FISHERY BULLETIN: VOL. 69, NO. 1 



where k specifies the year of the run, a/^. and 

 b); are constants, ./' is time in days, and Pkij) 

 is the cumulative proportion of the total catch. 

 In order to choose N, we defined the fishing 

 season to be those days for which .05 ^s P(j) 

 ^ .95 and the difference between the initial and 

 closing day was, to the nearest integral value, 

 set equal to N. 



We also used expression (16) to obtain Rij 

 from P'uU) = Pkii) — PkU-l) and then 



Ri 



Pk (J) Si, where Si is the number of fish 



of the /th entity in the run during the year 

 under consideration. The assumptions involved 

 in obtaining the Rij's are (1) the catch is pro- 

 portional to the "average" run as defined by the 

 fitted curve (16); and (2) Rij is proportional 

 to R.j. It should also be mentioned that our 

 Rij's are certainly different from the ti'adition- 

 ally used Rij's because the latter are based on 

 counts made several days after the fish enter 

 the fishery area. This, however, is not important 

 in the allocation whereas the relative daily size 

 of the run is important. We feel that these as- 

 sumptions are reasonable for the present model 

 until more accurate information can be obtained 

 on the behavior of the fish in the run. 



On some occasions, the catch in Bristol Bay 

 is limited by the cannery capacity. This ca- 

 pacity can be, of course, adjusted by the in- 

 dustry, but it appears that, according to Math- 

 ews (1966) , 1 million fish per day is a maximum 

 capacity and we used this value in constraint 

 equation (2). It would appear that the most 

 important assumption implicit in the nature of 

 this constraint is independence among the days; 

 that is to say, we assume that processing a cer- 

 tain number of fish on one day does not affect 

 the processing capacity on the next day. A sec- 

 ond assumption is that the maximum capacity 

 is not dependent upon the average size of the 

 fish processed. There is some question, when the 

 cannery operates near peak capacity, as to the 

 effect of overtime payments to cannery employ- 

 ees and to the effect of processing large numbers 

 of fish on the quality of the pack. 



For our example, of the 1960 run, the total 

 season constraint was not reached and so this 

 constraint had no effect on any of the examples 

 which we present. It is relevant to note, how- 



ever, that if this constraint is needed, then the 

 maximum number of fish ever processed (up 

 until 1969) in the Naknek-Kvichak system was 

 19.1 million. 



The next constraint is the escapement con- 

 straint. Desjjite the fact that the level of escape- 

 ment has, for salmon stocks, been the primary 

 management criterion, there is very little doc- 

 umentation on the proper escapement level for 

 many systems. The latest synthesis of the ex- 

 tensive work on the Naknek-Kvichak system is 

 addressed to the problem of optimum escapement 

 in that particular system as well as others 

 (Burgner, DiCostanzo, Ellis, Harry, Hartman, 

 Kerns, Mathisen, and Royce, 1969), but un- 

 fortunately no advice on optimum escapement 

 is given for the Naknek-Kvichak. Another 

 study implies, on the basis of limited data, that 

 escapements beyond about 10 million fish will 

 not result in any increased productivity of 

 smolts (Mathisen, 1969: Fig. 6), and thus if we 

 make a simplifying assumption and assume that 

 marine survival rates are independent of den- 

 sity, we can further assume that an increased 

 return will not accrue from an escapement larger 

 than 10 million fish (very roughly 20 billion 

 eggs). 



Despite the fact that we do not "know" the 

 optimal minimal escapement for the Naknek- 

 Kvichak system, it is clear that there must be 

 some level which is, in some sense, optimal. 

 This statement must, of course, be tempered 

 with our knowledge of the cyclical nature of the 

 run. 



Because of our uncertainty, we examined the 

 model under a variety of escapement conditions 

 but keeping under each set, what would appear 

 to be, according to studies by Mathisen (1962), 

 a conservatively high male-to-female sex ratio 

 of at least one male for every three females. 

 We found in general, as one might expect, that 

 as we increased escapement we decreased the 

 value of the catch. The objective function was 

 quite sensitive to this manipulation, focusing 

 again upon the need for a well-defined e.sca))e- 

 ment policy. What is not so obvious, however, 

 as we will see subsequently, is that as we change 

 the escapement level, we can obtain consider- 

 able changes in the imputed values of the var- 



124 



