ICES Journal of
Marine Science
ICES Journal of Marine Science (2014), 71(6), 1443– 1457. doi:10.1093/icesjms/fsu066
Contribution to the Symposium: ‘Gadoid Fisheries: The Ecology of Management and Rebuilding’
Original Article
Impacts of seasonal stock mixing on the assessment of Atlantic
cod in the Gulf of Maine
Jie Cao, Samuel B. Truesdell, and Yong Chen*
School of Marine Sciences, University of Maine, Orono, ME 04469, USA
*Corresponding author: tel: +1 207 581 4303; fax: +1 207 581 4990; e-mail: [email protected]
Cao, J., Truesdell, S. B., and Chen, Y. Impacts of seasonal stock mixing on the assessment of Atlantic cod in the Gulf of Maine. – ICES
Journal of Marine Science, 71: 1443 – 1457.
Received 26 November 2013; revised 17 March 2014; accepted 18 March 2014; advance access publication 23 April 2014.
Atlantic cod (Gadus morhua) in the Northwest Atlantic off NewEngland and southern Atlantic Canada exhibit acomplex population structure. This region hasthree
independently assessed stocks [Georges Bank, Gulf of Maine (GOM), and the 4X stock], all of which are known to mix with each other. Assessments of these stocks,
however, assume no interpopulation mixing. Using simulations, we evaluated impacts of ignoring mixing resulting from seasonal migrations on the GOM assessment.
The dynamics of the three stocks were simulated according to different scenarios of interstock mixing, and a statistical catch-at-age stock assessment model was fitted
tothesimulatedGOMdatawithandwithoutmixing.Theresultssuggestthat,whilemixingcausesmeasurablebiasintheassessment,undertheconditionstested,this
model stillperformed well.Ofthebiasthatdoesexist,spawning-stock biomass estimates arerelativelysensitiveto mixing compared with estimatesofrecruitment and
exploitation rate. The relative timing of seasonal migration of the three stocks plays a critical role in determining the magnitude of bias. The scale and trends among
years in the bias were driven by how representative the catch and survey data were for the GOM stock; this representation changed with the mixing rates.
Keywords: Atlantic cod, estimation bias, seasonal migration, stock assessment, stock mixing.
Introduction
Most fish populations exhibit complex spatial structure resulting
from larval dispersal, foraging, seasonal migration, and spatial variability in suitable habitats (Ying et al., 2011). However, the assessment of fish populations has historically relied on assumptions
that simplify these complexities. One important assumption is
that the individuals within the region of interest belong to a “unit
stock”, meaning that there is no immigration or emigration of
larvae or free-swimming individuals (Caddy, 1975; Hilborn and
Walters, 1992; Orensanz and Jamieson, 1998). However, geographical stock boundaries rarely guarantee that a population is strictly a
closed stock; boundaries are often politically and administratively as
well as biologically determined (Stephenson, 1999). Fish in many
populations move regularly (e.g. Lilly et al., 1998; Ames, 2004)
across stock boundaries, so the unit stock assumption is often violated. This research concerns the exchange of fish among populations regardless of the reason for the movement, and the terms
“movement” and “migration” are used synonymously to represent
this concept.
# International
Such fish movements are considered a potential problem for
stock assessment, and many researchers have attempted to address
the consequences of ignoring these patterns. For example, the probability of overexploitation as a function of managing a metapopulation as independent stocks or a unit stock was investigated by Ying
et al. (2011). Hart and Cadrin (2004) found model estimates for a
small, independently assessed stock of yellowtail flounder
(Pleuronectes ferruginea) next to other larger stocks to be highly sensitive to immigration into the stock area. These studies are not
unique; others (e.g. Cadrin and Secor, 2009; Frisk et al., 2010;
Kerr et al., 2010) have also discussed similar spatial problems in
the assessment of fish populations.
Seasonal movement among two or more areas occupied at different times during an annual cycle is typical of many marine fish
species as part of their life history. These are often migrations to
one location for spawning and to another for the remainder of the
year (Cushing, 1975; Lucas et al., 2001; Jennings et al., 2009). Such
migratory patterns have been observed in the Atlantic cod (Gadus
morhua) stock complex in the Georges Bank (GB)– Gulf of Maine
(GOM) –4X region (Ames, 2004; Robichaud and Rose, 2004).
