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Fishery Bulletin 101 (3) 



other fitting criteria are also given in Table 2. We used both 

 the AIC information criterion, A/C = KlniRSS)- K\nK+2m, 

 and the Schwartz-Bayesian criterion BIC = KlniRSS)- 

 (K-m )\n{K), where K is the number of data points, and 

 m is the number of parameters in the fitting equation 

 (Akaike, 1979). These tests of curve-fitting quality were 

 used to bring out any substantive difference between the 

 2-parameter and 3-parameter equations. The results show 

 that differences between the 2- and 3-parameter cases are 

 swamped out by the data, as might have been anticipated 

 from the disparity between the number of points (iiL'=211) 

 and the number of parameters. For the present data set, in 

 applying either of these criteria, one is essentially seeking 

 the smallest RSS. 



Individual models 



Logistic dose-response The logistic dose-response curve 

 (time-to-fishery estimate: 6.6 yr) 



f\Ji^=a/a+{J,/br) 



(3) 



(Hastings, 1997) fits our data the best of all the models 

 examined here. The curve fit (a=4.4, 6=12.9, c=6.8) with 

 RSS = 31.9, is a sigmoidal transition function (TableCurve 

 Windows, vers. 1.0, Jandel Scientific Corp., SPSS, Chicago, 

 IL). There is a transition between a fast-growing group 

 of sea urchins, which maintain a constant growth rate 

 {f{Ji)=annua\ AJbAA mm/yr up to about J(=8 mm), to 

 sea urchins growing .slowly at a rate that diminishes as 

 Jf increases beyond 16 mm. The inflection point is at J^ ^ 

 13 mm. There is considerable individual variation in both 

 data groups, but more in the fast-growing group than in 

 the larger slow-growing group. 



Gaussian The Gaussian function (time-to-fishery esti- 

 mate: 6.9 yr), although rarely if ever used in this context. 



f(J,) = Ae 



-{j,-nyil<r 



(4) 



fits the data about as well (RSS=32.8) as the logistic dose- 

 response model. It is a three-parameter model (Rogers, 

 1983) for which the parameters are maximum growth 

 (A=4.6 mm/yr), size at maximum growth (/i=5.8 mm), 



and standard deviation (a=5.6mm) of the distribution of 

 maximum growth versus size. Applied to the present data 

 set, the Gaussian function yields an initial annual growth 

 rate 4J=2.8 mm/jrr, and a time of entry into the fishery of 

 about 7.0 yr. A strength of the Gaussian model aside from 

 its good fit is that it provides a plausible growth model with 

 maximum AJ, not at settlement, but at a jaw size about 

 one third that of adults, and that the parameters are well 

 defined. In this model, annual growth is randomly distrib- 

 uted, according to jaw size, about the maximum in AJ. 



Tanaka The Tanaka equation (time-to-fishery estimate: 



8.2 yr) 



f(J,) = ^=\xh.G + 2.1c- + fa\ + d- J,. 



(5) 



where 



G = --^4-/ and E=exp(^(7, -</)) , 



can be obtained from its differential form (Tanaka, 

 Ebert, 1999) 



dJ 1 



dt ^f(t-c)-+a 



1982; 



(5a) 



by using a standard integral (Barrante, 1998). The param- 

 eters are a=0.0330, o'=15.7, and /■=0.0773. 



The Tanaka model shows an asymptotic approach to zero 

 growth at large Jf, allows for an early lag in growth, and 

 does not force a maximum growth rate on juvenile urchins. 

 The fit to our data set (RSS=39.6) requires three param- 

 eters (the parameter c in Eq. 5a drops out). This function 

 has been used to model red sea urchin growth (Ebert and 

 Russell, 1993). 



Ricker The Ricker function (time-to-fishery estimate: 9.2 

 yr) for population growth (Hastings, 1997) translated into 

 terms of urchin growth is 



flJ) = BJ^ e 



AW, 



(6) 



