FLETCHER: TIME-DEPENDENT SOLUTIONS AND EFFICIENT PARAMETERS 



presume that no practical technique of estimation 

 will have a precision of resolution better than 

 some assignable percentage of true stock size, and 

 let us reflect that practical uncertainty in our 

 analysis by expanding the asymptotic bound of 

 Equation (7) to a region of radius €'B::, around the 

 analytical value of the bound ( e being the measure 

 of the uncertainty). Whence, with B,j and Sj now 

 signifying initial and adjustment levels, and by 

 supposing thatF changes abruptly at time /„ from 

 value Fq to some new value F^, Equation (7) be- 

 comes 



(1± 6) = 



l + Cne~^^'"/^«-'''i^' 



the plus sign applying when Fj >F„ and the minus 

 sign whenFj <Fp. By setting initial time t^ arbi- 

 trarily at zero, 



r = ^ 



^0 



B^ — B(] 



Br 



and the transition time between initial level B„ 

 and the e-region at adjustment level 5i becomes 



'lag 



4m /Be 



(1 ± e)(Bi - Bo) 



i±eBo) 



(8) 



In few commercial fisheries do we expect to see 

 exploitation rates constant over intervals equal to 

 transition times ^i^g, and in any case we antici- 

 pate considerable variation in stock size along the 

 way, owing to chance events. Nevertheless, Equa- 

 tion (8) serves a purpose; it will give us some idea, 

 in a management strategy, of the time delays to be 

 expected in bringing a stock from one general 

 state of exploitation to another through the regu- 

 lation of mortality F. 



To illustrate the particularization of Equation 

 (8), we follow an adaptation by Ricker ( 1975: 312- 

 315) of Graham's work on demersal stocks of the 

 North Sea. To accommodate our formulation here, 

 parameters for Ricker's adaptation would be 



-1 



Boo = 220,000 tons, 

 m = 40,300 tons yr^ (the MSY of the model). 



With reference to Ricker's illustrations (1975: 

 312-315), we first calculate the time delay that 

 accompanies a reduction in mortality from Fq = 

 0.40 yr \ corresponding to a stock level of B^ = 

 100,000 tons, to a new mortality commencing at 



reference time zero, of F, = 0.20 yr~^ The adjust- 

 ment level to which B(t) will trend in the transi- 

 tion period is B, = 160,000 tons (by setting, in 

 Equation (6), B =0,F =F, and B =B^. If we now 

 specify the uncertainty in estimation precision as 

 being, say, 5% of true stock size, then Equation (8), 

 with Fj < F„ and e = 0.05, gives the estimated 

 delay in adjustment as 



^^^ 0.533 

 = 4.6 yr 



In 



(1 - 0.05) (160,000 - 100,000) 



0.05(100,000) 



When the model stock declines between similar 

 levels, the time delay is longer. That is, stock at 

 level Bq = 160,000 tons, corresponding to the 

 fishing mortality Fg == 0.20 yr \ declines to the 

 adjustment level B, = 100,000 tons following an 

 increase at t^ — to the new mortality Fj = 0.40 

 yr ^ Transition time t^^^ now becomes 



Mag 



0.333 

 = 6.2 yr 



In 



(1 + 0.05) (100,000 -160,000) 



-0.05(160,000) 



Yields from transient periods differ consider- 

 ably from the removals associated with equilib- 

 rium states. Obviously, an increase in fishing mor- 

 tality increases the yield temporarily, and a 

 decrease in fishing mortality decreases the yield 

 temporarily, but the ensuing trends of adjustment 

 will depend, in the context of the Graham system, 

 on the following relationships: 



F < 4m /Boo; stock size B(t)-^B,. (Figure 2), 

 which implies that Y-^'FB^. 

 F^4m/Bx; root B^<0 and B(t)-^0 (Figure 3) 

 which implies that Y->0. 



F = 2m/Bx; stock size B(t) implies p (p being 

 the biomass level Bx/2 where maximum latent 

 productivity occurs; Figure 1), which implies 

 that Y-^'m. Accordingly, we may identify 

 parameter m, in any of the rate equations here, 

 with MSY (which, we should remember, is it- 

 self a yield rate). 



Since, by Equation (4), instantaneous removal 

 varies in time as Y(t) = Fit)B(t), then over the 

 course of the adjustment interval that follows an 

 abrupt change in F, yield from a Graham stock 

 will accumulate as 



381 



