BIOLOGICAL BASIS 21 



relationships modeled by Baranov, and Thompson and Bell, in 

 simple differential equations, clarified the basic relationships in- 

 volved, and improved the methods of relating the various factors. 

 Schaefer adopted the logistic curve as the basis of his model, fitting a 

 parabola to the values of change in production from year to year 

 plotted against the size of population measured by the catch per 

 unit of effort. The apex of the parabola was interpreted as the point 

 of maximum sustained yield for the fishery. However, this point 

 varied widely ^vith different assumptions made in the model and 

 could not be determined with any exactitude, even for the halibut 

 fishery, in spite of the excellent records available. This inexactness 

 arose from ^\ ide variations of the observed data around the theoreti- 

 cal curve as ^\ ell as from variations in the location of that curve ^vith 

 different assumptions as to catch per unit of effort at maximum 

 population levels. 



Schaefer (1957) attempted to refine this model for yellowfin tuna 

 and plotted average annual values observed for the catch per unit of 

 effort against effort. These points were fitted by a straight line that 

 Schaefer called the estimated line of equilibrium catches. But here 

 again the point of maximum sustained yield Tvas only imperfectly 

 designated by the approach of the equilibrium line to one of the 

 curves of constant yield. (See Schaefer, 1957, Fig. 3.) Observed 

 points also showed ^vide variations around the theoretical curve. 

 Both Schaefer and Ricker concentrated upon production under 

 equilibrium conditions rather than upon year to year reactions of 

 catch to changing fishing pressure. 



Using the assumptions of constant recruitment, natural mortality, 

 and growth, Beverton and Holt (1957) incorporated the entire rela- 

 tionship into one differential equation. Their model differed from 

 those of previous authors in the use of Bertalanffy's gro^\ th equation, 

 and besides using separate values for natural mortality rates and 

 fishing rates, their model used different ages for recruitment onto the 

 fishing grounds, and recruitment into the fishery, as well as an as- 

 sumed maximum attained age. Beverton and Holt were interested 

 in demonstrating the consequences of using different mesh sizes in 

 trawl fishing for demersal or bottom dwelling fishes and dealt mainly 

 ^vith equilibrium conditions at different levels of fishing. 



These authors found that at any particular level of fishing inten- 

 sity and rate of natural mortality and gro^vth the catch rose to a 



