A CPUE-Soak time model
for the Bristol Bay red king crab fishery, 1991-1997
by
Geneviève Briand* Scott C. Matulich**, and Ron C. Mittelhammer***
*Graduate Research Assistant, Dept. of Agricultural Economics, Washington State University
**Professor, Dept. of Agricultural Economics, Washington State University
***Professor, Dept. of Agricultural Economics and Program in Statistics, Washington State University
Senior authorship is shared.
1
Abstract
Experiment-based CPUE literature is rooted in stock assessment, though a daily estimate
of CPUE based on actual fishery performance is an important in-season management tool. For
example, in-season CPUE (legal crabs per pot lift) is a key management tool for the extremely
valuable but biologically vulnerable Bristol Bay red king crab (Paralithodes camtschaticus). As
such, use of commercial fisheries data is critical to estimating a CPUE-soak time function that
captures behavioral responses of fishermen to changing biological and environmental conditions,
as well as the changing regulatory environment. We estimate an asymptotic catch model for
Bristol Bay red king crab using five years of commercial fishery data that was collected by the
management agency after season closure. A von Bertalanffy-type CPUE-soak time model is
nested within a more general specification that captures aggregate intra- and inter-seasonal
effects on the fishery. Data pooling guided by recursive estimation/hypothesis testing are shown
to be essential to overcoming data deficiencies characteristic of commercial fisheries data that are
not contemporaneously collected. The analysis provides insight into behavioral responses of
crabbers to changing conditions between 1991 and 1997, whether biological, ecological or
policy-induced.
2
Introduction
The U.S. Bering Sea crab fisheries developed rapidly throughout the 1970’s following the
decline of stocks out of Kodiak Island. The Bristol Bay red king crab (Paralithodes
camtschaticus) fishery collapsed after peaking at a 58,968 t harvest in 1980. The fishery was
closed for one year in 1983. Since then, Bristol Bay harvests have remained relatively low and
two additional closures occurred in 1994 and 1995.
The Bristol Bay red king crab fishery is a license-limited open access fishery, managed by
the Alaska Department of Fish and Game (ADFG). Management authority is delegated to ADFG
by a federal fishery management plan, the goal of which is to maximize the overall long-term
benefit to the nation of the Bering Sea king crab stocks. Annual trawl surveys conducted by the
National Marine Fisheries Service (NMFS) provide stock abundance data. ADFG analyzes the
data and in consultation with NMFS, recommends to the Alaska Board of Fisheries a preseason
quota or guideline harvest level (GHL), measured in pounds of crab, that the estimated biomass
can support. ADFG then uses in-season fishery performance, including legal crabs caught per pot
lift or CPUE (catch-per-unit-effort) and an estimate of cumulative catch to close the fishery at the
GHL (Herrmann and Greenberg 1997).
Fishery management has become increasingly difficult as depressed stocks and lower
exploitation/utilization rates shortened seasons. Managers implemented a uniform pot limit of
250 pots per vessel in 1992 in an attempt to elongate the season by mitigating fleet fishing
power. This policy was challenged by industry as discriminatory toward large vessels. A twotiered pot limit was implemented in 1993, where vessels less than or equal 125 feet in length
were allowed to fish 200 pots, while vessels over 125 feet were limited to 250 pots. Nevertheless,
season length compressed from nine days in 1991 to four days by 1996, when harvest exceeded
3
the pre-season GHL by 68% and the fishery experienced its largest CPUE (16 crab per pot lift)
since the fishery began to collapse in 1981. Managers reacted by halving the pot limit in 1997.
Harvest once again exceeded the GHL, this time by 25%. Herrmann and Greenberg attribute
under-estimation of CPUE as a principal cause of the excessive harvest.
Understanding the CPUE-soak time relationship is important to managing this vulnerable
but lucrative crab fishery (Pengilly and Tracy 1998). The purpose of this paper is to estimate a
CPUE-soak time relationship for the Bristol Bay red king crab fishery in order to gain regulatory
and in-season management insight into this high-valued but vulnerable fishery. Since the fleet is
not required to report daily catch statistics, the analysis is conducted using post-season
commercial fisheries data for the five years the fishery was open, 1991-93 and 1996-97. We will
show that while a reasonable statistical fit is obtained, limitations of currently available
commercial fisheries data impede using the estimated model as a forecasting tool that could
otherwise assist managers close this fishery at the GHL. The following analysis does, however,
provide important insight into a tendency for the pot limit policy to shorten rather than lengthen
seasons. More generally, the modeling framework developed below is useful in other contexts
where the use of commercial fisheries data is preferable to controlled experimentation, i.e.,
whenever the purpose is to gain in-season management insight or to assess the efficacy of
regulations intended to alter fishing strategies/behavior. This feature distinguishes our paper from
much of the CPUE-soak time literature that focuses on stock assessment. Fishing strategies and
thus, CPUE, adjust not only in response to biological and environmental factors, the object of the
experiment-based literature, but also in response to policy-induced considerations. Experimentbased studies cannot capture the influence of fishing strategies on CPUE that can be critical to
effective in-season management and regulatory analysis, particularly during short fisheries. Use
4
of commercial fisheries data, however, presents a variety of estimation challenges, especially
when the data are derived from neither a contemporaneous nor a scientifically designed sample
of the fleet. A statistical framework for dealing with such challenges is illustrated in this paper.
Methods
Model specification
The relationship between CPUE and soak time is extensively reviewed in Miller (1990)
primarily in the context of using CPUE as a method of stock assessment. Collectively, the
literature provides evidence that CPUE (generally measured as total catch of crabs or fish per
trap) increases asymptotically with soak time due to a progressive reduction in both bait
effectiveness and the density of the species being fished. Additional factors found in the
experiment-based literature to affect the shape of the asymptotic CPUE-soak time relationship
include: size and effectiveness of the gear (e.g., Munro 1974); inter- and intra-species behavioral
interactions (e.g., Miller 1978); local stock density (e.g., Sinoda and Kobayasi 1969); and general
environmental conditions such as temperature, time of day, and current (e.g., Bennett and Brown
1979).
An asymptotic catch model is particularly appropriate to describe the CPUE-soak time
relationship for short fishing seasons, where catches are not observed to decline much with soak
time (Zhou and Shirley 1997). Such is the case with the Bristol Bay red king crab fishery. The
study by Pengilly and Tracy specifically supports the use of an asymptotic catch model for
Bristol Bay red king crab. They conducted a controlled experiment to assess the effects of soak
time on catch of legal-sized red king crabs and by-catch of nonlegal crabs in the Bristol Bay.
While their study was descriptive and made no attempt to estimate an asymptotic catch model,
5
their analysis provided evidence of an asymptotic CPUE-soak time relationship for Bristol Bay
red king crabs.
Specification of an asymptotic catch model for the Bristol Bay red king crab fishery
theoretically should incorporate many of the factors found in the experiment-based literature
model that captures both intra- and inter-seasonal variations. Unfortunately, data limitations
prevent isolating likely factors that define CPUE-soak time relationships within and across
seasons. Accordingly, the approach taken in this study is to model CPUE-soak time in a way that
encompasses the basic asymptotic relationship within a more general specification that captures
aggregate or composite intra- and inter-seasonal variations, whatever the cause—biological,
environmental or behavioral. Three models, listed in order of hierarchical nesting, are specified
in equations (1), (2) and (3).
Equation (1), the most common asymptotic catch model, is a von Bertalanffy-type
equation:
 : beta

