Fisheries Research 72 (2005) 173–183
An application of the composite risk assessment method
in assessing fisheries stock status
Yan Jiaoa,∗ , Yong Chenb , Joe Wroblewskia,c
a
c
Biology Department, Memorial University of Newfoundland, Nfdl, Canada A1B 3X9
b School of Marine Science, University of Maine, Orono, ME 04469, USA
Ocean Science Center, Memorial University of Newfoundland, Nfdl, Canada A1B 3X9
Received 18 September 2003; received in revised form 2 November 2004; accepted 11 November 2004
Abstract
The status of a fishery stock is often determined by comparing an indicator reference point (e.g., current fishing mortality)
with a management reference point (e.g., F0.1 ). Both references are likely subject to large uncertainty. Thus, it is necessary to
incorporate such uncertainties in determining the status of fisheries. The composite risk assessment method, which is commonly
used in civil engineering, can be used to quantify uncertainty associated with both indicator and management references in
evaluating the status of a fishery. Ignoring either part of the uncertainty may result in overexploitation in fisheries management.
We applied this method to the Atlantic cod (Gadus morhua) fishery of eastern Newfoundland and Labrador. The results suggest
that the uncertainties in both indicator and management reference points can influence the evaluation of the fishery stock status.
Uncertainty can come from different sources and is difficult to quantify. We suggest conducting a sensitivity analysis to evaluate
the relative importance of uncertainties resulting from different sources. Such an analysis will enable one to identify key factors
influencing the assessment of stock status and management.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Composite risk assessment; Management reference point; Indicator reference point; Sensitivity analysis
1. Introduction
The collapse of many fisheries worldwide has been
attributed to overexploitation resulting from gross over∗ Corresponding author. Present address: Department of Zoology,
University of Guelph, Guelph, Ont., Canada A1G 2W1.
Tel.: +1 709 737 2502/519 824 4120x58555; fax: +1 519 7671656.
E-mail addresses: [email protected], [email protected]
(Y. Jiao).
0165-7836/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.fishres.2004.11.003
estimation of stock size, spatially extended fisheries,
increased fishing-fleet capacity, lack of management
control, and large scale bycatch, discarding at sea,
failed recruitment, and increased natural mortality due
to changes in ecosystems, which are all related to uncertainties of subjective or objective kind (Jackson et
al., 2001; Pauly et al., 2002; Christensen et al., 2003;
Baum et al., 2003). Ignoring or underestimating uncertainty in stock assessment and fisheries management
174
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
is an important factor that may contribute to the decline of fisheries (Hilborn and Walters, 1992; Ludwig
et al., 1993; Walters and Maguire, 1996). The decline
in stocks calls for fisheries management approaches
such as reducing fishing capacity, establishing marine
protected areas, and developing precautionary management plans and ecosystem-based management approaches (Pauly et al., 2002, 2003; Baum et al., 2003).
At the same time the widely discussed impacts of uncertainty on stock assessment and management calls for a
close examination of the role that uncertainty plays in
determining the status of a fish stock and in managing
fisheries to reduce the likelihood of overexploitation
(Ludwig et al., 1993; Myers and Worm, 2003).
The status of a fish stock is often determined by
comparing an indicator reference point estimated from
stock assessment (e.g., current fishing mortality, current stock biomass) with a management reference point
(e.g., F0.1 , Bmsy ; Caddy and Mahon, 1995). Replacing
a currently used management reference point with a
more conservative value to offset the impacts of uncertainty may bias the choice of management reference
points and cause fisheries stakeholders to distrust fisheries management plans and stock assessment. A better
approach would be to place emphases on risk analysis
and choice of risk tolerance, which incorporates the uncertainty in assessing the status of fish stocks (Shelton
and Rice, 2002). Uncertainty in assessing a fishery
may result from various sources such as measurement
error, process error, model error, and operating error
(FAO, 1996; Chen and Paloheimo, 1998; Patterson et
al., 2001). The above uncertainties can further be divided into two groups that are fundamentally different
from each other, i.e., the objective uncertainty arising
from variability of the underlying stochastic system,
and the subjective uncertainty resulting from not having
complete information about the system (Casti, 1990;
Ferson and Ginzburg, 1996). An approach commonly
used to incorporate uncertainty in assessment is to estimate the empirical probability distribution of indicator
reference points using methods such as bootstrap and
jackknife. The management reference points are calculated based on the fisheries data and life history parameters using per-recruit models, production models, or
stock-recruitment models and are often assumed to be
deterministic when they are compared with indicator
reference points. This assumption is rather unrealistic considering large variations that may exist in mod-
els, data and ecosystems (Fogarty et al., 1996; Prager
et al., 2003). Thus, management advisory statements
derived using such an approach of comparing deterministic management reference points with stochastic
indicator references do not reflect the fact that both the
indicator and management reference points are subject
to considerable uncertainty, and may yield erroneous
conclusions about the status of fish stocks. A general
approach should consider uncertainty in both indicator
and management reference points (Helser et al., 2001;
Chen and Wilson, 2002; Prager et al., 2003).
