58 
NOAA 
National Marine 
Fisheries Service 
Abstract— Many biological processes 
are described in terms of transitions 
between discrete stages. For ex- 
ample, crustacean larvae generally 
pass through a number of stages 
that are punctuated by transitional 
molting events. On the other hand, 
some continuous processes, such as 
embryo development, are frequently 
described in terms of discrete stages. 
Despite the widespread use of such 
conceptual models, a mathematical 
model that quantitatively describes 
the transitions between multiple 
stages has not been developed for 
crustacean larvae. I describe a model 
of multiple transitions between stag- 
es that can be fitted to such data 
and that holistically describes the 
processes and allows explicit, quan- 
titative comparisons among treat- 
ments or studies. The base of the 
model is the logistic equation that 
is frequently used to model a transi- 
tion between 2 stages. By summing 
together multiple logistic equations, 
one for each transition between 
stages, the model can accommodate 
multiple stages. Variance is modeled 
by treating each transition as a bi- 
nomial distribution and summing 
the variance from each transition. 
To demonstrate, I fitted the model 
to data on larval development of red 
and blue king crabs (Paralithodes 
camtschaticus and P. platypus). The 
model provides an excellent fit for 
these data and quantitatively de- 
scribes the process of larval develop- 
ment for these crab species. 
Manuscript submitted 3 February 2015. 
Manuscript accepted 4 November 2015. 
Fish. Bull. 114:58-66 (2016). 
Online publication date: 25 November 2015. 
doi: 10.7755/FB.114.1.5 
The views and opinions expressed or 
implied in this article are those of the 
author (or authors) and do not necessarily 
reflect the position of the National 
Marine Fisheries Service, NOAA. 
Fishery Bulletin 
established 1881 
Spencer F. Baird 
First U S. Commissioner 
of Fisheries and founder 
of Fishery Bulletin 
A new quantitative model of multiple 
transitions between discrete stages, applied 
to the development of crustacean larvae 
W. Christopher Long 
Email address for author: chris.long@noaa.gov 
Resource Assessment and Conservation Engineering Division 
Kodiak Laboratory 
Alaska Fisheries Science Center 
National Marine Fisheries Service, NOAA 
301 Research Court 
Kodiak, Alaska 99615 
Biologists often describe biological 
processes as discrete stages, either 
on the basis of a natural underly- 
ing stepwise process or to simplify 
a complex continuous process. For 
example, crustacean larval develop- 
ment generally encompasses discrete 
stages that are punctuated by molt- 
ing events (e.g., Costlow and Book- 
hout, 1959; Haynes, 1982). For sim- 
plicity’s sake, embryo development of 
various species is frequently divided 
into stages defined by particular 
characteristics, although develop- 
ment in some stages is considered 
continuous rather than discrete (e.g., 
Kimmel et al., 1995; Bas and Spivak, 
2000; Stevens, 2006). More broadly, 
diseases and communities are also 
described as transitioning between 
stages. Although such developmental 
processes are commonly described in 
the biological literature, no model 
has been developed that quantita- 
tively describes processes which in- 
volve sequential transitions between 
multiple discrete stages and allows 
explicit comparisons among treat- 
ments or species. 
Frequently, each stage is consid- 
ered independently (e.g., Paul and 
Paul, 1999; Andres et al., 2010; Wal- 
ther et al., 2010) and each measured 
or estimate variable, such as stage 
duration, is analyzed separately 
by using univariate statistics (e.g., 
analysis of variance [ANOVA] or t- 
tests). This approach is unsatisfac- 
tory because any comparisons among 
treatments require a large number 
of statistical tests, increasing the 
frequency of type-I errors — problems 
that are similarly caused by the 
use of a series of univariate statis- 
tics to analyze a multivariate data 
set (Quinn and Keough, 2002). In 
addition, in studies on larval devel- 
opment, the method used for deter- 
mining average interstage duration 
is often not defined (Paul and Paul, 
1999; Andres et al., 2010; Walther et 
al., 2010). The method is not defined 
because of the inherent difficulties 
in determining when a replicate con- 
tainer with many larvae has reached 
the next stage. Does the next stage 
occur the first larva transitions or 
when the last one does? Or does it 
occur on the first day when at least 
half have transitioned? 
Red king crab and blue king crab 
( Paralithodes camtschaticus and P. 
platypus) are commercially fished 
species in Alaska and have a wide 
and overlapping distribution (Somer- 
ton, 1985). In both species, mature 
females molt, mate, and extrude a 
batch of eggs in the spring and brood 
the eggs for about a year (Jensen 
and Armstrong, 1989; Stevens and 
