FISHERY BULLETIN: VOL. 86, NO. 1 



not specific to this one method. Model sensitivity 

 to a given derived value of M will be the same, 

 regardless of the method used to derive the value. 



General Patterns 



Sensitivity analyses of M in fishery models 

 have evolved through two phases. Earlier studies 

 noted the influence of M on estimates of maxi- 

 mum yield (Fmax) or maximum yield per recruit 

 {(Y/R)jnax\ and on F^ax (the fishing pressure re- 

 quired to produce maximum yield) in Beverton- 

 Holt yield models (Beverton and Holt 1957; Hen- 

 nemuth 1961; Francis 1974; Parks 1977; Bartoo 

 and Coan 1979; Bulgakova and Efimov 1982). 

 More recently, as cohort analyses have become 

 more popular, more attention has been directed 

 toward assessing the influence of M on age- 

 specific estimates of stock sizes (A^, ) and fishing 

 mortalities (F, ) produced by these models (Mur- 

 phy 1965; Pope 1971; Ricker 1971; Agger et al. 

 1973; Doubleday 1976; Ulltang 1977; Doubleday 

 and Beacham 1982; Pope and Shepard 1982; Sims 

 1982a, 1982b, 1984). A few other studies have 

 investigated the effect of M on estimates of maxi- 

 mum sustainable yield (MSY) or total biomass 

 (Francis 1976; Deriso 1982; Beddington and 

 Cooke 1983; Tyler et al. 1985). 



Most of these studies have used a single, invari- 

 ant value for M . Model sensitivity is then as- 

 sessed by comparing model results using some 

 "best" estimate of M, to results using one (or 

 rarely, more) pair(s) of M values some arbitrary 

 percentage above and below the best estimate. 

 Only a few studies exist of the effects of noncon- 

 stant M, where M varies in different groups of 

 fish within a given stock. These include Beverton 

 and Holt's (1957) example of density -dependent 

 M in plaice, and several investigations of age- 

 specific M (Parks 1977; Ulltang 1977; Bartoo and 

 Coan 1979; Sandland 1982; Bulgakova and Efi- 

 mov 1982; Caddy 1984; Tyler et al. 1985). 



No study to date has specifically addressed the 

 problems of estimating values of M for a full fish- 

 ery analysis, leading from cohort analyses (using 

 M to estimate F, , Ni , and recruitment R ) to esti- 

 mates of yield or yield-per- recruit using the same 

 M(s) and R subsequently in the Beverton-Holt 

 formulas. Also, no study to date has addressed the 

 possibility and consequences of differing patterns 

 of variability in M , although it has been sug- 

 gested in one case (Ulltang 1977) that random 

 variations will be unimportant if the rate is con- 

 stant (on average) over the fished ages. 



In general, the earlier analyses with yield mod- 

 els assuming a constant M show that higher esti- 

 mates of M lead to 1) lower estimates of y^ax oi" 

 iY/R)jnax (because fewer survive to be caught), 



2) higher estimates of Fj^ax 'yo^J must fish a bit 

 harder to catch a given amount of those left), and 



3) lower estimates of age at first capture it^.; be- 

 cause it pays to catch them before they die, rather 

 than waiting for them to grow bigger but less 

 abundant). 



Including density-dependence tends to exag- 

 gerate these trends, at least for plaice in the 

 North Sea (Beverton and Holt 1957). Including 

 age-structured M in yield models also affects the 

 estimates, but not necessarily in a straightfor- 

 ward manner. As described below in the section 

 on numeric results, change in model output for a 

 given change in M depends not just on the values 

 chosen for M, but also on those chosen for the 

 other parameters. M is not an independent 

 parameter in these models. 



Analyses with cohort or virtual population 

 models which assume a constant value for M 

 show that in general the effect of increasing M is 

 to increase estimates of N, (because the higher M 

 is, the more fish died in addition to those being 

 caught) and to decrease estimates of F,. The data 

 show only Z , which is the sum of M and F, . As- 

 suming Z has been constant, a decrease in F, 

 requires an increase in M. If Z has been variable, 

 the lower F, may be explained on the basis of 

 higher A^,, a smaller proportion of which (F,) 

 would account for the observed catch. 



The actual effect, particularly on estimates of 

 A^,, is not necessarily that simple. As with yield 

 models, a given change in M does not always 

 produce the same change in model output. The 

 result depends also on values chosen for other 

 parameters; M is not an independent param- 

 eter. 



In cohort analysis the results (estimates of A'^, 

 and F, ) are particularly sensitive to the relative 

 sizes of F and M (i.e., to the exploitation ratio 

 E = F/(F + M)). The effect of assuming an incor- 

 rect value (or series of values) of M tends to build 

 up as the analysis proceeds backward in time. 

 This is because with every time step backward 

 the catch (C) is inflated by the factor M in order 

 to estimate at that time the size of the entire 

 stock, not just the size of the catch. That is 



A^, =A^, + i + C,(F, +M)/F, 

 where F, satisfies the catch equation 



(1) 



30 



