254 



FISHERY BULLETIN OP THE FISH AND WILDLIFE SERVICE 



E. J. H. Beverton (unpublished manuscript) 

 used the model shown above to estimate popula- 

 tion parameters for the North Sea demersal 

 fisheries. I have used this model to develop for- 

 mulae for back-calculating estimates of fishing 

 rates and total populations for an anadromous 

 fishery. 



After the nth standard-fishing-unit day, there 

 remain q*N fish. These fish represent the es- 

 capement. The problem is to determine q, 

 then p, since p=l — q. From the 1951 tagging 

 data, we have estimates of total population, N, 

 and escapement, E. Table 4 shows that 2,589 

 standard-fishing-unit days of effort were expended 

 in 1951. Proceeding, 



q n N=E 



g 2589 (178,072) =77,105 



(2) 



gM8»=0.433 



and g=0.999675; p=l-g=0.000325. In one 

 standard-fishing-unit day 0.0325 percent of the 

 fish remaining are removed. In 1951 , for example, 

 where the population is treated as a whole, i. e., 

 $■=178,072 shad, in the first standard-fishing-unit 

 day, 58 fish are removed and 178,014 remain; in 

 the second standard-fishing-unit day, 58 fish are 

 removed and 177,956 remain; in the 2,589th 

 standard-fishing-unit day, 25 fish are removed 

 and 77,105 remain. 



Given that TJT is the fishing rate, then 

 {l — TJT) is the escapement rate. It can be seen 

 that q n =(T—T c )/T. In this case, the sampling 

 error in estimating 5 results from the sampling 

 error in T c . An approximation to the variance of 

 q is obtained from the expectation of (dq) 2 (Dent- 

 ing 1943). This gives 



dq- 



~n(T-T c ) 



(3) 



(dq) 2 =V(q)- 



Since 



q\dT c f 

 'n 2 (T-T c f 



(dT e f=V(T e )=T e (l-T c /T), 



A, v gT. 



V{q) ~n 2 T(T-T c ) 



For 2=0.999675, T c =359, T=633, n = 2,589, 

 p (g)=3.08X10~ 10 , and the standard deviation of 

 q equals 1.75 XIO" 5 , or 0.0000175. 



To determine the total catch, it is necessary to 

 add the numbers of shad removed in each standard- 

 fishing-unit day, as follows : 



C=pN+pqN+pq 2 N+pqW+ . . . + pq n ~ i N (4) 



= P N{l + q+q 2 +q*+ . . . +q n ~') 



The expression in the parentheses is the sum of the 

 first n terms of a geometric progression where the 

 first term takes on the value 1 and the common 

 ratio is q. The sum can be expressed by the 

 (1-9") 



formula, 

 Then, 



(1-ff) 



C= P N 



(l-<Z n ) 

 (1-2)' 



and 



N-- 



C 



V 



(l-g B ) 



(1-2) 



(5) 



(6) 



For any previous year, where the total catch 

 and the number of standard-fishing-unit days are 

 known, we can determine the total population of 

 shad in the river. For example, in 1950 when the 

 total catch was 77,090 shad and total effort 2,749 

 standard-fishing-unit days, the total population is 

 estimated as follows: 



77,090 



A 



N=- 



0.000325 



r i-0.999675 2749 "! 

 L 1-0.999675 J 



(7) 



The annual fishing rate for 1950 is estimated to 

 be 58. S percent, and the total escapement is 

 54,015 shad. 



Formula 6 has been used to estimate the total 

 population, annual fishing rate, and escapement 

 for each year that catch and effort data are avail- 

 able. These estimates are given in table 6 for 

 the period from 1935 to 1951. 



Some workers may question the validity of 

 treating the river run of an anadromous species 

 as a whole, since all the fish do not enter the river 

 simultaneously. However, if the 1951 run were 

 treated as a number of groups entering the river 

 throughout the season and the number of fish in 

 each group were known, it would be possible to 

 determine p provided catch figures for each group 

 were available and fishing effort were uniform 

 for each group. With uniform fishing effort, p 

 would remain constant throughout the season. 

 Further, this p value could be used to determine 

 total populations and escapements for other years 

 where we have (1) the same number of groups 