Composite risk analysis is a method of accounting for the risks resulting from various sources of uncertainty to produce an overall risk assessment for a
particular decision-making problem (Yen, 1986). The
composite risk analysis allows for the incorporation of
uncertainty in both indicator and management reference points in determining the status of a fishery stock.
Compared with the approaches of Helser et al. (2001)
and Chen and Wilson (2002), which decompose the
uncertainty of one reference point into a deterministic
value and a confidence interval, and then estimate the
risk at a certain confidence interval, composite risk assessment provides an easy and direct estimate of the
risk for overexploitation with full consideration of the
uncertainties. Prager et al. (2003) used a similar approach to developing the target reference point based
on the limit reference point. In Prager’s approach, the
risk assessment is regarded as a weighted integration,
while the weight is derived from the complementary
cumulative distribution function. We used a double integration function, which shows the exact scientific
meaning of composite risk. We developed a discrete
approach to estimate the composite risk, which is convenient to incorporate nontypical distributions and can
avoid the influence from covariance between the two
distributions of indicator and management reference
points. This approach in solving the risk calculation
is different from Prager’s approach of ratio-extended
probability approach to setting targets (REPAST).
For many fisheries there is a lack of understanding
about the uncertainty in input data, variations in life history and population dynamics of fish stocks. Thus, it
is important to evaluate the impacts of imprecise quantification of uncertainty of data in estimating reference
points on the risk assessment and to identify key parameters influencing the determination of fisheries status.
This calls for a sensitivity analysis, which refers to the
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
variation in output of a mathematical model with respect to perturbations in the values of the model’s input.
A sensitivity analysis attempts to provide a ranking of
the model’s input assumptions with respect to their contribution to model output variability or uncertainty. In a
broader sense, sensitivity can refer to how conclusions
may change if models, data, or assessment assumptions
are changed. A Monte Carlo simulation approach was
used in this study for the sensitivity analysis, which allows for the systematic evaluation of model output with
respect to uncertainty in input data. Uncertainty was
incorporated into the simulation process in the forms
of measurement and process errors. All uncertainties
in input data, and parameters were examined to determine their effects on the output distributions through
numerical experiments (scenarios). By comparing the
differences in biological reference points calculated under different uncertainty levels, we could evaluate how
a reference point responded to changes in a particular
life history process (growth, recruitment, or mortality).
This helps identify important pathways and parameters
through which assumptions about distributional functions contribute significantly to overall uncertainty and
aid in focusing data gathering efforts.
The composite risk analysis and associated sensitivity analysis were applied to the Atlantic cod (Gadus
morhua) fishery in NAFO divisions 2J3KL. The importance of incorporating uncertainty in both indicator and management reference points is discussed. Key
factors that may influence the composite risk analysis
were identified. This study provides an alternative approach for risk analysis in determining fisheries stock
status.
Growth parameters and their variations are calculated from the mean weight-at-age data obtained from
the Canada Department of Fisheries and Oceans (DFO)
(Lilly et al., 2001). The weight–age model, which
combines the allometric weight–length model and von
Bertalanffy length–age model, was used to estimate
weight at age Wt and their corresponding variation
(Quinn and Deriso, 1999),
b
Wt = W∞ [1 − e−k(t−t0 ) ] ,
where W∞ is the estimated maximum attainable
weight; k is the Brody growth parameter, and t0 is
the hypothetical age at which the weight is 0 (Ricker,
1975). A residual diagnostic analysis showed that the
residuals of Wt in modeling followed lognormal distribution. Since the decline in cod abundance, studies
have shown an increased variation in growth of the cod,
calling for the consideration of uncertainty in stock
assessment. The estimated parameter values based
on the weight-at-age data of cod are W∞ = 44.0597;
k = 0.0423; t0 = −0.9955; b = 2.5093.
