Long: A new quantitative model of multiple transitions between discrete developmental stages 
63 
The starting values for the £50 parameters are fairly 
easily estimated by simply examining the data for the 
times when the transitions are occurring; however, the 
s parameters are more difficult to estimate. Therefore, 
the use of an iterative process to determine reasonable 
starting values for these parameters may be helpful. 
When fitting the MT model or any model with a large 
number of parameters, it is highly recommended to fit 
the data under multiple sets of starting parameter val- 
ues, and it is imperative to graph the model and data 
together to ensure that the fit is optimal and realistic. 
In most cases, £ 50 will be a parameter of inter- 
est; however, there are times when the rapidity of 
the transition between stages may be relevant to the 
question posed by an investigator. For example, many 
crab species are cannibalistic, especially immediately 
after molting, when soft crabs are particularly vulner- 
able (e.g., Borisov et al., 2007). Therefore, to minimize 
cannibalism, a hatchery may find it valuable to deter- 
mine under what conditions molting is highly synchro- 
nous among individuals within a tank (because all the 
individuals transition within a short space of time). 
The s parameter, as stated previously, indicates how 
quickly the transition between one stage and another 
occurs. However, s values cannot be compared directly 
with each other without first normalizing them to the 
£50 values. The derivative of Equation 1 evaluated at 
tgo is 
This derivative demonstrates that the slope at £50 is 
dependent on both s and £50. 
In cases where comparisons in the rate of the stage 
transitions are important, it is necessary to calculate 
the ratio of s to £50 to make the comparison. For ex- 
ample, in Figure IB, the first transition, which has an 
s of -30, occurs more rapidly than the third transition, 
which has an s of -50. Interpreting the s values alone 
would indicate that the third transition should be the 
most rapid, and it is not. However, the ratios of s to £50 
for the first and third transitions are -3.0 and -0.8, re- 
spectively, and comparing the absolute values of these 
ratios allows an investigator to make a correct inter- 
pretation of the relative rapidity of transitions (Fig. 
IB). 
The data on larval development provide an example 
of how this ratio can be used to compare the rapidity 
of stage transitions. For both red and blue king crabs, 
the rapidity of the transition between stages decreases 
with each additional stage transition (Fig. 2, Table 2). 
The individual variance in developmental time leads to 
this decrease, which has been previously observed in 
both species (Stevens et ah, 2008; Persselin and Daly, 
2010), and it is reasonable to conclude that individual 
differences in feeding and growth rates would result in 
a larger spread in molting times later in development. 
In addition, the larvae of red king crab consistently 
had faster transitions between stages than did the lar- 
vae of blue king crab (Fig. 2, Table 2). This difference 
is most likely a result of the red king crab having been 
stocked in a single day, compared with the blue king 
crab, which were stocked over 3 days. 
The MT model presented in this article provides a 
flexible and holistic approach for a quantitative descrip- 
tion of complicated biological processes. This model is 
particularly well suited to crustaceans and indeed to 
arthropods in general, given that they develop though 
a series of transitional molts; however, any biological 
process that is divided into discrete stages (e.g., Kim- 
mel et al., 1995) can be modeled with this technique. 
Treating the process with a single model affords in- 
vestigators the ability to compare treatments by using 
model selection techniques (Burnham and Anderson, 
2002), while avoiding the increase in the type-I error 
rate inherent in analyzing a large number of response 
variables with univariate statistics (Quinn and Ke- 
ough, 2002). 
Metadata for the data produced in the study de- 
scribed in this article are available at InPort (website). 
Acknowledgments 
I thank the staff, particularly K. Swiney, A. Emley, S. 
Van Sant, R. Fields, and S. Dresdow, of the seawater 
laboratory complex of the Kodiak Laboratory, NOAA 
Alaska Fisheries Science Center, for assistance in rear- 
ing the larvae in this paper. Previous versions of this 
paper were improved by comments from D. Urban, J. 
Long, and R. Foy, and 5 anonymous reviewers. 
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