694 



Fishery Bulletin 97(3), 1999 



where K,„^,^., K,„i„,y,i,a^., and a are model parameters to be estimated or specified. Clearly, the functional form 

 ofK(a,t) serves its purpose well. This is because /r,„„,<K'„„„, K,„^,^=K,„,„. and K,„^,^>K,„,„ indicate, respectively, 

 positive, no, and negative effects of tagging on the growth of animals; and K„„„<0, K,„i„-0, and K„„„>0 suggest, 

 respectively, a shrinkage, cessation of growth, and a slower growth of tagged animals immediately after tagging. 



Age- and time-dependent growth models of von Bertalanffy (1938) type II 



If 



f(y^a,t}): 



y(a,t) 



y(a,t) 



Equations 1 and 2 become, respectively, 



oV(a,n dyiaj) 



+ - — — = K(a,t) 



via J) 



da dt 



P 



y(aj) 

 V (a,t) 



V V max ^ ' ' 



(7) 



dw \t) ^^ w (t) 



= Kit + c,t}— — 



dt 



wit) 



.y„,ax<^ + c,nj 



t>t.. 



(8) 



Solution of Equation 8 (a Bernoulli's equation) as an initial value problem with wjt) I ,^,^ = wjtj yields 



wit)' 



1 / f K{s + c,s) J 



-i-i/p 



wAty 



r K(s + c 

 J v„„(s + < 



+ I '■ e - ds 



3'max(S + C,s)'' 



t>t 



(9) 



If a-cto<^ then c<0, -c>0, then t^.=a,^-c-=t-a+aii; \fa~ti,)>t, then c>0, -c<0, then t^.=t(,. In other words. 



viaj)- 



y(aQ,t - a + a^ 



yit^+a-tj,,)" 



+ e • ds 



J v^.J.s + a-^s)" 



-i/p 



\^ 



is+a-t.s)ds I 



K(s + a-t,s) 

 + I e ds 



f K(s + a-t,! 

 •' y„,.„(s + a-/, 



- r A'(^+a-^;)^ 



-lip 



a -Oq <t 



a -a„>t 



(10) 



If p=l. Equations 7-10 and their special cases are reduced to age- and time-dependent growth models of 

 logistic (Verhulst, 1838) type. 



Ify,„„^.(.s+a-^,s)=v„m^-=constant in Equation 10, then 



yiaj) =  



1 



yL. y(a„,t-a + a„}" j 



1 



1 



1 



ylx l^max y^t„ + a-t,t„)'' j 



{ki^'o- 



\ Kis+n-l,s) 



I/p 



a -a,, <t 



-lip 



(10.0) 



