60 
Fishery Bulletin 1 14(1) 
This function is easily expanded to n stages with n - 1 
transitions: 
Stage = 1 + YJiZl — — — • 
, t ‘ ( 2 ) 
> *50, i , 
This function ranges from 1 to n and increases in a 
stepwise fashion (Fig. IB). Although I have used the 
power function in this study because the parameteriza- 
tion is convenient for interpretation, other sigmoidal 
functions could be substituted in the equation without 
otherwise altering the model. 
Modeling the expected variance employs a similar 
logic. Because the variance is expected to vary con- 
tinuously in this model, it is imperative to model the 
variance as well as the mean in a statistically valid 
manner (Bolker, 2008). When the logistic function (Eq. 
1 ) is used to describe the transition between 2 states, 
a binomial distribution is often assumed (e.g., Long et 
al., 2013a) and variance is given with the following 
equation: 
var = p(l - p), (3) 
where var - the variance; and 
p = the probability of the event occurring. 
This variance structure is appropriate because the 
variance is 0 at a probability of 0 (i.e., none of the 
population have made the transition), maximum at 
a probability of 0.5 (i.e., at t=t 50 , the point at which 
there is transition between the states), and 0 again at 
a probability of 1 (i.e., all of the popidation has made 
the transition; Fig. 1A). 
In the MT model, a pure binomial distribution can- 
not be assumed because the total number of states is 
greater than 2 ; however, a similar variance structure 
can be achieved by treating each of the transitions 
as a separate binomial distribution and summing the 
variances together. Therefore, variance for the expand- 
ed MT model (Eq. 3) can be given with the following 
equation: 
var — Cou(Xj,X k ) + ^''“ l 1 (l- p { ), (4) 
where Cov(X = the covariance between each combi- 
nation of stage transitions (where 
i^k) and p^ is the probability of 
an individual undergoing the i th 
transition. 
Because the covariance between any 2 stage transitions 
will be 0 if the stage transitions happen at different 
times (i.e., if only one of the transitions is occurring 
at the same time), this term is 0 under most circum- 
stances. If, however, there is substantial overlap be- 
tween 2 -stage transitions (i.e., if there are times when 
3 different stages are present at the same time), the 
covariance between those 2 stage transitions should 
be included in the model. This circumstance should be 
rare for the majority of uses for which this model is 
intended. 
Equation 4 has properties similar to those of Equa- 
tion 3 in that the variance is highest at values of t 
that are near one of the transitions but approaches 0 
at values between transitions when all the individu- 
als are expected to be in a single stage (Fig. IB). At 
a given variance, the binomial distribution can be ap- 
proximated by the normal distribution (Bolker, 2008). 
Although such an approximation is not as good at val- 
ues of p close to 0 or 1 , this approximation affects only 
the estimates of the tails of the error distribution and 
not the mean and, therefore, should not affect the fit 
of the model. By assuming normal distributions of er- 
ror with variances that change according to Equation 
4, the model allows more than 2 stages and therefore 
overcomes the 2 -state limit of the binomial distribu- 
tion. This approach allows the variance to change as if 
it were a binomial distribution, providing a good mech- 
anistic match to the data structure. 
Model applied to larval development 
In the winter of 2010, 9 and 11 ovigerous female red 
king crab and blue king crabs, respectively, were col- 
lected in baited commercial pots in the Bering Sea. 
Crabs were identified according to the methods of 
Donaldson and Byersdorfer (2005). Red king crab were 
transported to the Kodiak Fishery Research Center in 
the “live well” of a commercial fishing vessel, and blue 
king crab were transported in coolers by air cargo. In 
the laboratory, the crabs were held in flow-through sea- 
water supplied from Trident Basin, Kodiak, at ambient 
temperature and salinity and fed to excess on a diet of 
chopped frozen fish and squid. 
Larval rearing procedures were similar to those of 
Swingle et al. (2013). In brief, larvae were collected at 
hatching and larvae of red and blue king crabs were 
pooled and each stocked in a separate 2000-L tank. 
Larval red king crab were stocked at 50 larvae/L, the 
amount collected in a single day from 8 females that 
were hatching at the time. Because only 6 female blue 
king crab produced larvae simultaneously, larvae of 
blue king crab were collected over 3 days and were 
stocked at 30 larvae/L. Because of differences in ther- 
mal tolerances (Stoner et al., 2013), the red king crab 
were reared at 8 . 8 °C (standard deviation [SD] 1.0), and 
the blue king crab were reared at 6.5°C (SD 0.6). While 
the larvae were in the zoeal stages, they were fed a 
diet of DC DHA Selco 1 (INVE Aquaculture, Salt Lake 
City) enriched Artemia nauplii. The glaucothoe stage 
is a stage when larvae are not feeding (Abrunhosa and 
Kittaka, 1997a, 1997b); therefore, no food was provided. 
Each day, from stocking to the point when all of the 
larvae had molted to the first crab stage, 10 larvae 
from each species were removed, the developmental 
stage of each was determined, and the mean develop- 
mental stage of the 10 larvae was calculated. 
1 Mention of trade names or commercial firms does not im- 
ply endorsement by the National Marine Fisheries Service, 
NOAA. 