The information on proportion of cod caught at
age t that are discarded at sea, Dt , for 2J3KL was
from Kulka (1996). We modeled the relationship between age-specific discarding using the following logistic equation:
Dt =
(1)
NDt
1
=
,
d(t−d
50 )
Ct
1+e
(2)
where NDt is the number of fishes caught at age t but
discarded at sea, Ct is the total number of fishes caught
at age t including both landed and discarded catches,
d is the shape parameter, d50 is the age at which 50%
of fish are vulnerable to be discarded. Parameters d
and d50 were assumed to be time-independent. Estimated parameters for D were d = 4.05; d50 = 2.56. The
residual diagnostic analysis suggested that the residuals followed lognormal distribution in modeling. We
thus defined D in Eq. (2) following lognormal distribution with standard deviation shown in Table 1. D was
truncated to have values between 0 and 1 during the
random sampling process.
Age-specific selectivity was modeled using the following logistic equation:
St =
2. Materials and methods
175
1
,
1 + e−m(t−S50 )
(3)
where m is the shape parameter, S50 is the age at which
50% of the individuals are vulnerable to the fishing
gear. m and S50 were assumed to be time-independent.
Here, we used m = 1.5, S50 = 3.4 (Fu et al., 2001). We
also assumed that S followed lognormal distribution
with standard deviation shown in Table 1. S was truncated to have values between 0 and 1 during the random
sampling procedure.
Natural mortality (M) was assumed to follow lognormal distribution with a mean of 0.2 and standard
error of 0.1, 0.2 and 0.4 to show low, median and high
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Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
Table 1
Parameters and models with uncertainty considered in the F0.1 and Fmax estimate, and the simulation scenarios included in the study
Scenarios
1
2
3
4
5
6
7
8
Medium-variation scenario (most realistic scenario)
High-variation scenario
Low-variation scenario
High selectivity coefficient variation scenario
High growth-variation scenario
High discarding-variation scenario
High M-variation scenario
High M scenario (M̄ = 0.4)
Standard deviation of the parameters
M (year−1 )
St
Wt (kg)
Dt
0.2
0.4
0.1
0.2
0.2
0.2
0.4
0.4
0.2
0.4
0.1
0.4
0.2
0.2
0.2
0.2
0.28
0.4
0.15
0.28
0.4
0.28
0.28
0.28
0.4
0.6
0.2
0.4
0.4
0.6
0.4
0.4
See the text for the definition of F0.1 and Fmax .
variations of natural mortality (Table 1). The lognormal distribution is often used for many fish life history
parameters including growth and mortality parameters
(Hilborn and Walters, 1992; Shelton, 1992; Quinn and
Deriso, 1999).
The exploitation rate estimated based on tagging
data varied between areas and years. The averaged exploitation rates in 1999 and 2000 in 3KL area were
estimated to be 43 and 12%, respectively (Cadigan and
Brattey, 2002). These estimates were consistent with
the exploitation rates estimated from landings and survey data in 1999 and 2000 (i.e., 0.29 for 1999 and 0.12
for 2000). The average exploitation rates in different
tagging areas were weighted by experiment size (DFO,
2002). We used the weighted mean exploitation rate of
all the areas as the mean exploitation rate in the year.
The exploitation rate was assumed to follow a normal
distribution defined by N(0.24, 0.042 ) for 1999, N(0.10,
0.012 ) for 2000 (DFO, 2002). The N(µ, σ 2 ) above define a normal distribution with the mean of µ and variance σ 2 . The corresponding coefficients of variation of
the exploitation rates were about 10–16%. The estimate
did not consider variation in exploitation rate in the
unit tagging area and uncertainty resulting from natural mortality, tagging loss, and gear selectivity. We then
increased the coefficient of variation to 25–30%. The
corresponding fishing mortality would be drawn randomly from the distributions defined by N(0.30, 0.082 )
for 1999, and N(0.12, 0.032 ) for 2000 based on the tag
loss (2.8%), natural mortality rate (9.5%) and exploitation rates. For Scenario 8 of the simulation, a large natural mortality, 0.4, was used. The corresponding fishing
mortality rate was then drawn from N(0.34, 0.092 ) for
1999, and N(0.13, 0.032 ) for 2000 for Scenario 8.
2.1. Composite risk analysis
Let X represent a management reference point (e.g.,
F0.1 ) estimate and Y represent an indicator reference
point (e.g., current fishing mortality Fcur ) estimate;
then the corresponding probability distribution functions (pdf) of X and Y were f(x) and g(y), respectively.
