Restrepo et al : Monte Carlo simulation applied to Xiphias gladius and Cadus morhua 



743 



82 83 84 



Year 



Figure 5 



Distribution of swordfish Xiphias gladius recruitment esti- 

 mates, by year, from the sequential population analyses. Outer 

 lines are from the Monte Carlo simulations and show 95% con- 

 fidence bands (determined as the 2.5th and 97.5th percentiles 

 of the distribution resulting from 1000 simulations). Inner pair 

 of lines shows confidence bands obtained from the informa- 

 tion matrix after a single run of the ADAPT program using 

 actual data. Line with symbols gives median estimate for each 

 year from the simulations. 



equal to the observed proportions and sample size equal 

 to 1% of the annual catch. This model for the uncer- 

 tainty was purely heuristic rather than based on mea- 

 sured variances. 



Abundance (CPUE) indices The 11 available indices 

 from the longline fisheries were also assumed to be 

 lognormally distributed with a coefficient of variation 

 of 10%. We chose a value of 10% as a rough approx- 

 imation for all indices in all years. However, there is 

 no reason why each index could not have a different 

 coefficient of variation for each year depending on the 

 amount of data available. 



Results of swordfish simulations 



The simulations gave rise to 1000 sets of age- and year- 

 specific fishing mortality rates and population sizes. We 

 computed the coefficient of variation of these sets of 

 estimates for each age-year combination (Figs. 4a, b). 

 As expected, the coefficients of variation were highest 

 in the most recent year, 1988. Also, the age groups 

 which form the bulk of the catch (ages 3-5) were the 

 best determined. It is interesting to note that the coef- 

 ficients of variation of fishing mortality rates for ages 

 8 and 9 were consistently lower than those for pre- 

 ceding ages. This is due to the manner in which the 



estimates for ages 8 and 9 were determined: it was 

 assumed that Fgt = Fgj (subscripts refer to age and 

 year, respectively), and these were computed as a 

 weighted average of fishing mortalities for ages 5-7. 

 Thus, the uncertainty in the estimates of fishing mor- 

 tality for the last two age-groups is solely a function 

 of the uncertainties in the estimates for ages 5-7. This 

 underscores the fact that the simulation results are 

 conditional not only on the input-uncertainty distribu- 

 tions but on the formulation of the model being fitted 

 as well. 



The median recruitment (age 1) from the simulations 

 increased over time (Fig. 5). However, the 95% con- 

 fidence bands, defined by the 2.5th and 97.5th percen- 

 tiles of the 1000 estimates, are quite wide. The con- 

 fidence bands provided by the delta method for a single 

 run with the actual data are much narrower than the 

 ones obtained by the Monte Carlo approach. The 

 former confidence bands indicate there is no uncertain- 

 ty in the results for the converged part of the SPA in 

 contrast to the simulation results. This is because the 

 delta method results, based on the information matrix 

 of a single run, are conditional on the natural mortal- 

 ity rate, catch at age, etc., being known exactly where- 

 as the simulation accounts for uncertainty in these 

 inputs. For this reason, we believe the simulation 

 results are more reasonable. 



Note that there appears to be very little interannual 

 recruitment variability in the time-series (Fig. 5). This 

 is probably due to the fact that fish ages were estimated 

 from lengths deterministically by inverting the 

 Gompertz growth equation, and this tends to blur the 

 age-groups. 



The population of fish age 5 and above appears to 

 have declined rather steadily over time while the 

 weighted fishing mortality rate appears to have in- 

 creased (medians. Figs. 6a, b). Here, weighted fishing 

 mortality is defined as the mean of the fishing mortality 

 estimates for ages 5 through 9 + , computed with 

 weights proportional to the estimated population size 

 at age. Again, the confidence bands are very wide. 



It should be noted that for each run the estimates 

 of fishing mortality. Fat, and population size, Nat, are 

 highly correlated not only with each other but also with 

 the value of natural mortality, M, used in the simula- 

 tion run. For this reason, it is appropriate to examine 

 trends in an estimated quantity one run at a time. We 

 computed the ratio of the weighted fishing mortality 

 in a given year t to the weighted F in the base year 

 (taken to be 1978 in this example) for each simulation 

 run (Fig. 7). The distribution of the fishing mortality 

 ratio in 1979 was centered around 1.0; the ratio in 1986, 

 1987, and 1988 was >1.0 in 100% of the runs, thus 

 clearly indicating that fishing mortality has increased. 

 This result is not obvious from examination of Figure 