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J. Cao et al.
Figure 1. GOM – GB– Canadian 4X Atlantic cod stock complex with stock boundaries used for assessment.
This stock complex consists of three adjacent stocks that are currently assessed and managed independently: the GOM stock, the GB
stock, and the Canadian 4X stock south of Nova Scotia (Figure 1).
Some research has concluded that the between-area movement by
individuals within these populations is minimal (Robichaud and
Rose, 2004); however, tagging studies have revealed that there is at
least some exchange among them (Wise, 1963; Hunt et al., 1999;
Miller and Tallack, 2007; Miller, 2012). Thus, the “unit stock” assumption is violated, at least to some extent for each of the three independently assessed stocks.
Information on the movement of individuals like those in this
cod stock complex has regularly been collected by fishery biologists,
and model theory that includes movement has a long history (e.g.
Beverton and Holt, 1957), but this information is rarely integrated
into stock assessment or management (Caddy, 1999; Quinn and
Deriso, 1999; Goethel et al., 2011). Exceptions include Porch et al.
(1998), who used a two-stock virtual population analysis that incorporated mixing for bluefin tuna (Thunnus thynnus); Maunder
(2001), who discusses a tag-integrated catch-at-age analysis; Punt
et al. (2000), who used tagging data in assessing school sharks
(Galeorhinus galeus); and Hampton and Fournier (2001), who
extended the MULTIFAN-CL model of Fournier et al. (1998) to
include tagging data in their assessment of yellowfin tuna
(Thunnus albacares). Before shifting to a sophisticated spatially explicit assessment model, it would first be useful to determine the
extent of the mixing problem for a given stock.
This study uses the Atlantic cod stock complex in the GB –
GOM – 4X region to evaluate the potential consequences of assuming separate unit stocks when there is, in fact, exchange among
them. The specific nature of Atlantic cod movement patterns
within this region at our scale of interest is unknown. However,
there appears some seasonal directed migratory patterns associated with spawning, at least at a small scale (Bigelow and
Schroeder, 1953; Robichaud and Rose, 2004). Furthermore,
there is evidence of larger-scale movements of some individuals
(Wise, 1963; Hunt et al., 1999; Miller and Tallack, 2007; Miller,
2012). As such, we develop an assumption for the movement behaviour of the three stocks that involves some interstock mixing
during non-spawning periods and natal homing at the time of
spawning. Many alternative hypotheses exist, but given the lack
of certainty, we simply chose the one that was plausible.
Movement is described by instantaneous mixing rates among the
three stock areas estimated by Miller (2012) from tagging experiments (Table 1). Spawning time for each stock can vary, so that
the impact of assumptions such as asynchronous spawning by
stock may be tested.
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Impacts of seasonal stock mixing
Table 1. Annual instantaneous mixing rates among different stocks
estimated in Miller (2012).
To
From
GOM
GB
4X
GOM
–
0.114
0.167
GB
0.297
–
0.208
4X
0.088
0.041
–
Assuming no spatial mixing among stocks when exchange, in
fact, exists and plays an important role can cause problems in
stock assessment (Hart and Cadrin, 2004). If immigration and emigration are not accurately specified, biases may be introduced in
catch and survey abundance indices of the targeted stock, which
can confound the stock assessment analysis. This may result in
biased estimates of population size, fishing mortality, and other
population parameters. In practice, this error is usually unexplainable or unnoticed because the true values of the population
characteristics are unknown.
Simulation is a useful tool for investigating spatial problems
when actual states of nature are unknown or uncertain (Kerr
et al., 2010), and this research uses simulated scenarios with different
assumptions of stock-mixing patterns to identify mechanisms
regarding how the exchange of individuals can bias assessment
models. The simulation is parameterized to reproduce the
Atlantic cod complex in the Northwest Atlantic and includes seasonal migratory behaviour among the regions. The stock assessment
model ASAP (age-structured assessment programme), currently
employed for stock assessments of GOM cod (NEFSC, 2013), is
used to estimate population parameters from the output of the
simulation for the different scenarios. We use the GOM as the
case study for this research, so the ASAP model is run for that population only. The comparison between the ASAP model results and
the simulation values can offer clues as to how mixing can bias
stock assessment.
