(2) dm,/dt= am, (l-j^)- 



So the proportional 

 fishing) is: 



dirir 

 m,dt 



rate growth (with no 



'3)S,="Kt)- 



The second derivative of (1) is: 



Maximum absolute growth occurs when (4) 

 equals zero; that is, when m, = V2M (when 

 current biomass is one-half the potential 

 maximum). At that point, equation (2) shows 

 that the maximum growth, dm,/dt = aM/A. 



Suppose aM/4 were taken from the biomass 

 each year by fishermen: each year, the biomass 

 would grow by aM/4; biological growth would 

 just offset the amount taken by fishermen; and 

 there would be a steady-state equilibrium. 



The Theory of Yield 

 from a Given Biomass 



We now consider how yield responds to 

 effort when we abstract from changes in biomass. 

 Schaefer (1954) made the simple assumption 

 that the catch m, would be proportional to 

 effort, A: is the constant of proportionality, and 

 X, is effort: 



(5) y,/m, = kx,. 



Schaefer assumed that, with a given biomass, 

 there would be constant returns to effort; dou- 

 bling the effort would double the yield, tripling 

 the effort would triple the yield — and so on. 

 As a first approximation, this may be adequate 

 in many cases within the observed range of the 

 data. Schaefer and others have used it to make 

 many important estimates of maximum sus- 

 tainable yield; and as a basis for economic 

 controls. 



But we think that a more realistic catch 

 function is: 



(6) y,/m, = (l-z''), 



with <2 < 1, and with m, fixed. 



The rationale of (6) was explained by Carlson 

 (1969). Briefly, assume that the original biomass 

 is m, and that one unit of effort will catch pm,. 



leaving (\—p)m,; assume that the next unit 

 of effort will catch the same proportion of the 

 remaining biomass — that is, it will catch 

 p( 1— p )m , , leaving( \—p)'m,. The same reasoning 

 shows that « units of effort will catch (1 ■! !•<)"' . 

 In equation (6), we simply let z = \ — p. We 

 believe that on an a priori basis (6) is more 

 realistic than is (5). But probably there is no 

 magic mathematical formula that is exactly 

 right for all species and for all amounts of effort. 



yield, y 



effort, X 



Fi^re 2. — Two yield functions. (Based upon equations 

 5 and 6, assuniinpr that one unit of effort yields one- 

 half of the existing biomass.) 



Figure 2 compares the growth functions 

 represented by equations (5) and (6). Each 

 assumes that one-half the existing biomass was 

 caught with one unit of effort in some base 

 period. (The units are arbitrary. We find it 

 desirable to "normalize" both yield and effort by 

 dividing by the base-period data.) Note that 

 equation (5) would indicate that the entire 

 biomass would be caught with two units of 

 effort. But equation (6) would indicate that if 

 effort were increased indefinitely, the existing 

 biomass would be approached as a limit, but 

 never quite reached. Within the observed range 

 of historical data, it may not be easy to choose 

 between the two curves in Figure 2. But they 

 give far different results when they are ex- 

 trapolated to estimate the effects of large in- 

 creases in effort. This is especially critical where 

 one must make forecasts of the likely effect of 

 the expansion in fishing effort. 



74 



