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



697 



where iTo'^'ma-v-^- ^' ^^^ ^o ^^® model parameters to be estimated or specified. 



If K(s+a-t,s)=K„^^^-(K,„^,-K„„„)e-''-'+"-"a>'" if a-aQ<t and K<s+a-t,s)-K„^^^-(K„j^^-K„iJe'<'-'a>'o if a-aQ>t in 

 Equation 14.0, then (note that t-a+aQ-to=0, or t-t^)=a-aQ) 



v{a,t) = { 



^max 



,-'>'m.xi»-'Oi-'>"i'm.«-*'iii 



■'°-°"""^ll 



- '''msx "-'0 I-"' A'ma^ -A'm.n i|e-"-'0 "" -II 



a-Oa <t 



(14.3) 



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



Data requirements for parameter estimation 



Equations 6, 10, and 14, and their special cases are 

 segmented functions; they provide flexibility in the 

 analysis of growth data. Thus, by appropriately 

 choosing time t (which is a relative quantity), one 

 can use either segment {a-aQ<t or a-aQ>t) for an in- 

 dividual animal or for a group of individuals, or use 

 both segments io-a^^Kt and a-aQ>t) for a group of in- 

 dividuals. It is, however, more convenient to use only 

 one segment in a single analysis. Indeed, although 

 growth parameters can be estimated by use of either 

 segment of any of Equations 6.1, 6.2, 6.3; 10.1, 10.2, 

 10.3; 14.1, 14.2, 14.3, it is easier to use the segment for 

 a-ag<t, by letting time t start before the animals, whose 

 growth is to be modeled, are born, unless time is al- 

 lowed to take negative values. Use of the other seg- 

 ment, i.e., that for a-aQ>^ gives identical results, but it 

 is tortuous and requires first calculating ^(^^,+0-^^,,^ 



Data requirement for estimation of parameters in 

 a growth model is a function of the generality of that 

 model: the more general it is, the more data it gener- 

 ally requires. Equations 6, 6.0, 10, 10.0, 14, and 14.0 

 generally require knowledge of two ages o„ and a, 

 time t, and two sizes yfa^f-a-i-aj^j anAyia.t) lia-a^^Kt; 

 or knowledge of two times ^^ and t, age a, and two 

 sizes y(^Q+a-^^,,i and yCa,^^ if a-OQ>^ 



By contrast, use of Equations 6.1, 10.1, and 14.1 

 only requires knowledge of the difference between 

 two ages a-Qij, and two sizesy(0|-|/-a-(-a,y and y(a,^i; 

 or of the difference between two times t-t^^. and two 

 sizes y(fo"'"^~'''a'' and jv'(a,^i. Equation 6.1 has been 

 widely used to model tagging data, where a^ or t^^ is 

 interpreted as time at release, a or / as time at re- 

 capture, a-a^^ or t-t^ as time at liberty, y(a^-^t-a+a^J 

 or yit^+a-t, to) as size at release, and y(a,fi as size at 

 recapture. It has also been used extensively to model 

 size at age data (obtained, say, by ageing animals by 

 reading marks in their hard parts I, where o,^ or ^^ is 

 interpreted as age at birth, a or ^ as age, y{a^ t-a+a^y) 

 OT y(tQ+a-t,t^} as size at birth, and y(a.t> as size at 



age. However, it is rare to know an animal's two ages 

 and their corresponding sizes; what are commonly 

 measured are one age and its corresponding size. 

 Consequently, it is common practice to fit Equation 

 6.1 into such size-at-age data to estimate age at birth 

 a,) or t^^, as well as the gi'owth parameters, thereby 

 implicitly assuming, for all animals concerned, that 

 the size at birth yfaf^t-a+a^) or ydQ+a-tJ^) is zero 

 and that the age at birth Oq or t^ is the same. Exactly 

 the same argument applies to Equations 6.2, 6.3, 

 10.1, 10.2, 10.3, 14.1, 14.2, and 14.3. 



Data analysis 



Barramundi L. calcarifer is a protandrous fish found 

 in estuaries and other coastal areas of the Indo-West 

 Pacific (Griffin, 1987). Between August 1977 and 

 June 1980, 4933 barramundi with a body total-length 

 range of about 10-100 cm were captured by a combi- 

 nation of lure fishing, tidal trap, seine, and gill net. 

 They were measured to the nearest centimeter, 

 tagged with the then commonly used, but apparently 

 physically and physiologically damaging, Floy FT-2 

 dart tags for fish >35 cm and FD-67 anchor tags for 

 fish <35 cm, and released in rivers flowing into the 

 Van Diemen Gulf and the Gulf of Carpentaria of 

 northern Australia (Davis and Reid, 1982). Of those 

 tagged, 312 fish of a total length of 23-92 cm 

 (mean=60 cm, SE=13 cm) were recaptured, but only 

 308 are used in the analysis below owing to incom- 

 plete recapture information. The time at liberty 

 ranged from zero to 932 d, with a mean of 219 d 

 (SE=211 d), and the length increment from -2 1 to 35 

 cm, with a mean of 6 cm (SE=8 cm). Negative incre- 

 ments in length are often observed in a tagging ex- 

 periment because tagged animals can shrink in size 

 immediately after tagging. 



Let Qq or ^^ denote time at release, a or t, time at 

 recapture, a-cif^ or t-t^^, time at liberty, y{a^t-a+afy) 

 or y(t^+a-t,tf^), the length of a fish at release, yf'a.^i, 