: epsilon
(1)
   ST 

i 
2
   i
Ci   0 1  e 


where: C i denotes CPUE observations, i  1,..., n ; STi denotes observations on soak time,
measured in days;  0 and  2 are unknown parameters (scalars) representing, respectively, the
asymptotic catch and the rate at which it is reached; and  i denotes an error term.
Theoretical considerations suggest equation (1) should be generalized so that asymptotic
catch depends upon the initial stock. It should also reflect the possibility that the rate at which a
6
pot fishes toward its maximum may depend upon localized stock depletion throughout the
fishing season. That is, a particular CPUE-soak time relationship, on a given day of the season, is
a function of residual biomass and thus, past fishing behavior. Data limitations prevent
estimating such a specification. Instead, equation (2) incorporates an aggregate intra-seasonal
variable, drop time (DT), that is defined as the day within the season when a pot is set into the
water. This variable proxies for all factors contributing to changing catch as the season
progresses. It accounts for the fact that pots dropped at the beginning of the fishing season are
exposed to a more abundant, denser stock of crabs than pots dropped at the end of the fishing
season.
(2)
   1  DT  ST 


i
i  
2
3

  i
Ci   0 1  1 DTi  1  e 



The new parameters represent the effect of drop time on the asymptotic catch level,  1 , and the
effect of drop time on the rate at which a pot fishes toward this asymptote,  3 . Interpretation of
 0 and  2 are unchanged. Note that the elementary asymptotic catch model given by equation
(1) is embedded in the asymptotic catch model expressed by equation (2), i.e., equation (1) is a
special case of equation (2), where 1  0 and  3  0 .
A principal cause of inter-seasonal variation in the asymptotic catch model is due to
different initial biomasses across years. Numerous other inter-seasonal factors conceivably
impact both the catch rate and the asymptote. Spatial distribution of crab varies across years, as
do tidal conditions or other environmental factors that affect recruitment into the gear.
7
Regulatory conditions under which the Bristol Bay red king crab fishery operated also changed
throughout 1991-97. Fishery managers imposed pot limits in 1992 and reduced the exploitation
rate from 20% to 10% in 1996 and 1997. Fishing strategies adjust to all of these factors.
Equation (3) introduces a fishing season indicator variable (FS) into equation (2) in order
to capture generic, aggregate seasonal effects.
  97



