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



623 



for model selection; however both the AIC information cri- 

 terion (Akaike, 1979) and the Schwartz-Bayesian criterion 

 confirm the ranking of the models (Table 2). Model selection 

 is discussed in detail elsewhere (Bumham and Anderson, 

 1998; Quinn and Deriso, 1999). 



The Gaussian, Tanaka, and Ricker models yield both AJ 

 vs. J, and J,^; vs. J, curves that are concave downward 

 and that visually conform with the data. These models 

 are preferred over the Richards curve, which is concave 

 upward, and the von Bertalanffy function, which is linear. 

 The logistic and Gaussian models fit our data better (Table 

 2 ) than the other models examined. It is not surprising that 

 the logistic and Gaussian models fit our data well, given the 

 maximum or plateau visible in the data set for AJ vs. J,. 

 The Gaussian mean at J, = 4.6 mm, with a standard devia- 

 tion of 5.7 mm shows that a data set with J, > 4.6 -i- 5.7 = 

 10.3 mm represents urchins that are one standard devia- 

 tion (a) above the mean of the growth curve, i. e., 16% of the 

 growing population. Urchins with J, > 17.0 mm, are greater 

 than 2(7 above maximal growth, i. e. 2.5% of the growing 

 population. Therefore, a growth curve fit only to adult ur- 

 chins with J, > 17.0 mm represents only small subset of the 

 total growing population and is not representative of the 

 total population. This demonstrates that data from a lim- 

 ited size range can generate erroneous growth parameters 

 and shorten estimates of time to enter the fishery. 



Variation in growth 



The plateau in growth rate implied by the logistic dose- 

 response curve or the maximum in growth rate for juve- 

 nile urchins well after settlement implied by the Gaussian 

 curve suggests that urchin growth is not at its maximum 

 when sea urchins first settle. It is realistic to imagine that 

 a sea urchin will be at its maximum growth rate sometime 

 after the first year or two. 



In this study we found high individual variation in sea 

 urchin growth. Growth in juveniles was especially variable, 

 despite the fact that the juvenile urchins that were stocked 

 were full siblings. We found no evidence for an increase 

 in dispersion as sea urchins grow larger. Data from many 

 sources suggest individual variation in juvenile growth is 



high. Full sibling red urchins (rt=200) reared in the labora- 

 tory under identical food, temperature, and light regimes 

 varied in test diameter from 4 to 44 mm at one year ( Rogers- 

 Bennett, unpubl. data). Similarly, cultured purple urchins 

 (S. purpuratus). varied from 10 to 30 mm at one year 

 (Pearse and Cameron, 1991) — a trend observed in other 

 commercially cultured marine invertebrates (Beaumont, 

 1994) and fishes (Allendorf et al. 1987). 



Our data contain broad distribution in the region of the 

 small size classes, which is consistent with high individual 

 variation in growth iK). Varying the growth constant, K, 

 e.g. in the Ricker model (cf Sainsbury, 1980), produces dis- 

 persion at the smaller size classes. Our urchin growth data 

 also show a wide array of large sizes as well. Models have 

 been used to examine the impact of this type of individual 

 growth variation. In the von Bertalanffy model, if final size, 

 J^, is varied 10%, this results in a broad distribution of the 

 largest size classes (Botsford et al., 1994). We see a broad 

 distribution in the largest size classes in our data, with 

 animals larger and smaller than the estimated final size 

 J^. Many of the animals smaller than J^ could be at their 

 final size. The biological interpretation of this broad distri- 

 bution at the largest sizes is an open question. There may 

 be a wide array of final sizes because of independent values 

 of /iTand J, (cf Sainsbury, 1980) and each individual hits 

 its own final size abruptly or at an asymptotic approach to 

 final size (cf Beverton, 1992) also known as indeterminate 

 growth (cf Sebens, 1987). 



We suggest that the composite model presented in the 

 present study (Eq. 9) may be useful for a wide array of 

 invertebrates and fishes especially those with a broad ar- 

 ray of final sizes. 



Spatial patterns in growth 



In our study, we found no evidence for spatial patterns in 

 growth. To observe spatial patterns this would have to be 

 detectable above the background of individual variation. 

 Sea urchins from the shallow and deep sites at Salt Point 

 had measurable differences in gut contents, food availabil- 

 ity, and oceanographic conditions; however these did not 

 translate into significant differences in growth between 



