PiB) = CfB + C2 5", 



(2) 



with exponent n the additional parameter, but 

 with the signs of the coefficients now dependent on 

 the range of definition of n. As before, the com- 

 bined terms describe, at any stock size B, the 

 stock's latent capacity for productivity. With n 

 undetermined (its determination being a part of 

 the empirical demonstration), solutions of Equa- 

 tion ( 2 ) constitute infinitely many growth laws. By 

 setting n = 2, and with Cj >0, C2<0, Equation (2) 

 reduces to the Graham equation (Equation (1)). 

 PellaandTomlinson (1969) attribute Equation (2) 

 to Richards ( 1959). For a detailed analysis of (2) as 

 a general growth form, see Fletcher (1975); the 

 anticedents of this analysis appear there. 



In either of the two systems, exploitation enters 

 the formulation for productivity by the direct dif- 



• • • 



ference P - Y, with Y signifying the rate of 

 biomass removal owed to exploitation and P the 

 latent productivity of the stock. Wherefore, in 

 writing 



B(B) = P(B) - Y(B), 



(3) 



we interpret B(B) as being the resultant produc- 

 tivity, at stock size fi, that nets to the stock for its 

 growth. The net may be positive, negative, or zero 

 accordingly as P and Y vary with B. That is 



P > Y implies B > 0: the stock's latent 

 productivity exceeds the rate of exploitation; a 

 positive net productivity remains to the stock 

 and the stock so tends to a higher level of 

 biomass. 



• • • 



P < Y implies B < 0: the rate of biomass re- 

 moval exceeds the stock's capacity for growth; 

 the stock adjusts to the deficit in net productiv- 

 ity by tending to a lower level of total biomass. 



• • • 



P = Y implies B = 0: the exploitation rate just 

 balances latent productivity, and biomass 

 trajectory B(t) exhibits an extremum. Should B 

 = over finite time, stock biomass remains 

 stationary and the state called "equilibrium" 

 prevails. 



Although the detailed time course of any real 

 stock biomass is actually determined by varia- 

 tions in renewal, survival, member growth, and 

 the age- or size-dependent probabilities of capture, 

 such effects are not usually separated in the mod- 

 els of interest here, and yield rate Y customarily 

 takes the form 



378 



FISHERY BULLETIN: VOL. 76, NO. 2. 



Y(t) ^F(t)'B(t), (4) 



with the implication that all fish of the fishable 

 stock are presumed to share, in equal measure, the 

 force of fishing mortality F, irrespective of age or 

 size. By admitting Equation (4) into Equation (3), 

 our general form for net productivity becomes 



B =P - F'B, 



(5) 



where the time variation of F is usually prescribed 

 by average effort f on the assumption that F = 

 qfiT, quantity q being the individual probability of 

 capture per unit of effort and r the averaging in- 

 terval measured in fractions of the dimensional 

 time unit of F. 



ANALYSIS OF 

 THE GRAHAM SYSTEM 



Figure 1 illustrates the phase-plane graph of 

 Equation (1), the latent productivity of a Graham 

 stock. Maximum productivity m occurs at stock 

 sizep. And regardless of the conventions employed 

 in the formulation of Equation (2), essential 

 parametric control in the equation resides spe- 

 cifically with its nonzero root By~ and with coordi- 

 nate m of the critical point (p, m). Parameter m 

 and Bx constitute a complete, minimum set of 

 analytically independent parameters for latent 



^cxmtrcHJUxbLe 



parcuneter- 



m = Pm^ 



Figure l. — Latent productivity P as a function of stock size B, 

 the Graham model. See Equations (1) and (la). 



