Katakura et al.: The relation between otolith and somatic growth over the lifespan of walleye pollock (Theragra chakogromma) 



449 



/;'(g,)=/;ii (<?,). 



(7) 



When Equation 5 is substituteci for Equations 6 and 7, 

 the following equations are obtaineci: 



a,(7f'+c, =a,^i(7,''->+c,^i 



6,-1 



(8) 

 Solving Equations 8 and 9 simultaneously yields 



. «i^ 3,-&,. 



q? ""' 



c,+i=ai9,' 



1-: 



^+1 , 



+ c,. 



(10) 



(11) 



The functions of //x) and /^+i(.v) can be smoothly con- 

 nected at the inflection point if Equations 10 and 11 are 

 equal. The formula of the allometric smoothing function 

 y is shown as follows: 



^ = X/',U) = £5,U)(a,x'''+c,). 



(12) 



Fitting the OL-FL equations 



The allometric smoothing function (Eq. 12) is fitted by 

 using the maximum likelihood method. In the fitting, the 

 sample distribution around the dependent variable was 

 assumed to have a normal distribution. The estimated 

 standard deviation (SD) for the dependent variable was 

 used to calculate the weighted likelihood. The fitting 

 procedure is shown as follows (see Appendix Table): 



FL=f,iOL^} = a,OL''j+c,+£j iq,_^ <0L, <q,), (13) 



where FL , = the calculated FL for individual j: 



OL^ = the measured OL of individual /; and 

 f = the error for individual /. 



Equation 13 is validated between the inflection points 

 (q,_,SOL,<q,). 



The distribution of f, is assumed to have a normal 



J - 2 



distribution N (mean, variance) = N (0,crp^ ): 



(T^i^ =dOUj+f, 



(14) 



where o^, = the estimated SD of the FL of individual 



j; and 

 d, e, and f = parameters. 



The variable <t is assumed to fit the general equations 

 (Eqs. 3 or .^4). 



To fit FL to the general equations (Eqs. 1-3), the 

 following procedures are used. For Equation 1, the pa- 

 rameters in Equation 13 are fixed as 6, = 1 and c, = 0; 

 for Equation 2, the parameter is fixed as &, = 1; and for 

 Equation 3, the parameter is fixed as c- = 0. 



A likelihood of measured FL is calculated by the fol- 

 lowing equations: 



2;r cr. 



exp 



(FLj-FLj)' 



26 



FL, 



and 



LL = £lnL^, 



j=i 



(15) 



(16) 



where L = likelihood (the probability density) of FL; 



FL = the measured FL of individual /; and 



J jy 



LL = a log-likelihood. 

 LL is maximized by changing the parameters. 



Determination of the OL-FL equations 



The equation with the minimum AIC was selected: 



AIC = -2MLL + 2p (17) 



where MLL = the maximum LL and 



p = the number of parameters. 



