738 



Fishery Bulletin 101(4) 



ism growing from one size class to another size class in a 

 given unit of time (Sullivan et al., 1990; Sullivan, 1992). 

 In practice, two approaches can be used to incorporate a 

 growth-transition matrix into a stock assessment: one is 

 to incorporate the growth-transition matrix and simul- 

 taneously estimate matrix parameters with parameters 

 that describe other biological processes in the fishery 

 (Sullivan et al., 1990), and the other approach is to esti- 

 mate the growth-transition matrix independent of other 

 stock assessment models (Chen et al., 2000). The former 

 considers covariance among different processes by esti- 

 mating all parameters simultaneously, but makes the 

 analysis more complicated. The latter approach reduces 

 the complexity of modeling but does not consider the 

 covariance of growth and other biological processes. Be- 

 cause size-structured models are often complicated and 

 have many parameters to be estimated, the estimation 

 of a growth-transition matrix outside the main modeling 

 process may be preferable (Chen et al., 2000). In either 

 case, the quality of the growth-transition matrix can 

 greatly influence the quality of the stock assessment. It 

 is thus essential to develop a growth-transition matrix 

 for the Maine sea urchin stock that can capture the 

 variations in growth increments among individuals. 



The information required in estimating a growth- 

 transition matrix includes the mean growth increment 

 in a given unit of time and its associated variation for 

 sea urchins of different sizes. Because growth rates of 

 sea urchins vary with size, growth increments also vary 

 with size, and this variation in growth with size is rarely 

 constant. This has been implicit in the statements of 

 model assumptions in many papers (e.g. Sullivan et al., 

 1990; Sullivan, 1992, Quinn and Deriso, 1999). However, 

 because the variance for growth increments is difficult 

 to estimate, it is often assumed to be constant for organ- 

 isms of different sizes (Quinn and Deriso, 1999). Such 

 an assumption of constant variation in growth incre- 

 ment is rather unrealistic and may introduce biases in 

 estimating a growth-transition matrix. Thus, for the 

 Maine sea urchin we need to develop an approach that 

 can explicitly consider nonconstant variances for gi"owth 

 increments of sea urchins of different sizes. 



Growth of the sea urchin along the Maine coast has 

 not been studied extensively and the data are limited. 

 The data we used for this study were from Vadas et al. 

 (2002) who collected size-at-age data on sea urchins in 

 two habitats (barren and kelp) from three areas along 

 the coast of Maine. 



Methods and materials 



Previous studies have indicated that many environmental 

 variables might influence the growth of the sea urchin 

 (Meidel and Scheibling, 1998; Russell, 1998). Sea urchins 

 in favorable habitats, feeding on preferred seaweeds, grow 

 faster than those feeding on less favorable algae and 

 mussels, and sea urchins on barren grounds grow slower 

 Even in the same habitat, different rates of growth were 

 identified (Vadas, 1977). Previous studies divided the 



LU 

 Q. 



o 



1985 



1990 



1995 



2000 



2005 



80 



60 



40 



20 



I : . ^ - - - ' 1 



1993 1994 1995 1996 1997 1998 1999 2000 2001 



180 



160 



140 



120 



1993 1994 1995 1996 



1997 

 Year 



1998 1999 2000 2001 



Figure 1 



Observed catch measured in metric tons, effort measured in 

 diver-hours, and catch per unit of effort measured in pounds 

 per diver-hour for the sea urchin fishery in management zone 2 

 in Maine. Zone 1 has a similar temporal pattern. 



coast of Maine into three regions, northeast, center, and 

 southwest (Vadas et al., 1997). For each region, sea urchin 

 samples were randomly taken from two habitats, barren 

 and kelp. Size-at-age data were collected in 1997-98 for 

 sea urchins in each habitat and area (Vadas and BeaP). 

 Detailed descriptions about the derivation of size and age 



'' Vadas, R. L., and B. F. Beal. 1999. Temporal and special vari- 

 ability in the relationships between adult size, maturity and 

 fecundity in green sea urchins: the potential use of a roe-yield 

 standard as a conservation tool. Report to the Maine Depart- 

 ment of Marine Resources, Augusta, Maine 04333. 



