FISHERY BULLETIN: VOL. 76, NO. 4 



APPENDIX 



The linear, logistic, and Gompertz functions were expressed in terms of size at release age, size at 

 recapture age, and change in age (time at large) following the rationale presented by Fabens ( 1965) for 

 the von Bertalanffy function (as follows). 



Each individual was of some unknown age (a,„) upon the date marked (t,„) and released. Upon the 

 recapture date it^) the individual was of an unknown older age (a,) so that the difference between the 

 release and recapture date (AO is equivalent to the increase in age (Aa) of that individual: 



/sa = /st = t — t = a —a . (Al) 



r m r m 



That equality can be substituted into the von Bertalanffy function when expressed in terms of the size at 

 recapture (S^) and the age at recapture. Therefore the von Bertalanffy equation 



S^ = S^[l-6exp(-/2a^)] (A2) 



becomes S = S^{1 — b exp{—ka ) exp[—k{Aa)]). (A3) 



The equation, when expressed in terms of the size when marked (S,„ ) with rearrangement, is: 



bexp{-ka ) = 1-(S /S ). (A4) 



That expression is substituted into Equation (A3) to yield the required function: 



S = S -{S -S )e-fe(^«). (A5) 



r "o ^ CO ui ' 



That form can then be employed to estimate the equation parameters k andS from mark-recapture data. 

 The final parameter ib) can be calculated directly by first rearranging terms of the original function: 



S = S {1-be-'''') (A6) 



a °° 



SO that b = [l-{SJSJ]le-'''' (A7) 



where S,, is the size at age a. If the size at birth, i.e., at age 0, is known, then: 



b = 1-iSJSJ (A8) 



where S-^_ is estimated from Equation (A5) and the size at birth (S/, ) is derived from life history studies. 

 That same rationale was applied to the logistic function. The size of a recaptured individual is 

 expressed: 



S^ = S^/[l + b exp{-ka^)] . (A9) 



Since Ur = a + a,n substitution gives: 



S^ - S^I{l + bexp[-k(Aa)]bexpi-ka^)]. (AlO) 



Expressing the logistic equation in terms of the size marked and rearrangement of terms gives: 



bexpi-kaj + [{S^-SJISJ. (All) 



834 



