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



617 



Jaw size versus test size 



ANCOVA analysis indicated that the slopes of 

 the natural log of test diameter as a function of 

 the natural log of the jaw size are homogenous 

 (P=0.101 ), but that the adjusted means are signifi- 

 cantly different (P=0.017) — urchins in the deeper 

 habitat having larger jaws (Table 1). Therefore, 

 we constructed two allometric equations, one for 

 urchins from the shallow Salt Point site and a 

 second for urchins from the deep Salt Point site. 

 However, the two equations were so similar that 

 they generated identical test diameters for a given 

 jaw size; therefore we pooled our data from the 

 shallow and the deep sites. 



We used a larger independent data set of /i=384 

 from wild and cultured urchins to generate the 

 allometric equation relating jaw size to test size. 

 There is a strong relationship (r^=0.989, df=382) 

 between test diameter (D) and jaw length (J) de- 

 scribed by 



£> = 3.31Jii5 (1) 



where D = test diameter (mm); and 

 J = jaw length (mm). 



Equation 1 predicts that urchins of legal size in 

 northern California (test diameters £89 mm) have 

 jaw lengths alT.S mm. 



A comparison (using the allometric relationship 

 [Eq. 1|) of the measured test size at the time of 

 recapture with the predicted test size revealed no 

 bias in the conversion. Although individual values 

 of measured and predicted test diameters are not 

 identical, the sum of the differences between the 

 two reveals no strong directional bias. The sum 

 of the differences between the measured and 

 predicted values equals 41 mm for 139 urchins, 

 resulting in an average discrepancy of 0.30 mm 

 per urchin. This discrepancy is smaller than the 

 initial error in the measurement of test size (see 

 "Materials and methods" section). 



The von Bertalanffy model 



For many organisms, annual growth rate decreases as size 

 (age) increases. This process is frequently modeled by using 

 the von Bertalanffy equation (von Bertalanffy, 1938) 



J,^i =J,+ J J 1 - e'^) - J,( 1-e-''') 



J = JA1- 



-Kl 



), 



(2) 



(2a) 



which leads to a linear decrease in growth rate as a func- 

 tion of size. We make the point here, that J,^j and J, refer 

 to a discrete data set, whereas J is a smooth, continuous 

 function of ^ 



Our data and, quite possibly, much of the data collected 

 in similar studies, are not well represented by the von 



Bertalanffy equation. How is it then that the deficiencies 

 of this well used equation have not come to light? The 

 answer, not surprisingly, lies in the cancellation of errors 

 within data sets that only incompletely cover the critical 

 growth period. 



Our data ( Figs. IB and 2) show three features of sea ur- 

 chin growth that are inconsistent with the von Bertalanffy 

 model: 1) annual growth, AJ = J,^, -J, , for juveniles that 

 is lower than anticipated from the model; 2) a maximum 

 or plateau in the growth function, AJ = fiJf), for urchins 

 near jaw size J( = ^ mm (test diameter 20 mm); and 3) an 

 asymptotic approach of 4 J to zero (Figs. IB and 2), which 

 may be ascribed to indeterminate growth for adults of all 

 sizes or to dispersion of final adult urchin sizes (Sainsbury, 

 1980). 



There is a good deal of individual variation in growth 

 rate as a function of J^ , which prevents unequivocal selec- 



