FISHERY BULLETIN: VOL. 81, NO. 3 



Intensity of Removal (F) 



^MKp 



o 



O 



O 



Intensity of Removal (F) 



K ?K 



Abundance 



FIGURE 1. — The Graham-Schaefer production model scaled accord- 

 ing to the "potential yield" formula (K is virgin abundance or carry- 

 ing capacity, M is rate of natural mortality). 



dance is at one-half the virgin level, and at this point 

 the rate of fishery removals (F) is equal to the rate of 

 natural mortality of adults (M). While many criti- 

 cisms can be leveled at the potential yield formula, in 

 actual practice it is often the only available basis of 

 fishery management (e.g., North Pacific Fishery 

 Management Council 1979a, b; Pacific Fishery 

 Management Council 1982). 



The production model corresponding to the poten- 

 tial yield formula is shown in Figure 1. The assumed 

 relationship between equilibrium abundance (JV,,) 

 and rate of removal (F) is linear, giving the usual 

 parabolic yield curve. Equilibrium abundance is at 

 carrying capacity (K, denoted JV,', in this paper), when 

 there are no removals. From inspection of Figure 1, 

 the model predicts that equilibrium abundance falls 



to zero when the rate of removals is twice the rate of 

 adult natural mortality (M). Thus, based on linearity, 

 we have 



*.-«(! ~sr> 



(12) 



which predicts long-term equilibrium abundance in 

 the presence of removal rate F. In parallel with our 

 treatment of short-term impact, the long-term equi- 

 librium abundance of a harvested or impacted stock 

 relative to the abundance of the virgin stock is de- 

 noted R,„ and is given by 



R„ = NJK 



(13) 



and according to Equation (12), we have the ap- 

 proximation 



Re= (!—&)■ 



2M 



(14) 



E quation ( 1 4) says that we can estimate the impact of 

 all removals, given the total rate of removals and the 

 rate of adult natural mortality. 



The total rate of removals (F) includes entrainment 

 (E) and impingement (F p ) by all power plants operat- 

 ing in the area of the stock, and all fisheries (F f ) ex- 

 ploiting the stock: 



ZF + lE p + lE f . 



(15) 



The removal rate due to entrainment (E) is estimated 

 by the method presented in the previous section, 

 i.e., 



E = I£,t r (16) 



Impingement (F p ) and fishing (F f ) mortality rates may 

 be estimated by a method similar to Equation (7): 



and 



F P = I/N* 



F f =C/N> 



(17) 

 (18) 



where / is the number of adults impinged in a year, C 

 is the annual fishery harvest, and N* is the mean 

 abundance of the stock over the year. 



The natural mortality rate is a difficult parameter to 

 estimate. Ricker (1975) reviewed many of the meth- 

 ods. In some cases it may be necessary to assume a 

 value of M, based on comparison with better known 

 species. One useful method, based on comparative 

 growth and environmental parameters, has been de- 

 scribed by Pauly (1980). If a mortality rate can be es- 



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