for each sex and centimeter size class over all possible statistical 

 subareas. 



Length-weight relationships. — Determinations of length- 

 weight relationships were made to enable estimates of population 

 numbers using Method 1 (Equations (15)-(18)), and to compare 

 the body form and condition (weight-at-length) of individuals be- 

 tween sexes and geographical areas. 



The basic relationship between the length and weight of in- 

 dividuals of fish species k can be described by the power function 

 (Ricker 1975): 



weight = a- (length)*, or 



log (weight) = log (a) + b [log (length) ]. 



(22) 



(23) 



Length and weight data for each species were pooled in different 

 combinations (cases) of otolith areas (geographical areas of collec- 

 tion) and sex. For each case, the length (cm) and weight (g) data 

 were first transformed by taking logarithms (base 10). The data 

 were then fitted to a straight line (Equation (23)) by a regression 

 of "log(weight)" on "logflength)," and the values of a and b were 

 determined. The least squares method of linear regression was 

 used (Dixon and Massey 1969). 



Analysis of covariance was used to evaluate the extent of differ- 

 ences between the length-weight relationships shown by data 

 within the different test cases. The question asked by this analysis 

 was, "For each fish species, did the relationship between length 

 and weight significantly vary between sexes and between different 

 geographical areas of the Bering Sea?" 



The purpose of analysis of covariance was to 1) test whether 

 one regression line could be used for each pooling of observations 

 (case), if the slopes (b) of the regression lines within individual 

 data sets were the same, and 2) test if the individual data sets had 

 common adjusted mean values (Dixon and Massey 1969). 



Analyses of covariance were performed on the same logarithmic- 

 transformed length and weight data as used in the linear regres- 

 sion analyses. 



Age composition and growth. — The age-frequency distribu- 

 tions of populations vulnerable to the trawl were estimated by 

 proportioning the computed population length-frequency distri- 

 butions to ages using age-length keys (Ricker 1975). For each case, 

 length-age data were selected by species, sex, and area of collec- 

 tion (otolith area, Fig. 4) to construct a rectangular array (age- 

 length key) of the number of observations at each age (column) 

 and length (row). The estimated number of individuals within the 

 population at each length CP^/ m of Equation (21)) was then pro- 

 portioned to age groups (years) based upon the percentage of each 

 age, among fish of the same length (i.e., row), within the array of 

 actual length-age observations. 



The selection of data for age-length keys was based on two 

 principal considerations: 1) Data were selected for the same, or 

 closely neighboring, geographical area as that of the length- 

 frequency distribution; and 2) data were pooled between areas, 

 when necessary, to provide adequate representation of length 

 classes. 



The proportion of females was determined for each age group 

 following the estimations of age-frequency distributions. 



Population growth characteristics were described by compar- 

 ing the mean lengths of age groups within the expanded length- 



age array consisting of the estimated number of individuals in the 

 population (vulnerable to the trawl) at each length and age. For 

 each pooling of data by species, sex, and geographical area, a 

 decaying exponential growth curve (von Bertalanffy 1938) was fit- 

 ted to the vector of mean-lengths-at-age: 



/,=/,«,(! -g-^t-to)), 



"4, 



where /, is the length (cm) of individuals at time 1 (year). L a is a 

 mean asymptotic length (cm), K is a constant describing growth 

 completion rate (per year), and / is the hypothetical age (year) of 

 zero length. The method of Fabens (1965) was used for the 

 mathematical fitting of each growth curve, involving an iterative 

 least squares approximation of K and L tx . 



Two von Bertalanffy growth curves were fitted to each set of 

 data. The first curve was fitted using the complete and unaltered 

 vector of mean-lengths-at-age of all age groups represented within 

 the data set. The second curve was fitted to an adjusted vector 

 from which mean-lengths-at-age based upon < 10 length-age 

 determinations were excluded, and an artificial data point (0,0) 

 was added. In nearly all cases, the second curve fitting to the ad- 

 justed data resulted in a substantial reduction of the mean 

 square deviation of sample points from the regression line in the 

 approximation of K and L^, and was considered the "best" 

 description of growth characteristics. 



Reproductive condition. — The reproductive condition of the 

 population of walleye pollock within the survey area (and 

 vulnerable to the trawl) was assessed using the same computa- 

 tional procedures as the estimation of age-frequency distributions. 

 Observations of gonad condition were made on individuals col- 

 lected as subsamples of the length-frequency samples within 

 otolith areas A, B, and D (Fig. 4). The reproductive condition 

 (maturity stage) of each individual was coded on a scale of 1 to 5 

 (Table 7) based upon the visual appearance of the gonads (Holden 

 and Raitt 1974). For each sex, the length-maturity observations 

 were then organized into a rectangular array (length-maturity key) 

 of the numbers of observations at each length (rows) and stage of 

 gonad condition (column). The estimated number of individuals 

 within the population at each length (P jklm of Equation (21)) was 

 then proportioned to stages of reproductive condition based upon 

 the percentage of each stage, among fish of the same length (i.e., 

 row), within the array of actual length-maturity observations. 



Table 7. — The five-point scale for stages of gonad condition applied to walleye 

 pollock during the ApriWune 1976 Bering Sea survey. 



Code 



Gonad 

 condition 



Description 



Immature 



Developing 



Spawning 

 Spent 



Inactive 



Sexual organs very small, situated close to 

 vertebral column; ovaries pink or translu- 

 cent; testes translucent with slight leafing. 



Ovaries small to about one-half length of ven- 

 tral cavity; transparent and/or opaque ova 

 visible to naked eye; testes with increased 

 leafing and swelling. 



Roe and milt run under slight pressure. 



Ovaries and testes flaccid and empty; ovaries 

 may contain remnants of disintegrating 

 ova; testes bloodshot. 



Adults with gonads firm, shaped, and empty. 



11 



