PARRACK: ASPECTS OF BROWN SHRIMP GROWTH 



Substitution yields: 



S = S /(I + ((e~''^'''^)(S -S )IS )]. (A12) 



Since S,., S,„, and Aa were all directly observable from mark-recapture data, the logistic equation 

 parameters S^ and k may be estimated from the data set. The remaining equation constant was calculated 

 by rearrangement of terms: 



b = {(SJSJ-l) le-"^"'> . (A13) 



From life history studies the size at birth, Sf,, was determined. Since at birth age is zero (a = 0) the 

 expression can be written: 



b = {SJSJ-1. (A14) 



The Gompertz function was likewise expressed in terms of the mark-recapture data. From Equation 

 (3a), the size at the time of marking is: 



S^ = S. exp[G(l-exp[-^(a^-G.)])] (A15) 



and by substitution becomes 



S^ = S. exp[G -G(exp[-^(a^^^ -c.)]) (exp[-^(Aa)])] . (A16) 



Writing in terms of the size at recapture and rearrangement of terms gives: 



exp(-G exp[^(a^ -a.)]) = S^ /(S. exp G). (A17) 



Substitution yields the expression required to estimate the constants G and^ from the mark-recapture 

 data: 



S^. = [S.exp(G)][S^/(S. exp(G))J^''P[ '^^"^^ (A18) 



where S, was the smallest size observed in those data. The remaining equation constant, a,, was then 

 calculated by writing Equation (3a) in terms of the size at birth: 



S^ = S. exp[G( l-exp[-g(a-a.)])]. (A19) 



Since at birth age is zero (a = 0) the expression can be written as: 



a. = ln(l-[ln(S^/S.)/G]/^) (A20) 



where S , the size at birth, was determined from natural history studies. The linear function: 



S = b + ka (A21) 



a 



requires a much simpler derivation. Expressed in terms of the size at recapture: 



S = b + ka . (A22) 



r r 



Substitution gives: 



S = b + k {t.a+a ) . (A23) 



r ni 



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