 97


 
 1
 DT  ST  


FS

FS




i



i
,
j
i
,
j
2
,
j
3
,
j





 j 91

 



  j 91
i





(3) Ci    0, j FSi , j 1    1, j FSi , j  DTi  1  e  
 i




j 91
j

91







97
97
The indicator variable FSi , j attributes pot lifts to the fishing season in which they occurred, j =
91, 92, 93, 96 and 97, i.e., FSi , j  1 if pot lifts i occurred in season j , 0 otherwise. The
principal attribute of model (3) is that it introduces flexibility by allowing the coefficients  0 ,
 1 ,  2 and  3 to take on year-specific values. For example, 2 could take the value ̂ 2,91 in
1991, ̂ 2,92 in 1992, ̂ 2,93 in 1993, and so on. Equation (3) specifies year-specific asymptotic
CPUE-soak time models that account for intra-seasonal variations similar to equation (2). In the
absence of any parameter restrictions, estimating equation (2) separately for each year is
equivalent to estimating equation (3). Estimation of both models (2) and (3), together with an
examination of statistically admissible parameter restrictions, was pursued in order to determine
whether and what type of asymptotic catch model applied to all five fishing seasons.
8
Data considerations
The commercial fishery data used in this study are drawn from ADFG after-season
dockside interviews with available catcher vessels and from contemporaneous weekly trip
summaries completed by observers on catcher-processors (CPs). These surveys provide
information on vessels’ daily performances. Variables of interest include: day fished, the number
of pots lifted on that day, the total number of crabs caught, and the average soak time of the pots
lifted. 22% to 48% of the catcher vessel fleet was surveyed during the five seasons between 1991
and 1997. The number of CP vessels—all surveyed—ranged from 0 to 24. Missing data on the
variables of interest rendered 7% to 57% of all surveys unusable, depending on the year. The
sampled data are not the result of a statistical design that was intended to represent a randomized
cross section of a heterogeneous fleet. Nor do most of the usable surveys provide information on
vessels’ actual daily performances; 47% to 86% of the observations across the seasons are not
contemporaneous records of daily fishing activities, but appear to be seasonal averages.
Depending on year, 20% to 31% of the usable catcher vessel surveys recorded data for only a
single day or recorded the same catch every day. An additional 27% to 59% of the usable surveys
do not reflect different levels of catch every day, as would be expected of contemporaneous catch
records. Surprisingly, 37% to 88% of the usable CP surveys also report identical catch records on
adjacent days, despite the fact that CP data are collected by onboard observers.
These data deficiencies pose problems for model estimation. Rather than eliminating noncontemporaneous data, data from all complete surveys were utilized after they were first
aggregated. The aggregation was done on a year-by-year basis such that observations with
identical soak times on the same day were aggregated together to form a single, final observation.
Thus, year-specific CPUE performances were averaged out over observations of pots pulled on a
9
given day that soaked the same length of time. This data aggregation scheme isolated all reported
day-soak time combinations, thereby retaining all intra-seasonal CPUE-soak time variation
Tables
1&2
near
here
across the surveyed portion of the fleet.
The aggregation process is illustrated in Tables 1 and 2. The aggregated data
corresponding to Table 1 are shown in Table 2. In Table 1, vessels X and Y both recorded pulling
pots on day 2, and both utilized a two-day soak. These two observations are collapsed into one,
indicated as observation number 1 in Table 2, and reflect the total number of pots lifted and crabs
harvested on day 2, given a two-day soak. In this illustration, 2400 crabs were caught by 200
pots. The drop time variable is derived by subtracting soak time from the day fished; CPUE is
calculated as the aggregate number of pots lifted divided by the aggregate number of crabs
caught. For example, the first observation in Table 2 is based on pots dropped at the start of the
fishing season (drop time 0), and is calculated to have an average CPUE of 12 crabs.
The final aggregated data set totaled 235 observations over the five open seasons, 199193 and 1996-97. Aggregated CPUE observations are based on daily catch records of 1 to 69
vessels. An observation represents the average CPUE from as few as 12 and as many as 11,816
pots that were dropped on a particular day and which soaked an identical amount of time.
The range and variation of soak times over which catches are observed in commercial
fishery data are dictated by fishing conditions and strategies. This potentially adds further
difficulty to the estimation process because neither the data range nor variation may be large
enough to uncover an asymptotic catch model, even if true.
Table 3
near
here
The soak time frequency distribution reported in the available data is presented in Table
3. The range of soak times covered by the data varies across years but is especially short in 1997,
when 81.7% of the pots soaked at most one day and no pot soaked longer than two days.
10
Variation in soak times was the smallest in 1997, with a mean soak time of 0.96 days and a
standard deviation of 0.29. It follows that the estimation of a CPUE-soak time model based on
individual year data such as equation (2) is likely to fail for 1997. An alternative estimation
approach capable of coping with this common characteristic of commercial fisheries data is to
pool data across years.
Data pooling shares information between 1997 and other seasons, increasing the range
and variation in soak times observed in the expanded data set, thereby enhancing the likelihood
of identifying an asymptotic catch model for that year. Data pooling is accomplished by imposing
one or more cross-year parameter restrictions on model (3). For example, the rate at which a pot
filled in 1997 might be statistically identical to that of 1996, allowing the coefficient 2 to share a
common value in 1996 and 1997. In such a situation, the following restriction might be imposed
on model (3):  2,96   2,97 . Of course, any such restrictions would need to be tested for statistical
validity.
Estimation
Data pooling in the context of model (3) can have negative consequences. There is always
the risk that restrictions imposed by data pooling might be incorrect (Type II error). Moreover,
the more restrictions tested based on the information in one data set, the greater the potential for
compounding Type II errors. Sequential statistical tests of the restrictions are, thus, not sufficient;
a benchmark is also needed to evaluate the validity of the final restrictions imposed on model (3).
Model (2) provides such a reference point.
Theoretically, year-specific parameter values estimated in model (2) should be identical
to those of model (3), in the absence of data pooling. Accordingly, model (2) parameter
estimates, for years in which asymptotic catch models are identified, are compared with
11
corresponding model (3) parameter estimates. The restrictions imposed on model (3) are
pertinent only if they are not statistically rejected. Moreover, they must result in parameter
estimates that make logical sense compared to what is already known a priori about the CPUEsoak time relationships, as well as what is learned from unconstrained and significant parameter
estimates generated from model (2). That is, estimates of both models (2) and (3) should be close
and consistent with one another for those parameters that are statistically significant and precisely
estimated. In the final analysis, which of the models provides the better empirical representation
of the CPUE-soak time relationship involves considerations of mean squared error, bias-variance
tradeoffs, and interpretability of the estimated relationship.
Two sources of prima facie heteroskedasticity were corrected prior to estimation: one
source relates to unequal intra-seasonal variances and the other relates to unequal inter-seasonal
variances. Average CPUE observations are based on different numbers of pots pulled, which
generally introduces prima facie intra-seasonal heteroskedasticity across observations. In
particular, CPUE observations based on averages of a large number of pots pulled generally have
smaller variances than CPUE observations based on a small number of pots pulled, all other
things being equal. This group-induced heteroskedasticity problem is a common artifact of
combining individual data into aggregated averages. Heteroskedasticity arises when the data set
contains an unequal number of observations across groups (Johnston 1984). For purposes of
parameter estimation, the standard group-induced heteroskedasticity correction is introduced;
data observations in eqs. (2) and (3) are weighted by the square root of the number of pot lifts in
each observation. Specifically, both sides of the equations are multiplied by mi , where mi is the
number of pots lifted for observation i .
12
(2.1)






