FISHERY BULLETIN: VOL. 71, NO. 2. 1973 



the effects on M^ , F^, and Ni of constant, non- 

 trivial values of natural mortality, M^, on all 

 recruited age groups {Mo > during T — t^. 

 in our notation). The effect of offshore fishing 

 on hatchery stocks in terms of age-specific 

 escapements to the river was computed (with 

 F — 0} for different values of M^. 



The model yields no point estimators and 

 therefore no equations for bias, but is important 

 here for several reasons. First, it utilizes known 

 offshore catch data for the first time in the 

 indirect approach. Second, it emphasizes the 

 need for direct sampling of the offshore catch 

 for maturity by origin (i.e., after marking 

 smolts): the resulting nl would give nominally 

 unbiased estimates of all parameters but F and 

 M'z in Equations 1-8. Third, the calculated 

 maturity schedules changed little with changes 

 in Mq after recruitment to offshore fishing 

 (Cleaver, 1969; Henry, 1971); this agreement 

 with Fredin's (1964) result, based on varying 

 the time distribution of natural mortality 

 during all of ocean life for sockeye salmon, 

 evidently is quite general for the 3-4 returns 

 these three authors treated. Finally, reliance of 

 the Cleaver model on one-sided limits (Equa- 

 tions 15-17) led directly to development of the 

 last model. 



Limit -Mean Model 



Given the foregoing development of the 

 indirect approach, it is natural to consider 

 1) devising two-sided limits on (say) Z^ or 

 Si initially and 2) taking for each parameter a 

 value intermediate between these limits as a 

 nontrivial estimator with unknown bias. The 

 opposite and usual procedure in population 

 problems is to derive nominally unbiased 

 estimators and their variances (thus confidence 

 limits) from statistical theory (e.g., the change- 

 in-ratio estimators reported definitively by 

 Paulik and Robson, 1969). 



Offshore Catch Unknown 



In this case C may be considered "unknown" 

 because either 1) no offshore fishery exists 

 (Zj = Mj) on the target population and we 

 deal actually with the M, as Ricker (1962) and 



Fredin (1964) did, or 2) an offshore fishery 

 exists {Z2 includes F > as in our coho 

 salmon) but techniques to evaluate C > are 

 not applied. The survival product, .s'iX2, is the 

 target parameter. Limits on the .s/ are devised 

 initially with the help of Assumption 1 {s^y = 

 ^2L ~ ■''') ^'^^ Assumption 2 {S21J < 1.0). 



Assumption 1 is based on the evidence cited 

 earlier which demonstrates almost incontro- 

 vertibly that M, > M-,. Now f, - fo (Equation 1) 

 is shorter for coho salmon emphasized here — 

 6-9 mo at the outside — than the T — ti ^ 12 

 mo of ocean life remaining in the stocks south 

 of British Columbia (Shapovalov and Taft, 

 1954; Godfrey, 1965; Drucker, 1972). It is 

 most unlikely that .s-i > .so, however, even when 

 the intensive offshore fishery during T — ty 

 ^ 4-5 mo is considered. Assumption 1 gives 

 from Equation 5 the quadratic : 



s2 - {EJN^)s - EJNq = = s,^S2, 

 = (SiS2)l • 



(19) 



Only one root is possible and the solution of Equation 

 19 is: 



- 0.5 {ejN^ + [(^i/A^o)' +4£2/A^o]"^}- (20) 



Areasonablevaluefor Assumption 2 (§2 (J < 1.0) 

 depends on the stock and species. Murphy's 

 (1952) often cited point estimates for coho 

 salmon during their last year of ocean life 

 serve as a guide for later numerical illustration. 

 His estimates are 16-51% for 12 successive 

 smolt classes (1938-49); only three estimates 

 exceed 50% , of which two are 51% . Today's more 

 intensive offshore fishery probably reduces 

 .S2 below that in 1947 to which his estimated 

 maximum, .^2 = 57%, applied. [In the next 

 section I use S2y ~ 50% but assign no value 

 here. Other estimates and limits warrant 

 mention: 68% by Parker (1962) as a point 

 value grossly averaged for various smolt 

 classes, stocks, and species; 51-80% as point 

 estimates for sockeye salmon by Fredin (1965) 

 and Mathews (1968); and 39-73% as upper 

 limits by Cleaver (1969) for chinook salmon 

 of ages 3-5 taken along with coho salmon 



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