474 SCHAEFER AND BEVERTON [CHAP. 21 



interpreted as the maximum number of fish which the food supply and other 

 characteristics of the environment can allow to survive to the age of recruit- 

 ment. 



Ricker (1954), on the other hand, has proposed a dome-shaped curve, with 

 recruitment increasing to a maximum at some intermediate level of adult stock 

 and then declining, as providing the best general interpretation of the relation 

 between stock and recruitment. A relation of this kind has been found in 

 certain experimental populations, such as Drosophilu, maintained under 

 restricted conditions of space, where above a certain abundance of adults the 

 medium becomes fouled and so causes the survival of young to be sharply 

 reduced. A tendency towards reduced recruitment at the higher stock levels 

 has also been detected in certain salmon populations, where again the environ- 

 mental conditions for spawning and for the early life of the young fish are 

 highly restricted compared with those found in the sea. It is, however, a 

 measure of the difficulty of interpreting the data available so far for marine fish 

 populations that no conclusive evidence of a declining recruitment at higher 

 stock levels has yet been demonstrated, remembering that a given set of rather 

 scattered points can often be fitted rather better by a paraboidal-shaped curve 

 than by a simpler asymptotic one. In this connection the reader will find it 

 instructive to read the discussion by Clark and Marr (1955) on the interpreta- 

 tion of the data of stock and recruitment for the Pacific sardine. In the present 

 state of knowledge, the two hypotheses can perhaps best be reconciled by 

 supposing that if the stock size were to increase far enough it would indeed give 

 rise to conditions which caused something like a catastrophic mortality of young 

 fish to occur, but that before that stage was reached the stock-recruit curve 

 would have become nearly flat over a considerable range of stock size. 



A dome-shaped stock-recruit relation gives rise to rather complex stability 

 characteristics. On the left of the maximum the population has essentially the 

 same kind of stability as that implied by the asymptotic relation (13), in which 

 a change in the external restraints, such as the amount of fishing, causes the 

 population to increase or decrease to a new steady state. On the descending 

 right-hand limb of the curve, however, self-induced oscillations are set up, the 

 amplitude of which may either gradually decrease with time or remain constant 

 or increase, depending on the particular shape of the curve. Theoretical models 

 of a fishery in which the relation between stock and recruitment is dome- 

 shaped have not yet been developed, but Beverton and Holt (1957) have 

 examined the properties of models in which the relation is of the kind formu- 

 lated by (13). If the other rates are treated as density-independent, they found 

 that certain important characteristics of the relations between equilibrium 

 catch and F or t c predicted by the simple model with constant recruitment are 

 scarcely changed, notably the amount of fishing or the gear selectivity required 

 to produce the greatest steady catch, although the magnitude of that catch 

 was markedly affected. If, in addition, growth was also made to vary with 

 density, more profound departures from the properties of the simple model 

 were found, since introducing the variation of recruitment with stock size 