   1   DT ST   


3
i
2
i  

mi Ci  mi  0 1  1 DTi  1  e



  i



  97
 


 97



 





 97
     2 , j FSi , j  1   3, j FSi , j  DTi  ST   
  97




  j  91

i
j  91






 
(3.1) mi Ci  mi    0, j FSi , j 1    1, j FSi , j  DTi  1  e
 i




 j 91



j

91

 






Non-prima facie sources of intra-seasonal heteroskedasticity also may exist. For example,
CPUE may exhibit smaller variance as it becomes less dependent on local crab density at
increasing soak times and drop times. A Breush-Pagan-Godfrey heteroskedasticity test
(Mittelhammer et al. 2000) was applied to equation (3.1) and thus, implicitly on the nested
equation (2.1). In particular, an auxiliary regression was specified expressing estimated squared
residuals as a linear function of soak time and drop time. The hypotheses of homoskedasticity
relative to variations in soak time and drop time could not be rejected at typical levels of Type I
error probabilities (2-sided t-distribution probability values were 0.21922 and 0.2009,
respectively, for soak time and drop time).
CPUE observations are also subject to unequal inter-seasonal variances due to the
different levels of biomass to which pots were exposed across years. Other potential seasonal
effects (e.g., tides and weather) also may contribute to year-specific CPUE variances. Equation
(3.1) is further transformed to correct for this prima facie inter-seasonal heteroskedasticity. The
 : sigma correction was implemented by multiplying both sides of the equation by
:
phi
13
2
1
, where ˆ i , j is the
ˆ i, j
2
estimated variance of the residual terms for fishing season j . ˆ i , j 
 eˆi2
i j
nj k
and ei  C i  Ci ,
where C i is the CPUE value calculated from model (3.1),  j denotes the set of observations
corresponding to season j, n j is the number of observations for fishing season j , and k is the
number of parameters (Johnston 1984). The weighted model form is given in eq. (3.2).
(3.2)
  97
 