While these results were produced by approximating conditions
for this particular stock complex of Atlantic cod, the findings
regarding the potential effects of mixing are relevant to many
stocks of non-sedentary species that have multiple, contiguous assessment areas with boundaries across which fish may move [e.g.
yellowtail flounder or winter flounder (Pseudopleuronectes americanus) in the northeastern United States; NEFSC, 2011, 2012].
Material and methods
Metapopulation simulation
The metapopulation model consists of three populations: GB,
GOM, and 4X stocks (Figure 1). We refer to the system as a metapopulation in a general sense, meaning that it is a group of populations
among which individuals have a limited exchange over some of their
life history stages. Mixing among these three stocks is applied during
a predefined non-spawning period of the year (see the Simulation
scenarios section) using instantaneous mixing rates estimated
from tagging studies (Table 1; Miller, 2012). The age-structured
simulation operates at 96 discrete time steps within each year; and
at each time step during the mixing period, cross-boundary movement occurs. The discrete structure allows for migration during any
subset of the 96 time steps; so, stock migration can be modelled for
any defined time period within a year. Given our assumptions
regarding natal homing, individuals mix only during the non-
spawning periods of the year. When spawning of the three stocks
is not synchronized, there can be one-way exchange from one
stock to another. For each time step, the simulation follows quantities for each of the three stocks: (i) total number of fish alive at
the beginning of the time step; (ii) total number of fish removed
by the fishery within the stock region; and (iii) total number of
natal fish removed by fisheries in all stock regions. Technical
details of the simulation can be found in the Supplementary data.
Two different sets of catch data for the GOM stock, the assessment region of interest for this study, were generated: (i) apparent
GOM catch, referring to the total number of fish removed by the
GOM fishery; and (ii) simulated GOM catch, referring to the
actual number of natal GOM fish that are removed. Apparent
catch data, therefore, represent the catch as it would be observed
in actual landing statistics, and while these quantities are assigned
to the GOM region, they may represent cod belonging to all three
areas (depending on the mixing rates). The simulated GOM
removal data would not be observed in the actual case, but are available from the simulation. Those data represent the sum of landed
fish that belong to the GOM stock, no matter which area they are
removed from. The “simulated” and “apparent” terms are also
used to describe the concept as related to survey data.
The timing of three important processes needs to be defined in
the simulation: (i) time of spawning, which is used to calculate
the spawning-stock biomass (SSB); (ii) survey timings, which
are the time of year when surveys occur that determine the abundance indices; and (iii) time-blocks of natal homing for each of
the three stocks, which are used to define the stock-specific time
period when they do not migrate and reside in their natal area.
The SSB in year y (SSBGOM,y) is calculated as
SSBGOM,y =
A
SSB
NGOM,y,a
wGOM,y,a Maturity GOM,y,a ,
(1)
a=1
SSB
is the number of GOM age a fish at spawning time
where NGOM,y,a
in year y, wGOM,y,a the mean weight of age a fish in the GOM stock in
year y, and MaturityGOM,a the proportion of mature fish in age class
a for the GOM stock.
The simulated survey index is the product of total abundance in
a region (at the particular survey time), survey selectivity, and
survey catchability. Survey catch compositions are also generated.
Parallel records for GOM stock abundance at each survey include:
(i) apparent abundance, which is the number of fish in the GOM
stock region, including fish from the other two stocks; and (ii)
simulated GOM stock abundance. The “observed” survey data
were generated from the apparent abundance, as was the case for
the catch data.
Simulation scenarios
Three scenarios regarding mixing were simulated: (i) no mixing,
(ii) consistent mixing, and (iii) mismatched mixing. The no-mixing
scenario is consistent with the unit-stock assumption of the assessment model, and is a base model for comparison with the mixing
scenarios. The consistent-mixing scenario assumes that the three
stocks have the same mixing period within a year (in this example,
the second half of each year). The mismatched scenario assumes
that the periods of natal homing for the three stocks (when they
do not migrate) within a year are different. We assumed for this
scenario that natal homing periods were March, April, and May
for the GOM stock; December, January, and February for the GB
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J. Cao et al.
Table 2. Summary of biological and fishery information used in the simulation.
