FISHERY BULLETIN: VOL. 83, NO. 3 



or two cohorts. The environmental factors which 

 lead to such huge variations in recruitment are as yet 

 imperfectly understood for most species. In order to 

 predict future population sizes, the year-to-year 

 variation could be incorporated into the variances of 

 the effective fecundity terms in the first row of the 

 Leslie matrix. This will lead to enormous (but 

 realistic) confidence limits for predicted future 

 population sizes of such stocks. A more fruitful use of 

 the results of this paper, however, would be to 

 separate recruitment uncertainty from survival 

 uncertainty and to calculate a confidence interval on 

 future population size given recruitment success for 

 a particular cohort. Among harvested species such a 

 conditional forecast could be used to incorporate 

 environmental variation into management recom- 

 mendations. 



In keeping with the fact that applied management 

 may often depend on very elementary quantities, we 

 also calculate a particularly important special 

 statistic of the distribution of projections- the prob- 

 ability that the population will increase under the 

 specified conditions. In the first example, the prob- 

 ability of an increased population was found to be 

 about 0.8. In the second example, the fur seal popula- 

 tion projection, there is a higher probability that the 

 population will increase. Starting with the female 

 population of 100,000, the calculations indicate 99% 

 certainty that the population will have increased to 

 between 126,410 and 171,930 in 5 yr. 



Our simulations of stochastic population growth 

 differ from previous efforts by Boyce (1977) and 

 Slade and Levenson (1982) by allowing all vital rates 

 to vary simultaneously, rather than only one at a 

 time, and by permitting correlations among the vital 

 rates to be specified. In the stochastic growth models 

 of Cohen (1977, 1979a) and Tuljapurkar and Orzack 

 (1980), at each time step the population finds itself in 

 one of several possible environments. Within each 

 environment vital rates are fixed. By contrast, here 

 we model a single variable environment whose condi- 

 tions, as reflected in the population's vital rates at 

 any point in time, are never precisely duplicated. The 

 results of Example 1 verify the results for the mean 

 and variance of future population vectors and show 

 that the mean and variance for total ultimate popula- 

 tion size can be computed from Sykes' formulae. Our 

 results confirm the conclusions of Cohen (1977), Tul- 

 japurkar and Orzack (1980), and Slade and Levenson 

 (1982) that the expected mean value of a stochastic 

 population projection with no serial correlation in 

 vital rates is equivalent to the value projected deter- 

 ministically from mean vital rates. Cohen (1979a, b) 



and Tuljapurkar (1982) address the more general 

 question of serial correlation in vital rates. 



All of the work cited above has been concerned 

 v^dth the state of the population at a time in the 

 future much greater than will generally be useful in 

 management. In this paper we have examined the 

 stochastic behavior of the population at a shorter 

 time in the future. Example 1 has verified that the 

 distribution of ultimate population sizes from 

 stochastic population projections is approximately 

 lognormal (Tuljapurkar and Orzack 1980). From the 

 perspective of fitting the tails of this distribution for 

 a small number of time steps t, however, it appears 

 better to assume that the 1/tth power of the distribu- 

 tion is normally distributed. In either case the distri- 

 bution of ultimate population sizes is skewed (with 

 long tails at the higher values), and the skew 

 becomes more pronounced as t increases. An impor- 

 tant property of such a distribution is that the most 

 likely or modal population value will always be 

 smaller than the mean. How much smaller depends 

 on the number of time steps t, and on the variances 

 and covariances among the survival and fecundity 

 rates. 



An interesting theoretical and practical problem is 

 to find a descriptor of population growth under 

 stochastic conditions which characterizes the skewed 

 distribution of ultimate population size. Cohen 

 (1979a) has proposed two measures of long-run popu- 

 lation growth: A, the ensemble average of realized 

 factors of increase; and pi, the factor of increase need- 

 ed to realize the ensemble average of final population 

 sizes. The first is a measure based on growth rates, 

 while the second is based on population sizes (Cohen 

 1979b). The average realized factor of increase calcu- 

 lated here is analogous to A. If the Leslie matrix of 

 mean vital rates is known, pi is easily calculated as 

 the dominant eigenvalue of that matrix. The prob- 

 lem, as we have seen, is that under stochastic condi- 

 tions the mean of the population sizes is not very in- 

 formative and may, in fact, be misleading. Tul- 

 japurkar (1982) has proposed a growth measure a 

 which leads to the approximate median population 

 size. The two measures proposed here -namely, E 

 [(iV^/A^o)'"] a.ndE[\og{N,/N(f)]-are close approxima- 

 tions to the rate of growth leading to the modal 

 population size. As such, they may loosely be said to 

 describe the most probable trajectory of the popula- 

 tion under stochastic conditions. 



ACKNOWLEDGMENTS 



The work was supported by a National Research 

 Council Fellowship to the first author and by NOAA 



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