 97




 






 97
     2 , j FSi , j  1   3, j FSi , j  DTi  ST    

 j  91

  97



  j  91

i
1
1 






 

mi C i  
mi   0, j FS i , j 1     1, j FS i , j  DTi  1  e
 i


 j 91
ˆ i , j
 i, j 


j

91












Model (2.1), applied individually to each year, and model (3.2), applied to all years
simultaneously, are estimated by non-linear least squares using the GAUSS and NewtonRaphson algorithms contained in the GAUSSX statistics program (GAUSSX Econometric
Software 1996). For details regarding the GAUSS and Newton-Raphson algorithms, see
Mittelhammer et al. (2000).
Results
Initial estimation of model (2.1) revealed evidence of an asymptotic catch model for all
years but 1997. However, the results also indicated evidence of specification error in the sense of
insignificant  3 parameter estimates for three years. Equation (2.1) was re-specified without the
Table 4
near
here
 3 parameter in these cases, and re-estimated for the affected seasons, 1991-93 and 1996. Final
estimated models (2.1) are presented in Table 4. Table 4 shows well-defined asymptotic CPUE-
14
soak time relationships for years 1991-93 and 1996. Not surprisingly, estimates for 1997—the
year of little soak time variation—reveal no significant statistical evidence of an asymptotic catch
model for that year.
Equation (3.2) was estimated in a three-step process with the goal of identifying an
asymptotic CPUE-soak time relationship for 1997, while also maintaining comparable model
structure for 1991-93 and 1996 to that presented in Table 4. The first step involved estimating
model (3.2) with no data pooling, i.e., without shared-parameter restrictions. The second step
involved testing for data pooling restrictions. The third step involved re-estimating model (3.2),
specified according to the non-rejected restrictions. Then the estimated coefficients of model
(3.2) for the years 1991-93, and 1996 were compared to the corresponding estimates obtained
from model (2.1) (Table 4).
Initial unrestricted estimates of model (3.2) revealed evidence of model misspecification
for years 1991-92 and 96, similar to the initial estimation of model (2.1). This finding suggested
that any re-specification of model (3.2) required, at the least, a restriction on the 3 parameters
such that  3,91   3,92   3,96  0 . This restriction is used to delimit potential specifications of
model (3.2) that might reveal an asymptotic catch model for 1997. All of the potential restrictions
Table 5
near
here
that were tested on model (3.2) are presented in Table 5, stated from the least to the most
restrictive.
The null hypotheses were tested to determine which of the possible restrictions on the
parameters, if any, should be imposed on the final model. The final pooled model (3.2)
specification was chosen to be the least restrictive, statistically significant specification, in an
attempt to retain as much of the year specificity outlined by model (2) estimates as possible. Final
Table 6
near
here
15
estimates are presented in Table 6, which indicate year-specific asymptotic catch relationships for
all five fishing seasons.
Overall model fit is good. All the explanatory variable effects are statistically significant
at standard levels of Type I error probabilities except one (2-sided t-probabilities of 0.000 for all
but two variables that have 0.003 and 0.119 t-probabilities), and the overall explanatory power of
the model is high (R-squared of 0.9306). The -8.25% mean percent error (MPE) indicates the
model somewhat overestimates CPUE values observed in the Bristol Bay red king crab fishery
from 1991 through 1997. Out of 235 final aggregated observations, 50.6% are overestimated and
49.4% underestimated by the catch-soak time model. The model averages a 19.74% mean
absolute percent error (MAPE) in fitting observed CPUE values (i.e., the model errors by an
average of 1.59 crabs per pot lift). It should be noted that this MAPE is inflated by three
prediction errors that are very small in nominal terms but very large in percentage terms—a
consequence of some pots catching very few crabs. These three observations account for 5% of
the 19.74% error and represent only 319 pot lifts (0.1% of all observed pot lifts). The three
observations were omitted and the model was re-estimated. Similar parameter estimates, P and
R-squared values were obtained. Other goodness of fit measures improved to: 14.73 for MAPE,
