Thompson Variations on a simple dynamic pool model 



721 



Substituting Equation 12, 13, or 14 into Equation 

 18 thus gives the q value that sets F MSY under a 

 Gushing stock-recruitment relationship equal to F MSY 

 under a Beverton-Holt relationship. Assuming that both 

 stock-recruitment curves are parametrized to pass 

 through the same pristine stock-recruitment point 

 (B(0),6(0.a r )), this q value implies a second intersec- 

 tion at some lower biomass level. It turns out that this 

 lower level is always less than about 20% of B(0) (Fig. 

 1A) and greater than about 20% of b(0,a r ) (Fig. IB). In 

 other words, Kimura's (1988) placement of a lower 

 intersection at 50% of B(0) would always cause the 

 Cushing model to overestimate F MSyi Placing the lower 

 intersection at a biomass level less than 20% of B(0), 

 however, might result in either an over- or under- 

 estimate. 



For example, one rule of thumb (Clark, 1991) holds 

 that F UI (the fishing mortality rate that reduces the 

 slope of the yield-per-recruit curve to one-tenth of the 

 slope at the origin), F as „ (the fishing mortality rate 

 that reduces the level of spawning biomass per recruit 

 to 35% of the pristine level) and M should be approxi- 

 mately equal. In the base model, this rule of thumb 

 holds exactly at if '=1.5 (Thompson, in press). In the 

 base model with Beverton-Holt recruitment, then, 

 F MSY =F 01 =F 35 r=M at Q'=4 and K"'=1.5. These param- 

 eters imply a stock-recruitment curve in which recruit- 

 ment is reduced from b(0,a r ) by exactly 1/11 when bio- 

 mass is reduced to 50% of B(0), and in which 

 recruitment is reduced from b(0,a r ) by exactly 50% 

 when biomass is reduced to 1/11 of B(0). 



In the base model with Cushing recruitment, on the 

 other hand, F SIS ^F t =F 35% =M at q=2/7 and K = 1.5, im- 

 plying a stock-recruitment curve in which recruitment 

 is reduced from b(0,a r ) by about 18% when biomass is 

 reduced to 50% of 5(0), and in which recruitment is 

 reduced from 6(0,a r ) by about 50% when biomass is 

 reduced to 1/11 of S(0). Thus, in the "rule of thumb" 

 case, the form of the stock-recruitment curve (Cushing 

 or Beverton-Holt) has virtually no impact on the re- 

 sulting estimate of F MSY so long as the curve passes 

 through (S(0),6(0,o r )) and (B(0)/ll,rj(0,a,.)/2). 



More generally, to cause Cushing and Beverton-Holt 

 curves to intersect at (S(0),6(0,a,)) and at some frac- 

 tion p of 5(0), set 



Alternatively, the lower intersection can also be de- 

 fined in terms of relative recruitment (as opposed to 

 relative biomass). To cause Cushing and Beverton-Holt 

 curves to intersect at (B(0),6(0,a r )) and at some frac- 

 tion Hot 6(0,a r ), set 



9 = 1- 



ln( 9) 



\n(0) -\n[(l - B)(K" + 1)Q'+ 0] 



(21) 



andp as in Equation 20. 



Biases resulting from placement of the lower inter- 

 section at 50% and 10% of S(0) are compared in Fig- 

 ure 2, A and B, respectively. Note that use of the 50% 

 value (as in Kimura's [1988] example) causes large 

 and uniformly positive biases in the base model's esti- 

 mate ofF MSY . On the other hand, use of the 10% value 

 constrains bias to the +/- 10% range over a large por- 

 tion of parameter space. 



Biases resulting from placement of the lower inter- 

 section at 90% and 50% of b(0,a r ) are compared in 

 Figure 2, C and D, respectively (the 50% reference 

 point has been suggested by Mace 1 and Myers et al. 2 ). 

 As with the relative biomass reference level, use of 

 Kimura's (1988) relative recruitment reference level 

 (90%) causes large and uniformly positive biases in 

 the base model's estimate of F MSY over a large portion 

 of parameter space. On the other hand, use of the 50% 

 value constrains bias to the +/- 10%' range over a siz- 

 able region. 



As Figure 2 illustrates, then, there is reason to be- 

 lieve that the form of the stock-recruitment curve 

 (Cushing or Beverton-Holt) may not be particularly 

 important in terms of the resulting estimate of F MSY so 

 long as the candidate curves intersect at a fairly low 

 level. In other words, fishery managers need not al- 

 ways view an estimate of F MSY as being critically de- 

 pendent on the form of the stock-recruitment curve. 



A general growth function 



The linear growth function used by Thompson (1992) 

 may be viewed as a special case of the following more 



ln(p[(^"+l)Q'-l] + D-lnlQ'l-lni/T+l) 

 q = l - (19) 



and 



ln(p) 



M w<JT+l>Q'-h 



(^TlX 



i -i 



(20) 



'Mace, P. M. 1993. Relationships between common biological refer- 

 ence points used as thresholds and targets of fisheries management 

 strategies. Dep. Commer., NOAA. Natl. Mar. Fish. Serv., 1335 East- 

 West Highway. Silver Spring. MD 20910. 



-Myers, R. A., A. A. Rosenberg. P. Mace, N. Barrowman, and V. 

 Restrepo. 1993. In search of thresholds for recruitment overfishing. 

 Dep. Fisheries and Oceans. St. John's. New Foundland. Unpubl. 

 manuscr., 14 p. 



