2o8 J. A. GULLAND 



It remains to put these into a suitable mathematical form, and to add them 

 to give the total. Provided that the fishing effort can be expressed in the right 

 units, then the numbers caught per unit time will be proportional to the 

 number in the stock at the time, multiplied by this fishing effort, i.e. in 

 mathematical terms : 



\ at /(fishing) 



where q = constant. 



/= fishing effort, in some suitable units. 



F = fishing mortahty coefficient. 

 Similarly the rate of death due to other (natural) causes can be expressed as 





(natural) 



where M = natural mortality coefficient 



from which the numbers at any time t can be expressed in terms of the 

 numbers Nq at some previous time Iq as 



Nt = No e-(^+^) ('-^o> 



Various expressions have been used for the weight of the individual fish, 

 but only a limited number can be combined with the expression above for 

 numbers to give an expression which can be handled without too difficult 

 mathematics. These include the exponential (W = ke^^) (Bicker, 1948) and 

 the von Bertalanffy W= W^{i - e'^^f (Beverton & Holt, 1957). It is not 

 intended to discuss here the relative theoretical merits of various growth 

 equations for which there exists an already over-voluminous literature. 

 However it does seem unlikely that a single formula will give an exact and 

 meaningful fit to growth during the major part of a fish's life, both mature 

 and immature, and in which there are likely to be major changes in diet, as 

 well as marked seasonal changes. For the present purpose the requirements 

 for a growth curve are that, over the range considered it does give a satis- 

 factory fit to the observed growth, and that mathematically it combines 

 with the other expressions easily. In some practical apphcations, the require- 

 ments for a satisfactory fit to growth data are not so much that the theoretical 

 weight at age agrees with the observed weight at age, but that the theoretical 

 and observed growth rates agree, particularly over some critical range. For 

 instance in considering the results of an enlarged trawl mesh size the vital 

 question relating to growth is the time interval between reaching the sizes 

 retained by the old and new mesh size. Using the exponential growth curve 



