690 



Abstract.— Most growth models are 

 age-dependent only. Although their 

 modifications can be used to consider, 

 implicitly, the seasonal growth of ani- 

 mals and the effects of tagging, a gen- 

 eral framework is unavailable for ex- 

 plicitly incorporating time and time- 

 dependent factors (i.e. ambient tem- 

 perature and food availability I in age- 

 dependent growth models. In this pa- 

 per, I derived general age- and time-de- 

 [J&ndent growth models for animals and 

 gave a comprehensive list of special 

 cases for age- and time-dependent 

 growth models of von Bertalanffy, lo- 

 gistic, and Gompertz types. Such mod- 

 els 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 incre- 

 ments (e.g. length increments at age 

 and time increments data from tagging 

 studies), the effects of tagging, and the 

 effects of many population character- 

 istics. I also examined their data re- 

 quirements, their independence of the 

 start of time and adjustment of esti- 

 mates of parameters essential for en- 

 suing applications, and concluded that 

 age- and time-dependent growth mod- 

 els are useful for subsequent applica- 

 tions, if and only if they are indepen- 

 dent of the start of time or time-homo- 

 geneous and if estimates of their pa- 

 rameters are properly adjusted. A 

 scheme for such an adjustment is pro- 

 posed and demonstrated. Finally, I used 

 nine special cases of these general mod- 

 els to analyze tagging data on a centro- 

 pomid perch (Lates calcarifer (Bloch)). 

 Such analyses suggested that tagging 

 is antagonistic to fish growth and leads 

 to a shrinkage of size and that L. 

 calcarifer exhibits a strong seasonality 

 in growth, namely its length grows fast- 

 est at the start of autumn, grows less 

 until a full stop in the middle of win- 

 ter, shrinks until the middle of spring, 

 and resumes a positive growth for an- 

 other cvcle. 



General age- and time-dependent 

 growth models for animals 



Yongshun Xiao 



CSIRO Division of Fisheries 



GPO Box 1538, Hobart, Tasmania 7001, Australia 



Present address SARDI Aquatic Sciences Centre 



2 Hamra Avenue 



West Beach, SA 5024, Australia 

 E-mail address xiaoyongshunapi sa govau 



Manuscript accepted 11 August 1998. 

 Fish. Bull. 97:690-701 ( 1999). 



Most growth models relate an 

 animal's size to its age alone, are 

 independent of time, and are meant 

 to be useful at all times. Some fac- 

 tors (e.g. ambient temperature and 

 food availability) that are known to 

 affect the growth of animals vary 

 with time, however. Consequently, 

 time has been incorporated in age- 

 dependent growth models implic- 

 itly, to consider seasonal (Pitcher 

 and Macdonald, 1973: Appeldoorn, 

 1987; Smith and McFarlane, 1990; 

 Pauly et al., 1992; Pauly and Ga- 

 schlitz' ) and biphasic (Soriano et al., 

 1992) growth of animals, and the 

 effects of tagging (Xiao, 1994). Xiao's 

 (1996, equations 3.0-4.2, p. 1676- 

 1677) deterministic extensions of 

 the classical von Bertalanffy ( 1938 ), 

 logistic (Verhulst, 1838), and Gom- 

 pertz (1825) growth models also 

 serve these purposes. Similarly, 

 Wang (1998) derived a set of age- 

 and time-dependent growth models 

 for a special case of the von Bertal- 

 anffy (1938) growth equation and 

 even constructed distribution-free 

 and consistent estimating functions 

 for estimating their parameters. 

 Although these implicit age- and 

 time-dependent growth models can 

 describe a set of data better than 

 age-dependent growth models, a 

 general framework is unavailable 

 for an explicit incorporation of time 

 and time-dependent factors. 



However, an explicit entry of age, 

 time, and time-dependent factors 

 into growth models is essential for 

 studying the effects of many char- 



acteristics of a population (e.g. its 

 age composition, size composition, 

 density, and size- or age-specific 

 mortalities) on the growth of its in- 

 dividuals (Moulton et al., 1992; 

 Walker et al., 1998). Indeed, much 

 insight can be gained by examining 

 density-dependent growth alone. 

 This is because density-dependent 

 growth can be effected by 1) com- 

 pensatory decreases in natural mor- 

 tality, which may result from a de- 

 crease in predation, cannibalism, 

 competition or diseases; 2 ) compen- 

 satory increases in fecundity when 

 food is more readily available or fe- 

 tal mortality decreases; and 3) com- 

 pensatory increases in growth 

 rate when more food induces ear- 

 lier maturity and greater fecundity 

 for each age class (Holden, 1973). 

 For these studies to be feasible, 

 equations for the sizes of individual 

 animals at age a at time ^ in a popu- 

 lation must be coupled with those 

 of their numbers at age a (or size) 

 at time t. 



Just as an increase in dimension 

 can reveal new horizons, an explicit 

 incorporation of time and time-de- 

 pendent factors in age-dependent 

 growth models can be of great use 

 and promise. It also poses interest- 

 ing philosophical and practical 

 problems. Indeed, in general, time- 

 dependency makes age- and time- 



' Pauly. D., and G. Gaschutz 1979. A 

 simple method for fitting oscillating length 

 growth data, with a program for pocket cal- 

 culators. ICES Council Meeting 1979/ 

 G:24, 26 p. 