If indicator Y is deterministically quantified without
uncertainty, then
y
P(Y > X) =
f (x) dx.
(4)
−∞
If X is deterministically quantified without uncertainty,
then
+∞
P(Y > X) =
g(y) dy.
(5)
x
Eq. (5) is commonly used in assessing the status of the
stock (Caddy and Mahon, 1995).
If both X and Y are not deterministic, the probability
of Y > X can be calculated as:
+∞ +∞
P(Y > X) =
g(y) dy f (x) dx
=
=
=
−∞
+∞
−∞
+∞
−∞
+∞
−∞
= 1−
x
G(y)|+∞
x f (x) dx
{1 − G(x)}f (x) dx
f (x) dx −
+∞
−∞
+X
−∞
G(x)f (x) dx
G(x)f (x) dx,
(6)
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
where G(y) is the cdf of Y. This can be explained as a
double integral as in Calculus.
+∞ The probability of Y > X
is first integrated along Y x g(y) dy by assuming that
X is deterministic, and then integrated along X again by
considering the uncertainty of X. This equation shows
the risk of overexploitation, or the probability of the
indicator (i.e., Fcur ) being larger than the management
reference point (i.e., F0.1 ). In most of the cases the
distribution of X and Y do not follow a standard distribution, and they are not continuous as shown in the
above equation, because both of them or one of them
are usually estimated from Monte Carlo simulations.
In these cases, the above equation can be written as a
discrete type.
+∞
+∞ P(Y > X) =
g(y) y f (x) x
x
−∞
=
+∞
G(y)|+∞
x f (x) x
−∞
=
+∞
ing from variations in biotic and abiotic environment.
In this study some parameters and their variations (e.g.,
natural mortality, fishing selectivity) used in estimating
F0.1 and Fmax were not derived from field data. Thus,
the realism of these values could be questioned. Thus,
we need to run a simulation, which involves running
a large range of values for the model parameters in
deriving F0.1 and Fmax .
A commonly used discrete YPR model can be written as
Y=
tl
{1 − G(x)}f (x) x
= 1−
+∞
G(x)f (x) x.
(7)
−∞
Replacing X and Y with F0.1 and Fcur in the above equations, we can estimate the composite risk. Because both
the F0.1 and Fcur were estimated as a sample of a random distribution generated from the Monte Carlo simulation (i.e., empirical distributions), we estimated the
composite risk with Eq. (7).
2.2. YPR model
F0.1 and Fmax are estimated using the yield-perrecruit (YPR) model, which calculates the average
yield to be expected under a given pattern of fishing
mortality over the life span of a cohort of fish. The YPR
model is defined by parameters defining life history and
fishery processes including growth, natural mortality,
gear selectivity, and discarding. The uncertainty in estimating F0.1 and Fmax using the YPR model may come
from the following sources: uncertainty in the model
parameters, which are often estimated from other studies, and natural variability of life history process result-
Ct Wt (1 − Dt ),
(8)
t=tr
where Y is the attained yield, tr is the age of entry into
the fishery, and tl is the maximum age of fish that could
contribute to the fishery. The tr was set at 3, and tl = 20
for the cod fishery. Given natural mortality, fishing mortality, and selectivity coefficients, catch-at-age Ct can
be calculated from the catch equation as:
Ct = Nt
−∞
177
St F
(1 − e−St F −M ),
St F + M
(9)
where F is the fishing mortality. Nt is the number of
fish at age t at the beginning of the year. Nt is estimated
from recruitment R as follows:
−
Nt = Re
t−1
j=tr
(sj F +M)
.
(10)
Combining the Eqs. (8), (9) and (10), we get the YPR
model as:
tl
Y
=
R
t=tr


t−1




−
(sj F +M)
W s F

t t
−st F −M
j=tr
×
)e
(1−Dt ) .
(1−e


s F +M


t

(11)
The F0.1 can be estimated from:
∂(Y/R) ∂(Y/R) =
0.1
F =0 .
∂F F =F0.1
∂F (12)
The Fmax can be calculated from:
∂(Y/R)
= 0.
