An analysis of weekly fluctuations in 

 catena bili ty coefficients* 



Steven M. AtrarT* 

 Joseph G. Loesch 



School of Marine Science, Virginia Institute of Marine Science 

 College of William and Mary, Gloucester Point, Virginia 23062 



Analyses of time series of commer- 

 cial catch statistics are usually 

 made with an implied assumption 

 that catchability remains constant. 

 Although the annual catchability 

 from year to year may remain fairly 

 constant, this assumption is rarely, 

 if ever, valid for catchability within 

 a season. Changes in catchability, 

 abundance, and fishing all contrib- 

 ute to fluctuations in the catch from 

 a fish stock (Clark and Marr, 1956; 

 Pope and Garrod, 1975). Behavioral 

 changes due to size or age may 

 cause variations in catchability 

 (Morrissy and Caputi, 1981). By ex- 

 amining the within-season changes 

 in catchability, it may be possible to 

 discern properties of a stock that 

 are not apparent when only annual 

 time intervals are examined. 



The objective of this study was 

 to develop a means to estimate 

 weekly within-season catchability 

 coefficients of a stock and to dem- 

 onstrate how examination of these 

 short-term fluctuations might be 

 useful in a stock analysis. Atlantic 

 menhaden, Brevoortia tyrannus, 

 was selected as a model because of 

 the availability of a time series of 

 weekly catch-at-age data for this 

 species. 



Methods 



Data 



Weekly menhaden catch-at-age (in 

 numbers) and vessel-landings data 

 from 1968 to 1982 were made avail- 



able by the Beaufort Laboratory, 

 National Marine Fisheries Service 

 (NMFS), Beaufort, North Carolina. 

 Migrating menhaden stratify by 

 age and size (Nicholson, 1971b); 

 therefore, the stock was divided 

 into age groups to eliminate differ- 

 ences in catchability due to age-spe- 

 cific (and size-specific) migration 

 patterns. 



Calculation of weekly 

 abundances 



Abundance estimates with con- 

 stant time intervals of one week 

 were needed to allow between-year 

 comparisons of weekly catchability 

 and to conform to the Beaufort 

 Laboratory's system of reporting 

 catches. Such short time intervals 

 usually resulted in consecutive in- 

 tervals of zero catches commonly 

 occurring near the beginning and 

 end of a sequence of weekly land- 

 ings data for a given year and age 

 group. 



Murphy (1965) developed a 

 method for estimating abundance 

 and fishing mortality rates on a co- 

 hort offish when catches are known 

 within time intervals and when an 

 estimate of instantaneous fishing 

 mortality for one time interval and 

 natural mortality for all time inter- 

 vals are available. A restriction on 

 this method is that the time inter- 

 vals must be of equal duration and 

 that each time interval must con- 

 tain catches. Tomlinson ( 1970) pre- 

 sented a generalization of Murphy's 

 method, which allowed for variable 



time intervals and zero catches, 

 provided that the first and last time 

 intervals each contain catches and 

 that two or more consecutive zeros 

 do not occur. 



The normal method for ensuring 

 that consecutive zero's do not occur 

 in the catch data is to pool time in- 

 tervals containing zero catches 

 with adjacent nonzero intervals. 

 In this study it was desirable to 

 keep the time intervals fixed, even 

 if it results in consecutive zeros in 

 the catch data. A modification of 

 Tomlinson's method was used, 

 which allows for any number of con- 

 secutive zeros, provided that the 

 first and last time intervals contain 

 catches. 



Extension to Tomlinson's 

 model 



A catch ratio, i?-, can be constructed 

 between catch in the current and 

 subsequent time interval. The ra- 

 tio for interval i is given by (Eq. 4 

 in Tomlinson, 1970) 



R, 



C, E, 



 (1) 



where C i ,C i j = number of fish 

 caught in time in- 

 tervals i, f+1; 



F- - instantaneous fish- 

 ing mortality in 

 time interval i; 



M ( . = instantaneous nat- 

 ural mortality in 

 time interval i; 

 E { , E i+l = exploitation rate 

 in time intervals i, 



' Contribution 1932 of the College of Will- 

 iam and Mary, Virginia Institute of Ma- 

 rine Science, School of Marine Science, 

 Gloucester Point, Virginia 23062. 



** Present address: Gulf of Mexico Fishery 

 Management Council 

 5401 West Kennedy Boulevard, Suite 331 

 Tampa, Florida 33609-2486 



Manuscript accepted 13 February 1995. 

 Fishery Bulletin 93:562-567 (1995). 



562 



