APPLICATION OF MATHEMATICAL MODELS TO FISH POPULATIONS 207 

 its constituent parts, so that in the simple formulation of Russell (1942) 



S,= S^ + {A+G)-{C + M) 



i.e. the stock at the end of the year [S^ is equal to the stock at the beginning 

 of the year {S^ plus the weight of young fish entering the stock — the 

 recruits — {A) and the growth of fish in the population (G), less the £sh 

 removed from the population by fishing (C) and other 'natural' causes of 

 death (M). 



400 



West of Scotland Hake 



200 



o 



^ 100 



O 0'\ 0-2 0-3 



Averaqe amouni" of fishing (million hoursj 



Fig. I. — ^The relation between stock abundance of hake to the west of Scotland (as measured by 

 the catch in 100 hours' fishing by English steam trawlers), and the amount of fishing. The 

 amount of fishing used is the average amount up to and including the year of observation. 



The mathematical exposition is made easier by considering the growth 

 and deaths of individuals of a single year-class. In any steady state the yield 

 from a single year-class during its whole life-span is the same as that in a 

 single year from all year-classes present. The yield during the entire time in 

 the fishery may be computed as the sum of the yield during small time 

 intervals, within which conditions do not alter. This yield in the small time 

 interval is equal to the product: 



Rate of fishing X Number in the stock X Average weight 

 of the individual fish 



