448 



Fishery Bulletin 105(4) 



680 mm) to data for three stages: larval stage (4.6- 

 14.0 mm TL), juvenile stage (11-96 mm TL), and young- 

 adult stage (88-680 mm TL). Zeppelin et al. (2004) 

 adapted a quadratic equation for a fish size range of 

 49-530 mm FL. These previous equations allowed re- 

 searchers to characterize the growth patterns of walleye 

 pollock by regression analysis (the least-squares method), 

 but they have several shortcomings because somatic 

 length is not estimated across the whole life span of 

 the fish. First, the equations, each of which represents 

 a different life stage, do not facilitate comprehension of 

 the continuity of each life stage. The equations are fit- 

 ted to each segment of the data separately by inflection 

 points that are derived from empirical data or visually 

 from the scatter plots. Second, the quadratic equation 

 has limitation in the shape of its curve which does not 

 show the inflection point. The complex growth patterns 

 are not adequately reflected in the equation. Third, the 

 least-squares method does not allow for the incorporation 

 of increasing variance with increasing fish length. When 

 the sample distribution is biased, the calculated equation 

 is largely influenced by the range of fish lengths from 

 the largest number of samples. Fourth, previous OL-FL 

 equations were not considered objectively in the selection 

 of an adequate equation. No attempt has been made to 

 apply information criteria such as Akaike's information 

 criterion (AIC: Akaike, 1974), which is an operational 

 way of trading off the complexity of an estimated equa- 

 tion against how well the equation fits the data. 



To overcome these problems, we developed a new 

 OL-FL equation for the whole lifespan of walleye pol- 

 lock using a proposed allometric smoothing function 

 to describe the relation between OL and FL. We also 

 derived three distinctive allometric smoothing functions 

 to establish the relationships between the short otolith 

 radius (SOR: from core to the tip of rostrum) and FL, 

 between the long otolith radius (LOR: from core to the 

 tip of postrostrum) and FL, and between the FL and 

 body weight (BW: wet body weight). 



Allometric smoothing function 



A new OL-FL equation was developed by using a math- 

 ematical smoothing method based on an allometric 

 equation with a constant term. The assumption of the 

 allometric smoothing function was to have a common 

 tangent at the inflection point to reflect the variable 

 allometric growth smoothly. A composite of two or more 

 allometric smoothing functions was defined as follows: 



<5,(*) = 



1 ((?,_, <x<9,) 

 {x < q^_-^,q^ < X) 



(5) 



Materials and methods 



General equations 



The general equations in this analysis are linear equa- 

 tions (Eqs. 1 and 2), an allometric equation (Eq. 3), and 

 an allometric equation with a constant term (Eq. 4): 



y = ax 



y = ax + c 



y = a.r* 



y = a.r* + c 



where x = the independent variable; 



y = the dependent variable; and 

 a, b, and c = parameters. 



(1) 

 (2) 

 (3) 

 (4) 



f,ix) = 5^(x)(a,x''- +c,), 



where d^ix) = switch function; 



q^ = a value of j: on the inflection point, here 



9o=0; 



f](x) = a number ; of function; and 

 a,, 6,, and c, = parameters for the function oft. 



f^ix) is validated between the inflection points iq^_iSx<q^) 

 which depend on the d^Lv). 



We assumed that for the smooth integration of f^ix) 

 and f,^i{x) (the function on the next order of i), both 

 functions must pass through the inflection point (x, 

 y) = (9,, f,iq,) = fi+i^Qi'i'i and have a common tangent at 

 this point (Fig. 1). To satisfy the above conditions, the 

 following two equations must be equal. 



/',<9,'=/,.i'9,) 



(6) 



