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Appendix 



Some combinatoric terms 



In order to incorporate Equations 22 or 23 into the 

 model, it is helpful to define a few concepts taken from 

 combinatorial theory (the notation used here follows 

 Riordan |1980]i. First, the number of permutations of 

 n objects taken k at a time is given by 



with {n ) k defined as zero for k<0 or k>n. 



The number of combinations (i.e., permutations with- 

 out regard to order) of /; objects taken k at a time is 

 given by the binomial coefficient 









kUn-k)! 



(A2) 



with ( I) defined as zero for k<0 or k>n. 



The number of ways in which an n -element set can 

 be partitioned into k subsets is given by Stirling num- 

 bers of the second kind, written 



S(iuk) 



■(s)^"(a)*- 



(A3) 



The coefficients of the polynomial expansion of Lr)„ 

 are given by Stirling numbers of the first kind, written 



£* //i-l+Ax /2n-k \ 



sin, k)= L (-D 1 / 11 1 S(n-k+\, A). iA4 



\n-k+kJ\n-k-\J 



Polynomial solution for a generalized 

 growth function 



K=0 Beginning with the simpler case where K=0 

 (Equation 23), stock biomass per recruit can be 

 written 



BPR(F) = [ w r ( a ~ ar Y e- M,i + F "°-°rda 

 J ° \a r -a n J 



(A5) 



(w r \ y (n) l! K" k 



In general, stock biomass in any simple dynamic 

 pool model with Cushing recruitment can be written 

 as the following function of biomass per recruit: 



m , )= (EBE^ 



(A6) 



Substituting Appendix Equation 5 (A5) into Appen- 

 dix Equation 6 (A6), multiplying throttgh by MF, and 

 differentiating gives the following expression: 



dY(F) 

 dF 



\a-q)(l+F)"* 2 ) 



( X [(/! )kK" k a-kF')a+F' >" - *] - 



q X [(n) k K" k (1 + F')« + '-"]). 



(A7i 



(n) k 



[n-k)\ 



(All 



