54 S. J. HOLT 



concerning the use of equations with unlimited growth, which is that if 

 they are incorporated into yield equations without provision for an arbitrary 

 upper limit to the fishable life-span, their extrapolation beyond the limits of 

 the original data can lead to quite anomalous predictions. It seems for the 

 moment generally more satisfactory, therefore, to use equations which lead 

 to fmite upper asymptotes offish-size, provided of course that they represent 

 the observations reasonably well. It is worth noting, in this comiection, that 

 predictions of changes in catch resulting from changes in the intensity or 

 selectivity of fishing of the order of magnitude normally considered in 

 practice, do not appear to depend at all critically on the precision with which 

 the growth function used fits the observations of size at age. For many 

 purposes it is permissible even to approximate growth by treating weight as 

 linear with, or even proportional to, age over the middle range of sizes. On 

 the other hand computations are commonly performed by numerical 

 means using the original observations, instead of smoothing them by fitting 

 a growth equation. This latter practice, while serving the purpose of 

 empirically obtaining predictions for particular stocks and situations, seems 

 to eliminate the possibilities of predicting density-related changes in para- 

 meter values or of comparing different stocks and situations in order to 

 deepen understanding of the population processes and simplify methods of 

 analysis. 



Richards (1959) has generahzed the von Bertalanffy growth function in 

 such a way as to include von Bertalanffy's own extension of the simple 

 equation, and that by Ricker (1958). Richards shows that if in general 



Jf^ = 7]w^ — Kw (where ?/, m, and K are constants) 

 dt 



then 



j^(i-m) = I^^(i-»^) — he-^^ 



where 



WJ"^-^) = TjlK 



k = (i - m)K 



If the size at the inflexion is denoted by Wi then 



and the relative growth rate at this point is 



dw , I 

 — p = km 

 wdt 



