FISHERY BULLETIN: VOL. 83, NO. 3 



APPENDIX 1. 



Calculation of the mean and variance of the realized factor 

 of increase, assuming it is normally distributed. 



Let A, the realized factor of increase, be defined as the ^th root of the ratio of the 

 population size at time t to the initial population size: 



nJ 



or 



Let ^ be the mean and v the variance of A. The mean and variance of A' are given by 

 formulae in the Methods section. The problem is to find the mean and variance of A. 

 Let F (/^,v) be a function which gives the ^th moment of A: 



F(m,v) = E(A'). 



Similarly let G{^a,v) be a function which gives the variance of A' in terms of the ^th 

 and 2tth moments of A: 



G(m,v) = F(A2')- [F(A')]2. 



Now assume that A is normally distributed. Appendix 2 gives a recursive algorithm 

 which allows any moment of a normally distributed variate to be calculated. From 

 the tth and 2tth moments of A, the functions F and G can be computed from the 

 equations above. Generally, F and G will be tth and 2tth order polynomials in fu and 

 V. Then, with F and G known, we have a system of two equations 



F(m,v) - Eil^) = 

 G(m,v) - Var (A' ) = 



in two unknowns. Given initial estimates of ^ and u, a two-variable version of 

 Newton's method, or any similar iterative technique, can be used to converge on a 

 simultaneous solution. 



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