SECT. 4] FISHERY DYNAMICS — THEIR ANALYSIS AND INTKRPRETATION 477 



and the equilibrium catch at the level of population in year i is estimated as 



Y Y + APi 



Yei = li+ ^r 



The value of c required for estimating the values of F from X was obtained, 

 in each case, from the data of tagging experiments. The equilibrium catches. 

 Y e . i, thus estimated can be plotted against population size, P t , regardless of 

 any hypothesis as to the form of the relationship between them, and if a long 

 enough series of data, over a wide enough range of values of Pi, is available, an 

 empirical average relationship can be drawn. If, however, the model of (20) is 

 assumed, a parabola can be fitted to estimate the relationship. 



Likewise, designating Yt/Xi (catch-per-unit-of-effort) by Ui, and noting 

 that Ui = cPt, we have from (19) for year i : 



cXi = k[ L 

 where 



AUj _ Yj+i/Xj+i— Yj-i/Xj-i 

 ~At " 2~~ 



(IjUt) AUt/At being an estimate of (I /Pi) APijAt, and where e is a population- 

 independent random variable, to take account of variable environmental 

 conditions. 



From a sufficiently long series of data on catch and catch-per-unit-of-effort, 

 the parameters, including c, may be estimated by statistical procedures, 

 without reference to tagging data. Schaefer (1957) applied this method to the 

 data of the yellowfin tuna fishery of the Eastern Pacific, and thus estimated 

 for the model of (19) and (20) the average relationships between fishing effort, 

 catch-per-unit-of-effort, and total catch for steady states. He rearranged the 

 terms in (21), accumulating some constants for simplicity of notation, and 

 obtained 



MrSr-**-**-**? < 22) 



Summing over n years, 



1 n 1 ATI- " " 



- c I^=I^-U>)-lX t , (23) 



since the expectation of 2 e is zero. 



Since positive values of AU are, on the average, associated with positive 

 values of the right side of (22), and conversely, he also obtained, summing 

 over m years, 



= ^[aiM-U^-Xtl-^iaiM-U^-Xt]. (24) 



1 m 



\1 



Ui At 



AUi > AU{ < 



