56 S. J. HOLT 



where M' = MjK 



and Ic and /r = length offish at ages tc and U respectively. 



Beverton & Holt (1957) and Parrish (1957) examined the effect on long-term 

 catch predictions of variations in values of growth and mortality parameters, 

 and particularly of K It should be noted thatH^oo and R appear as factors of 

 proportionahty, and therefore do not have any effect on the shapes of curves 

 relating Y to F and to tc, nor therefore on catch predictions which would 

 normally be made in relative terms as fractions or multiples of observed levels 

 of past catches. 



The number of parameters in the yield equation can thus be reduced by 

 one or two (I^oo» -R) if predictions are to be made in relative terms and if it 

 can be assumed that W^ and R will not change with changes in the intensity 

 or selectivity of fishing and hence in stock size and density. This may not be 

 a reasonable assumption to make in general, but perhaps can be made in 

 particular circumstances. The number of parameters can again be reduced 

 by one by writing the coefficients of mortality always as numerators of ratios 

 the denominator of which is K, i.e. putting as before 



M' = MjK 

 andP' =F/K 

 and Z' = ZjK 



This arrangement makes it practicable, as Holt (i957^) found, to prepare 

 tabulations of the yield equation and thus avoid much laborious computation. 

 Tanaka (1958) has indeed pubhshed such tables, for Z' values from o to 15-0 

 and K{tc — t^ from o to 6-o. In his tables t^ is in effect put to infinity, but 

 catches ifor f^, ^ 00 can be obtained by subtracting one tabulated value from 

 another. Jones (1957) showed that the same yield equation can be expressed 

 in terms of incomplete beta-functions, which are of course already tabulated, 

 so that the computation of further tables may not now be necessary. The fact 

 that the yield can be expressed in terms of the ratios F' and M' has an 

 important implication even in cases where age determination is possible but 

 difficult. Thus one kind of error in age determination results in fish aged t 

 being nearly always assigned an age t + x where x is an unknown constant 

 integer, often + 1 or — i. Both K and the mortality rates will, in this case be 

 correctly estimated from age compositions. If, however, there is also 

 involved an error such that true age t is given as yt because the time interval 

 between annuh is not sure, then there will occur the same error in estimates 

 of the mortality coefficients and of K\ their ratios will nevertheless be 

 correctly estimated, provided that the method of estimation does not involve 

 the use of addition data, such as tag returns, for which the true time-scale 



