APPLICATION OF MATHEMATICAL MODELS TO FISH POPULATIONS 213 



DENSITY DEPENDENT RECRUITMENT 



In the simpler forms of most of the mathematical models described, the 

 numbers of young fish, the recruits, entering the fishery each year are assumed 

 to be the same whatever the size of the parent stock. This is by no means so 

 simple an assumption as that of constant growth or natural mortality and in 

 effect assumes that the mortaUty of the young is, at some stage, directly 

 related to the number present, increasing at higher abundances. In a contribu- 

 tion to this Symposium Beverton has discussed the various stages at which 

 this effective *gate* allowing only a fixed number through, can occur, and 

 argues that the most likely mechanism is shortage of food in the larval stages. 



While some such factor or factors undoubtedly exist, and probably form 

 the mechanism whereby the population as a whole is kept in balance, their 

 control is likely not to be precise and width of the 'gate' is variable. This is 

 obviously true at very low parental abundances, when the normal recruit- 

 ment could not be reached even if all the young survived, though with the 

 very high numbers of eggs produced by many marine fish, the same number 

 of recruits could be produced by a wide range of sizes of parent stock. For 

 instance a single female cod {Gadus morlma) may produce 10 miUion eggs in 

 her life; perhaps i per cent of young cod entering the fished stock survive 

 to maturity. Thus in a stable stock only 200 recruits need be produced out of 

 10 million eggs and the same recruitment could be produced by an adult 

 stock one-hundredth of the size, if the egg-to-recruitment survival is 200/ 

 100,000, i.e. a mortahty of 99- 8 per cent compared with the probable present 

 value of 99*998 per cent. 



The potentially extremely variable number of recruits obtainable from any 

 one size of mature stock means that the relation between stock and recruit- 

 ment can in turn be highly variable. The relation may be most easily des- 

 cribed by means of a plot of recruitment against parent stock such as that 

 used by Ricker (1954) (Fig. 3). In this diagram the two simplest assumptions 

 are shown, that of constant recruitment (except at the lowest stock abund- 

 ance) (curve a), which is the assumption made so far, and of recruitment 

 proportional to stock (curve h). More likely relationships are of the form of 

 curve c, with the descending right-hand Hmb, as favoured by Ricker, or 

 curve d which though curved is always increasing and is the form favoured 

 by Beverton & Holt (1957). The important implications of the precise form 

 of this curve in the stabihty of the population have been discussed at length 

 by both Ricker and Beverton & Holt. The only position possible where the 

 stock is neither expanding nor decreasing is the point where this curve giving 

 the number of recruits produced by a certain adult stock, cuts the constant 

 proportional line, giving the size of stock produced by a given number of 



