SECT. 4] FISHERY DYNAMICS THEIR ANALYSIS AND INTERPRETATION 467 



where fishing is in progress, and so become liable to encounters with fishing 

 gear, and the number which survive to an age at which they have grown large 

 enough to be retained by the particular kind of fishing gear in use. In this way 

 it is possible to develop a model which can be used to predict the effect on the 

 equilibrium catch of a change in the selectivity of the gear as well as in the 

 amount of fishing, a requirement which is of special importance in the trawl 

 fisheries of the North Sea and adjacent waters with which the authors were 

 primarily concerned. 



Beverton and Holt's simple model can be derived most conveniently by 

 considering the catch obtained from one year-class throughout its life, since in 

 the steady state, with the same number of recruits entering the exploited area 

 each year, this catch is the same as the average steady catch obtained each 

 year from the whole stock. Suppose E fish are recruited to the exploited area 

 at age t r , but are not retained by the gear until they have reached some greater 

 age t c . During this early "pre-exploited" period of their life their numbers will 

 be reduced by natural mortality only, and with a constant coefficient M, the 

 rate of decrease of numbers during this period is 



dN/dt = -MN. (4) 



The solution of this equation gives the number surviving to enter the catch at 

 age t c as 



Re-mh-t r ). ( 5 ) 



Thereafter, their numbers will be reduced by the combined operation of both 

 fishing and natural mortality, according to the equation 



dN/dt = -(F + M)N, (6) 



and from this and (5) the number surviving to any age t greater than t c will be 



N t = [Re- M «c-tr)] e-VF+wv-tc). (7) 



If w t is the average weight of fish at age t, the total weight of the year-class at 

 age t will be N t -Wt, and the rate of catch in weight as a function of age becomes 



dYJdt = FN r wt, (8) 



where F is the fishing mortality coefficient. In the first instance a linear in- 

 crease of weight with age was used for the function iv(t) (Hulme, Beverton and 

 Holt, 1947), and the same relation has also been used by Doi (1951). This gives 

 a reasonable representation of part of the growth curve but not the whole of it, 

 and does not lead to a model which is particularly convenient for computation. 

 As giving the best simple representation of the general growth pattern of fish, 

 Beverton and Holt later adopted the equation developed by von Bertalanffy 

 (1934, 1938), namely 



w t = JToo(l-e-*<*-«o))3 3 (9) 



in which Woo is the asymptotic weight to which the fish tends with increasing 