∂F
(13)
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Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
We used the following procedure to estimate the uncertainty for the F0.1 and Fmax : (1) identify the parameters
that are likely to have uncertainties in the YPR model;
(2) identify the magnitude and nature of errors for each
model parameter identified in step (1); (3) randomly
sample each parameter from its probability distribution; (4) apply the sampled model parameters to the
YPR model to calculate F0.1 and Fmax (Eqs. (12) and
(13)); (5) repeat steps (3) and (4) for N times to yield N
estimates of F0.1 and Fmax and (6) estimate the probability distribution of F0.1 and Fmax using the results
derived in step (5).
Eight simulation scenarios were considered in this
study (Table 1). Latin hypercube sampling was used
to avoid unrealistically large or small values. The first
three scenarios had medium, high, and low levels of
variations for model parameters, respectively (Table 1).
Scenarios 4 through 7 included high variations for
selectivity coefficient, growth, discarding and natural
mortality (the magnitudes of the corresponding parameters were the same as those for Scenario 2), while variations for other parameters were the same as those for
Scenario 1. The difference in estimating F0.1 and Fmax
between these scenarios and Scenario 1 illustrated the
importance of variability in those parameters. Because
of the speculations of a higher cod natural mortality
possibly resulting from climate changes and seal predation, a high natural mortality scenario was explored to
identify possible impacts of increased natural mortality
on composite risk analysis (i.e., Scenario 8; Table 1).
The difference in the estimates of F0.1 and Fmax between the eighth and first scenarios thus reflected the
relative importance of natural mortality in estimating
uncertainty for F0.1 and Fmax .
For the first seven scenarios, 800 simulation runs
were used to derive the stable probability distribution
function for F0.1 and Fmax . For the high-natural mortality scenario (i.e., Scenario 8; Table 1), 1000 simulation
runs were needed to yield stable probability distribution
functions for F0.1 and Fmax . For each simulation run,
the same randomly sampled values of model parameters from assumed distributions were used. Boxplot
was used to show the estimates of F0.1 and Fmax in the
simulation. It produced a box and whisker plot for F0.1
and Fmax estimates out of the 800 or 1000 simulation
runs. The box has lines at the lower 25 quartile, median,
and 75 upper quartile values. The distance between the
top and bottom of the box is the interquartile range (1.5
here). The whiskers are lines extending from each end
of the box to show the extent of the rest of the data.
The plus symbol at the top of the plot is an indication
of an outlier in the data. An outlier is a value that is
more than 1.5 times the interquartile range away from
the top or bottom of the box in this plot. If there are no
data outside the whisker, a dot is placed at the bottom
whisker.
3. Results
The mean of the F0.1 estimates using the data with
medium level of variation (i.e., Scenario 1) was 0.146.
The probability distribution of F0.1 in this scenario
shown as boxplot type had 90% probability intervals
ranging from 0.109 to 0.198. The probability distributions for Fcur and F0.1 did, however, share some overlaps. The composite risk assessment suggested that the
probability of the current fishing mortality being higher
than F0.1 was 0.97 in 1999.
The distribution of F0.1 derived for the highvariation scenario (i.e. Scenario 2) differed from that
for the medium-variation scenario with long tails and
different mean. Compared with those for the mediumand high-variation scenarios, variation in F0.1 was
lower for the low-variation scenario (i.e., Scenario 3;
Fig. 1).
The high selectivity-variation, high growthvariation and high discarding-variation scenarios (i.e.,
Scenarios 4, 5 and 6, respectively) had a level of
uncertainty for selectivity, growth and discarding as
high as those in high-variation scenario (i.e., Scenario
2), while other parameters had uncertainty levels same
as those for medium-variation scenario (i.e., Scenario
1; Table 1). The probability distributions of F0.1 for
Scenarios 4, 5 and 6 were similar to that for Scenario
1 both in modes and in tails (Fig. 1).
The level of uncertainty for natural mortality in Scenario 7 was the same as that for the high-variation scenario (i.e., Scenario 2), while other parameters were
the same as those for the medium-variation scenario
(i.e., Scenario 1). The probability distribution of F0.1
for this scenario was different from that for Scenario
1, but similar to that for Scenario 2 (Fig. 1). These
results suggest that the uncertainty in natural mortality is the main contributor to the estimated uncertainty
for F0.1 and uncertainties of other parameters (growth,
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
179
Fig. 1. Boxplot of the estimated F0.1 and Fmax in the simulation study of the eight scenarios. See text for the definition of F0.1 and Fmax .
discarding, and selectivity) had limited influence on the
estimated uncertainty of F0.1 .
