FISHERY BULLETIN: VOL. 86, NO. 1 



pended on the values chosen for the other 

 parameters. 



Pope and Garrod (1973) present another exam- 

 ple of sensitivity in MSY to values chosen for M. 

 They describe briefly the consequences of using 

 an incorrect constant for M of cod stocks when 

 estimating the F required to generate MSY 

 (Fmsy*- Underestimating M by 507r (assumed 

 M = 0.1 year^^; true M = 0.2 year"M leads to a 

 choice of Fmsy that is 67% too high. Overestimat- 

 ing M by 50% (assumed M = 0.3 year ^ true 

 M - 0.2 year"M underestimated Fmsy by 50%. 



The simulations described above tested the ef- 

 fects of choosing alternative constant values for 

 M. Choosing a vector alternative can also have 

 significant effects; again, the magnitude of the 

 effect depends on the values chosen for other 

 parameters. Beverton and Holt (1957) showed 

 that incorporating density-dependence in M for 

 plaice decreased (y/i?)max by 12%, when holding 

 tc constant at 3.72 years and letting F vary. Con- 

 versely, holding F constant and letting t^, vary 

 decreased {Y/R )max by about 37%. 



Age-dependent values for M were compared 

 with age-constant values by Bartoo and Coan 

 (1979), Bulgakova and Efimov (1982), and Tyler 

 et al. (1985). In their analysis of Atlantic yel- 

 lowfin tuna stocks, Bartoo and Coan found that 

 replacing an assumed constant M of 0.8 year"^ 

 with an age-structured M increasing from 0.1 

 year~^ at age to 1.2 year~^ at age 7, increased 

 (Y/R )max by 17% (from 6 to 7 kg). 



Estimating total yield (7,) rather than {Y/R ) 

 and estimating R as a function of constant versus 

 age-specific M in analysis of catch curves for rela- 

 tively unexploited stocks of Pacific ocean perch 

 and Oregon hake, Bulgakova and Efimov (1982) 

 found that replacing a constant (age-averaged) M 

 with age-variable M tended to increase estimated 

 Yi when fish recruited fairly late to the fishery, 

 but decreased Yf if the fish recruited early. This is 

 because of the interaction between the values as- 

 sumed for M (constant or age-variable) and the 

 value calculated for R from each type of mortality 

 curve. 



Starting with a given value for recruitment at 

 age 6 years (from Efimov 1976), they calculated R 

 twice for ages 4 and 8 years — once with age- 

 averaged M and once with age-specific M. Be- 

 cause in this set of data the age-averaged M was 

 generally higher than the age-specific M at the 

 tested ages of recruitment (ages 4, 6, or 8 years), 

 back-calculations with age-averaged (i.e., con- 

 stant) M predicted fewer recruits than back- 



calculations with age-specific M. With fewer re- 

 cruits and generally higher M , potential yield at 

 later ages obviously must drop. Differences in 

 predicted potential yield ranged from about 

 -30% at ^4 (age-specific estimate lower than age- 

 averaged estimate, when fish were assumed to 

 recruit to the fishery at age 4 years) to +15% at 

 ^6 (age-specific estimate higher) and to +60% at 



Tyler et al. (1985) tested (among other things) 

 the effects of ignoring "true" age-structure in M 

 and using instead a constant value in estimating 

 stock biomass using Deriso's (1980) delay- 

 difference model. They did the tests on catch data 

 generated by Walter's (1969) age-structured sim- 

 ulation model of cod, using three different (input) 

 age structures for M in Walter's model. After gen- 

 erating "catch data" from Walter's model, they 

 analyzed the simulated data set using Deriso's 

 model with constant M (= 0.5 year~^). The age 

 structures tested were 1) mortality increasing 

 and then decreasing with age (Walter's original 

 mortality vector spanning ages 3 to 12 years; age- 

 averaged M = 0.55 year~\ range = 0.33 to 0.70 

 year~^), 2) mortality increasing with age (ages 7 

 to 12 years; average M = 0.5 year~^, range 0.3 to 

 0.7 year"^) and 3) mortality decreasing with age 

 (ages 7 to 12 years; average M = 0.5 year"\ 

 range 0.7 to 0.3 year M. In all three cases Deriso's 

 model with constant M misestimated the "true" 

 biomass generated by Walter's model (with age- 

 structured values for M). The differences were 

 relatively small, however: —13% for the increas- 

 ing and then decreasing series, +19% for the de- 

 creasing series, and +4% for the increasing 

 series. These differences were due to the differ- 

 ences in M, and not the differences in model 

 structure; generating and analyzing biomass 

 with the same constant M in both models led to a 

 discrepancy of only 0.5%. 



By analogy to life history patterns in other 

 adult animals, M (after recruitment into most 

 fished stocks) is more likely to increase with age 

 than to cycle or decrease. By implication, the sim- 

 ulation results from the increasing series are 

 probably most realistic. If so, the effects of ignor- 

 ing age-structure in favor of using a constant M 

 may be relatively small (5 to 20%), at least for the 

 cod stock simulated in this study. But the results 

 obviously depend again not just on correctly 

 choosing the values for M, but on the values cho- 

 sen for the other parameters. In this case, Tyler et 

 al.'s (1985) results imply that age-structure in M 

 can be relatively unimportant, at least when the 



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