FISHERY BULLETIN: VOL. 83, NO. 3 



RESULTS 



Example 1. 



The results of the stochastic projection by program 

 SPP are presented in Table 3. The second column 

 shows the expected (mean) population vector for 

 each time step. The mean population vector is obtain- 

 ed by projecting w^ith the mean vital rates. The co- 

 variance matrix for the population vector gives, on 

 the diagonal, the variances of each age class and, 

 above the diagonal, the covariances between age 

 classes. 



The calculations using Sykes' formulae concur with 



the results of the Monte Carlo simulation of a sto- 

 chastic population projection, taking the entries of 

 the life table as time-varying random variables (pro- 

 gram SLT). In Tables 4 and 5 the results of the simu- 

 lation are presented. The means and covariances of 

 the vital rates actually achieved on this particular 

 run of program SLT are shown in Table 4 and are 

 close to the specified rates given in Table 1. By com- 

 paring the results in Table 5 with those of time step 6 

 in Table 3, we see that the results of the simulation 

 (SLT) and the analytic solution (SPP) agree closely. 

 The distribution of the ratio of the final population 

 size to the initial population size is shown as a histo- 

 gram in Figure 1 A. The curve is skewed to the right, 



Table 3.— Results of the stochastic projection of the population, given in Table 1, through 6 time steps 

 (program SPP). The columns labeled "95% C.L." give the lower and upper 95% confidence limits for total 

 population size and for the realized factor of increase relative to the initial population. The last column 

 gives the probability P that the final population size will be greater than the initial population size. 



Expected 

 Time population 

 step vector 



Covariance matrix 



Total population 



Lower 95% C.L. 

 Mean Upper 95% C.L. 



Factor of increase 



Lower 95% C.L. 

 Mean Upper 95% C.L. 



230 



252 



250 



259 



264 



271 



277 



215 

 289 



197 

 307 



193 

 335 



187 

 356 



184 

 378 



182 

 400 



1.096 



1.041 



1.039 



1.033 



1.031 



1.029 



0.934 

 1.258 



0.925 

 1.156 



0.934 

 1.134 



0.950 

 1.116 



0.957 

 1.105 



0.961 

 1.096 



0.8764 



0.7545 



0.7856 



0.7808 



0.7910 



0.7990 



Table 4.— Means, variances, and covariances of vital rates achieved 

 during a Monte Carlo projection of the population given in Table 1 (pro- 

 gram SLT). Values were computed on the basis of 30,000 vectors of 

 vital rates. F refers to fecundity rate, P to survival rate, and numbers 

 to age classes. Values in this table should be compared with the 

 "target" values in Table 1. 



Covariance matrix 



Mean 



F^ 



F2 



F3 



PI 



P2 



0.00505 



0.00453 

 0.00502 



0.00100 

 0.00099 

 0.00501 



0.00092 

 0.00091 

 0.00414 

 0.00435 



P3 



F1 0.10016 0.00101 0.00202 0.00201 0.00049 0.00045 0.0000 



0.0000 

 0.0000 

 0.0000 

 0.0000 

 0.0000 





210 



