VETTER: NATURAL MORTALITY IN FISH STOCKS 



assumed constant is evenly bracketed by the 

 "true" age-structure in M . 



Further simulations by Tyler et al. (1985) using 

 a wide range of constant values for M (0.4 to 1.4 

 year M and the growth rate parameter rho 

 (mean Ford growth coefficient for the fishable 

 stock; 0.46 to 1.6) showed that incorrect guesses of 

 M (and rho) could produce errors up to 1,000^^ in 

 estimated biomass. More realistic ranges for the 

 two parameters (0.4 to 0.8 year" ^ forM, 0.6 to 1:2 

 for rho), extending about 509r above and below 

 the "true" values for these parameters, induced 

 much lower error in biomass estimates (about the 

 same order of magnitude, 50 to 100% below and 

 above the "true" biomass). As before, changes 

 (errors) of a given amount in M (expressed as 

 fraction or percentage of the original value) ap- 

 pear to produce about the same amount of change 

 (expressed as percent of original value) in simple 

 estimates of yield, depending on the conditions of 

 other parameters in the model. 



Chatwin (1958) compared estimates of Yj^ax 

 from lingcod populations. Rather than compare 

 constant and age-variable values for M, he as- 

 sumed several different values for an average 

 (constant) M in adults, but assumed that M in- 

 creased from the assumed average for adults to 

 higher values in both juveniles and senescent 

 fish. He reports no quantitative results but states, 

 as found above, that increasing the average M, 

 for a given F, considerably decreased l^max' that 

 decreasing M increased Yrna-x' snd that size at 

 first capture changed relatively little with those 

 changes in M. 



These comparisons between age-structured 

 versus constant M, or between different constants 

 have demonstrated that effects on results can be 

 large for some combinations of parameters yet 

 small for others. Alternative choices drawn from 

 apparently realistic parameter values lead to rel- 

 atively small differences in estimates of M. 

 Specific amounts of change depend strongly not 

 only on the values chosen for M, but also on the 

 value of M relative to values chosen for the other 

 interacting parameters in the yield models. For 

 most choices of parameter values, sensitivity of 

 output is roughly equal to perturbation of input. 



Cohort Analyses 



Effects of interactions between changes in M 

 and values chosen for other parameters is even 

 more obvious in stock reconstruction analyses 

 (e.g., cohort analysis and virtual population anal- 



ysis (VPA)). These analyses are used to "recon- 

 struct" estimates of stock abundance during pre- 

 vious years, based on catch data and assumptions 

 about the value(s) of M during those previous 

 years. Studies of sensitivity to M in Beverton- 

 Holt types of yield or biomass assessments were 

 usually empirical, based on analyses of catch data 

 from specific fisheries. Studies of sensitivity to M 

 in VPA and cohort analysis include both theoret- 

 ical and empirical studies; i.e., simulations using 

 totally contrived data sets (e.g.. Agger et al. 

 1973), analyses of specific data sets (e.g.. Pope 

 1971; Doubleday and Beacham 1982) and combi- 

 nations of analytical evaluations and analysis of 

 specific data sets (e.g., Doubleday 1976; Ulltang 

 1977; Sims 1982a, 1982b, 1984). 



Simple analyses of sensitivity to M, in which M 

 is varied but all else is held constant, include 

 1) Pope's (1972) analysis of Atlantic yellowfin 

 tuna, in which he found that replacing constant 

 M with age-structured M (higher Ms for older 

 fish) produced lower estimates for fishing mortal- 

 ity {F, ) in the later ages, but had little effect on 

 estimates for the younger ages, and 2) Doubleday 

 and Beacham's (1982) statement that I07c error 

 in constant M translated into 9 to 149f error in 

 estimates of i? (at age 3) for cod in the Gulf of St. 

 Lawrence. 



Somewhat more complicated analyses are pre- 

 sented by Ulltang ( 1977 ) and Sims ( 1982a, 1982b, 

 1984). Ulltang evaluated the effects on model pre- 

 dictions of F, and A'^, , of several types of variation 

 in M. These included no variation (uniformly con- 

 stant M), M constant within years but varying 

 randomly between years, M varying with age, 

 and M varying with season. Sims evaluated the 

 effects of choosing various constants for M on esti- 

 mates of A'^,, and derived an analytical expression 

 relating variance in M to expected variance in 

 estimates of abundance. 



In Ulltang's simulations, increasing (decreas- 

 ing) a constant M by 50*^ (from 0.2) decreased 

 (increased) F by about 207c ("true" F's ranging 

 from 0.4 to 0.8). Creating a data set with M vary- 

 ing randomly from one year to the next, then an- 

 alyzing those data with an assumed constant M, 

 Ulltang (1977) found that the Z calculated from 

 the constant-M model was on average the same as 

 the "true" Z from the random-M model. He con- 

 cluded that random fluctuations in M will cancel 

 out during analysis and so can be ignored. Ull- 

 tang assessed the influence of age-dependent M 

 compared with constant M by generating a catch 

 curve with age-variable M (decreasing curvilin- 



33 