Parameters
Number of years
Number of ages
Number of surveys
CV of surveys
CV of total catch in weight
ESS of catch composition
ESS of survey composition
Initial stock
abundance-at-age
Maturity-at-age
Weight-at-age for catch
Weight-at-age for SSB
Recruitment
Natural mortality
Fishing mortality
Spawning time
Survey time
Survey catchability
GOM
27
9
3
0.2 (low); 0.6 (high)
0.15 (low); 0.5 (high)
75
30; 30; 15
Estimates from ASAP base model
(NESFC, 2013)
ASAP base run input (NESFC, 2013)
ASAP base run input (NESFC, 2013)
ASAP base run input (NESFC, 2013)
Estimates from ASAP base model
(NESFC, 2013)
0.2
Estimates from ASAP base model
(NESFC, 2013)
GB
27
9
–
–
–
–
–
Estimates from VPA base model
(NESFC, 2013)
–
–
–
Estimates from VPA base model
(NESFC, 2013)
0.2
Estimates from VPA base model
(NESFC, 2013)
April
April; October; April
0.9199; 0.5349; 0.1621
–
–
–
stock; and October, November, and December for the 4X stock.
These time periods of natal homing were assumed for different
stocks to demonstrate what might happen when mixing timing differed among the stocks and did not necessarily represent the true
case. Using these scenarios, we aim to examine the general effects
of mixing, including whether differences in the temporal migration
pattern among the three stocks would result in estimation biases
beyond that of the simple mixing scenario. Mixing rates used in
the three scenarios are constant over time and identical across
ages. For each simulation scenario, the catch data (i.e. total catch
and catch composition) and survey data (i.e. abundance index and
survey composition) were generated using the simulation model.
The simulation runs from 1982 to 2008. Except for mixing, all parameters and auxiliary information used in the simulation were
identical among the three simulation scenarios (Table 2).
4X
27
9
–
–
–
–
–
Estimates from base model (Clark and
Emberley, 2009)
–
–
–
Estimates from base model (Clark and
Emberley, 2009)
0.2
Estimates from base model (Clark and
Emberley, 2009)
0.2
–
–
–
Figure 2. Stock abundance for each of the three regions over the time
series. Abundance is from the baseline scenario with no mixing, but the
mixing scenarios follow the same trends.
Stochasticity
Besides using deterministic data for each scenario, random errors
were added to the simulated apparent catch and survey data in
some cases so that we could evaluate the relative importance of
data quality and ignoring stock mixing in the assessment. We
added lognormally distributed observational errors to annual
total catch and survey weight to yield an “observed” annual total
catch and an “observed” survey index. Random errors, assumed
to follow a multinomial distribution and defined by effective
sample size (ESS; e.g. Lauth et al., 2004; Table 2), were also added
to the age composition data derived from the simulation to yield
the “observed” age composition data. For each scenario, two levels
of random errors were added to the simulated catch and survey
data (Table 2). We refer to data as “high” error, when catch and
abundance index data were subject to lognormal error with a CV
of 0.6 and 0.5, respectively, and as “low” error, when the catch and
survey index were subject to errors with a CV of 0.2 and 0.15 for
survey and catch (Table 2). Therefore, the subscenarios for each
migration scenario are deterministic data, low CV data, and high
CV data.
Estimation
An assessment model usually assumes no mixing (Hilborn and
Walters, 1992; Guan et al., 2013). Thus, the no-mixing scenario
is consistent with the assessment model assumptions. However,
for the mixing scenarios, the assessment model cannot account
for the mixing that is built into the simulation and can only use
the apparent catch and survey data. For each of the stochastic
simulation scenarios, 100 sets of “observed” data were generated and used as input data for the ASAP model. The datasets
where the assessment model failed to converge were excluded.
The ASAP parameter settings are consistent with the base
model used in the 2011 Atlantic cod stock assessment (obtained
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Impacts of seasonal stock mixing
from Mike Palmer, NEFSC, Woods Hole, MA, USA) and with the
simulation.
Relative error (RE) was used as a metric to measure the disparity
between the simulation and assessment models for a particular set of
“observed” data:
N
1
RE= 100 ×
2 n=1
Estn − Simn 2
.
Simn
(2)
For a given year within a particular simulation run, relative bias (RB)
was used:
RBn = 100 ×
Estn − Simn
,
Simn
(3)
where Estn is the point estimate of the quantity in year n from the
assessment model, Simn the simulated quantity for that year, and
N the number of years. The quantities we considered in this study
were SSB, recruitment, and exploitation rate.
