470 SCHAEFER AND BEVERTON [CHAP. 21 



whether any increase in the amount of fishing beyond the contemporary level 

 will result in an increased steady catch — is relatively insensitive to the degree 

 of accuracy with which certain parameters of the model can be estimated but 

 is critically affected by that of others. In such cases the properties of the 

 analytical model, as well as establishing in quantitative terms the reliability 

 of the answer, may also serve as a valuable guide to the direction in which 

 future research on the fishery should be conducted. 



Ricker (1944), in reviewing the analytical type of population model, pointed 

 out that the assumption that the numbers of recruits, the coefficient of natural 

 mortality and the rate of growth are all independent of population density, as 

 in the above model, cannot be valid over a large range of population size, nor 

 hence over a wide range of fishing effort or gear selectivity. In some circum- 

 stances, for example when dealing with the regulation of an established fishery, 

 this may not be a serious drawback, because for economic reasons regulation 

 has to be introduced in stages, each involving only a relatively small change in 

 the fishing activity and hence in population size. Thus the effect of the first 

 steps in regulation itself provides the opportunity of establishing whether the 

 population parameters are being influenced by population density, and if they 

 are, the prediction of the effect of the next step towards the final objective of 

 regulation can be made that much more accurate in the light of the secondary 

 effects which regulation may be having on the dynamic characteristics of the 

 stock. Nevertheless, although it may be sufficient for certain purposes to use 

 an analytical model with density-independent parameters, it is clearly im- 

 portant for a more general understanding and interpretation of the dynamics 

 of exploited fish populations that these secondary effects should be measured 

 and incorporated into the theoretical formulation of the fishery. 



Beverton and Holt (1957) have considered at some length the problems in- 

 volved in the analysis of density-dependent effects. Generally speaking, the 

 effect of population density on growth would usually be expected to be the 

 easiest to detect, partly because changes in growth can be measured with some 

 accuracy where routine age-determination is possible, and partly because 

 growth is directly related to the average amount of food available per fish. 

 The authors have, in fact, detected a relation between growth and density in 

 both the plaice and haddock populations of the North Sea, and have shown in 

 the latter species that it can be adequately represented by a linear relation 

 between the asymptotic length L& [corresponding to the asymptotic weight 

 Woo of (10)] and population numbers. There is also some justification from the 

 physiological basis of the von Bertalanffy growth equation for regarding the 

 asymptotic size as being more likely to be influenced by density than the co- 

 efficient K, although this is a question which has yet to be properly resolved. 

 Now, an expression for population numbers can readily be derived in the same 

 way as before, except that reference to size of fish is omitted, and is 



il—o-M(t e -t r ) e -M(t e -t r )\ 



