measured by location of the hauls or isothermal 

 contours. 



The first approach regards the value of Q as q/A. 



Since catch per unit of effort is related to the 

 mean density of the fish stock in the fishing 

 ground, as well as to the efficiency of individual 

 gear, this quantity should be adjusted by an 

 apijropriate measure (such as ai'ea of fishing 

 ground) to yield the relative stock size and the 

 virtual mortality coefficient (GuUand, 1955; Bev- 

 erton and Holt, 1957). Watt (1956, p. 629) pointed 

 out that the catch per unit of effort, obtained by 

 dividing the total catch by total effort in a season, 

 is not realistic when availability changes within 

 the season. He, therefore, compared the stock 

 sizes in 2 years by the catch per unit of effort in a 

 particidar month. 



If a fishery is operated in several localities, there 

 is no reason to believe that rate of availability 

 (accessibility and vulnerability) is common for the 

 total range of the fishery. Each locality should be 

 studied, especial^ if the stocks in different areas 

 are composed of different subpopulations. 



Such studies require that the calculations be 

 carried on for shorter periods and over smaller 

 areas. Because the mean of the last term of 

 equation (9) does not reduce to zero for such a 

 situation, we have to assume the following: 



a. AI and q fluctuate around their means at 

 random, and their variances are so small that these 

 quantities may be regarded as constant. 



b. There is a mean of /- that gives a mean of the 

 second term for any given set of q, f, and xA. 



c. The ratio of availabilities in 2 adjacent years 

 fluctuates around a logarithmic mean at random. 



d. The availability and gear efficiency are 

 common for two adjacent age groups. 



On the basis of these assumptions, the virtual 

 mortality coefficient of a certain age group of fish 

 during the tih season, Z',, in an area is 



Z;, = il/ -log {rrex-p (-f,qjA,) + {l-r,) } 



+ (logr, -log /■,_,) (10) 



and the mean of the virtual mortality coefficients 

 of the age group, Z', is 



Z' = M-log {r exp {-fq!A) + il-r)} +d (11) 



where ? and y are estimated averages of /■ and q 

 in the age group, which give the mean of the 

 second term in the formula (10) for a given set of 

 / and .4, and fZ=log {rJro)/m—l. 



The parameters, M, y, r, and d, may be esti- 

 mated by the least-squares method if appropriate 

 data which were taken in at least 6 successive years 

 and a suitable computer are available to make the 

 calculations. 



Putting a,=e.xp { — f,q/A,), and 6, = log fa, 

 -f(l— f)-(-fZ, the expected virtual mortality co- 

 efficient, E{Z't), equals M—b,. The differences 

 between observed and expected mortality coeffi- 

 cients, A's, are: 



Ai=log {a,n + (l-n)}+(logr,-log/-2)-6,, 



A2=log {a2''2+(l — '•2)} — (log /■2— log /•a) — 62, 



A5=log {a,r,+ {l—r,)}-{\og r,-log )\ + 5d)-bs 



(12) 



These equations give the rates of availability for 

 these 5 years. Repeating this procedure for each 

 successive 6-year period, we may obtain the run- 

 ning averages of the parameters on which a more 

 advanced discussion can be made. Wlien A is not 

 accurately measured, the calculation of availabil- 

 ity, based on the constant Q=q/A, may give some 

 clues for estimating vulnerability. When fish show 

 differential distribution by age, the mortality rate 

 in the entire population may be estimated from 

 the summation of the stock size of each age group 

 in each locality. Before calculation, division of the 

 area should be reexamined, such as by areal varia- 

 tion in fishing season and relative size of sub- 

 populations (perhaps by scale characters as well 

 as serological research). 



Finally, it should be noted that this type of 

 analysis does not provide absolute values of 

 availability for a whole population. Estimates 

 may differ from each other for availability of an 

 age group in a season by six different series. If 

 relative values of the estimated rates for successive 

 seasons are comparable for all of the six series, 

 however, the absolute rate may be surmised from 

 information on the distribution of stocks and 

 independent from fisheries, such as an egg census. 

 As a matter of fact, estimates of availability rates 

 in this type of analysis coidd be compared with 

 geographic distributions of egg stocks. This com- 

 parison is based on the fact that the distribution 

 of the parent stock of the Pacific sardine was 

 represented by egg distribution for the 5 years 



ANALYSIS OF CATCH CURVE OF PACIFIC SARDINE 



595 