and -3.34 for MPE. In the end, we decided to retain all usable data because we had no way of
knowing whether these three observations were true outliers or simply reflective of setting gear
Figure 1
near
here
on no crab.
Graphical representation of model 3.2 is presented in Figure 1. Year-specific CPUE-soak
time curves depict predicted catch from pots dropped on the first day of the season ( DT  0 ).
The slopes of CPUE-soak time curves attenuate as DT progresses.
16
Discussion
CPUE-soak time relationships for Bristol Bay red king crab were modeled as elementary
von Bertalanffy-type equations nested within more general specifications that capture aggregate
intra- and inter-seasonal variations. Data pooling and recursive hypothesis testing were utilized to
address limited data variability and other commercial fisheries data deficiencies. The estimated
model suggests that at the beginning of the season, fishers could potentially catch, on average,
22.81 crabs per pot in 1991 and 1997, 8.21 crabs per pot in 1992, 12.26 crabs per pot in 1993,
and a high of 34.39 crabs per pot in 1996. That high-CPUE year was the year in which the
smallest fleet (196 vessels, down from the five-year average of 265 vessels) harvested 68% more
crabs than expected in just a 4-day season. A total of 3,810 t of crabs were harvested, while the
GHL was set at only 2,268 t. The effect of drop time on the maximum catch was more important
in 1996 compared to the other years. The maximum potential catch per pot decreased by 20%
after one day in 1996, and by 7% in 1991-1993 and 1997, suggesting stock aggregations may
have been more dense in 1996. A dense stock of crab increases catchability but contributes to a
more rapid decline in the remaining stock available to harvest.
The basic CPUE-soak time relationship changed during the 1991-97 period. In 1991, a
one-day soak achieved 66% of the catch that would result from soaking a pot two days. The rate
at which the pot fished toward its two-day potential increased in subsequent years. In 1992, pots
achieved 87% of a two-day soak catch, after a single day. That rate increased to 96% in 1993-97.
(Rates at which pots fill were derived from model (3.2) using calculated CPUE values at
different soak times.) These observations together suggest a structural change might have
occurred in the fishery, though inadequate data prevent isolating the components of change.
Nevertheless, it lends insight into a change in fishing behavior. It is rational for individual
17
crabbers to shorten soak time in order to increase their seasonal catch. Of course, when this
behavior is extended across the entire fleet, individual shares remain relatively constant but the
season shortens.
Crabbers adjust fishing strategies not only in response to biological and environmental
factors, the object of the experiment-based literature, but also in response to policy-induced
considerations. During the 1991-97 period, pot limits were introduced and then halved in 1997 in
an attempt to slow the fishery and better facilitate in-season management/fishery closure. Yet, the
data indicate average soak time shortened from as many as two days in 1991, to less than one day
when the most restrictive pot limit was implemented. While there was an incentive to reduce
soak time from two days to one day, even in 1991 when a one-day soak yielded twice as much
crab as a two-day soak, this did not occur. It would appear crabbers may have learned to shorten
soak times, possibly because they were restricted in the amount of available gear beginning in
1992. It wasn’t until 1997, however, that the pot limit policy left crabbers with so little gear that
they had to pull all pots every day or remain idle to maintain longer soak times. It follows that if
the observed shift from a two-day soak to a one-day soak were due primarily to the pot limit
policy, the policy was counter-productive. Behavioral response from the crabbers may have
defeated the policy intent and exacerbated the in-season management problem. Ironically, the pot
limit policy may have educated fishers that short soaks are effective ways to increase seasonal
catch during short-season, open access fishing derbies like that of Bristol Bay red king crab.
Such a policy inference should be regarded as tentative because the empirical analysis
was based on after-season dockside surveys that lacked scientific experimental design and, in
