835 



Abstract— In the original von Berta- 

 lanfty gi'owth equation, the rate of 

 change in body mass of an individual 

 is assumed to result from two oppos- 

 ing biological processes: anabolism and 

 catabolism. Because this differential 

 equation cannot be solved analytically, 

 some of its analytically solvable special 

 cases are commonly used, despite their 

 restrictive assumptions. In this study, 

 I used a generalization of the original 

 von Bertalanffy growth equation and 

 some of its commonly used special cases 

 to estimate parameters from a set of 

 tagging data on times at liberty, lengths 

 at release, and lengths at recapture 

 of a centropomid perch iLates calcari- 

 fer) and provide a method for deter- 

 mining the anabolic and catabolic rates 

 of animals in their natural environ- 

 ment. Fitting the original von Berta- 

 lanffy growth equation to the tagging 

 data suggests that a 1% increase in 

 body mass of the fish corresponds to 

 a 0.8721'7r increase in anabolic rate 

 and a 1.0357% increase in catabolic 

 rate. Alternatively, L. calcarifer may be 

 interpreted as exhibiting a strong sea- 

 sonality in growth: it grows fastest in 

 length at the start of autumn, grows 

 less until a full stop in the middle 

 of winter, shrinks until the middle of 

 spring, and then resumes a positive 

 growth for another cycle. Consequently, 

 it is unnecessary to use the analyti- 

 cally solvable special cases of the origi- 

 nal von Bertalanffy growth equation in 

 data analysis, unless their assumptions 

 are validated. I also explain why Pau- 

 ly's index of growth performance is ade- 

 quate and propose an index of catabolic 

 performance. 



Use of the original von Bertalanffy growth model 

 to describe the growth of barramundi, 

 Lates calcarifer (Bloch) 



Yongshun Xiao 



SARDI Aquatic Sciences Centre 



2 Hamra Avenue 



West Beach 



Southern Australia 5024, Australia 



E-mail address yongshun. xiaoia'bigpond com 



Manuscript accepted 15 March 2000. 

 Fish.Bull. 98:835-841 (2000). 



Information on the growth of animals is 

 important for studying their population 

 dynamics, physiology, and biochemistry 

 (Peters, 1983; Calder, 1984; Schmidt- 

 Nielsen, 1984; Reiss, 1989; Xiao, 1998). 

 Many empirical inodels have been 

 developed to describe the growth of 

 animals macroscopically, including the 

 Gompertz (1825) and logistic growth 

 models (Verhulst, 1838). By contrast, 

 von Bertalanffy ( 1938 ) proposed a some- 

 what mechanistic growth model for 

 body mass W(a)>0 of an individual of 

 age a, of the form 



dWia)/da = AW(a )« - CWia )^, 



where A, S, C and Z> = positive biolog- 

 ical constants; 

 AW(a)^ = the rate of ana- 

 bolism (build- 

 ing up of body 

 mass! at age a; 

 and 

 CWia)" = the rate of cat- 

 abolism (break- 

 ing down of body 

 mass) at age a. 



Thus, in this model, the rate of change 

 in body mass of an individual dWfa)/c?a 

 at age a is assumed to result from two 

 opposing biological processes (anabo- 

 lism and catabolism). Although the 

 underlying mechanisms may be too 

 complicated for dW(a)/da to be approx- 

 imated or even interpreted as such, this 

 differential equation has opened up a 

 line of thought for integrating the mac- 

 roscopic growth of animals with certain 

 physiological and biochemical processes 

 (Pauly, 1981). Also, it is fairly general, 

 includes almost all previous determin- 

 istic growth models as its special cases. 



and forms a basis for identifying the 

 "right" growth models from amongst all 

 its special cases. Consequently, some 

 work has been done to estimate para- 

 meters A, B, C, and D to determine 

 the anabolic and catabolic rates of fish 

 (Ursin, 1967; Pauly 1981). 



However, because the differential 

 equation cannot be solved analytically, 

 its analytically solvable special cases 

 are so commonly used that one simple 

 special case has become known in the 

 fisheries literature as the von Berta- 

 lanffy growth equation (Xiao, 1996). 

 Nonetheless, assumptions for its vari- 

 ous analytically solvable special cases 

 can be very restrictive. Indeed, although 

 assumed to take a value of % in that 

 simple special case (Pauly, 1981), con- 

 stant B can take any value from % to 

 'Vt, because B is often assumed to satisfy 

 /3(B-1)-h1=0 and because B in Equation 

 2 below is known to take any value from 

 2V2 to 3V2. It is also possible that cat- 

 abolic rate is not proportional to body 

 mass (i.e. D^l). In any case, it is best 

 not to make any assumptions about the 

 values of A, B, C. and D. 



Like most growth models, the von 

 Bertalanffy (1938) growth equation is 

 age-dependent. Although it can be mod- 

 ified to consider, implicitly, the sea- 

 sonal growth of animals and the effects 

 of tagging, a general framework was 

 not available for explicitly incorpo- 

 rating time and time-dependent fac- 

 tors (i.e. ambient temperature and food 

 availability) in age-dependent growth 

 models. This prompted Xiao (1999) to 

 derive general age- and time-dependent 

 growth models for animals and to give 

 a comprehensive list of their commonly 

 used special cases. Such models explic- 

 itly incorporate age, time, and their de- 



