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



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dependent growth models depend on the start of time 

 and thereby renders them useless, unless the start 

 of time is known. Of course, the start of time (if it 

 did start at all) is unknown ( although some may settle 

 for the Big Bang) and remains a subject of philosophi- 

 cal debate. It is obvious, then, that, for practical pur- 

 poses, workable age- and time-dependent growth 

 models must be independent of the start of time. But, 

 under what conditions are they so? How should esti- 

 mates of their parameters be adjusted to make these 

 models useful for subsequent applications at all 

 times? To answer these questions, both age and time 

 must enter a growth equation, explicitly. 



In this paper, I derive general age- and time-de- 

 pendent growth models for animals and give a com- 

 prehensive list of special cases for age- and time-de- 



pendent von Bertalanffy (1938), logistic (Verhulst, 

 1838), and Gompertz (1825) growth models. Such 

 models explicitly incorporate age, time, and their 

 dependent factors and are useful for modeling growth 

 at age and time (e.g. from length-at-age data), incre- 

 mental growth at age and time increments (e.g. from 

 length increments at age and time increments from 

 tagging studies >, the effects of tagging, and the ef- 

 fects of population characteristics. I also examine their 

 data requirements, their independence of the start of 

 time, and adjustment of estimates of their parameters 

 for ensuing applications. Finally, I use nine special cases 

 of these general models to analyze data on length in- 

 crements at age and time increments from a tagging 

 study of a centropomid perch (Lates calcarifer (Bloch)) 

 in the Northern Territory, Australia. 



General age- and time-dependent growth models 



Just as a formal derivation of age-dependent growth models necessitates use of ordinary differential equa- 

 tions, a formal derivation of age- and time-dependent growth models entails use of partial differential equa- 

 tions. This is because both age and time must be taken into explicit account. Readers unfamiliar with first 

 order partial differential equations may wish to skip immediately to Equations 6-6.3, 10-10.3, and 14-14.3, 

 with little loss of comprehension. 



Now, let 0<y{a,t)<oo^ —oo<aQ<a<°°, 0<tQ<t<oo^ denote the size of an individual of a species of animal of age a at 

 time t, with an arbitrary reference age Qq and an arbitrary reference time t^. Suppose that the change in its 

 size at age a at time t in a small time interval of length At is proportional to a function of y(a,/^ and At, such 

 that 



y(a + ^,t +At)-y(a,t) = K(aJ)f(y(aJ))At, 



where K(a,t) is its instantaneous rate of growth in size at age a and time t, and can capture the effects of age, 

 time, and their dependent factors. Dividing both sides of this equation by At, Taylor series expansion of 

 y(a+Aa,t+At) in the neighbourhood ofia.t) as 



y(a + Aa,t + At) = y(a,f) + 



dv(a,t) , dv(a,t) ^^ _, ,^^ 

 ■Aa+^-, At + 0(At) 



da 



dt 



passing to the limit At^O, assuming that 



and assuming further that 



limO(An^0, 



da _ 

 ~dt~ 



yield a first order partial differential equation 



'dy(a,t) dyia.t) 



da 



dt 



K(a,t)f(y{a,t) 



(1) 



To solve this equation, I use the following approach. Suppose that the solution o{y{a,t) is known. Let a=t+c, 

 or c=a-t, then 



iv^it) - y(t+c,t) t > ti.= max{tQ,aQ-c} 



