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



621 



(Ricker, 1954). This model yields a maximum in the growth 

 function and an asymptotic approach to zero that charac- 

 terize the data set (Fig. 2D). The empirical fitting param- 

 eters are maximum growth rate constant (fi=3.15/yr) and 

 K= 0.252 /mm, a constant that controls decrease in growth 

 rate as the animal matures. Fitted to the present data set, 

 it gives RSS = 48.7. Initially, Jf is small and AJ = BJ^. At 

 larger J^, annual AJ passes through a maximum as the 

 negative exponential becomes important. Growth, though 

 never zero, will eventually be too small to measure over a 

 one-year period. This model requires an arbitrary specifica- 

 tion of the jaw size at settlement which is not well known 

 and to which the resulting /IJ^ ) curve is quite sensitive. 



Richards The Richards function (time-to-fishery estimate: 

 6.1 yr) incorporates the von Bertalanffy and logistic (as 

 distinct from the "logistic dose-response") models 



fU,) = \j:""{\-e-'') + j;""e 



(7) 



and has an additional "shape parameter" n allowing for an 

 inflection in the curve of J versus t (Richards, 1959; Ebert, 

 1980a) 



J = JJl-6e-Kt)- 



(7a) 



When n = -1, this equation is the von Bertalanffy model, 

 and when /! = 1, it is the logistic model. Minimization of the 

 fitting parameters leads to J^ = 21.2 mm, K = 0.239/yr, and 

 ;; = -0.747 (unitless) with RSS = 59.4. In general, there is 

 another parameter, 6, to be determined: 



b = 



(J. -J,. 



where Jggnig is the jaw size at settlement. In the present 

 case, Jggttig is very small in relation to J^; therefore b is 

 essentially 1. 



Minimization can be difficult owing to the singularity at 

 n = 0. Minimization of the Richards function from small 

 negative values of n and reasonable guesses as to J^ and 

 K leads to a pseudo von Bertalanffy curve with diminish- 

 ing slope as Jf increases (Fig. 2E). The SSE is better than 

 it is for the true von Bertalanffy model (Fig. 2F) because 

 there is one more fitting parameter. Approaching the n =0 

 singularity from positive values of n does not produce the 

 desired logistic curve. Rather the fitted n value becomes 

 very large, leading to the Gompertz case (see also Ebert, 

 1999, chapter 11). The equation with n tending to <=<= does 

 not appear to represent any real case and will not been 

 considered further. 



Von Bertalanffy Currently, the most widely used growth 

 model is the von Bertalanffy or Brody-Bertalanffy model 

 (time-to-fishery estimate: 5.9 yr) 



f{J,) = JJ1- e-^') - J,( 1 - e--^') 



(8) 



(Brody. 1927; von Bertalanffy, 1938), which produces a 

 decreasing linear function of AJ vs. Jf (Walford, 1946) 

 with a slope of-(l - e''^) and 



J(t) = JJl-e-'^'), 



where J^ = the limiting jaw size at long time t. 



(8a) 



The model predicts that the smallest individuals have 

 the fastest growth and yields the shortest time-to-fishery 

 (however, see "Discussion" section). Fitting parameters 

 (4J=5.45 - 0.261 Jf ) for the present data set yield J^ = 

 20.9 mm and K = 0.303/yr, and an RSS = 62.14 mm. 



Growth curves AJ = fit) 



Having A J = fiJf) , one can assume a small (essentially 

 zero) initial size at settlement and determine the size 1, 

 2, 3, . . . years after settlement by a recursive calculation. 

 Six growth curves J = f(t) can be generated from our six 

 models from different functions for AJ = /It/p. We provide 

 a single function 



J = A(B-e-C'}+Dt 



(9) 



encompassing the entire group of six models, which differs 

 only in the fitting constants A, B, C, and D given in Table 

 3. Parameters A and C, with B = 1.0, lead to a first-order 

 growth curve familiar from chemical kinetics (Atkins, 1994). 

 When Bs^l.O, the curve is no longer first order but shows 

 deviation near t = 0, typically a short delay or induction 

 time. Parameter D indicates growth after "final" growth is 

 achieved (indeterminate growth); in our study it is approxi- 

 mated by a small increase of constant slope. This is used to 

 add growth during the indeterminate growth phase. 



Sensitivity to changes in the parameters 



We examined the robustness of each of the parameters in 

 the six models by changing them ±10%, then noting the 

 behavior of the model. Results are given in the last two 

 columns of Table 3. In the first two models, ±10 % variation 

 in the parameters yields a change in the estimate of years 

 to fishery of less than 1 year. Other models gave estimated 

 time-to-enter-the-fishery variations over the range shown. 

 See Schnute (1981) and Ebert et al. (1999) for discussions 

 of parameter sensitivity. 



Discussion 



Our results show that red sea urchins in northern Califor- 

 nia are slow growing animals. The six models we used to 

 generate growth predictions yielded estimates of the time 

 to enter the fishery (89 mm test diameter) averaging 7.2 

 ±1.3 years and a range of estimates from 5.9 to 9.2 years. 

 The robustness of this result is important for its use in fish- 

 ery management. These six growth models, applied to the 

 same data set, give similar growth curves, J versus. <( Eq. 9, 

 Fig. 3) differing mainly at small J. Ranking these diverse 

 models according to goodness of fit with our large data set 

 shows that three models represent the data well, but that 

 the von Bertalanffy model, the most widely used model, 

 describes the data least well (Table 2). 



