722 



Fishery Bulletin 91(4), 1993 



0.25 



E 

 o 



-O 



a 

 o 



c 

 o 



o 

 a. 

 o 



0.10 



2 3 4 5 6 7 



Scaled recruitment parameter 0' 



2 3 4 5 6 7 8 



Scaled recruitment parameter Q' 



Figure 1 



Location of the zero-bias intersection of the Gushing and Beverton- 

 Holt stock-recruitment curves. Horizontal dashed lines represent 

 limiting values, obtained in the limit as F MSY goes to zero. Proceed- 

 ing from left to right along a given horizontal dashed line, the curves 

 which intersect the line correspond to K" values of 2.0, 1.5, 1.0, and 

 0.5, respectively. Given values of the composite parameter K" and 

 the scaled Beverton-Holt recruitment parameter Q'. fixing the lower 

 intersection of the recruitment curves at the point identified by the 

 appropriate locus in this figure causes the base model to give the 

 same value for F[, SY as the Beverton-Holt modification. iA) Zero-bias 

 intersection defined in terms of relative stock biomass. Points below 

 and to the left of the curves result in negative bias, while points 

 above and to the right of the curves result in positive bias. (Bl Zero- 

 bias intersection defined in terms of relative recruitment. Points 

 below and to the right of the curves result in negative bias, while 

 points above and to the left of the curves result in positive bias. 



general form: 



w(a i 



(\ _g-«(o-o )-K K V< 



(22) 



where K is Brody's growth coefficient, K' = KIM, and n 

 is a positive integer. In different parametrizations. 

 Equation 22 corresponds to (or is a special case of) a 



number of general growth functions, including 

 those of Richards (1959, see also Fletcher, 1975), 

 Savageau (1980), and Schnute (1981). Schnute 

 described his parametrization of Equation 22 as 

 "generalized von Bertalanffy growth" (although 

 he did not restrict n to integer values). When 

 n=3. Equation 22 corresponds to the common 

 ("specialized") von Bertalanffy (1938) curve, and, 

 when n = l, the "monomolecular" curve of Putter 

 ( 1 920 ) and Brody ( 1928 ) is obtained. 



In the limit as K approaches zero, Equation 22 

 gives an nth-degree polynomial in age: 



w(a) = 



w r ( u - u "\ 

 \a r -a J 



(23) 



which has been used to describe growth (though 

 not always in weight) by Mendelsohn (1963), 

 Dethlefsen et al. (1968), Knight (1968), Rafail 

 (1972), Roff (1980), Geoghegan and Chittenden 

 (1982), Standard and Chittenden (1984), and 

 Chen et al. (1992). Equation 1 thus represents 

 the special case of Equation 22 where K ap- 

 proaches zero and n=l. 



Polynomial solution 



The polynomial solution for this model is parti- 

 tioned into two cases (K=0 and K>Q) and derived 

 in the Appendix. 



When K=0, the solution for F,'„„, can be written 

 as a polynomial of degree n, and the solution for 

 F' MSY can be written as a polynomial of degree 

 n+\. When K>0, the solution for F' max can be writ- 

 ten as a polynomial of degree 2«, and the solu- 

 tion for F' MSY can be written as a polynomial of 

 degree 2n + l. As with the base model, the solu- 

 tion in either case indicates that maintaining an 

 F' MSY value of 1.0 requires an inverse relation- 

 ship between q and K" (which, as in the base 

 model, can be written explicitly). Likewise, the 

 upper limit to the ratio B(F| /sv )/B(0) is the same 

 as in the base model ( lie) in both cases. 



The polynomial solution can be manipulated 

 easily to show how it varies across the range of 

 possible K', q, and K" values. For example, sev- 

 eral limiting values of F'„ mx and F' MSY are shown 

 in Table 1. 



Bias resulting from the assumption of 

 linear growth 



Assuming that growth is linear when it actually fol- 

 lows a generalized von Bertalanffy form can lead to a 



