COMPARATIVE POPULATION STUDIES IN FISHERY BIOLOGY 53 



parative ecological work also in the other lines of approach — the studies of 

 the properties of eco-systems, and of population growth patterns. 



A YIELD EQUATION AND ITS TRANSFORMATIONS 



Under steady state conditions the annual catch in weight (Y) offish from a 

 stock to which there is an annual number of recruits (R) is given by the 

 integral, over the fishable life-span, of the product of an instantaneous 

 fishing mortahty coefficient (F), an expression giving the number of fish 

 surviving to age t as a function of R and of a coefficient of total mortahty 

 (Z = F + M), and an expression for the body weight (w) of a fish in terms 

 of its age. 



An expression for body growth which has been rather frequently used in 

 recent years is that of von Bertalanffy (1957), in which weight is taken as 

 proportional to the cube of the length and 



^-l = E-Kl 

 dt 



from which 



and 



ti^t = W^{i - e- Kit-to)}^ 

 where 



Loo = E/K = upper asymptote of length 

 W^ = upper asymptote of weight 

 K = parameter expressing the relative rate of approach to I ^^^ 

 tQ = scale constant; the theoretical age at which h and Wt = 



This gives a curve for length at age without an inflexion; the corresponding 

 curve for weight at age has an inflexion at 0-29 of the asymptotic weight. 

 Parker & Larkin (1959) have recently questioned the suitability of such an 

 equation for representing the growth of some fish, and have suggested a 

 'parabolic' function which gives an inflected weight at age curve but does 

 not have a finite upper asymptote. It is significant that these authors fitted 

 their curve to data for the growth of young fish, and although there does not 

 in fish appear to be a size or age at which growth ceases entirely, nevertheless 

 in only a few species is a substantial rate of increase in size maintained 

 throughout life. Even in these cases there is sometimes doubt concerning 

 the rehability of age-determination in the larger specimens, which distorts 

 the growth curves, and in fish with average life-spans artificially shortened 

 by the pressure of fishing or other predation it may be difficult to identify 

 an upper asymptote, though one may exist. There is a further difficulty 



