Kim and Bang: Simulation of walleye pollock distribution in Shelikof Strait 



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dCi a2c, a^c, ac, 



dt 3x2 ^ Qy2 gj. 



- V - rCi 



dy 



(1) 



where C, = larval concentration of t-th cohort 

 (number/m^), 

 X and y = along- and cross-strait coordinates, 

 K^ and Ky = along- and cross-strait turbulence 

 diffusion coefficients, 

 u and V = along- and cross-strait velocity 

 components, 

 r = instantaneous daily mortality of 

 larvae, and 

 Ti = starting time of i-th cohort larval 

 production. 



The solution using Laplace transform is 



C,{x,y,t) = ^= f ^^i^ent-T) 



(^-K 



iKAt-T) 



(y-B,f 





AKJt-T) 



 dT (2) 



where C,{x,y,t 



larval concentration of i-th cohort 

 at time t at point {x,y), 

 P,{T) = i-th cohort's larval production 

 rate at time T {T^<T<t) and at 

 source point (xn, ^o). 

 A„ = (-1)I"IA-* + a{x,L-x.l^ + 



mod(|a|,2) (x+i + x_i ) 



Bf, = {-lf\Y* + 6(i/,z.-2/-L) + 



mod{\b\,2){y^L + y_i^) 



where a and b = indices of symbol summation 

 (2) 

 if there is no boundary, a or 6 = 



if there is negative-side ( - L) boundary only, 

 a = -1, 0; 6 = -1, 



if there is positive-side ( + L) boundary only, 

 a = 0, 1; b = 0, I, 



if there are boundaries at both sides, 



a = -oo -(-<» 



b = -oo, . . . , -i-°o 



X* = Xq + u(t-T) 

 Y* = y^ + v{t-T) 



^+L, x~i, y+i, and y_i = positive and 

 negative side boundaries in .r and y 

 coordinates. 



mod(|a|,2) = remainder of |a| divided by 2, and 

 mod(|6|,2) = remainder of |6| divided by 2. 



The area of Shelikof Strait used in the computer 

 simulation was divided into 162 lOx 10 km grid areas 

 with boundaries along the Alaska Peninsula and Kodiak 

 Island (Fig. 1). The grid scheme was used to obtain con- 

 tour patterns of results and to compare the simulated 

 values with observations. To obtain larval abundance 

 for each grid area from Equation (2), numerical integra- 

 tion was used, since the integration could not be han- 

 dled by further analytic approach. For time integra- 

 tion, an 8-point Gaussian Quadrature is used (Scheid 

 1968), and for spatial integration an error-function in- 

 finite series is used (Gradshteyn and Ryzhik 1980). 



Once all parameters were selected for the diffusion- 

 advection equation, the simulation program was run 

 for 50 days starting on 5 April (Julian day 95) and con- 

 tinuing until 24 May (Julian day 144). Larvae were pro- 

 duced near the central strait every day. They then were 

 advected and diffused in the grid area according to 

 given values of the parameters. Finally the concentra- 

 tion obtained from the model for each size group dur- 

 ing the 5-day period of 20-24 May was averaged to 

 show the expected larval abundance of each size group 

 in late May 1981. 



The mortality parameter in Equations (1) and (2) was 

 derived in Kim and Gunderson (1989) from observed 

 data during two consecutive surveys (i.e., 26-30 April 

 and 20-24 May, 1981) assuming that larvae stayed in 

 the survey area and that the larval population decreas- 

 ed exponentially with time. If, however, some of the 

 larval cohort that existed in the sampling area during 

 the first survey drifted out of the area by the second 

 survey (called out-fraction in this paper), then the mor- 

 tality rate must be an overestimate. The out-fraction 

 of larval abundance— which is the ratio of the larval 

 abundance evicted from the simulation box in Figure 

 1 to the total larval abundance assuming no disper- 

 sion—is computed after simulation, and this computa- 

 tion can be used to estimate a new mortality value in 

 an exponentially decreasing population with time: 



Z*(L) 



1 , JMWI 



> + ln(l-/) 



AT 



(3) 



where Z*(L) = revised instantaneous daily larval 

 mortality, 

 AT = time difference in days (24 days) 

 between surveys, 



