566 



Fishery Bulletin 90(3). 1992 



Fmelsy/Fmsy 



1 V 



0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 



Scaled variance v 



Figure 4 



Ratio of F'jiELSY ^° F'^jy under different combinations of 

 parameter levels. The ratio tends to decrease as K" decreases 

 or as m or v' increases. 



Equation (20) also implies that F'melsy falls to zero 

 whenever v' reaches a critical value v'o defined as 



Scaled variance v 



00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 



Expected value of q (m) 



Figure 5 



Limiting values of v'. The solid curve shows v'„ , the locus at 

 which F'j,E.Lsv=0- The dashed curve shows v', , the locus 

 limiting the parameter subspace for which F'melsy ^^^ ^^~ 

 ceed 1. For (m, v' ) combinations below the v', curve, F'j,elsy 

 can take any value, depending on K'. For (m, v' ) combinations 

 between the two curves, F'^^lsy can range between and 1. 

 again depending on K". For (m, v' ) combinations on or above 



1-m 



Vo = 



(23) 



■m 



By Equation (19), v'o corresponds to a /3 value of 1. 

 Whenever p< 1, the right-hand tail of the beta distribu- 

 tion fails to reach zero, implying a non-zero probabil- 

 ity that q= 1. When q= 1, any positive F value causes 

 the stock to go extinct. Given the preservationist at- 

 titude implicit in the logarithmic loss function, any 

 possibility of extinction is unacceptable, so F'melsy 

 drops to zero in this case. Note that F'melsy is never 

 positive for values of v' greater than 0.5. 



Just as Equation (2) could be solved to determine the 

 locus of parameter values under which F'msy takes on 

 the special value of 1 (Eq. 3), Equation (20) can be 

 solved to determine the following locus of parameter 

 values under which F'melsy = 1- 



K" 



l-2v' 

 m(l-v') 



(24) 



In the certainty equivalent case, Equation (24) 

 reduces to Equation (3). As K" approaches zero, Equa- 

 tion (24) defines an upper limit on v' (v'l) for the 

 special case where F'melsy = 1: 



Vi = 



l-2m 

 2-2m' 



(25) 



Under Equation (3), F'msy could exceed 1 only if q 

 were less than 0.5. While Equation (25) implies essen- 

 tially the same property (replacing F'msy with F'melsy 

 and q with m), it adds a similar restriction on v', namely 

 that F'melsy can exceed 1 only if v' is less than 0.5. 

 [Note that this is a weaker version of the restriction 

 implied by Equation (23). Equations (23) and (25) are 

 compared in Figure 5.] 



Biomass at MSY compared 

 with biomass at MELSY 



Dividing Equation (1) through by F gives equilibrium 

 stock biomass. By substituting Equations (20) and (2) 

 into this expression and setting q = m, the ratio of stock 

 biomass at MSY to stock biomass at MELSY is given 



by 



B(Fmsy) 

 B(Fmelsy) 



(26) 



/F' 



MELSY 



+ V 



K"-!- F'msy + 1 



F' 



with limits 



MSY 



-1-1 



iK"-hF' 



MELSY 



r + l 



1 

 1-m 



