Long: A new quantitative model of multiple transitions between discrete developmental stages 
59 
Figure 1 
Relationships of stage transitions and predicted variance to 
time, measured in arbitrary units, from the (A) single-stage 
transition model (time at which 50% of the individuals have 
made the transition [^so,l] = 15, slope parameter at the transition 
that describes the abruptness of the transition [si]= -8) and the 
(B) Model of multiple transitions between stages with 4 stages 
and 3 transitions (£504=10, £50,2=25, £50,3=40, si=-30, S2=-20, 
s 3 =-50). 
Swiney, 2007). The species have a similar size-fecundi- 
ty relationship (Herter et al., 2011; Swiney et al., 2012; 
Swiney and Long, 2015), but blue king crab reproduce 
only once every 2 years whereas red king crab produce 
a clutch annually (Jensen and Armstrong, 1989). 
The larvae of both species are planktonic for 2-3 
months before the glaucothoe settle into benthic habi- 
tats (Shirley and Shirley, 1989; Stevens et al., 2008). 
Because newly settled king crabs are highly vulnerable 
to predators (Stevens and Swiney, 2005), the glauco- 
thoe typically remain planktonic until they find a com- 
plex habitat suitable for settlement (Stevens and Kitta- 
ka, 1998; Stevens, 2003; Tapella et al., 2009). Red king 
(Pirtle and Stoner, 2010) and blue king (Daly and Long, 
2014a) crabs are vulnerable to predation from both 
conspecifics (Stoner et al., 2010; Daly and Long, 2014b) 
and other predators (Daly et al., 2013), but predation is 
reduced in complex habitats such as cobble, shell hash, 
and macroalgae (Stoner, 2009; Long et al., 2012; Long 
and Whitefleet-Smith, 2013). In red king crab, individ- 
uals transition into podding behavior as they 
grow too large to be cryptic (Powell and Nick- 
erson, 1965; Dew, 1990), but nothing is known 
about blue king crab at this age. Both species 
mature at a carapace length of about 90 mm 
(Somerton and Macintosh, 1983; Blau, 1989), 
although size at maturity varies among popu- 
lations (Pengilly et al., 2002). 
In this study, I present a simple model that 
describes such stepwise processes. It is flexible 
enough to be expanded to multiple stages and 
allows for explicit comparisons among species 
or treatments in a holistic way. Throughout 
this article I refer to this model of multiple 
transitions between stages as the MT (multiple 
transitions) model. To illustrate this model, I 
fitted larval development data from laboratory- 
reared red and blue king crabs. Larvae of both 
species pass through 4 zoeal stages (ZI-ZIV) 
and 1 glaucothoe stage (G) before they meta- 
morphose to the first benthic crab stage (Cl) 
(Sato and Tanaka, 1949; Hoffman, 1968); there- 
fore, these larval stages provide an opportunity 
to explore the utility of this model. 
Materials and methods 
Description of the multiple transitions model 
The basis of the MT model is the logistic fam- 
ily of equations, which are frequently used to 
describe a transition from one stage to another, 
for example, from life to death as a function of 
time (e.g., Long et al., 2008) or from immature 
to mature as a function of size (e.g., Somerton, 
1980). For the power-function version used to 
describe the transition between 2 stages of de- 
velopment, the equation would be parameter- 
ized as follows: 
Stage = 1 H — , (1) 
' \*1 
, * 50,1 , 
where t = the independent variable (time); 
£504 = the time at which 50% of the individuals 
have made the transition; and 
si = the slope parameter at the transition that 
describes the abruptness of the transition. 
This equation could be simplified as Stage=l+p 1, where 
pi is the probability of an individual having undergone 
the transition. Larger absolute values of s indicate a 
more rapid transition between states. The lower and 
upper limits for this function are 1 and 2, respectively. 
This equation has the desired properties of the func- 
tion being 1 at values of t far below £50,1, rising sigmoi- 
dally to 1.5 at £504, and rising toward an asymptote of 
2 as £ increases above £504, with the amount of time 
both stages are present being a function of si (Fig. 1A). 
