FISHERY BULLETIN: VOL. 83, NO. 3 



puter programs for stochastic population projections 

 which can serve as research and management tools. 

 Here we illustrate the utility of these programs with 

 numerical examples, compare our results with recent 

 theoretical analyses, and discuss the implications of 

 these results to the management of living resources. 



METHODS 



Sykes (1969) presented three models for incor- 

 porating stochasticity into population projections. He 

 concluded that the observed variability in human 

 demographic projections was best described by his 

 third model, in which the elements of the Leslie 

 matrix (the effective fecundity rates and the survival 

 rates) are random variables, each with a specified 

 mean and variance, and with specified covariances 

 between them. The model does not allow for serial 

 covariance in vital rates between successive time 

 periods. 



Let Ut be a population vector of co age classes at 

 time t. The stochastic projection model is 



n,^i = {A + A,)n„ t = 0,1,2, .. . 



where i4 is a constant projection matrix of mean vital 

 rates and A, is a matrix of random deviations whose 

 elements have a specified covariance structure 

 {Cov(A,,Aj)} but which are uncorrelated in time. Let 



N, = ^ Ufj be the total population size at time t. It 



is convenient to normalize the projected population 

 to the initial population size and consider the distri- 

 bution of the ratio N,INq. The mean and variance of 

 this ratio are given by 



EiN./N,) = EiN,)INo 



and 



Yar (NtlNo) = VsLriN,)/Nl 



= 11 Gov {nt„ntj)/Nl 

 1=1 j=i 



From Sykes (1969, equations 19 and 20), the mean 

 and variance of the population vector are given by 



E(n,) = {E{n,,)} = A'no 



and 



Var(7i,) = {Cov(%,n(^)} 



t-i 



k=0 Vo=l fl=l 



Cov(A,„, A,„) 



[Gov (nfc„,n^p) -i- Ein,,JE{n,,p)] [ A ' 



't-l-k 



where A ' is the transpose of A and where the curly 

 brackets indicate that the expression inside them is 

 the tth element of the vector or the ijth element of 

 the matrix considered. 



Tuljapurkar and Orzack (1980) predict that for 

 large t, Nf/N^ will be lognormally distributed. The 

 mean and variance of the normally distributed 

 variable log {Nf/No) are calculated from the mean 

 and variance of the lognormally distributed variable 

 NtIN, by 



E[\og(N,/No)] = \og[EiN,/No)] - 



par[log(iV,/i Vo)] 



} 



and 



Var 



[log(^,/^o)] = log|'^^^^^l! 



(Aitchison and Brown 1957). We have found in simu- 

 lations that the distribution of the realized factor of 

 increase (Nf/NQf" is approximately normal. Based 

 on the assumption that the realized factor of increase 

 is normally distributed, the mean and variance of 

 (Nt/NQf" are computed from the mean variance of 

 Nf/NQ by methods given in Appendices 1 and 2. 



Using the formulae of Sykes (1969), the mean and 

 variance of each age class in the future population 

 can be computed analytically. Confidence intervals 

 for the total population size and for the realized fac- 

 tor of increase, and an estimate of the probability 

 that the future population will be larger than the 

 starting population, are computed based on the 

 assumption that either log (NflNo) or (A/^(/iVo)^" is nor- 

 mally distributed. 



We can also simulate the growth of an age-struc- 

 tured population under fluctuating environmental 

 conditions. At each time period, a new set of fecun- 

 dity and survival rates, the elements of the Leslie 

 matrix, are chosen and used to project the popula- 

 tion. Each fecundity and survival rate is a normally 

 distributed random variable with specified mean, 

 variance, and covariance with every other fecundity 

 and survival rate. The projection, starting from the 



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