697 



Abstract — Growth of a temperate reef- 

 associated fish, the purple wrasse (iVo- 

 tolabrus fucicola), was examined 

 from two sites on the east coast of 

 Tasmania by using age- and length- 

 based models. Models based on the 

 von Bertalanffy growth function, in 

 the standard and a reparameterized 

 form, were constructed by using oto- 

 lith-derived age estimates. Growth 

 trajectories from tag-recaptures were 

 used to construct length-based growth 

 models derived from the GROTAG 

 model, in turn a reparameteriza- 

 t ion of the Fabens model. Likeli- 

 hood ratio tests (LRTs) determined 

 the optimal parameterization of the 

 GROTAG model, including estima- 

 tors of individual growth variability, 

 seasonal growth, measurement error, 

 and outliers for each data set. Growth 

 models and parameter estimates were 

 compared by bootstrap confidence 

 intervals, LRTs, and randomization 

 tests and plots of bootstrap param- 

 eter estimates. The relative merit of 

 these methods for comparing models 

 and parameters was evaluated; 

 LRTs combined with bootstrapping 

 and randomization tests provided 

 the most insight into the relation- 

 ships between parameter estimates. 

 Significant differences in growth of 

 purple wrasse were found between 

 sites in both length- and age-based 

 models. A significant difference in 

 the peak growth season was found 

 between sites, and a large differ- 

 ence in growth rate between sexes 

 was found at one site with the use 

 of length-based models. 



Estimates of growth and comparisons 



of growth rates determined from 



length- and age-based models for populations 



of purple wrasse (Notolabrus fucicola) 



Dirk C. Wetsford 



Jeremy M. Lyle 



University of Tasmania 



Tasmanian Aquaculture and Fisheries Institute 



Marine Research Laboratories 



Nubeena Crescent 



Taroona, Tasmania 7053, Australia 



E-mail address (for D. C Welsford) Dirk Welsford g utas edu au 



Manuscript submitted 25 May 2004 

 to the Scientific Editor's Office. 



Manuscript approved for publication 

 10 April 2005 by the Scientific Editor. 



Fish. Bull. 103:697-711 (2005). 



Methods for estimating growth in wild 

 fish stocks derive largely from two 

 sources: 1) age-based models, such 

 as the von Bertalanffy growth func- 

 tion (VBGF), from data for length- 

 at-age. where fish ages are known 

 or estimated from scales, otoliths, 

 and other hard parts; and 2) length- 

 based models, from recapture data 

 from tagged fish to describe a growth 

 trajectory over time at liberty (e.g., 

 Fabens, 1965), or analysis of modal 

 progressions in length-frequency data 

 (e.g., MULTIFAN, Fournier, et al., 

 1990). Many of these models seek to 

 characterize growth of the population 

 in terms of the three standard von 

 Bertalanffy parameters, viz. l x , the 

 theoretical asymptotic mean length; k, 

 the growth rate coefficient; and r , the 

 theoretical age at length zero. 



Despite its wide use in descriptions 

 of fish growth, the standard VBGF is 

 often criticized because the function's 

 parameters may represent unreason- 

 able extrapolations beyond available 

 data and hence lack biological rele- 

 vance (e.g.. Knight, 1968; Roff, 1980; 

 Francis, 1988a; 1988b), estimates of/, 

 produced by standard length- and age- 

 based versions of the model lack math- 

 ematical equivalence (e.g., Francis, 

 1988b; 1992), the statistical properties 

 of the parameters make comparisons 

 between samples difficult (Ratkowsky, 

 1986; Cerrato, 1990; 1991), and indi- 

 vidual variability introduces biases 

 in parameter estimates (Wang, et al., 

 1995; Wang and Thomas, 1995; Wang, 

 1998; Wang and Ellis, 1998). 



These criticisms have led to various 

 reparameterizations of the VBGF (see 

 Ratkowsky, 1986; Cerrato, 1991 for 

 examples). Analyses of reparameter- 

 izations for age-based VBGFs indicate 

 that the inclusion of parameters that 

 are expected lengths-at-age, for age 

 classes drawn from the data set, dra- 

 matically improve the statistical prop- 

 erties of the model (Cerrato. 1991) and 

 also result in parameters that have 

 direct biological interpretation. Repa- 

 rameterizations that fit this criterion 

 include the reparameterization of the 

 Francis (1988b) model for length-at- 

 age data, and GROTAG, a repara- 

 meterization of the Fabens model from 

 tagging data with expected growth 

 rates for length as parameters (Fran- 

 cis. 1988a). GROTAG in particular 

 has the advantage of being readily 

 parameterized to include seasonal 

 growth terms, and, through the ap- 

 plication of a likelihood function, can 

 include estimators of measurement 

 error, individual growth variability, 

 and the proportion of outliers in a 

 data set. It has been used to produce 

 growth estimates for cartilaginous 

 fishes (Francis and Francis, 1992; 

 Francis, 1997; Francis and Mulligan, 

 1998; Simpendorfer, 2000; Simpendor- 

 fer, et al., 2000), bony fishes (Francis, 

 1988b; 1988c; Francis, et al.. 1999), 

 and bivalve mollusks (Cranfield, et 

 al., 1996). Fitting of any growth mod- 

 el with maximum likelihood methods 

 also permits straightforward appli- 

 cation of LRTs in order to compare 

 parameter estimates, and to deter- 



