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Figure 3 



Growth rate K(a.t)=Kf^+A cosy <t-tj as a function of time t in equa- 

 tions 6.2 (•••!, 10.2 ( — ) and 14.2 ( — ), with estimates of parameters 

 Kg, A. and 1^, in Table 1 for L. calcanfer. and 7'=365.25 d. 



Finally, notice my use of the length at 

 recapture rather than length increment of 

 a tagged individual as the independent 

 variable in relevant models and in the data 

 analysis. This was to avoid error propaga- 

 tion. The variance of the length increment 

 is 



V(L.,~L,) = V(L.,) + V(L,)- 



2p(L,,L,)^V(L,)V(U) , 



where p( L j , L., ) is the correlation coefficient 

 between lengths Lj and L.,. 



A higher value of noise to signal ratio is 

 expected if length increments are used; the 

 higher value of noise to signal ratio helps 

 mask patterns in the data and makes their 

 analysis difficult. 



lengths at release and recapture and at times at lib- 

 erty in the above application. It should be noted, 

 however, that the information from a tagging experi- 

 ment is limited. It might not be possible to estimate 

 all the parameters in the model reliably, as in the case 

 of the L. calcarifer data. Such a limitation also applies 

 to extracting environmental signals from growth data. 



More importantly, all the above models can be used 

 to study the population dynamics of some species of 

 animals, simply by letting yfa,^) denote the number 

 of individuals of a species of animals of age a at time 

 t. Indeed, similar models in studies of population 

 dynamics also lead to partial differential equations 

 (e.g. Nisbet and Gurney, 1982). 



It is interesting that L. calcarifer is a tropical and 

 subtropical species of fish and yet exhibits a strong 

 seasonal growth. For all three models (Equations 6.2, 

 10.2, or 14.2), its growth rate K{a,t) reaches its maxi- 

 mum on 3 or 4 March (i.e. at the start of autumn), 

 slows down to zero on 17 July (i.e. in the middle of 

 winter), reaches its minimum on 2 or 3 September 

 (i.e. at the start of spring), returns to zero on 19 or 

 20 October (i.e. in the middle of spring), and comes 

 back to its maximum rate on 3 or 4 March (i.e. at the 

 start of autumn) (Fig. 3). Thus, its length grows fast- 

 est on 3 or 4 March (i.e. at the start of autumn), grows 

 less until a full stop on 17 July (i.e. in the middle of 

 winter), shrinks until 19 or 20 October (i.e. in the 

 middle of spring), and resumes a positive growth for 

 another cycle. Thus, L. calcarifer does not grow in 

 length for three months in a year, from 17 July (i.e. 

 in the middle of winter) to 19 or 20 October (i.e. in 

 the middle of spring). Such a strong seasonality in 

 growth seems related to the seasonal availability of 

 food and seasonal changes in water temperature. 



Acknowledgments 



I am most grateful to Grant G. Thompson (Alaska 

 Fisheries Science Center, NMFS) and an anonymous 

 referee for their valuable comments on this work that 

 have substantially improved its presentation and for 

 their encouragement. I also wish to thank You-Gan 

 Wang (CSIRO IPP&P Biometrics Unit) for stimulat- 

 ing discussions; Yong Chen (Memorial University), 

 Tony Fowler (SARDI Aquatic Sciences Centre), 

 Charlie Macaskill (School of Mathematics and Sta- 

 tistics, University of Sydney) for commenting on an 

 earlier version of the manuscript, and Xiaoxu Li 

 (SARDI Aquatic Sciences Centre) for producing Fig- 

 ure 1. Roland K. Griffin (Northern Territory Depart- 

 ment of Primary Industry and Fisheries) and Tim L. 

 O. Davis (CSIRO Division of Fisheries) kindly sup- 

 plied the bates calcarifer data. John D. Stevens and 

 Anthony D. M. Sm.ith (CSIRO Division of Fisheries) 

 are thanked for supervising this project. The work 

 was funded by the Australian Fisheries Management 

 Authority. 



Literature cited 



Appeldoorn, R. S. 



1987. Modification of a seasonally oscillating growth func- 

 tion for use with mark-recapture data. J. Cons. Int. 

 Explor Mer 4.3:194-198. 

 Davis, T. L. O., and D. D. Reid. 



1982. Estimates of tag shedding rates for Floy FT-2 dart 

 and FD-67 anchor tags in barramundi, Lates calcarifer 

 (Bloch). Aust. J. Mar Freshwater Res. 33:1113-1117. 

 Fabens, A. T. 



1 965. Properties and fitting of von Bertalanffy growth curves. 

 Growth 29:26.5-289. 



