NOTE Atran and Loesch: An analysis of fluctuations in catchability coefficients 



563 



If catch in time interval i+1 is zero, then the ratio R t 

 is zero and the subsequent ratio R i+l is undefined. 

 In this case, the ratio can be constructed between 

 the current time interval and the second subsequent 

 interval. This ratio (R l+1 ) is given by Equation 5 in 

 Tomlinson, 1970, as 



!+1 * C " E, 



(2) 



The above equation is Tomlinson's extension to 

 Murphy's catch equation. This can be further extended 

 to include any number of intermediate time intervals 

 with zero catch. A generalized form of the catch ratio 

 between any two time intervals i and i+k, where the 

 catches for all intermediate time intervals is zero, is 





(3) 



Given an estimate of natural mortality rate and fish- 

 ing mortality rate for the final time interval, F's for 

 the previous time intervals can be solved by estimat- 

 ing E { x expU-(.F+M-)]. This can be estimated from 

 E i+k by rearranging Equation 3 as 



E,e 



t,(F+M,) 



C E e~ (w * f, ' +l) ""'~ H "*- lAf '' + *- l) 



- l i - h 



(4) 



J i+k 



where C is nonzero, and all catches between t- and 

 t i+k are zero. F t may be found by iteration after in- 

 serting the result of Equation 4 in the following equa- 

 tion (Equation 9 in Tomlinson, 1970) 



E,e 



t,(F l+ M,) _ E t {e 



t,(F,+M, ) 



1) 



F+M, 



(5) 



Once the F.'s have been estimated, the F ( 's can be 

 estimated by inserting the value of the above equa- 

 tion into the following: 



(the value of Equation 5) 

 h i~ jJfTm-) < b > 



After calculating the F-'s and F 's, the population size 

 at the start of each time interval (AT) can be esti- 

 mated from 



N t =- L , where E, <>0. 

 E, 



To demonstrate the use of this extension to 

 Tomlinson's method for solving the catch equation, 

 the computer program MURPHY (Abramson, 1971), 

 which implements Tomlinson's model, was modified 

 to incorporate the extension for consecutive zeros to 

 estimate weekly abundance and fishing mortalities 

 for the Atlantic menhaden purse-seine fishery. The 

 catch data were broken up into weeks and into age 

 groups within a week. 



Constant parameters used 



In addition to catch-at-age data, virtual population 

 analysis (VPA) requires estimates of instantaneous 

 natural mortality for all time intervals and an esti- 

 mate of instantaneous fishing mortality for one time 

 interval. Natural mortality was assumed constant 

 and a weekly value of 0.0087 (annual M=0.45) was 

 adopted on the recommendation of the Beaufort Labo- 

 ratory Estimates of fishing mortality for the final 

 week of landings data in each year were obtained 

 from Table 13 of Broadhead et al. 1 for the years 1968— 

 76 and for age groups 0-5. For age groups 6-8 the 

 values for age 5 were used. For the years 1977-82, 

 the average values for the years 1968-76 for each 

 age group were used (1968-75 for age group 0). In 

 each case, the annual value of F from the table was 

 divided by the number of weeks in the year that had 

 landings data to obtain a weekly F. Instantaneous 

 fishing mortality values were probably overestimated 

 because catch generally declined at the end of the 

 season. However, in the backward solution to the 

 catch equation, the value for F tends to converge to- 

 ward its true value for a given M. Therefore, the er- 

 ror in abundance estimates due to this overestima- 

 tion of F should be minor at the beginning of each 

 year's landing data, although it may result in the 

 underestimation of abundance toward the end. 



Defining effort 



An index of fishing effort was needed to calculate a 

 catchability coefficient. The number of vessels with 

 landings in a given week (vessel-week) is commonly 

 used as the unit of fishing effort in studies of the 

 menhaden purse-seine fishery and was the unit used 

 in this study. Menhaden vessels generally operate 

 continuously throughout all or part of the fishing 

 season, fishing every day, as weather permits, un- 



(7) 



1 Broadhead, G., C. Grimes, J. Loesch, W. Nelson, G. Sakagawa, 

 and K. West. 1980. Report of the Atlantic menhaden popula- 

 tion dynamics subcommittee to the Atlantic menhaden scien- 

 tific and statistical committee on the status of the Atlantic men- 

 haden stock and fishery. Unpubl. manuscr.. 68 p. 



