Kendall and Picquelle Egg and larval distributions of Thersgrs chslcogramrm 



137 



general pattern of occurrence of eggs and larvae as the 

 season progressed, this time period was divided into 

 five intervals (12-28 March, 29 March-13 April, 14-29 

 April, 30 April- 15 May, and 16 May-2 June), and 

 samples collected during each interval were grouped 

 for analysis regardless of cruise or year (Fig. 2). In- 

 tervals of a 16-18 day duration were chosen because 

 the egg incubation period is about this duration at am- 

 bient temperatures, and lai^vae would grow about 3 mm 

 in this interval (at an observed growth rate of about 

 0.2 mm per day [Kendall et al. 1987]). Thus eggs or lar- 

 vae within a 3-mm length increment from each time 

 interval can be considered a cohort when comparing 

 their distribution with that of adjacent time intervals. 



Similarly, since sampling patterns and positions were 

 different for each cruise, the Shelikof Strait area was 

 divided into a number of sectors, and the samples that 

 were collected in each time interval and each geo- 

 graphic sector were grouped so they could be analyzed 

 regardless of cruise or year. The grid of sectors was 

 laid out with axes parallel and perpendicular to the axis 

 of Shelikof Strait (Fig. 1). Within Shelikof Strait prop- 

 er, where heaviest concentrations of stations and abun- 

 dances of eggs and larvae occurred, the sectors were 

 10 X 10 miles (18.5 x 18.5 km); southwest of this area 

 the sectors were 20 x 20 miles (37 x 37 km). 



The geographic distributions of walleye pollock eggs 

 and larvae were summarized by mapping their cen- 

 troids of distribution (see Appendix 2). Centroids were 

 computed for total walleye pollock eggs and larvae by 

 time interval regardless of year, and by egg age-group 

 and 3-mm larval length-interval (3.0-5.9, 6.0-8.9, 

 9.0-11.9, 12.0-14.9, 15.0-17.9 mm SL). The centroid 

 can be considered the center of mass of the distribu- 

 tion and is computed as the weighted mean location of 

 the sectors, where the weights are the estimated total 

 number of eggs or larvae for each sector (Koslow et al. 

 1985): 



Centroid = (A', f ) 



I A^, • X, 

 where A' = , 



I AT, 



2 A^, • Y. 



Y 



I AT,. 



N, = estimated total number of pollock in 

 sector (■ 



A,  J.N.., 



A, = area of sector i (in units of 10 m-) 

 A'^,, = number of pollock per 10 m- for sample 

 j in sector (', 

 w, = number of samples in sector i, 

 X, = center position of sector i in the east- 

 west axis 



X,i = position of sample J in sector ; in the 



east-west axis, 

 Y, = center position of sector / in the north- 

 south axis 



I N,j  Y,j 



J 



and Y, , = position of sample j in sector ; in the 

 north-south axis. 



The distribution was further analyzed by rotating the 

 A' and Y axes so that one axis was in the direction of 

 the greatest geographic range of walleye pollock eggs 

 or larvae in the study area. This axis is called the prin- 

 cipal axis. The second axis is orthogonal to the prin- 

 cipal axis. 



An ellipse was drawn around the centroid that is one 

 standard deviation away from the centroid along the 

 rotated_axes. The ellipse is defined as having the center 

 at (A', }') and the major axis parallel to the principal 

 axis. The major and minor axes are each two standard 

 deviations long. The ellipse is the two-dimensional 

 analogue of a mean and standard error bars; it shows 

 the center and the orientation of the animal's distribu- 

 tion in space and the amount of dispersion about the 

 center. If the distibution of animals in space follows 

 a bivariate normal distribution, then the ellipse is a 40% 

 confidence contour. 



Total abundance in the Shelikof Strait area was 

 estimated for eggs and larvae for each time interval. 

 Total abundances were also estimated for the six egg 

 age-groups and the five larval length-increments for 

 each time interval. The sectors were treated as strata 

 to correct for the higher density of samples in areas 

 of high egg and larval abundances. Not all sectors were 

 sampled in each time interval, so the estimate of total 

 abundance was adjusted to the total area. This was ac- 

 complished by grouping the sectors into two regions, 

 (one included all the 10 x 10 mile sectors, and the other 

 included all the 20 x 20 mile sectors), estimating a 

 stratified mean density for each region, then multiply- 



