466 SCHAEFER AND BEVERTON L CHAP - 21 



by Hjort, Jahn and Ottestad (1933), Graham (1935, 1939) and others. It has 

 since been further developed by Schaefer (1954, 1957), and is denoted here as 

 the "Schaefer" approach. 



Historically, the development of these two methods has been determined 

 very much by the type of fishery with which the investigators have been 

 concerned and by the amount and kind of data available. In what follows, an 

 account is given of each of them separately, but it is important to note at this 

 stage that the difference between them lies essentially in the kind of assumptions 

 and procedures that are adopted to overcome the lack of certain kinds of 

 information. The objective of both is the same, namely to provide the best 

 means of utilizing the available data to understand and predict the effect of 

 fishing on a stock. Some further comments on the relation between the two 

 approaches, both in concept and application, will be found in the concluding 

 remarks. 



2. The Beverton-Holt Approach 



The pioneer work along this line was that of Baranov (1918), although this 

 was not subsequently taken up in Russia, and did not come to the attention of 

 western scientists until the 1930's. Baranov constructed what amounts to a 

 particular case of the general model represented by (1). He assumed that a 

 constant number of recruits, B, enters the fished stock each year [which 

 implies r proportional to 1/P in (1)], and that the natural mortality coefficient, 

 M, is constant. He incorporated the age-specificity of growth by assuming 

 that the length of a fish increased in proportion to its age and that its weight 

 varied as the cube of length. With these assumptions Baranov derived an 

 equation expressing the length-frequency of the stock corresponding to any 

 given values of the above parameters and also to any value of the fishing 

 mortality coefficient, F ; and from this equation it is possible to compute the 

 equilibrium catch for various values of F. By making similar assumptions 

 about recruitment and mortality, Thompson and Bell (1934) calculated arith- 

 metically the stable-age composition of halibut corresponding to various rates 

 of fishing, and, by introducing the observed relation between age and weight 

 for that species, they computed the equilibrium relation between catch and 

 fishing rate. 



The methods developed by Beverton and Holt combine, in effect, the formal 

 mathematical treatment of Baranov with the age-composition treatment of 

 Thompson and Bell. For what they have called their "simple" model (Beverton, 

 1953), Beverton and Holt retained Baranov's assumption of constant recruit- 

 ment and a constant natural mortality rate but used a different growth func- 

 tion, because the assumption that the length of a fish increases in proportion to 

 its age is, in fact, valid over only a small part of the growth span, and this 

 assumption causes serious discrepancies when Baranov's model is used to 

 predict the catch at low fishing rates. They also distinguished between the 

 number of young fish comprising each year-class when they first enter the area 



