62 
Fishery Bulletin 114(1) 
Table 2 
Mean estimates, with standard errors in parentheses, of the 1 50 parameter (the time at which 50% of indi- 
viduals have made the transition) and the s parameter (the slope at the transition) for the transitions be- 
tween each stage in the larval development of red and blue king crabs (Paralithodes camtschaticus and P. 
platypus). The stages are the 4 zoeal stages (Zl-IV) and the glaucothoe stage (G). The stage given in the 
first column indicates that the transition is from that stage to the next one (e.g., “G” indicates the tran- 
sition from the glaucothoe stage to the first crab stage). The estimates for 1 50 are given in degree-days. 
Red king crab Blue king crab 
Stage 
*50 
s 
-s/t^o 
^50 
s 
-s/tso 
ZI 
57.0 (SE 0.0002) 
-1743 (SE 2929) 
30.57 
61.5 (SE 0.0004) 
-1434 (SE 58) 
23.31 
ZII 
107.2 (SE 0.6) 
-642 (SE 1208) 
5.99 
122.5 (SE 0.1) 
-129 (SE 1.2) 
1.05 
ZIII 
165.7 (SE 0.1) 
-225 (SE 4) 
1.36 
180.9 (SE 0.3) 
-113 (SE 1.3) 
0.62 
ZIV 
263.9 (SE 0.1) 
-158 (SE 0.1) 
0.60 
265.6 (SE 0.1) 
-101 (SE 0.1) 
0.38 
G 
450.6 (SE 0.2) 
-286 (SE 6 ) 
0.64 
438.9 (SE 0.8) 
-158 (SE 6 . 8 ) 
0.36 
were low for several stages because of the rapidity of 
the transition. 
Discussion 
In this article, I describe a new method for modeling 
biological processes that involve multiple transitions 
between discrete stages. The summing of simple lo- 
gistic equations, which are frequently used for stage- 
transition models, is analogous to commonly used 
time-series analyses that model periodic phenomena 
as a sum of multiple cosine waves (e.g., Linhart and 
Zucchini, 1986). The method provides a concise, math- 
ematical description of such transitional processes, is 
mechanistically sound, and provides easily interpreted 
parameters. The £50 for each transition is used easily to 
determine the length of time between stages, is objec- 
tive and quantitative, allows for explicit comparisons 
among studies, and avoids problems with qualitative 
determinations, such as the time when molts are ob- 
served (e.g., Swingle et al., 2013). The s parameter, 
which is proportional to the rate of change between 
stages, may also be of interest to investigators. 
The MT model provided an excellent fit to the data 
for larval development of red and blue king crabs and 
provided estimates of the standard error (SE) in the 
estimates of parameters that allow comparisons among 
studies. It is worth noting that I used these data as 
example data (the original purpose for rearing the lar- 
vae was to produce crabs for use in other experiments), 
and no conclusions can be drawn about reasons for the 
differences between the red and blue king crabs in this 
experiment because there was no replication and mul- 
tiple factors (e.g., species, stocking density, and tem- 
perature) differed between the tanks. However, the 
estimates for development time can be compared with 
those of other studies, and they agree well with them. 
Kurata (1960) compiled results from a number of 
experiments on larval rearing of red king crab and 
reported a range of 260.4-397.8 degree-days (mean: 
325.0) from hatching to the G stage and a range of 
392.4-514.8 degree-days (mean: 462.8) from hatching to 
the Cl stage. My estimates of 263.9 degree-days (from 
hatching to G) and 450.6 degree-days (from hatching 
to Cl) fall within both ranges from that earlier study. 
Similarly, larvae of red king crab from the Barents Sea 
had stage durations of 66.0, 68.7, 69.3, and 79.1 (284 
total) degree-days for the Z1-Z4 stages (Kovatcheva et 
al., 2006) compared with my estimates of 57.0, 50.2, 
58.5, and 98.2 (263.9 total). Blue king crab have been 
studied less than red king crab, but our estimate of 
265.6 degree-days for hatching through the G stage 
and 438.9 degree-days for hatching through the Cl 
stage are very similar to the 254.4 degree-days (from 
hatching to G) and 439.4 degree-days (from hatching 
through Cl) averages found by Stevens et al. (2008). 
In general, the estimates for the s parameters were 
good, but on a couple of the transitions, particularly 
the first 2 transitions for the red king crab, the es- 
timates had poor precision (Table 2). In these cases, 
there were 0-1 observations of the actual transition, 
and, therefore, the MT model could not precisely esti- 
mate the rapidity of the transitions. Values of s that 
approach infinity are possible given the data; therefore, 
the SE in the parameter estimate is high. If the esti- 
mate of s is of particular interest, then the precision 
of the estimate can be increased by increasing the fre- 
quency of observations. 
Theoretically, there is no limit to the number of 
stages that can be modeled with this approach. I orig- 
inally developed this technique to model embryo de- 
velopment in golden king crab ( Lithodes aequispinus) 
and was able to obtain a good fit for a 13-stage model 
(Long and Van Sant, 2016). However, as the number 
of stages and the number of parameters increase, it 
becomes more difficult for the algorithms to find the 
global minimum in the log-likelihood surface (Bolker, 
2008), and the model fitting becomes more sensitive 
to the starting values for the parameters (Appendix). 
