Long: A new quantitative model of multiple transitions between discrete developmental stages 
61 
Table t 
Ranking of models of larval development in red and blue king crabs ( Para - 
lithodes camtschaticus and P. platypus ) with the use of Akaike information 
criteria, corrected for small sample sizes (AIC e ). Common parameters (species 
the same) or different parameters (species different) were used in the models. 
K=number of parameters. Likelihood=likelihood of each model relative to all 
the models considered. 
Model 
K 
AIC c 
AAIC c 
Likelihood 
AIC c weights 
Species different 
20 
-5974 
0.00 
1.00 
1.00 
Species the same 
10 
-2249 
3725 
0.00 
0.00 
there was no support for this model 
(Burnham and Anderson, 2002). 
In terms of degree-days, larvae 
of red king crab molted to the ZII, 
ZIII, and ZIV stages earlier than 
larvae of blue king crab, both spe- 
cies molted to the G stage at about 
the same time, and blue king crab 
molted to the Cl stage earlier than 
red king crab (Fig. 2, Table 2). The 
stage transitions of red king crab 
were more rapid than those of blue 
king crab (Fig. 2, Table 2), although 
the precision in the estimates for s 
The mean stage on each day was fitted to the MT 
model (Eq. 2) in R vers. 2.14.0 (R Development Core 
Team, 2011), by using maximum likelihood (mle func- 
tion, stats4 package, vers. 3.1.0) and by assuming a 
normal distribution of errors with a variance struc- 
ture defined by Equation 4. Time was expressed in de- 
gree-days (a measure that accounts for both time and 
temperature) to control for the difference in rearing 
temperatures (e.g., Stevens, 1990; Long et al., 2013b; 
Swiney et al., 2013) and calculated as 
DD m = ZT=o T em Pi x 1 day, ( 5 ) 
where DD m = the degree-days on day m; 
t = the time in days; and 
Tempi = the temperature (in Celsius) on day t. 
Two models were fitted, one in which the parameters 
were common between red and blue king crabs and one 
in which parameters differed between the species. Mod- 
els were compared with the Akaike information crite- 
ria, corrected for small sample sizes (AIC c ): 
A/C c = -log(L) + 2id - I, (6) 
{ 1 — K — 1 ) 
whereL = the likelihood of the model; 
K = the number of parameters in the model; and 
n - the sample size; and 
where the AIC c was used to select the best model 
(Burnham and Anderson, 2002). Normality of the er- 
rors was checked through examination of the standard- 
ized residuals of the best model. 
Results 
The model of larval development with independent 
parameters for red and blue king crabs was the one 
in which red and blue king crabs provided the best fit 
(Table 1) with a coefficient of determination (r 2 , calcu- 
lated with the raw data) for red and blue king crabs of 
0.98 and 0.97, respectively (Fig. 2). The model in which 
they did not differ had a AAIC c of 3700, indicating that 
Figure 2 
Larval development of (A) red king crab ( Paralithodes 
camtschaticus) and (B) blue king crab (P. platypus) 
from the first zoeal stage through the first crab stage. 
Points represent the mean stage as determined each 
day throughout development. Error bars are one stan- 
dard deviation; note that on days when all the larvae 
were at one stage the standard deviation was 0. Lines 
represent the best-fit stage-transition model for each 
species. The larval stages shown are the 4 zoeal stages 
(ZI-ZIV), the glaucothoe stage (G), and the first ben- 
thic crab stage (Cl). r 2 =coefficient of determination. 
