Watanabe and Yatsu: Interannual variation in length at age of Scomber /aponicus 



199 



face temperature according to Millar 

 and Myers, 3 who nvestigated three 

 formulations of the modified von Berta- 

 lanffv equations: 1) a reversible effect 

 on the growth constant k; 2) a revers- 

 ible effect on the asymptotic length L r ; 

 and 3) an irreversible effect on L x or k. 

 We tested two of the models, 1 and 2, 

 to investigate the effect of population 

 density and SST. We did not test model 

 3 because we did not consider that the 

 environmental effects on growth were 

 permanent. Mean length at age i of 

 year-class y was estimated with the fol- 

 lowing formulas: 



Model 1: reversible environmental 

 effect on k 



L, v =L„(l-e"*°'-'°) (8) 



4v = A-lv + ( L ~ ~ A-i.v X 1 - e~*  ) ( 9 ) 

 k lv =k + P l T, fv +P 2 D :v . 110) 



Model 2: reversible environmental 

 effect on L 



L ,, = L^ v (l-e-*'») 



L lv = L,_ ly+ (L^-L,_ lv n-e- 



Year 



Figure 3 



Interannual fluctuations in mean fork length (FLl at age 0, age 1, and age 2 for 

 chub mackerel iScomberjaponicus) in 1970-97. Horizontal lines show the 28 year 

 mean FL at age 0, 1, and 2, respectively. Vertical bars show standard deviations. 



(Ill 

 (12) 

 (13) 



We ran the models with all possible combinations of 

 explanatory variables (T, D, T, and D), and compared AIC 

 with that obtained with the base parameters (L,, r , k). 



Results 



where tr, 



XL, V 



D. 



= the age at length (year); 

 = the asymptotic length; and 

 = the growth coefficient; 

 = L, at age i of year-classy; 

 = k at age ;' of year-class y; 

 = the sea surface temperature in year i+y; and 

 = a population density presented by the number 

 of stock at age i of year-class y. 



These variables were z-score standardized. The model 

 parameters a x and /3 2 were estimated to represent the 

 effects of T l+v and D I v on k or L v . 



The parameters were estimated by maximizing the like- 

 lihood function which is represented by 



and 



L(i,y) = L : v +£,, 

 f, -MO.cr), 



UL,,k,t ,p v P 2 ,o'f) = 



{L(/y)-L, v } 2 



nnM'-p 



2a; 



(14) 

 (15) 



(16) 



3 Millar, R. B., and R. A. Myers. 1990. Modeling environmen- 

 tally induced change in growth for Atlantic Canada cod stock. 

 ICES CM 1990/G:24. 



Fork length at age 



Mean FL at age varied substantially over the time series 

 examined. For example, it ranged from 16.9 (Sd ±3.0) cm 

 in 1975 to 25.9 (Sd ±1.0) cm in 1989. The mean FL for the 

 28 years period was 21.7 (±2.1) cm (coefficient of variation: 

 CV=9.8%, Table 1, Fig. 3). The FL-at-age-0 values were 

 smaller than the 28-year mean FL for the 1970s, varied 

 around the mean in the early and mid 1980s, reached a 

 maximum in 1989, and were at about 22-24 cm in the 

 1990s (Fig. 3). 



Mean FL at age 1 was similarly variable; it ranged from 

 24.3 (±1.9) cm in 1976 to 31.6 (±1.4) cm in 1995. The 28- 

 year mean FL was 27.7 (±1.6) cm (CV=5.6%,Table 1). The 

 trend in interannual variability was similar to that in age 

 0, i.e. it was smaller in the 1970s and larger in the 1990s 

 (Fig. 3). In age-2 fish the 28-year minimum FL of 29.1 (±1.8) 

 cm was observed in 1986 and the maximum of 34.5 (±1.3) 

 cm was observed in 1990 (the 28-year mean FL=31.1 (±1.5) 

 cm, CV=4.7%, Table 1, Fig. 3). 



In fish age 3 and older, mean FL varied year-to-year in a 

 manner similar to that found in the younger ages ( Table 1 ). 

 Annual mean FLs for 3-, 4-, and 5-year-olds were 33.7 

 (±1.3) cm (3.8%), 36.2 (CI ±1.4) cm (CV=4.0%), and 38.5 

 (CI ±1.5) cm (CV=3.8%), respectively (Table 1). The mean 

 FLs for ages 0-5 of each year were significantly different 

 among different years (one-way ANOVA, P<0.01 ). 



