726 



Fishery Bulletin 91(4), 1993 



Solving Equations 25, 26, 33, and 34 simultaneously 

 gives 





i 



(35) 



The fact that F' appears in the exponent in Equation 

 35 complicates the solution for F' MSY somewhat, increas- 

 ing the degree of the polynomial solution to four: 



\(K" f K" m + 2K';+2K';„ + 3) &-j£ta + 

 (2K:, + 3)q+ K}-l]F„ s ,? + 

 [(K; + UK"„, + l)(^r-^r) 9 + VS" f + 3W m + 1 

 K" f K" m - K" m - 2]f; ;sv - (K" f + 1 )(«-; + 1X1 - q) = 



(36) 

 k/ + 



0. 



The solution for F' nmx is the same as in the base 

 model (Equation 7). 



Equation 36 contains F'„ SY =1 as a special case, ob- 

 tained when the following relationship holds: 



<7 = 



K" + 2 



iK" t +2 )(K';„+2 ) tj^- ^-\+K" l iK"„,+ 1 )+3K" m +4 



(37) 



As with the base model (Equation 8), the above ex- 

 pression implies an inverse relationship between q and 

 K" f . Likewise, the upper limit to the ratio B(Fl 1SY )IB(Q) 

 remains the same ( lie). 



Bias resulting from the assumption of a f =a 



When a t >a m , Equation 6 tends to underestimate F' MSY . 

 Loci of -10% bias are shown in Figure 4A for four 

 values of K" m . Parameter combinations above a par- 

 ticular curve and below the horizontal line K" f = K" m 

 result in an F' MSY estimate that is within 10% of the 

 value given by Equation 31. Note that the base model's 

 solution is fairly sensitive to K" f when K" m is low. For 

 example, when K"„ =0.5, almost any value of K" f <K" m 

 will result in Equation 6 underestimating F' MSY by more 

 than 10%. At higher K"„ values (e.g., K" m >\), the base 

 model's solution is less sensitive. 



The results for the case where a l <a Tn are similar, 

 except that here Equation 6 tends to overestimate 

 rather than underestimate F' VSY . Loci of +10% bias are 

 shown in Figure 4B for four values of K"„. Parameter 

 combinations below a particular curve and above the 

 horizontal line K"f=K" m result in an F' MSY estimate that 

 is within 10% of the value given by Equation 31. Again, 

 the base model's solution is fairly sensitive to K'j when 

 K" m is low (e.g., ^,=0.5), while at higher values (e.g., 

 K" m >l), the base model's solution is less sensitive. 



Finite maximum age 



As defined by Thompson (1992), all simple dynamic 

 pool models exhibit mortality and growth rates which 

 are independent of age (above the age of recruitment). 

 This implies that there is no maximum age. However, 

 in more complicated dynamic pool models, it is com- 

 mon to specify a maximum age above which all re- 

 maining fish die in knife-edge fashion. As noted by 

 Fletcher (1987), misspecification of maximum age 

 can introduce significant bias into some models. When 

 the base model is modified so as to exhibit a finite 

 maximum age (a,„,„), Equation 5 will tend to overesti- 

 mate true equilibrium stock biomass, which can be 

 written as 



i-i 



(38) 



where K" ma =ll[M(a max - a,,)], after Equation 3. The dif- 

 ference inside the exterior parentheses in Equation 38 

 is proportional to the difference between two calcula- 

 tions of stock biomass per recruit in a population with 

 infinite maximum age, where the first calculation be- 

 gins the integral (over age) at age a r and the second 

 begins at age a ma . Subtracting the second term from 

 the first adjusts for the assumption of a maximum age 

 at a=a„ mx . 



Because of the presence of F' in the exponential 

 term in Equation 38, it is not possible to solve for F' MSY 

 explicitly in this modification. 



Bias resulting from the assumption of infinite 

 maximum age 



Note that as K", mx becomes small (e.g., as c,„„, becomes 

 large), the proportion surviving to the maximum age 

 (the exponential term in Equation 38) goes to zero and 