general, were not taken from contemporaneous skipper logbook records. Nevertheless, the fact
that after-season dockside surveys continue to be collected suggests the managers believe the
18
data reflect general fishery performance. If one accepts the veracity of the survey data as
“representative” of fleet behavior, then the results of our analysis may be presumed to
approximate the underlying CPUE-soak time relationship for the fishery. At the very least,
additional research into the impact of pot limits on season length seems warranted.
Lack of either daily observations on fleet-wide activity or a scientifically designed sample
of daily fleet activity complicated the statistical analysis and ultimately required the use of
indicator variables to capture only the aggregate presence of inter- and intra-seasonal CPUE-soak
time variation. Specific causal mechanisms identified in the experiment-based literature and by
industry as factors contributing to inter- or intra-seasonal variation (e.g., initial and daily
biomass, or general environmental conditions like water temperature, weather or tides) could not
be investigated because they require more data-intensive model specifications. Accordingly,
while the results of this model may be useful as a descriptor of past behavioral responses to
conditions affecting the fishery, usefulness for forecasting future fishery performance and for
enhancing in-season management is likely quite limited.
Richer, more data-intensive specifications conceivably could have been estimated had
real-time, daily fishery performance observations been collected from the entire fleet. Such data
collection capability now is both technologically feasible and relatively inexpensive. Complete
enumeration of daily, vessel-by-vessel activity during 1991-97 would have yielded approximately
9,400 daily vessel observations compared to the 235 aggregated observations used in this study.
This 40-fold increase in observations should capture much greater variation in vessel
performance and accommodate added model parameterization that facilitates forecasting. The
potential to both forecast and hindcast fishery performance should be a compelling impetus for
19
fishery managers and industry, alike, to collect real-time data that could assist in-season
management of this very valuable but vulnerable fishery.
Acknowledgements
This research benefited from the assistance of many individuals, especially Alaska
Department of Fish and Game, Division of Commercial Fisheries staff, and many members of
industry. Our gratitude is also extended to the reviewers of this manuscript who helped sharpen
our focus.
References
Bennett, D.B. and C.G. Brown. 1979. The problems of pot immersion time in recording and
analysing catch-effort data from a trap fishery. Rapp. P.-v. Réun. Cons. Int. Explor. Mer
175: 186-189.
Herrmann, M. and J. Greenberg. 1997. Potential economic implications of more restrictive pot
limits in the Bristol Bay red king crab fishery. Report to the Alaska Department of Fish
and Game.
Johnston, J. 1984. Econometric methods, 3rd ed. McGraw-Hill, N.Y.
Miller, R.J. 1978. Saturation of crab traps: reduced entry and escapement. J. Cons. Int. Explor.
Mer 38: 338-45.
Miller, R.J. 1990. Effectiveness of crab and lobster traps. Can. J. of Fish. Aquat. Sci. 47: 12281251.
Mittelhammer, R.C., G.G. Judge and D. J. Miller. 2000. Econometric foundations. Cambridge
University Press, N.Y.
20
Morrison R., R..B.Gish, M. Ruccio, and M. Schwenzfeier. 1998. Annual Management Report for
the Shellfish Fisheries of the Bering Sea. Dutch Arbor Office, Dutch Harbor, Alaska.
Munro, J.L. 1974. The mode of operation of antillean fish traps and the relationships between
ingress, escapement, catch and soak. J. Cons. int. Explor. Mer 35(3): 337-350.
Murphy, M.C. and K.L. Griffin. 1996. Adak red and brown king crab pot Limits: a report to the
Alaska Board of Fisheries. Alaska Department of Fish & Game, Juneau, Alaska.
Pengilly, D., and D. Tracy. 1998. Experimental effects of soak time on catch of legal-sized and
nonlegal red king crabs by commercial king crab pots. Alas. Fish. Res. Bul. 5: 81-87.
Sinoda, M. and T. Kobayasi. 1969. Studies on the fishery of zuwai crab in the Japan Sea - VI.
Efficiency of the toyama kago (a kind of Crab Trap) in capturing the beni-zuwai crab.
Bul. Japan. Soc. Sci. Fish. 35: 948-56.
