the instantaneous rate of fishing mortality; M is the instan- 

 taneous rate of natural mortality; G is the somatic growth 

 rate. 



Fish stock biomass is decreased by death and increased 

 by growth, so the rate of biomass change (decrease) is 

 F + M - G. The biomass of the fish stock at the beginning 

 of the year is B Q or initial biomass, or spawning biomass for 

 anchovies. That biomass decreases at_the rate F + M - G. 

 The average biomass during the year, B, is: 



B = B Q x ( 1 - e -( F+M " G > ) / (F+M-G) (1) 



Then the estimated weight of fish that die naturally, or 

 forage available to predators, is MB. 



Without a fishery there is no fishing mortality. In that 

 case, the estimate of predation of natural mortality is de- 

 rived in the same way as the previous case, except that there 

 is no fishing mortality, so F = 0. Then estimated predation 

 is MB where: 



B = B x (1 - e " (M_G) ) / (M-G) (2) 



The difference in the two estimates of MB, or total annual 

 natural mortality, in the cases with and without a fishery, is 

 the hypothetical loss of forage due to the fishery. Values for 

 M and G are the same in both cases . The value for F is a 

 variable, being greater than zero in the case with a fishery, 

 and zero in the case without a fishery. Values for B Q are the 

 same for considering short term changes in forage, and differ- 

 ent for considering long term changes when variation in average 

 spawning biomass becomes a factor. 



Calculations of total annual natural mortality with 

 various levels of B Q and F for the northern anchovy fishery are 

 summarized in Table F-2. The short term loss of forage avail- 

 able to predators , under the proposed fishery level based on 

 the 0Y formula, for years with a high spawning biomass (e.g. , 

 3.6 million tons in 1976) is estimated at 420,000 tons. Simi- 

 larly, for years with a low spawning biomass, short term loss 

 is estimated as 160,000 tons. Since there is considerable nat- 

 ural variation in the anchovy population, and consequently in 

 available forage from year to year, these values may not be 

 significant. Of greater importance is the change in forage re- 

 sulting from the long term decrease in anchovy density or 

 average spawning stock that is expected to result from the 

 proposed fishery, case C in Table F-2. On the basis of expect- 

 ed median population values with and without the proposed 



F-14 