The error and bias in estimates of SSB, recruitment, and exploitation rate were expressed as a percentage. Exploitation rate
was calculated as the total catch in weight divided by the total
biomass at the beginning of the year. Boxplots were used to summarize the results for each year, with bold horizontal lines at the
median RE among the 100 simulations, boxes displaying the first
and third quartiles (i.e. inter-quartile range), and whiskers extending to 1.5 times the difference between the third and first quartiles
(McGill et al., 1978).
Results
Simulated metapopulation dynamics
Figure 3. Net migration into or out of the GOM by year.
The three simulated stocks, parameterized to generally follow the
observed trends in these populations, show an overall downward
trend in abundance over the modelled years. GB stock abundance
Figure 4. SSB, recruitment, and exploitation rate under the no-mixing scenario with no stochasticity for the simulated population and the
ASAP estimates.
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J. Cao et al.
Figure 5. SSB, recruitment, and exploitation rate under the consistent-mixing scenario with no stochasticity for the simulated population and the
ASAP estimates.
Figure 6. SSB, recruitment, and exploitation rate under the mismatched-mixing scenario with no stochasticity for the simulated population and
the ASAP estimates.
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Impacts of seasonal stock mixing
was high in the early years and dropped dramatically to the lowest
abundance level among the three stocks after the mid-1990s.
Abundance of the Canadian 4X stock was higher than the GOM
stock until the late 1990s, after which the two stocks maintained a
similar size until recently. In recent years, the 4X stock showed a
mild increase, while the GOM stock decreased slightly. The greatest
disparity in abundance among the three stocks occurred before
1990, especially before 1985 when the GB and 4X stocks were high.
The metapopulation dynamics from the simulation are nearly
identical under the three mixing scenarios (Figure 2), suggesting
that these mixing rates do not greatly influence the population dynamics. The only process in this simulation that can alter a stock’s
population dynamics under mixing scenarios is if a portion of the
stock is subject to a fishing mortality rate that is different from
within its natal region, and those rates do not differ dramatically
(Table 2).
Assumed migration rates from the GOM and 4X stocks to the GB
stock are higher than those from the GOM and GB stocks to 4X
(Table 1). However, the impact of migration (i.e. immigration –
emigration) on a given stock is also dependent on the relative sizes
of the stocks. While the mixing rates were unchanged over the
course of the simulation, net migration for the GOM stock
changed over time (Figure 3). During the early years in the simulation, there was more immigration than emigration; then from the
mid-1990s to 2009, emigration exceeded immigration, until the
two most recent years when net migration again became positive
(Figure 3).
and harvest rates are relatively robust to this misspecification; in
both mixing scenarios, the REs are ,5%.
The temporal migration patterns of the metapopulations largely
dictated the bias in the assessment results, especially in the estimation of SSB. The RE for SSB in the mismatched scenario is considerably higher in the consistent-mixing scenario (Table 3), suggesting
that SSB is more biased when the migration timings are offset. SSB
Figure 7. Simulation and apparent catch under the consistent- and
mismatched-mixing scenarios with no stochasticity. The largest
deviation between simulated and apparent values occurs at the
beginning of the time series under the mismatched scenario (where the
migration timing between stock units is offset).
Deterministic scenarios
An assessment using simulated data without error should produce
estimates identical with the simulation output. However, model
misspecification (e.g. ignoring mixing in this study) results in differences between the simulated and assessment-estimated values. As
expected, the estimated SSB, recruitment, and exploitation rates
are almost identical with the simulated values in the no-mixing
scenarios (Figures 4 –6). The REs are 0.1% or less (Table 3), suggesting that the discrete simulation accurately approximates the
assessment model, which is continuous. In the consistent-mixing
scenario, the GOM REs for recruitment, exploitation rate, and
SSB are 2.4, 0.8, and 5.4%, respectively. However, for the mismatchedmixing scenario, the REs for recruitment, harvest rate, and SSB
increase to 4.2, 3.9, and 17.5%, respectively. SSB estimates are
sensitive to ignoring mixing in the assessment, while recruitment
Table 3. Relative error (RE) for estimated recruitment (R), SSB, and
exploitation rate (ER) for each scenario.
