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



693 



yiaj)-- 



- \Kls+a-t,s)ds 

 3'max - [Jmax " V^A) + « " ^ ^() )]« '° ' O - O,, > f 



(6.0) 



If K"(s+a-^sJ=K()=constant in equation 6.0, then 



y(a,t)- 



\yn 



y(a,|,/ -a -i-a„)Je ° 



a - a„ < t 

 a - a,, > t 



(6.1) 



which is the age- and time-dependent von Bertalanffy ( 1938) growth model, or (if o-Oq or t-tQ is interpreted as 

 time at liberty) Fabens (1965) growth model, with parameters K^ and y,„av- 



Since many factors (e.g. ambient water temperature and food availability) vary seasonally, the instanta- 

 neous rate of growth of many animals K(a,t) fluctuates seasonally. If data are available on K(a,t) as a function 

 of these factors, their relationships can be hypothesized. In reality, however, few such data are available. 

 Nonetheless, one can still hypothesize about a temporal trend in K(a,t) and attribute it to the combined effects 

 of all responsible factors. For example, as a first approximation, K(a,t) is seasonal because of seasonal changes in 

 ambient water temperature and food availability and can be approximated by a sine or cosine curve. Thus, if 



K{s + a-t,s) 



Tjr A 2;r 



Ao+^COS — (S-^,,) 



in Equation 6.0, an application of the trigonometric function-difference relation 



gives 



Viaj)- 



sin(a)-sin(^) = 2cos —{a + P) sin —ia-p) 



Vm.« - [ Vnu« -yia„,t-a+ a„ )]e 



AT n 2nl 1 ] 



" n T ° T\ ° 2 "I 



AT n 2ff/ 1 1 



l^.t't-t„\ sin— 1(-(,. cos — (-/j-- (-(„) 



' " n T " t[ ' 2 " ) 



a -Qq <t 



(6.2) 



where KQ,y,„^^,A, T, and t^ are model parameters to be estimated or specified. 



Many species of animals are tagged for a variety of purposes. Tagging can affect the growth of some animals 

 positively, neutrally, or negatively. Indeed, some animals may slow down their growth, cease their growth, or 

 even shrink in size after tagging. A proper functional form of K(a,t) is needed to infer these consequences of 

 tagging. lfK(s+a-t,s)=K,„„,-(K„,a^-K,„Je-'-'^-t*''^o^'" i{a-ao<t and K(s+a-t,s)=K,„„,-(K,„^,-K„„„)e-''-'-'o^/» if a-ao>t 

 in Equation 6.0, then (note that ?-a-^-ao-^l=0, or t-tQ=a-a()) 



