FISHERY BULLETIN: VOL. 79, NO. 2 



Wl 



R, +R2 +R3 



M'(l+Si +Sl) 



(2) 



the number tagged and released, 

 adjusted for tagging loss, 



where M ' 



Si = e~^ the constant annual survival 



Rn 



rate, and 



number of adjusted recoveries 



for years-at-large n. 



For release groups with an estimate of constant 

 Z for years 2-4, called here Z2, the constant exploi- 

 tation rate ( U2) had to be estimated by trial, since 

 the rate for the first season could not be assumed a 

 constant for later years (Type C error condition). A 

 trial value of Zi for the first year-at-large also 

 provided first year trial values of Si and Ai (total 

 annual mortality rate); «2 was estimated for the 

 second and later years-at-large by (Ricker 1975): 



XV2 ' R3 "I" R4 



SiM'{l+S2 +Sl) 



(3) 



and also by: 



"2 



(Z2 -z, + 



M' Ai 



). 



(4) 



The trial value of Zi was adjusted until Equa- 

 tions (3) and (4) converged on virtually equivalent 

 estimates of U2. 



The annual instantaneous rate of fishing mor- 

 tality {F) was estimated by: 



uZ 

 F = — (5) 



A 



where A = I — e~^, the total annual mortality 

 rate, and u,Z,F, and A are subscripted 1, where Z 

 was constant through years 1-3, and 2 where Z was 

 constant through years 2-4. 



The annual instantaneous rate of natural mor- 

 tality ( M) was estimated by: 



M 



(6) 



where, as above, Z and F are subscripted 1 for 

 constant mortality years 1-3 and 2 for constant 

 mortality years 2-4. 



The central area had the greatest variation in 

 parameter estimates between years, while the 

 western area had the least (Table 5). I averaged all 

 estimates of M and u for each tagging loss rate 



examined to obtain unweighted means (Table 6). A 

 change in the estimate of tagging loss resulted in a 

 slightly greater than 1:1 fractional change in the 

 estimates of M and u. 



Table 5. — Estimated annual instantaneous natural (M) and 

 fishing (F) mortality rates and exploitation rates ( u) for tagged 

 adult Gulf menhaden in the Gulf of Mexico, assuming a 20% 

 initial tag loss to shedding and mortality. 



Table 6. — Estimates of mean annual rate of exploitation (u) 

 and instantaneous natural mortality rate (Af) for Gulf menha- 

 den in the Gulf of Mexico, estimated from recoveries of tagged 

 adults, with varied levels of tagging mortality. 



Since the earlier estimates of each Z were from 

 weighted regressions with only three data points 

 and hence only one degree of freedom for estimat- 

 ing confidence intervals (CD, I conducted a com- 

 bined analysis of scaled data for the six sets of 

 release-recovery data for 1969 and 1971 (which 

 consist of 18 data points and, hence, 16 df) to de- 

 termine the stability of the estimate of M relative 

 to the variance about the estimate of Z. After 

 scaling the data, each release-recovery set sum- 

 ming to 10,000, I estimated the unweighted pre- 

 dictive regression slope estimate of Z2 and its 95% 

 C.I. by standard methods (Draper and Smith 

 1966). Assumptions on which years represented 

 periods of constant total mortality were the same 

 as for the earlier analysis. Equations (3) and (4) 

 were used to estimate «2 from the scaled estimate 

 of Z2 and for the high and low extremes of the 95% 

 C.I. F2 and M were estimated by Equations (5) 

 and (6) as before. The resulting estimate of M and 

 its approximate 95% C.I., although slightly lower 

 than the earlier estimate, represent very similar 

 values, and relative to the variance about the es- 

 timate of Z2, M is quite stable (Table 7). 



To determine if the arbitrary selection of 3 yr of 

 recovery data for analysis, adjustment for Type C 

 error, and use of weighted regressions had a major 

 altering effect on the resulting estimate of M, I 

 estimated M using all recovery data points for the 



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