2i6 J. A. GULLAND 



While the preceding paragraphs have shown the importance of the stock- 

 recruitment relationship, no mathematical model which could be used in 

 analysis has been suggested. What appears to be required is some function 

 of the form: 



R =/(s), 



which could be fitted to the available data and then used in the earher 

 simple models, together with the relevant values of growth and mortality. 

 As a special case there is the original assumption 



R =f{S) = constant. 



In fact the data usually available from any marine fishery are likely to be 

 highly variable, and the stock-recruitment plot has a very high scatter. Thus 

 even if the form of the curve is known, fitting it to the data is usually difficult. 

 From a different angle this implies that it is equally difficult to determine 

 whether any given form of curve, having been fitted, is indeed a satisfactory 

 description of the data. Thus is it is not surprising that Beverton & Holt and 

 Ricker, having developed from different but reasonable assumptions two 

 distinctly different forms of curve {R = i/(fl + biS) which tends asymptoti- 

 cally to a maximum and R = cSe^'^ with a pronounced peak respectively), 

 should be able to fit them satisfactorily to some of the same sets of data. 



It appears therefore that this type of model describing the stock-recruitment 

 relationship is not sufficient for most fish; in some ways they are akin to the 

 simplest models of fish populations described at the beginning. Just as a 

 better understanding of the fish population can be achieved by studying it 

 in more detail — e.g. the various growth and mortaUty rates — so here 

 more details will have to be examined. In particular if indeed food supply is 

 hmiting, then the dynamics of the population of food organisms must 

 themselves be investigated. 



It is not intended to construct a mathematical model for this analysis, 

 especially as the details of such a model are likely to vary from one fish 

 population to another, depending on the range of foods eaten. However, 

 some points may be mentioned. 



If food supply is the factor causing higher mortaHty among more numerous 

 broods, then not only must shortage of food cause higher mortality, either 

 directly or by lengthening the period of special vulnerability e.g. to predation, 

 but also the higher abundance of young fish must cause a shortage of food. 

 This could be caused by interference or confusion between the fish, but this 

 seems unlikely in the sea and a more probable cause is a direct reduction in 

 the abundance of the food population. An initial guide to the possibility of 

 control by food supply is given by a comparison of the abundance of food 



