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



the following sequence from relations given 

 previously: 



Estimate Relation 



Bias 



Equations for bias in Z* and in other 

 estimates can be written for the cases of C 

 unknown and known. The equations are much 

 longer than for the Ricker model (e.g.. Equa- 

 tions 12-14), however, because estimators from 

 the limit-mean model involve so many substitu- 

 tions. The equations are therefore not written 

 out but the next section gives numerical 

 examples of bias. 



NUMERICAL EXAMPLES OF BIAS 



Cleaver's model gives no unique estimators 

 as noted earlier, and it is evident that bias 

 from the Ricker and Fredin models cannot 

 be compared fairly with bias from the limit- 

 mean model when offshore fishing is absent. 

 Thus Equations 10 (Ricker model) and 15 

 (Fredin model) utilize the information that 

 M, 5= My > M2 (Ricker, 1962; Parker, 1968), 

 hence Mj = Mg ^ as mentioned with Equation 10. 

 For the limit-mean model without offshore 

 fishing, however, it is interesting to see how 

 relative bias differs among the estimates of 

 mortality or survival, maturity, and abundance. 



With offshore fishing present and C > un- 

 known, we expect Z2 - Z^ or Z* - Z* to be less 

 from the Ricker and Fredin models than 

 Mo - M2 or M^ - Mrj, without offshore fishing. 

 For comparisons with bias from the limit-mean 

 model, I use Z* = Z2 and the resulting .s, as 

 the initial estimate for the Ricker model; 

 Z* and .sv,2, for the Fredin model. With C > 

 known I illustrate bias for the limit-mean model 

 only — again to avoid possibly unfair compari- 

 sons with the Ricker and Fredin models. 



Data published by Johnson (1970) guided 

 selection of certain hypothetical values for 

 input parameters: the resulting values of 

 EilNo are intermediate between those he re- 

 ported for small and large smolts (1964 brood), 

 and in that respect may be typical of coho 

 salmon reared at hatcheries on the lower 

 Columbia River. The value of F assigned here 

 is close to that calculated recently by K.A. 

 Henry of the Northwest Fisheries Center, 

 National Marine Fisheries Service, NOAA, 

 and myself for the 1965-66 brood coho salmon 

 reared at a group of Columbia River hatcheries. 

 All time intervals are accurate to within 1.0 

 mo and probably to within 0.5 mo as actually 



1.00 



Spawning migration reduces 

 population left at sea 



> 



a: 



=) 



o 



s 





.05 - 



.03 - 



.01 



Without offshore 

 fishing 



With offshore 

 fishing 



6 12 18 



TIME SINCE OUTMIGRATION OF SMOLTS (months) 



Figure 1. — E.xtinction of a hypothetical smolt class of 

 coho salmon with and without offshore fishing during 

 the last year at sea. Natural mortality is Mi = 0.40/mo 

 during the first n - /o = 7 mo and lO^t of the population 

 matures as jacks: M2 = 0.06/mo during the last T - i\ = 

 12 mo with or without offshore fishing: and an offshore 

 fishing rate of F = 0.30/mo during the last T - t,. = ^ 

 mo gives a final population {L-ilNu. includmg the in- 

 shore catch in both cases) about 22*"^ of that with F = 0. 



520 



