622 



Fishery Bulletin 101 (3) 



JL(t) 



2a 



ia- 



Jg(») 



20-- 



10-- 



Jt(<) 



2a- 



ia- 



jRi(t) 



2a- 



10-- 



jRk(t) 



20-- 



10-- 



JvBC) 



20-- 



10-- 



10 

 t 



20 



Figure 3 



Jaw size (mm) versus time (y) obtained with Jf^i Jf = J = fiJf) onto J = flt). J^^ = 

 logistic dose response, Jf. = Gaussian, Jj. = Tanaka, J^^ = Ricker, jjj, = Richards, J,,g = 

 von Bertalanffy. Note: A. B, C, and D in Equation 9 characterizing these curves are 

 constants of the model and are not equal in number to the fitting parameters of the 

 curves in Figure 2. 



Our results suggest another important caveat: gaps in 

 the data influence parameter estimates and time-to enter- 

 the-fishery estimates. Many studies of sea urchin growth 

 are based on a limited range of urchin sizes, primarily those 

 of slow growing adults. Growth information from juveniles 

 is difficult to obtain because recapturing tagged juveniles 

 is problematic owing to high movement or mortality rates 

 (or both). If growth information from adult urchins is used 

 exclusively, inappropriate models can be fitted to the data 

 AJ vs. J, (J, al6 mm, Figs. 1 and 2) because there is no 

 information at smaller -7,. In our example, lack of growth 

 information from juveniles produced an overestimate of 

 early growth and, as a consequence, an underestimate of 

 the time to enter the fishery (Fig. lA). Bias toward faster 

 growth rates could lead to more liberal fishing policies 

 and less precautionary management compared with bias 

 toward slower growth rates. This problem has been noted 

 by other researchers (Yamaguchi, 1975; Rowley, 1990; 

 Troynikov and Gorfine, 1998). 



Model selection 



Model selection has always been an important aspect in 

 applying growth modeling. In our case, with a data set from 

 a broad range of size classes, we find that the six models 

 yield similar growth curves, J versus t, indicating that our 

 results of time to enter the fishery are robust to model 

 selection. The unique features of the models show why the 

 estimates are either longer or shorter than the mean we 

 derive from all six models. Our composite model (Eq. 9) 

 allows for the prediction of the most probable growth tra- 

 jectory. The error terms (Table 2) describe dispersion about 

 the most probable trajectory. 



Both the logistic dose-response and the Gaussian func- 

 tions qualitatively fit our sea urchin growth data better 

 than the von Bertalanffy or Richards functions (Fig. 2). 

 A comparison of the sum of the squared residuals (RSS) 

 confirms this observation, ranking these models first and 

 second, respectively. We use RSS as our primary criterion 



