696 



Fishery Bulletin 97(3), 1999 



Thus, to obtain equations under the assumption corresponding to Equations 7-10.3, one can either take limits 

 of equations 7-10.3 forp— >0 or solve Equation 12 directly. I chose the latter, without resort to applying the 

 L'Hopital's rule to log-transformed quantities to evaluate these limits. Solution of Equation 12 as an initial 

 value problem with wjt) I ,^i^. =wjt^.) yields 



wAt) = wUf 





(.5+c.s)log^(.v^a^(s+c.s))e 



e 



t>t,. 



(13) 



l?-a-aQ<t, then c<0, -c>0, then t,.=a„-c=t-a+ai)\ if a-a,,S/, then c>0, -c<0, then ^,.=<o- In other words. 



y{a,t) = 



J '" ' ' K{s+a-t,s)\og^[y^^is+a-t,s})e ' 



[fiil^a-li'd^ 



y(t^+a-t,tj 



In J 



■h 



*'og,.(.Vma,<s+a-f,sl)e ' 



a -Qf^ <t 



a - Og > ^ 



(14) 



If y„,Q^(s+a-<,Syl=y,„m.=constant in Equation 14, then 



y(a,n 



I Ki>*a-l.tidt 



^max 



Y(a,,,t -a + a,, ) 



v(^,i +a -t,t„) 



a-a^^ <t 



a -Qn^t 



(14.0) 



If iiL('s+a-^,s7=/Co=constant in Equation 14.0, then 



yictj)- 



V 



»^ max 



via,,,t -a + o„ ) 



vl^u +a -t,t„) 



.^max 



a - a„ < t 



a -an>t 



(14.1) 



which is the age- and time-dependent Gompertz (1825) growth model, with parameters Kfi and_v,„„j-. 

 If 



K{s + a-t,s) = Kq -I- Acos — is-t^) 



in Equation 14.0, then 



y(a,t) = 



^ni'a> 



t-O0'"»^l '-'«--'o-ao' 



AT n 2«f 1 ,1 



a - a,, < t 



a - a,>t 



(14.2) 



