Katakura et aL: The relation between otolith and somatic growth over the lifespan of walleye pollock (Therogra cholcogramma) 453 



smoothing function (Eq. 12), and an allometric smooth- 

 ing function without a constant term in the first function 

 (Cj=0). The minimum AIC was the three-phase allome- 

 tric smoothing function without c^. The relation between 

 FL (mm) and o^^ (g) is shown as follows: 



BW = 2.01x10-5 FL2" (0.00<FL<70.0) (25.1) 



BW = 6.61x10-6 FL3.02 + 0.21 



(70.0<FL<431,2) (25.2) 



BW = 4.17x10-6 f 13.09 + 13 89 (431.2<FL) (25.3) 



tTBi,,= 1.06x10-5 FL2 63. (26) 



The coordinates (FL, BW) of the two inflection points 

 were seen at (70.0, 2.6) and (431.2, 586.2). 



Discussion 



The allometric smoothing function 



The best equation to describe the relation between OL 

 and FL in walleye pollock throughout the entire life- 

 span of the fish was depicted by a four-phase allometric 

 smoothing function with three inflection points. In our 

 preliminary analysis, a quadratic equation was applied 

 for the OL and FL relationship, and the resulting AIC 

 was 21,743. This value was smaller than that derived 

 from general equations, but was higher than the value 

 derived from any of our allometric smoothing functions 

 (see Table 2). The general equations and the quadratic 

 equation do not adequately reflect the variable otolith 

 and somatic allometric growth during the whole lifespan 

 of the species. 



Equations relating OL to somatic length have been 

 developed to represent complex growth curves. Bervian 

 et al. (2006) used an allometric equation transformed 

 from the logistic function for the OL-TL relationship 

 in whitemouth croaker (Micropogonias furnieri). Imai 

 et al. (2002) applied a Gompertz model to the relation 

 between otolith height and standard length in cyprinid 

 fish "Ukekuehi-ugui" (Tribolodon nakamurai). However, 

 these two models have limitations in both the shape 

 of the curve and the number of inflection points. If 

 the species in the model, such as walleye pollock in 

 this study, has more than two inflection points in the 

 derived curve, these models cannot represent the al- 

 lometric growth patterns adequately. Our present al- 

 lometric smoothing function has no such limitation in 

 the number of inflection points or the shape of curve 

 between inflection points and responds appropriately 

 to the growth pattern of the fish. 



Our allometric smoothing function has the ability 

 to satisfy both the needs for mathematical continuity 

 (see Fig. 1) and objectivity in the selection of an equa- 

 tion (see Table 2) while allowing for biological events. 

 The allometric smoothing function was developed by 

 using a mathematical smoothing method based on an 



allometric equation with a constant term. Among the 

 smoothing methods available, the moving average, au- 

 toregression, and spline curve proved to be useful for 

 fitting scatter sample plots, a type of plot that cannot 

 be properly fitted in a single function. Nevertheless, the 

 moving average requires that the modeler be subjec- 

 tive in determining the number of data points used to 

 calculate the average. In contrast, the autoregression 

 allows a measure of objectivity in selecting the equa- 

 tion; however steady growth conditions are assumed 

 with this method. Finally, the spline curve is based on 

 a multidimensional function developed by mathematical 

 procedure where biological events were not taken into 

 consideration. 



Until recently, back-calculation models for individ- 

 ual fish growth have been developed to estimate past 

 fish length and growth, under the assumption that 

 fish growth is proportional to otolith growth (Francis, 

 1990). However, many studies have recognized that fish 

 growth and otolith growth are uncoupled. "Growth rate 

 effect" and "age effect" are two of the most important 

 factors affecting uncoupling. The growth rate effect 

 occurs when otoliths from slow growing fish are larger 

 than those of fast growing fish, when these fish are 

 compared at the same somatic length (Reznick, 1989; 

 Campana, 1990; Secor and Dean, 1992). Adapting Cam- 

 pana's (1990) biological intercept method can reduce the 

 error inherent in back-calculated somatic length from 

 this growth-rate effect. Additionally, with the back- 

 calculation model developed by Morita and Matsuishi 

 (2001), the fact that age effect on otolith size increases 

 continuously during nongrowth periods (Mugiya, 1990; 

 Secor and Dean, 1992) can be taken into account. The 

 inclusion of these growth and age effects of individual 

 fish to our allometric smoothing function provides a 

 more accurate analysis of growth at the individual level 

 in the back-calculation model. 



Application of the allometric smoothing function 

 for walleye pollock 



Our best equation to describe the relation between OL 

 and FL was derived as a four-phase allometric smooth- 

 ing function with three inflection points (Fig. 4). In 

 Huxley's (1924) allometric equation (y=ax^), relative 

 growth rate was expressed by the relative growth coef- 

 ficient (allometric coefficient, b). Our allometric smooth- 

 ing function is based on the allometric equation with an 

 added constant term {y=ax^ + c). The superscript b in 

 our equation is not an allometric coefficient; it indicates 

 the relative growth between .r and y on the slope of the 

 curve between inflection points. 



The explicit changes in the shape of the curves and 

 the appearance of inflection points in our equation 

 imply that ecological and physiological changes are 

 associated with unique aspects of life history of wall- 

 eye pollock. In the first function (Eq. 19.1), somatic 

 growth is slower than otolith growth, whereas in the 

 second function (Eq. 19.2), somatic growth is faster 

 than otolith growth. Concerning these contrasting 



