692 



Fishery Bulletin 97(3), 1999 



for a fixed value of cei?. Because vra,^) satisfies Equation 1, we obtain 



dwjt) 



dt 



K(t + c,t)f{w^.(t)). t>t^. 



(2) 



Equations 1 and 2 are too general to be solved analytically. Now, I examined three of their special cases for 

 f(y(a.f)). For each of these special cases. Table 1 describes where to find equations corresponding to various 

 quantities of interest, and Table 2 describes where to find equations corresponding to various special cases of 

 the solution for y(a,t). 



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



If f(y<a.t))=y,„„Ja,t)-y(a,t), Equations 1 and 2 become, respectively, 



'• + ^^ = /C(a,n v„,„(a,n- viaj)] 



da dt '' "'^'^  ' 



K(t + c,t)[y^^Jt + c,t)-u\(t)]. t>t, 



dt 



(3) 



(4) 



Of its many interpretations, y,„^,/a.t) can represent the asymptotic size of an average individual as age ap- 

 proaches infinity. 



Solution of Equation 4 as an initial value problem with wjt> I ,^,, = wJt^J yields 



w,(t) = wAt)e ' + 



ii'(.s + c,s)_v„,„^(s + c,s)e ' ds. t>t^ 



(5) 



If a-o,|<^ then c<0, -c>0, then /,=ao-c=/-a+a„; if a-aQ>t, then c>0, -<?<0, then ^,.=/o- In other words. 



y[a,t): 



y(a^,t-a + a^|)e ' + K(s + a -t,s)y^^^(s + a-t,s)e ds a-a^^<t 



t-a*a„ (6) 



vit,, +a -t,t,,)e '•' 



+ Kis + a-t,s)y^^^Js + a -t,s)e 

 Ify,„„,.Cs+a-<,s,)=y„,„,.=constant in Equation 6, then 



ds. 



a-a„>t 



