742 



Fishery Bulletin 101(4) 



60 80 



Midpoint of size class (mm) 



Figure 5 



The variances of growth increment estimated for dif- 

 ferent sea urchin size classes by using Equation 8. 



ferent areas and habitats (Table 1). Large standard errors 

 were estimated for the VBGF parameters for sea urchins of 

 the same area and habitat, and large differences occurred 

 in the estimated VBGF parameters between different areas 

 and habitats (Table 1). The approach developed in the pres- 

 ent study considered observations made in both the fishery 

 and scientific studies and provided a systematic way to 

 incorporate the large variation in growth into the estima- 

 tion of a growth-transition matrix, and subsequently into 

 the sea urchin stock assessment. 



It should be noted that the algorithm developed for esti- 

 matmg the variance of growth increments is approximate, 

 and violations of the assumptions used in deriving the 

 algorithm may introduce errors in estimating a growth- 

 transition matrix. For example, large errors in estimat- 

 ing K and L„ will introduce errors in Equation 5, which 

 was derived by assuming small errors for the two growth 

 parameters. Nonnormal distribution of AL with its mean 

 defined by Equation 6 and variance defined by Equation 8 

 will also result in errors in developing a growth-transition 

 matrix. Other factors that may influence the quality of the 

 estimated growth transition matrix include errors in esti- 

 mating CVs for K, L^ estimated from Equations 10 and 11, 

 and omitting high order items in deriving Equation 8. 



Unlike most studies in which the variance for the annual 

 growth increment was assumed to be the same for all size 

 classes (Quinn and Deriso, 1999), our study explicitly sug- 

 gested that the variance for the annual growth increment 

 changed with size (Fig. 4). The differences in the variance 

 were large between size classes, and changed nonlinearly 

 with size. If a constant variance were used for all size 

 classes, the variance in growth increment would be se- 

 verely underestimated for large and small fish. This could 

 introduce large biases in a stock assessment. 



Size-dependent variation might better describe the 

 variation in annual growth increment. Fish in small size 

 classes tend to grow fast, but their growth tends to be more 

 susceptible to environmental variation than adult growth, 

 often resulting in large variation among individuals (Sum- 

 merfelt and Hall, 1987). Fish in large size classes (older 

 fish) have to divert some energy to reproduction but tend 



to have considerable variation in energy allocation strate- 

 gies among individuals. Differences among adults in the 

 ability to grow can also be considerable because of genetics, 

 specific growth patterns during juvenile stages, and differ- 

 ences in energy allocation between growth and matura- 

 tion during younger ages (Nikolskii, 1969). This difference 

 may cause large variations in growth for large and old fish 

 (Summerfelt and Hall, 1987; Chen et al., 1988). Compared 

 with old and young ages, growth rates for medium-size and 

 medium-age fish may be less varied (Nikolskii, 1969). This 

 pattern can be reflected realistically in the estimated varia- 

 tion by using the approach derived in our study. 



Although the choice of L^ was a bit arbitrary in our study, 

 it reflects observations from both the fishery and scientific 

 studies. The largest sea urchins observed in the different 

 scientific studies tend to be smaller than 100 mm, as in- 

 dicated by the estimated L^ values for different areas and 

 habitats (Fig. 1 ). The inability to observe larger sea urchins 

 in scientific studies may result from relatively small sam- 

 ple sizes, the focus of research (small areas), and the large 

 growth variations even in small spatial scales. The data 

 collected from the fishery were more extensive and covered 

 more areas. This, together with the tendency for taking 

 large individuals in the fishery, may suggest that large 

 individuals are more likely to appear in the fishery, rather 

 than in scientific studies. Thus, it may be reasonable to set 

 the expected value of L^ at 100 mm. Also, this higher value 

 corresponds more closely to the upper growth estimates for 

 green sea urchins from the northeast Pacific ( Vadas, 1977). 

 The CV was assumed to be 15% for L_, resulting in the 95% 

 confidence interval of L^ ranging from 70 mm to 130 mm. 

 This range was believed to be a reasonable estimate for the 

 maximum attainable length for green sea urchins on the 

 coast of Maine (Vadas, 1977). 



The approach developed in our study can be readily used 

 to incorporate the VBGF parameters estimated from dif- 

 ferent studies. This can be accomplished by rerunning the 

 regression analyses between K and L^ and between CVs 

 for K and L^. As more information about the growth of 

 sea urchins on the coast of Maine becomes available, the 

 growth transition matrix can be easily updated to reflect 

 the variation identified in newer studies. The flexibility and 

 ability to easily update and incorporate new information 

 makes this approach desirable to the Maine sea urchin 

 fishery, which is currently undergoing large changes in its 

 population size and has only limited growth data. 



The value of 100 mm chosen for L^ was rather arbitrary. 

 However, because we considered the negative correlation 

 between K and L_ in deriving the growth transition ma- 

 trix, a small error in the L^ estimate would not change the 

 growth-transition matrix greatly. In the future, however, 

 we can conduct a systematic sampling of the stock across 

 its geographical range and derive some forms of weighted 

 average size with a composite variance that captures the 

 range of sizes exhibited by the species. Such an approach 

 would provide us with a better estimate of L„. 



The growth-transition matrix developed in our study 

 summarizes the growth patterns of sea urchins along the 

 coast of Maine. It can be updated whenever new growth 

 data become available. It can be readily incorporated into 