Scenarios
No mixing
Consistent
mixing
Mismatched
mixing
Subscenarios R
SSB ER
R
SSB ER R
No error
0.0 0.1 0.0 2.4 5.4 0.8 4.2
Low CV
12.2 8.1 11.1 11.1 9.8 11.7 14.6
High CV
19.5 18.9 26.7 20.2 20.8 27.8 23.7
SSB ER
17.5 3.9
18.8 12.2
29.7 27.5
Notes: No error denotes no random errors added in catch and survey data.
Low CV denotes CV used in adding random errors to catch and survey data is
low. High CV denotes CV used in adding random errors to catch and survey
data is high. REs listed for low CV and high CV scenarios are the median value
of 100 runs.
Figure 8. Simulation and apparent stock abundance under the
consistent- and mismatched-mixing scenarios with no stochasticity at
the time of autumn (a) and spring (b) surveys. Consistent mixing is not
reported for the spring survey (b) because there is no mixing at that
time.
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J. Cao et al.
Figure 9. Per cent relative bias for SSB, recruitment, and exploitation rate under the no-mixing scenario with low observation error added.
No bias is apparent.
and harvest rate biases in both mixing scenarios occur in the early
years of the model time series; SSB is overestimated early on
(Figures 5 and 6). However, unlike SSB, the recruitment RE does
not change as much when the temporal migration pattern of the
metapopulation changes (from 2.4% in the consistent-mixing scenario to 4.2% in the mismatched-mixing scenario).
In the mixing scenarios, the estimation biases result from the
difference between apparent and simulated catch and survey data.
The disparity between apparent and simulated catch is large during
the first few years as a result of the relative difference in stock size
in the 1980s (Figure 7). However, the apparent catch in the early
years in the timing-mismatch scenario was further removed from
the simulated catch than that in the consistent scenario. This could
be one of the reasons why the mismatched-mixing scenario produced
higher bias than when the migration timings were held constant.
There are three GOM surveys simulated in this study: one conducted in April and the other two in October. In the consistentmixing scenario, because mixing was assumed to occur in the
second half of the year, there is no process error associated with
the spring survey. For the mismatched scenario, apparent abundance is higher than simulated abundance during both the spring
and autumn survey, although this disparity is larger during the
autumn survey than in the spring survey (Figure 8).
Stochastic scenarios
Random noise in the catch and survey data was incorporated in the
stochastic scenarios to examine the performance of the assessment
model under both observation error and process error. In the
no-mixing scenario, which has only low CV observation error,
the median REs for recruitment, SSB, and harvest rate are 12.2,
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Impacts of seasonal stock mixing
Figure 10. Per cent relative bias for SSB, recruitment, and exploitation rate under the no-mixing scenario with high observation error added. No bias
is apparent, though the range of errors is greater than in the low-observation-error scenarios.
8.1, and 11.1%, respectively, suggesting that recruitment and
harvest rate are more sensitive to observation error than SSB.
Under the consistent-mixing scenario when low observation
error is added, performance of the assessment model is close to
that of the no-mixing scenario with low CV (Table 3); the difference between these two scenarios is ,2% for all REs. However,
process error in the mismatched-mixing scenario with low observation error inflates the bias, especially for SSB estimates, which
were 10.7% greater than in the no-mixing, low CV scenario. The
same conclusion can be drawn from the high CV scenarios
(Table 3).
The year-specific RBs for recruitment, SSB, and harvest rate in
the no-mixing scenario with low CV are centred around zero
(Figure 9). Similar results were found in the high CV scenario,
only with more variation about zero (Figure 10). The CV used in
adding random error to the data is reflected in the variation of
RBs for each year (Figures 9 and 10).
For the mixing scenarios, biased patterns of SSB emerged, and
the inter-quartile range of SSB RBs for the early years did not
include zero (Figures 11 –14). This pattern is more obvious in the
mismatched-mixing scenarios (Figures 13 and 14). The process
error in the consistent-mixing scenarios does not cause any
obvious bias in recruitment (i.e. the medians of the RBs are close
to zero); however, for the mismatched-mixing scenarios, recruitment in recent years tends to be overestimated (Figures 13 and 14).
Discussion
Interstock-mixing resulting from fish movement has been recognized as a potential uncertainty associated with many fish stock
assessments (Stephenson, 1999; Fu and Fanning, 2004; Hart and
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J. Cao et al.
