other parameters in Schnute's growth model 

 were.Vi and Vg, the mean sizes at ages ij and r^ re- 

 spectively, where the value of Tj and r, are specified, 

 but usually chosen to be near the lower and upper 

 ends of the range of ages in the data. In this study, r, 

 and r., were set at 1 and 5 respectively. All growth 

 models were calculated by minimizing sums of 

 squares and using additive errors because variation 

 in size-at-age was similar for all ages of fish (see 

 "Results" section). 



Initially, a two-parameter model (y, and.v^) was 

 fitted to the data (case 4 in Table 1). Two types of a 

 three-parameter model [parameters were a,y^. and 

 Vp and ^2 (case 3)1 and a four- 



y.2 (case 2), and 6 



