Rogers-Bennett et al.: Modeling growth of Strongylocentrotus franciscanus 



619 



for groups 1 and 3 of the growth; consequently 

 they cancel, in whole or in part. In fact, all re- 

 ported data sets have many more observations 

 falling into group 3 than into group 1, which 

 is either swamped out by group 3 or does not 

 appear at all. This leaves gi'oups 2 and most 

 or all of group 3 to determine the slope of the 

 von Bertalanffy linear function. The average of 

 these two erroneous slopes may or may not be 

 a realistic approximation for urchin growth, 

 depending on the number of measurements in 

 each group. 



Alternative growth models 



Curves that rise to a maximum and then decay 

 asymptotically are very common in the physical 

 sciences and have been successfully modeled 

 for more than a century (e.g. Wien, 1896). Any 

 rising function multiplied into an exponential 

 decay, e.g. (.v) exp(-.x:), models such a curve 

 more or less well. The problem is not in find- 

 ing a model but in selecting from among many 

 possibilities. We compared several models in 

 our study and included a Gaussian model for 

 this data set because it has a small sum of 

 squared residuals and because it has well- 

 defined parameters in the arithmetic mean and 

 standard deviation. Here the arithmetic mean 

 merely serves to fix the position of the maxi- 

 mum on the Jf axis and the standard deviation 

 from f.1 gives the range, in units of J^, of actively 

 growing animals. The model is descriptive only; 

 it does not imply a mechanism of growth. 



We present results from six growth models, 

 the logistic dose-response, Gaussian, Ricker, 

 Tanaka, Richards, and von Bertalanffy mod- 

 els, in order of quality of fit (Fig. 2). Each 

 model is characterized by a different AJ = 

 fiJf), where f(Jf) is a function of annual 

 growth AJ versus size at tagging, J^. Equa- 

 tions 3-8 were input as user-defined functions 

 into a curve-fitting program (TableCurve, 

 Jandel Scientific, now SPSS, Chicago, IL), 

 either as f^Jf) or the equivalent J^ j^-^ - J^. 

 In certain cases, additive parameters that 

 make a negligible contribution to the final fit 

 were dropped. This curve-fitting program uses 

 the Levenburg-Marquardt procedure for find- 

 ing the minimum of the squared sum of devia- 

 tions. During the least-squares minimization, 

 local minima are occasionally found and must 

 be discarded in favor of the global minimum. 

 Matrix inversion is performed by the Gauss- 

 Jordan method (Carnahan et al., 1969). 



We present these models ranked by the fit- 

 ting criterion of the sum of squared residuals, 

 called "Error Sum of Squares" in the output 

 from the TableCurve fitting program, which 

 we have given the abbreviation RSS. Several 



