Shimada and Kimura: Seasonal movements of Gadus macrocephalus 



807 



Let 



r u = 



N, 



N e = 

 y 



s ij = 



"y = 



Ps(i) = 



the number of fish tagged in time period i in 

 areaj, 



the recoveries of tagged fish in time period i in 

 areaj, 



the number of tags at the beginning of time 

 period i in area j, after tags have been redis- 

 tributed according to estimates of their sea- 

 sonal distribution, 



the number of tags at the end of time period i 

 in areaj, 



s, = exp(-M - F sll) ) the survival of tagged fish in 

 time period i and areaj, 



Ul = F sil) (l.0-exp(-M-F sU) ))/(M + F sU) ) the 

 exploitation rate of tagged fish in time period i 

 and areaj, and 

 the areal distribution by seasonal time period. 



Here, the subscript s(i) refers to the season corre- 

 sponding to the time period subscript i. Therefore 

 s(i) could be w=winter, sp=spring, sw=summer, or 

 /"=fall. Each seasonal area-distribution vector (p w for 

 example) is a vector containing one element for each 

 of the three areas. 



There are 13 parameters to be estimated in this 

 model: the seasonal distribution vectors (p w , p sp , p su , 

 p f ); the seasonal instantaneous rates of fishing mor- 

 tality (F w , F s , F su , F f ); and the seasonal instanta- 

 neous natural mortality rate (M). The seasonal 

 area-distribution vectors each contain only two pa- 

 rameters to be estimated because they are probabil- 

 ity distributions that must sum to one. The model is 

 tied together by three simple equations: 



N^Nfc + Refii, 



r^Nfc + Rfr/2, 



N b 



Pjisii+m • 



Here, p jlsU+l)] represents the estimated proportion 

 of tags in areaj in season s (i+1). Note that -^s, and 

 u,/2 are the estimated survival and exploitation 

 rates, respectively, over half a season. 



Following Hilborn (1990) and Heifetz and Fujioka 

 ( 199 1 ), the model was fit by using maximum likelihood 

 and by assuming recoveries were distributed as Pois- 

 son random variables. That is, the parameters were 

 estimated by minimizing minus the log-likelihood: 



-L = ]T r tJ - r tJ log(/v ) + const . 



The probabilities in (p w , p sp , p su , p f ) were modeled as 

 expressions similar to exploitation rates following the 

 method of Heifetz and Fujioka (1991). Parameters 

 were estimated on the logarithmic scale and coeffi- 

 cients of variation were estimated by using the in- 

 verse Hessian of the minus log-likelihood and the 

 delta method. 



Our seasonal population dynamics model of the 

 tagged population was applied to the areas described 

 in Figure 3. For modeling purposes, we used only 

 fish tagged and recovered in these three areas. Thus, 

 9,318 releases, 



and 353 recoveries, 



V o 



were available for analysis. 



A problem associated with our population dynam- 

 ics model is the strong assumption that the F s(i) are 

 constant across areas. An attempt was made to use 

 existing commercial trawl and longline data to de- 

 termine recovery effort, but these data varied too 

 much in their seasonal coverage, and gears and ar- 

 eas were confounded. 



Nevertheless, the population dynamics model pro- 

 vides evidence that our tag data are representative 

 of the entire eastern Bering Sea population. In the 

 following sections we present evidence that our tag- 

 ging study probably suffered from significant tag loss, 

 tag mortality, or under-reporting of tag recoveries. 

 By comparing the estimated seasonal distribution of 

 the tagged population with the distribution from com- 

 mercial catches, we can verify that the tagged popu- 

 lation and untagged population were distributed 

 similarly. Since catch distribution should reflect 

 abundance in a heavily fished region such as the 

 eastern Bering Sea, we interpret this as meaning that 

 the behavior of the tagged Pacific cod population largely 

 reflected similar patterns in the entire Bering Sea popu- 

 lation. Commercial catch statistics were taken from the 

 Alaska and Pacific Northwest Historical Groundfish 

 Database (Berger 3 ), from which we calculated the ar- 

 eal trawl and longline catch distribution (for numbers), 

 by season, for the 1982-92 study period. 



Results 



Approximately 12,396 tagged fish were released be- 

 tween 1982 and 1990 (Table 1; Fig. 1). A total of 373 



3 Berger, J. Alaska Fish. Sci. Cent., Seattle, WA 98115-0070. Per- 

 sonal commun., January 1993. 



