FISHERY BULLETIN: VOL. 76. NO. 2. 



7(0 = (B^-B^) In 



B* = 



B 







i4mlB^-F 



..-1) 



Boo 



FB^ 



Am 



(9) 



m 



^ 6(1 -n) U1i/(i-"> 



the plus sign applying to Equation (10) and the 

 minus sign to Equation (11). 



ANALYSIS OF 

 THE PELLA-TOMLINSON SYSTEM 



As noted in the foregoing section, the maximum 

 latent productivity m of a Graham stock always 

 occurs at a biomass value exactly one-half the 

 unexploited maximum 5^. In turn, MSY of the 

 equilibrium model must also occur at the stock 

 level B^I2. So as to gain control over the locations 

 of those extrema, Pella and Tomlinson ( 1969) mod- 

 ify the Graham system by writing^the differential 

 equation for latent productivity P essentially in 

 the form of Equation (2), which, by the customary 

 treatment, has a troublesome, dual formulation 

 owing to the sign changes at /? =1 of coefficients 

 c, ,c.^. On the interval Q <n < 1 latent productivity 

 in the Pella-Tomlinson system takes the basic 

 form 



5 



aB" 



bB 



(10) 



(where, for the sake of emphasis, c, = -b,C2 = a, 

 with a and b positive), but on the interval n > 1 

 latent productivity takes on the basic form 



P = bB 



aB" 



(11) 



(where c\ = b, c.2 = -a, with a and b positive). In 

 either case, the bound B-^, the maximum produc- 

 tivity m, and the ordinate p (which governs the 

 biomass level where m occurs), all depend on the 

 numerical value assigned to exponent n. That is, 

 root By is given by 



Boo ~ 



l/(l-n) 



the ordinate p is determined by 



P = 



an 

 b 



i/(i-M) 



while maximum productivity m, by the conven- 

 tional casting of the model, must be determined 

 from the formula 



n > 1 



6«/^ </o < 6oo 



n<1 



0< p< 6«/e 



JZritixxd paint 



(p.m) 



Threshold&^/e_ 

 'cort*espondsM> n= 1 



P(5) 



m 



Figure 4.— Typical graph of Equation (12), latent productivity 

 P as a function of stock size B, the Pella-Tomlinson system. 



Coordinate p. in its location with respect to root 

 By., directly reflects the value assigned to expo- 

 nent n, as indicated by Figure 4. When n takes any 

 value between zero and unity, coordinate p falls on 

 the range between zero and B-^/e ( ~ 0.3679 B^), in 

 which case Equation (10) applies. When n takes 

 any value greater than unity, coordinate p falls on 

 the range between Byle and By, in which case 

 Equation (11) applies. But the coordinate m has no 

 essential dependence on exponent n, and its ap- 

 parent coupling with n ( as indicated by the formu- 

 lation above) is merely an inconvenient artifact of 

 the conventional analysis. With parameters m 

 and n uncoupled (see Fletcher 1975), the Equa- 

 tions (10) and (11) that govern latent productivity 

 in the Pella-Tomlinson system can be consolidated 

 into the single governing equation 



P = ym 



[£] --[£]"■ <- 



with y a purely numerical factor wholly prescribed 

 by n as 



382 



