no attempt to estimate the absolute value of rate of 

 availability, but rather I estimated the fluctuations 

 in it as related to season and age. Strictly speaking, 

 availability should be divided into two categories: 

 (a) accessibility, the fish stock (in terms of 

 numbers) accessible witliin the range of the 

 fishermen and (b) vulnerability, which dei)ends 

 on factors involving gear efficiency. Measuring 

 vulnerability in the commercial catch requires 

 more than the examination of age composition 

 and the amo\mt of effort. I comment on this 

 pn)blem later. In the absence of information to 

 separate availability into its components, their 

 separate effects are disregarded here and included 

 in the changes in availability and mortality. 



Two basic assumptions are requisite for the first 

 study of the relation of fluctuations to season and 

 age: first, that the total mortality coefficient 

 fluctuates randomly around a mean; and second, 

 that the rate of availability fluctuates around a 

 logaritlimic mean. Even though no fishery may 

 satisfy these assumptions, they may be accepted 

 for a first approximation when age and year class 

 are significant sources of variation in catch com- 

 pared to the interaction that is a measure of total 

 effects of changes in rates of mortality and 

 availability as well as survey errors. In otlier 

 words, changes in mortality and availability are 

 regarded as less important in determining the 

 catcli tluin age and year-class strength when the 

 lattei two are significant sources of variations. 



FORMULATIONS 



Without excejjlion the number of the ith year 

 class caught in the year when they were jth age, 

 Cij, is expressed as: 



0=0 a = tp 



(2) 



where: 5,^= annual rate of survival of the ith 

 year class at jth age, £'(^=rate of exploitation for 

 the available part of the ith year class at jth 

 age, (p= youngest age in maturing spawning 

 products, A'^,„= initial stock size of the ith year 

 class, A^'(„=stock size of the ith year class at 

 the beginning of the year when they were tp year (s) 

 old. 



Under conditions assumed above, it holds that 



log (7,=log N'-J: Z„+log r. + log E, (8) 



where, log Cj=logarithmic average of all catches 

 of jth age fish over year classes, log A'^„=log- 

 arithmic average of recruitment stock sizes over 



j'ear classes, Za= average mortality coefficient 

 at ath age over year classes, log /■j=Iogarithmic 

 average of availability rates at jth age over year 

 classes, log £'j= logarithmic average of exploitation 

 rates at jth age over year classes 

 and 



h.g <:/,+,=log m-± Z„+log r,+, + log £,+, (3') 



and then 

 logrj+,-logC;=— Z^ 



+ (log r,+,-log r,) + (log i?,+,-log Ej) (4) 



If jth age fish migrate with fish older by 1 

 year, formula (4) approaches —Zj for data avail- 

 able over a period of years, if there is no trend 

 in the rates of exploitation and availability, or if 

 the number of years is large. Thus, the logarithmic 

 means of several ages of fish give a "standard 

 virtual catch curve" for a given period under the 

 above conditions. The standard curve becomes 

 mure reliable in estimates of — Zj as the i)eriod 

 under consideration becomes longer and the catch 

 curve of each year class becomes more stable. 



Year-class means of log Cn, log C,, are 



log C,=log Ni-jz t«Z,.+ log r. + log E, (5) 



where ka={ti — a+l)l{U — tp-^\), and <i = tlie old- 

 est age in question. The last two terms are the 

 year-class means of the logarithms of rates of 

 availability and exi)loitation, respectively. 



These means may not give a good estimate of 

 year-class strength in the sardine because its life 

 span is too short to allow changes in rates of 

 availability and ex])loitation to cancel each other. 

 It may be regarded, however, as a measure of the 

 mean available stock size of each year class when 

 the year class is found to be a significant source 

 of variation. 



Since the logaritluuic menu of catches is 



logr=logA^;-Z; lcaZ„+\og r+\og E (6) 



we can construct an "exjjected catch," E (log 6\j), 

 when Zfa=Za is common for the entire year class 

 and r,a=ri and FJia=E, are common for all the 

 age groups of any year class. 



E (log <7,,)=log C<+log <7,-log C 



=iogiv;„-|(|]'z„ 



j-i 



~r y , kaZ^ia' 



-|-(logr,-flogr^— logr) 



+ (log£:<+log^,-log^) (7) 



J y ■ ka^a ) [ 

 a=(p /J 



5RA 



U.S. FISH A\U WILDLIFE SERVICE 



