476 SOHAEFER AND BEVERTON [CHAP. 21 



This logistic function, however, gives a relationship which is symmetrical, 

 with a maximum at exactly half the maximum average population size, i.e. at 

 L/2. There is reason to believe that the maximum will actually occur, for many 

 if not all populations, at a level of P lower than L/2. It will, therefore, be 

 desirable to take more terms than the first two in/(P) if the data of the fishery 

 are sufficient to estimate the parameters involved. On the other hand, the most 

 simple approximation may be adequate for the data available (see, for example, 

 page 478). 



We have, then, 



±^ = k(L-P)-F(X) (18) 



and, if we further assume that fishing mortality rate is proportional to fishing 

 effort, F = cX, 



±^ = k(L-P)-cX (19) 



and, at equilibrium, the catch, Y e , is 



Y e = cXP = JcP(L-P). (20) 



Obviously, if we have data on the abundance of the population (catch-per- 

 unit-of-effort, which is proportional to P) and the effort or catch at two different 

 steady states, we can from (20) compute the parameters in this equation, and 

 so arrive at an estimate of the relationship between Y e and P. Graham (1935) 

 applied this method, essentially, to the data of the demersal fisheries of the 

 North Sea. 



The number of instances in which two or more clearly defined steady-state 

 periods, corresponding to substantially different levels of fishing intensity, are 

 encompassed by the available data are, however, rare. To make the method of 

 wider application it is necessary to adapt it to the analysis of data of fisheries 

 which are not in a steady state. Schaefer (1954) has developed a technique of 

 this kind which enables an approximation to the equilibrium catch to be 

 obtained on a year-to-year basis, and has applied it to the data of the Pacific 

 halibut and the California sardine. The actual catch. Y. in each year is divided 

 by its fishing rate, F, to estimate the mean population during the year. The 

 level of population at the end of each year is taken as approximately the 

 average of the mean population of that year and the following year, and the 

 difference between the populations at the beginning and end of the year is 

 taken as the increase (or decrease) for that year. That is, the increase in year i 

 is estimated as 



A Pj Yi+i/Fj^-Yt-^Fi-i 

 At 2 