Zhou, S., and T.C. Shirley. 1997. A model expressing the relationship between catch and soak
time for trap fisheries. N. Am. J. Fish. Mgmt. 17: 482-487.
21
Table 1. Initial daily observations, illustrative sample.
Vessel
identification letter
X
X
X
Y
Y
Z
Day
fished
2nd
3rd
4th
2nd
3rd
2nd
Number of Number of
crabs caught pots lifted
1600
120
1500
120
900
80
800
80
900
80
1100
100
22
Pots soak time
(in days)
2
3
2
2
3
1
Table 2. Final aggregated observations, illustrative sample based on Table 1.
Observation
1st
2nd
3rd
4th
Day
Number of Number of
fished crabs caught pots lifted
2nd
2400
200
3rd
2400
200
th
4
800
80
nd
2
1100
100
23
Pots soak time
(in days)
2
3
2
1
Pots drop time
(in day)
0
0
2
1
Catch
per Pot
12
12
10
11
Table 3. Soak-time frequency distribution across fishing seasons.
Year
Number of
final
aggregated
observations
Final
sampled
vessels
(as % of
fleet)
Mean
CPUE
Mean
soak time
in days*
(st. dev.)
Number of pots lifted
with reported soak time in days of:
(number of pots lifted in % of total pots lifted
from usable data of that year)
1
1-2
2-3
3
1991
55
104
(34%)
12
1.98
(0.54)
10053
(13.0%)
58297
(75.2%)
7969
(10.3%)
1158
(1.5%)
1992
26
36
(13%)
5
1.28
(0.48)
20195
(70.7%)
8241
(28.8%)
74
(0.3%)
60
(0.2%)
1993
41
122
(42%)
8
1.63
(0.58)
37664
(41.1%)
50423
(55.1%)
2920
(3.2%)
536
(0.6%)
1996
52
70
(36%)
18
1.13
(0.30)
17639
(64.8%)
9506
(34.9%)
90
(0.3%)
0
(0%)
1997
61
50
(19%)
17
0.96
(0.29)
13633
(81.7%)
3054
(18.3%)
0
(0%)
0
(0%)
*Mean soak times are weighted by the number of pot lifts per observation.
Mean soak time in year j: ST j 
 m ST
i 1
nj
i
 mi
i
 m ST  ST 
nj
nj
; standard deviation (st.dev.): s j 
i 1
i 1
2
i
j
i
nj
m 1
i 1
i
where mi denotes the number of pots upon which observation i is based; n j denotes the number
of observations in year j .
24
Table 4. Final model (2.1) specification and estimates.
Parameter
Estimated coefficient
(standard error)
2-sided t-probability values
1992
1993
1996
Definition
1991
̂ 0
Asymptotic catch
23.4583
(3.4579)
0.000
7.6122
(0.8040)
0.000
12.4758
(0.4654)
0.000
34.5322
(3.1375)
0.000
1997
1
5.6497
(3.7145)
0.000
ˆ1
Effect of drop time
on the asymptotic
catch
-0.0761
(0.0095)
0.000
-0.0425
(0.0157)
0.013
-0.0770
(0.0059)
0.000
-0.2023
(0.0112)
0.000
0.0504
(0.0844)
0.552
ˆ2
Rate at which the
asymptotic catch is
reached
0.6293
(0.1929)
0.002
1.6134
(0.4993)
0.004
2.7559
(0.7030)
0.000
2.9879
(1.7600)
0.096
0.7702
(2.0921)
0.714
̂ 3
Effect of drop time
on the rate at which
the asymptotic
catch is reached
R-squared
-0.0473
(0.0329)
0.159
0.9281
0.8787
25
0.9579
2.8243
(8.4024)
0.738
0.8953
0.3292
Table 5. Model (3.2) potential restrictions.
Parameter
Null
hypotheses
0 , 1 , 2
3
H 0 :  q, j   q,97
H 0 :  3,91   3,92   3,96  0
H 0 :  q, j   q,l   q,97 ; j  l
H 0 :  3,91   3,92   3,96  0 ,  3,93   3,97
H 0 :  q, j   q,l   q, p   q,97 ; j  l  p
H 0 :  3,91   3,92   3,96   3,97  0
H 0 :  q,91  ,,92  ,q,93   q,96   q,97
where: j , l , p  91, 92, 93, 96 and q  0,1, 2
26
Table 6. Final model (3.2) specification and estimates.
Model specified by following restriction: (F-probability (df: 10, 219): 0.3028)
H0:
 0,91   0,97
1,91  1,92  1,93  1,97
 2,93   2,96   2,97
 3,91   3,92   3,96   3,97  0
Estimated Coefficient
(standard error)
2-sided t-probability values
1992
1993
1996
Parameter
1991
1997
̂ 0
Asymptotic catch
22.8094
(1.3955)
0.000
8.2095
(1.4423)
0.000
12.2619
(0.3401)
0.000
34.3919
(1.9936)
0.000
22.8094
(1.3955)
0.000
ˆ1
Effect of drop time
on the asymptotic
catch
-0.0747
(0.0044)
0.000
-0.0747
(0.0044)
0.000
-0.0747
(0.0044)
0.000
-0.2029
(0.0125)
0.000
-0.0747
(0.0044)
0.000
ˆ2
Rate at which the
asymptotic catch is
reached
0.6586
(0.0920)
0.000
1.8575
(1.1761)
0.116
3.1289
(0.6312)
0.000
3.1289
(0.6312)
0.000
3.1289
(0.6312)
0.000
̂ 3
Effect of drop time
on the rate at which
the asymptotic
catch is reached
-0.0604
(0.0200)
0.003
R-squared
Mean absolute percent error (MAPE)
Mean percent error (MPE)
Mean absolute deviation (MAD)
0.9306
19.74
-8.25
1.59
MAPE, MPE and MAD are weighted by the number of pots each observation is based upon:
Ci  C i
Ci  C i
100 n
100 n
100 n
MPE

m
MAD

mi Ci  C i , where
;
;
MAPE  n
m



i
i
n
n
Ci
Ci
i 1
i 1
i 1
 mi
 mi
 mi
i 1
i 1
i 1
C i is the CPUE value calculated from final model (3.2) estimates.
27
Figure 1. CPUE-soak time relationships: model (3.2) estimated at DT  0 .
35
30
25
1991
1992
1993
1996
1997
CPUE
20
15
10
5
0
0
0.5
1
1.5
2
Soak time (in days)
28
2.5
3