Figure 11. Per cent relative bias for SSB, recruitment, and exploitation rate under the consistent-mixing scenario with low observation error added.
Positive bias is apparent for SSB in the first few years, though not for recruitment or exploitation rate.
Cadrin, 2004). Because of limited information regarding mixing
and the complexity of integrating movement explicitly into assessment models, many routine fish stock assessments assume mixing
to be negligible, including the assessment of GOM Atlantic cod
(NEFSC, 2013). We used this case as an example to investigate
how stock-mixing resulting from seasonal migration affects the performance of the assessment model. Previous studies have examined
the consequences of ignoring mixing and indicated that this can lead
to misleading estimates of population status or ineffective management strategies (Fabrizio, 2005; Taylor et al., 2011; Ying et al., 2011).
However, few studies have investigated the impacts of within-year
variability in mixing timing on stock assessment. In this study, we
developed a metapopulation simulation that can incorporate seasonal mixing among stocks. The results suggest that ignoring
mixing will not bias assessment models greatly except in specific
cases, especially when the timing of movement or the relative size
of stocks differ considerably. Naturally, biases that do result are magnified with increased observation error. In cases where there was
bias, SSB was more sensitive to mis-specified mixing than to recruitment and exploitation rate. For a given mixing rate, the seasonal migration pattern of different stocks plays a critical role in determining
the degree of bias.
Non-specified stock mixing is translated to assessment model
bias through the apparent catch and survey data. This is true for
any process or observation error that biases these data, such as
changes in fish distribution or unreported catches. These systematic errors cause the catch and survey data to be non-representative,
which, in turn, results in assessment model bias. Our analyses
found that, in terms of stock mixing, bias usually worsens with increasing differences in the population dynamics of adjacent stocks
Impacts of seasonal stock mixing
1453
Figure 12. Per cent relative bias for SSB, recruitment, and exploitation rate under the consistent-mixing scenario with high observation error added.
SSB bias is apparent at the beginning of the time series, and some minor recruitment bias is evident at the end of the time series.
in parameters such as population size or fishing mortality rate.
These findings support those of previous researchers. Hart and
Cadrin (2004) found that a large difference in stock size for yellowtail flounder made the stock assessment sensitive to assumptions
regarding dispersal of larvae or adults. Lima (2012) found that
the magnitude and direction of the bias caused by mixing
depends strongly on the exploitation history of the stock.
Differences in adjacent stock areas explain why the mismatchedmixing scenarios in this paper have higher relative bias than the
consistent-mixing scenarios (Figures 7 and 8). In the consistentmixing scenarios, some of the interstock mixing is cancelled
because fish are exchanged (e.g. some GB fish move to GOM, and
some GOM fish move to GB), but under mismatched mixing at
certain times of the year, the full effect of migration from one
stock to another is realized.
Mixing probably causes some bias in the GOM cod stock assessments, as the apparent catch and survey data that are influenced by
mixing produced a biased view of the simulated population. We ran
sensitivity analyses of mixing rate for the consistent-mixing deterministic scenario. As the mixing rate increased (while maintaining
the same relative rates among all stocks), the RE (using simulated
and ASAP estimates) of SSB increased at a decreasing rate
(Figure 15). The RE derived from simulated and apparent catch at
different mixing rates is similar to the RE derived from the simulated
SSB and that estimated by ASAP. This demonstrates that the SSB RE
is directly related to error in assessment model indices such as catch
(Figure 15).
The impacts of mixing investigated in this study are restricted to
the specific seasonal migration patterns and natal homing behaviour
that we assumed. The conclusions we reached do not include the
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J. Cao et al.
Figure 13. Per cent relative bias for SSB, recruitment, and exploitation rate under the mismatched-mixing scenario with low observation error
added. Positive bias for SSB is apparent in the first few years and for recruitment in the last few years. The bias is greater than under the consistent
mixing, low CV scenario.
possibility that there is some permanent connectivity between the
three stocks; fish might not return to the area where they were
born. If this is the case, the apparent catch observed is more representative of the population, though the stock assessment model
would still have to contend with immigration and emigration.
Observation errors are certain to exist when data are collected.
