Xiao: General age- and time-dependent growth models for animals 



695 



If /iL'fs+o-?,si=Ko=constant in Equation 10.0, then 



y(a,t)- 



( 1 



-K„ta-a„l 



Lvn,ax yiao^t-a + a^)" j 

 (^ 1 



Up 



-A',,ir-r„ 



Up 



a-a„ <t 



a - a,, > t 



(10.1) 



with parameters Kq and Vnmx- 

 If 



in Equation 10.0, then 



K{s + a-t,s) = K„ + Acos — (s-t^) 



via J) 



( 1 



AT n 2lti \ \ 



-Kr,ia~a,.f sin— (a-Qfi (COS— - t-t^ — ^a-a:.} 



y^ax Uma-x ViO^J-a+a,)' j 



{ 1 



-1/p 



a - a,, < t 



jLx yih+a-t^,f 



-«■„(<-(„ l-^sm^'(-(„lcosy^[(-(,-|l/-(„)) 



-,-1/p 



(10.2) 



where /fo'-^mav^' ^' ^^id t^ are model parameters to be estimated or specified. 



If /frs+o-^s)=/C,„Q,-f/s:„,„,-A'„„Je-'s-'+<'-°o'/« if a-ao<^ and Xfs+a-^s)=/i:,„„,-('ii'„,„,-/r„„Je-'s-'&'/« VL a-a^>t in 

 Equation 10.0, then (note that t-a-^aQ-t^-^^, or ^-<o=a-ao* 



y{a,t) 



1 1 



1 



..nax vXax .V<a,„ ^ " Q + O,, )" J 



- '^'max '"-'■O '-"* Kma^ -A'njm M~' "'"^ "" -W 



Vp 



( 1 



1 



UvLx .v(fo+a-^^o) 



-''■max"-'0'-f"'i'max-'i''iiim 'If' 



I'p 



a-a^^<t 



a-a„>t 



(10.3) 



where K„j^^, K,„i,j,y,„a^, and a are model parameters to be estimated or specified. 



Age- and time-dependent growth models of Gompertz (1825) type 



If 



f(y{a,t)) = lim- 



p^O p 



1- 



y(a.t) 



^v(a,n[log,(3'„,,(a,n)-log,(.y(a,n)]. 



Equations 1 and 2 become, respectively, 

 <9y(a,n dytaj 



da dt 



= /^(a,nv(a,n[log^,(Vmax'o-'»)-log.(v(a,n)] 



dwAt) 



K(t + cJ)w^t)[\ogXy^Jt + c,t))-\ogXu.\{t))]. t > t^ 



(11) 



Notice that Equation 12 can be written as a linear ordinary differential equation for log,,(u',,(^i'*. ie. as 

 d\ogXii\(t)) 



dt 



-K{t + c,t)\QgXwM)) + K(t + c,t)\ogXy^^it + c,t)). 



