SECT. 4] FISHEKY DYNAMICS THEIR ANALYSIS AND INTERPRETATION 473 



in weight, these observations suggest that a pronounced degree of compensation 

 occurs in the early life of plaice which is absent in haddock. In fact, the fluctua- 

 tion of recruitment in plaice is sufficiently small for it to be possible to say that 

 recruitment is effectively independent of the parent population over the range 

 of size of the latter which has so far been observed. In the haddock, on the other 

 hand, although there is a slight tendency for the larger recruitments to come 

 from the larger parent stocks, the relation is not significant and the scatter of 

 the recruit values is so great that, from the data available so far, no firm state- 

 ment can be made about the form of the relation between stock and 

 recruitment or even whether the two are related at all. 



The theoretical consequences of introducing a relation between parent stock 

 and subsequent recruitment lead to some important conclusions concerning 

 the requirements for a population to be able to stabilize itself under the in- 

 fluence of fishing. Moran (1950) was the first to investigate this problem, 

 various aspects of which have since been discussed further by Ricker (1954) 

 and Beverton and Holt (1957). It has been shown that postulating a propor- 

 tional relation between stock and recruitment leads to a highly unstable 

 population ; the slightest disturbance from an initial steady state, such as a 

 very small change in the amount of fishing, causes successive generations either 

 to decrease in numbers until the population is extinct, or to increase without 

 limit. In practice, as Ricker (1954) has shown, the superimposition of a large 

 random component of fluctuation in recruitment could cause this progression 

 to be delayed for many generations. Nevertheless, the impression gained from 

 the history of even the very long established fisheries, including those for the 

 Arcto -Norwegian cod and the Atlanto-Scandian herring, of which there are 

 records dating back many hundreds of years, is that although there may have 

 been cycles of abundance, the major fish stocks are broadly stable and are not 

 tending either to extinction or to an ever increasing size. 



The essential requirement for stability is that the relation between stock and 

 recruitment should curve away from proportionality, so that the rate of 

 increase of recruitment with stock declines throughout as stock increases. 

 Beverton and Holt (1957) have shown that postulating a linear relation between 

 the natural mortality coefficient M at one or more stages in the pre-recruit life 

 of the fish and their density leads to the simplest form of relation between 

 stock and recruitment which gives stability, namely a curve which bends over 

 continuously and tends to an asymptotic limit of recruitment with increasing 

 stock. Denoting the total egg production of the population by E, which to a 

 first approximation can be regarded as proportional to the biomass of mature 

 fish, the equation is 



* - s+W (13) 



in which a and jS are coefficients defining the effective mortality rate and its 

 dependence on density during the pre-recruit phase of the life-history. The 

 limiting recruitment, as E tends to infinity, is seen to be i?= 1/a ; this can be 



