Xiao: Use of the original von Bertalanffy growth model to describe the growth of bariamundi, Lates cakanfer 



837 



dL{a,t) _ 1 

 dt ~ p 



A' + 6cos| ^(t't^,) 



K 



■Ha,t)\, (7) 



the solution of which as an initial value problem, with Lia, 



t) I ,^t^^ = L{t„ + a - t, ?„) for a - «„ > t-tf^ or with Lia, <) | ,, ^ ,,^^ 

 = LioQ, a„ + ^ - a ) for a - a,, < t-tg. yields 



the solution of which as an initial value problem, with 

 L(n. ^1 1 ,^, = ^('i, + a - t, /,,) for o - Og > t-tf., or with Lia. 

 t)\^^^ = Lia^.a^i + / - a) for a -Qg < /-^y, yields 



UaJ) 



a'' -i^a'" 'Ua.,J-a + aJ 

 \ K 



T . Ik 



— sin — lo - a, 



, A' bT 



xexp|--m-a„.-- 



xcosl -— - / -(, ( 



T \ 



a-a„<t-t„ 



(8) 



K \K 



+ a-(,/„l exp 1?-',|) ^, , 



"I 'I j} " a - a„ > I - 1„ 



Pk \T " ) [T\ '2 



Similar age-dependent models have been given, inter 

 alia, by Appeldoorn (1987), Pauly et al. (1992), Fontoura 

 and Agostinho (1996), and Xiao (1999). Also, notice that 

 6T(in Appeldoorn (1987) and Pauly et al. (1992), b=C and 

 T=l) is a dimensionless quantity and is useful for inter- 

 specific comparison of the strength of seasonal growth 

 oscillations (Pauly, 1984, 1985, 1990). 



If i3(B-l)-i-l=0 (i.e., B-l=-l//3), /3(I»-l)-i-l = l (i.e., D=l), 

 and 6=0, Equation 5 becomes 



Ua.t)- 



K 



y_„vi> L{a^,t-a + aJ 

 KK 



xexp|-(a-a„)- — sm[-.a-a„) 



T I 1 



 COS I — t - t, ia - a,,) 



K 



Y ^ VII L(t„ +a-t.t„) 

 \ K 



Y vbT . n 

 xexpj -^U-t,,)-- sin —U-tJ 



2nt 1 



xcosl — t -t, — it -t,, 



a - a,, <t - tr, 



(11) 



a-a,>t-t„ 



a reparameterization of the seasonal logistic growth equa- 

 tion (Xiao, 1999). 



If piB-\)+l=\ (i.e., B=l), li(D-\)+l=2 (i.e., D-l=l//3), 

 and 6=0, Equation 5 becomes 



UaJ) 



K \K 



(a„,t - a + a„)\ 



K 



xexp ia - a,,) 



K [k 



a - a,, < t - t„ 



(9) 



a-a>t- f„ 



xexp it - t,,) 



a reparameterization of what is commonly called in the 

 fisheries literature the von Bertalanffy (1938) growth 

 equation. 



UpiB-l}+l=0 (i.e., fi=l) and /3(D-1)+1=2 (i.e. D-l=l//3). 

 Equation 5 becomes 



LiaJ) = 



a -a,, < t -t„ 



(12) 



a-a„>t- t„ 



a reparameterization of the logistic growth equation (Xiao, 

 1999). 



Data and analysis 



Equations 8. 9. 11, and 12 are segmented functions (Xiao. 

 1999); they provide flexibility in analysis of growth data. 

 (10) Thus, by appropriately choosing the value of time t (which 



- is a relative quantity), one can use either segment (o - Oq 

 < t-tff or a - a„ > t-t„) for an individual animal or for a 

 group of individuals, or use both segments (o - a„ < ?-/,, and 