Two levels of observation error reflecting the data quality were
used in this study. The results suggest that: (i) the combined effect
of observation and process error is less than the addition of those
two when they act independently (Table 3); (ii) when the seasonal
migration pattern of stocks is consistent among stock areas, the estimation bias is largely determined by data quality, i.e. process error
is less important (at least considering the migration rates and observation error CVs that we used); and (iii) when the seasonal
migration pattern of stocks is mismatched, process error derived
from movement patterns and data quality are both important.
Therefore, in addition to knowledge of mixing rates, it is also important to understand the seasonal migration patterns within a
metapopulation, as the effects of mixing can be magnified if this
timing differs substantially between stocks. In any case, good data
quality helps to alleviate some of the estimation bias.
If the consistent-mixing scenario resembles the actual circumstances describing the metapopulation dynamics and fisheries of
the GOM, GB, and 4X stocks, the current assumption (i.e. no
mixing) in the routine assessment of GOM stock is likely to have
only a minor impact on the assessment results. If the three stocks
behave closer to the mismatched-mixing scenario, bias will be
larger. However, in our analyses, most of the error occurs in the
Impacts of seasonal stock mixing
1455
Figure 14. Per cent relative bias for spawning stock biomass, recruitment, and exploitation rate under the mismatched-mixing scenario with high
observation error added. Major bias in SSB is evident at the start of the time series, and some minor bias is apparent in recruitment at the end of the
time series.
early part of the time series (possibly because the differences in
population size were more substantial); in more recent years, the
degree of bias is still fairly low (Figure 6). Possible ways to address
this that are implied by this study are: (i) adjust the apparent
fishery and survey catch based on knowledge of stock composition;
and (ii) give less weight to surveys conducted at times when mixing
is highest.
Stock identification, necessary for (i) or (ii) above, can be inferred by population parameters such as length-at-age, tagging,
morphometrics and meristics, analysis of calcified structures including microchemistry, parasite assemblages, and genetics,
among other techniques (Ihssen et al., 1981; Cadrin and Secor,
2009). Pursuing a model of stock membership can, under some circumstances, prove critical to stock assessment accuracy. Specifically,
this paper demonstrates that in cases where movement severely
biases the catch or survey stock composition, it must be emphasized to identify stock components. Besides a direct adjustment
of survey indices and catch (as noted above) so that they offer a
more representative description of the stock dynamics, another
option is to use a more complex assessment model. While assessment models that incorporate mixing explicitly have been developed and tested (see Goethel et al., 2011), model performance
under within-year variability in mixing is unknown (Taylor
et al., 2011), and this study suggests that seasonal variability can
be important. Another important issue for GOM cod stock assessment is that, in addition to the mixed-stock structure of the three
stocks, there is further population structure within the GOM
(Ames, 2004).
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J. Cao et al.
References
Figure 15. Sensitivity analysis for mixing rate. As mixing rate increases
(i.e. the instantaneous migration rates are multiplied by a scalar),
relative error in spawning stock biomass and catch both increase,
though at a decreasing rate. This shows how the input data can drive the
results of the assessment model.
In this study, we did not assume age-specific mixing rates for each
of the stocks, because there was no available information, though
age-varying migration probably occurs. Juveniles are less likely to
migrate as far as adults, but they are also less likely to be removed
by the fishery due to selectivity. Therefore, we do not expect that agespecific mixing would alter the apparent catch very much, although
surveys could be affected more, given that they typically use smaller
mesh sizes. This could affect estimates of fishery selectivity and some
biological parameters. Another assumption we made regarding
mixing is that the mixing rates do not change over time. We lack information to set up year-varying mixing rates; however, our sensitivity analysis that incorporated stochasticity in mixing rates
among years did not show much difference in bias, so time-varying
mixing is not expected to change our conclusions.
It would be informative to examine bias in reference points and
relative stock status as a function of mixing, as this might also be of
interest to stock assessment scientists and managers. However, there
is no standard method for reference-point calculation that has the
capability of considering mixing when calculating reference
points. Consequently, this study does not report the impacts of
mixing on reference-point calculations, but this should be a topic
of future research.
Supplementary material
The following supplementary material is available at the ICESJMS
online version of the manuscript: details of the discrete
age-structured simulation.
Acknowledgements
Financial support for this study was provided by the Maine Sea
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Handling editor: Emory Anderson