Kim and Bang Simulation of walleye pollock distribution in Shelikof Strait 



307 



Shelikof Strait. Estimates of u and v, from regression 

 analysis (see Table 1 in Kim and Kendall (1988) for 

 data) were 4.2 and 1.3 km/day in April, and 2.7 and 

 0.4 km/day in May, respectively (Table 2). Because the 

 variance of the horizontal distribution is a suitable 

 measure of the spread of the substance (Bowden 1983, 

 Okubo 1971), the change in variance (i.e., Sf and S^ 

 for along- and cross-strait directions) with time pro- 

 vides a reasonable measure of the diffusion coefficient; 



K, 



K„ 



2 dt 



2At 



(4) 



(5) 



where S?,o. •Sjn- Sjjo, and Sjn are the spatial vari- 

 ances at time ^0 and time il in along- and cross-strait 

 directions, and M = tl - tO. The variances of two 

 dominant larval cohorts were calculated as in Kim 

 (1987). Variances of larval distribution were much in- 

 creased in a month, and along-strait components were 

 dominant compared with those in the cross-strait direc- 

 tion (Table 3). The estimated Kj. for these two cohorts 

 were 79.6 and 50.7 km-/day (average 65.2). For Ky, 

 the values were 1.0 and 6.3 km-/day (average 3.6) 

 (Table 2). 



Simulation results 



Spatial distribution and abundance of larvae 



The model distribution of larvae in late May after 50 

 days of simulation was similar to that observed. Two 

 major larval cohorts, 8-9 mm and 9-10 mm size groups, 

 were selected for comparing the model and observed 

 distributions (Fig. 2). In general, the simulated cen- 

 troids of distribution were close to the densest patches 

 of the larvae, and the area of larval distribution and 

 contour levels of the larval concentrations were not 

 very different from those observed. The simulation 

 demonstrated the elliptical pattern of distribution, 

 which was elongated in the along-strait direction, and 

 the southwesterly movement of centroids from the 

 main hatching area. The first (8-9 mm) and second 

 (9-10 mm) cohorts drifted 92 km and 114 km, respec- 

 tively, from the source point after hatching. 



The maximum size of larvae was 12 mm, and the 

 abundance of larvae and the out-fraction in each size 

 group were computed. Comparing the simulated values 

 with the observed ones, we found that the results were 

 very close (Table 4). Also, as expected, the effect of 

 out-fraction on larval distribution was more important 

 for the larger size group than the smaller group. This 



was caused by the larger larval size having a higher 

 out-fraction value. Negligible amounts of small larvae 

 were advected from the simulation box, but over 50% 

 of the large larvae were removed. Among total larval 

 abundance, about 20% were flushed out of the simula- 

 tion area. This concept of out-fraction due to diffusion 

 and advection might change the larval abundance and 

 mortality previously reported by Kim and Gunderson 

 (1989). Even though their derived estimates agreed 

 well with observed ones, their results should be recon- 

 sidered because the areas involved for abundance esti- 

 mates were not the same. Our study revealed that their 

 sampling area, which was similar to our simulation box, 

 was only part of the area of larval occurrence in Sheli- 

 kof Strait. Therefore the total larval abundance should 

 be higher than they observed, and the simulated abun- 

 dance within the simulation box should be close to that 

 observed. In Table 4, the reason for the smaller larval 

 abundance in the simulation box (3.37 x 10'-) than 

 observed (4.15 x 10'^) might be due to their overesti- 

 mate of the mortality rate. 



Re-estimation of larval mortality 

 and abundance 



The first approximation of larval mortality has been 

 recalculated using the out-fraction in Table 4. By ap- 

 plying the out-fractions of the two major cohorts to 

 Equation (3), we computed revised instantaneous daily 

 mortalities of 0.081 and 0.059 from the first and sec- 

 ond cohorts, respectively (average 0.070). Assuming no 

 significant change in diffusion coefficients, the revised 

 mortality rate was used for the second computer simu- 

 lation of the diffusion-ad vection model. Figure 3 re- 

 vealed that the second simulation resulted in a 50% 

 increase in the total larval abundance (6.21 x 10'-) in 

 Shelikof Strait, compared with the first simulation 

 (4.23 X 10'- from Table 4). The size-specific abundance 