When the higher natural mortality scenario was
used, the probability distribution of F0.1 for this scenario was different from the first seven scenarios both
in the central tendency and in the tails of the distributions (Fig. 1). The estimated mean of the F0.1 was
0.34, which was more than twice the values estimated
for other scenarios.
The risks of Fcur > F0.1 in 1999 were near to 97–98%
in Scenarios 1, 4, 5 and 6 (Fig. 2). They were near to
92–93% in Scenarios 2 and 7. This implied that the
overexploitation risks were lower in the high-variation
scenario and high M-variation scenario in this year.
It also implied that the risk of overexploitation was
sensitive to the variation of natural mortality.
The average of Fcur in 2000 was smaller than the
mean and median of F0.1 estimated for the first seven
scenarios when natural mortality was assumed to be
0.2. The composite risk assessment suggested that
there was a 37–45% of probability of overexploitation
(Fig. 3). In the high-natural mortality scenario the F0.1
estimate was larger, so the P(Fcur > F0.1 ) was only 10%.
The probability distribution of Fmax summarized by
boxplots in different scenarios showed similar trends
as F0.1 . The probability distribution of Fmax for Scenarios 1, 4, 5 and 6 were also similar. The probability distribution of Fmax in Scenarios 2 and 7, both
of which have larger variations, were similar (Fig. 1).
When higher natural mortality scenario was used, the
probability distribution of Fmax was different from
the first seven scenarios both in central tendency and
in the tails. The P(Fcur > Fmax ) estimated using composite risk assessment suggested that when natural
mortality was assumed to be 0.2, the P(Fcur > Fmax )
(see the plus symbols in the Fig. 2 and Fig. 3) was
about 70–80% for 1999 (Fig. 2); and 8–25% for 2000
(Fig. 3). Among them Scenario 3 resulted in the risk of
8%, the smallest and mostly approximate the result of
P(Fcur > F̂max ) because of the lower variation of Fmax
in this scenario and the larger difference between F̂max
and F̂cur .
Ignoring the uncertainty of management reference
point resulted in errors in risk assessment. In 1999,
when F0.1 was used as the management reference point,
ignoring uncertainty of F0.1 resulted in overestimation
of risk for all the scenarios (P(Fcur > F̂0.1 ), see the
cross symbols in Fig. 2); when Fmax was used as the
management reference point, ignoring uncertainty of
Fmax resulted in overestimation of risk in Scenarios 1,
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Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
Fig. 2. Estimated risk in 1999. (a) When F0.1 was used as the management reference point. Plus sign: P(Fcur > Fmax ); cross: P(Fcur > F̂0.1 );
circle: P(F̂cur > F0.1 ). (b) When Fmax was used as the management reference point. Plus sign: P(Fcur > Fmax ); cross: P(Fcur > F̂max ); circle:
P(F̂cur > Fmax ). The risk when Scenario 8 is used is not shown here because of the scale difference with the other scenarios. It is stated in the
text.
Fig. 3. Estimated risk in 2000. (a) When F0.1 was used as the management reference point. Plus sign: P(Fcur > F0.1 ); cross: P(Fcur > F̂0.1 );
circle: P(F̂cur > F0.1 ). (b) When Fmax was used as the management reference point. Plus sign: P(Fcur > Fmax ); cross: P(Fcur > F̂max ); circle:
P(F̂cur > Fmax ). The risk when Scenario 8 is used is not shown here because of the scale difference with the other scenarios. It is stated in the
text.
4 and 7, but underestimation of risk in Scenarios 2, 5
and 6 (P(Fcur > F̂max ), see the cross symbols in Fig. 2).
In 2000, either when F0.1 or when Fmax was used as
the management reference point, ignoring uncertainty
of F0.1 and Fmax resulted in underestimation of risk in
all scenarios (see the cross symbols in Fig. 3).
Ignoring the uncertainty associated with indicator
reference point could also result in errors in risk assessment. In 1999, when F0.1 was used as the management reference point, ignoring uncertainty of indicator reference point Fcur resulted in overestimation of
risk for all scenarios (P(F̂cur > F0.1 ), see the circles
in Fig. 2); when Fmax was used as the management
reference point, ignoring uncertainty of Fcur resulted
in overestimation of risk in Scenarios 1, 3, 4 and 6,
but underestimation of risk in Scenarios 2, 5 and 7
(P(F̂cur > Fmax ), see the circles in Fig. 2). In 2000,
either when F0.1 or when Fmax was used as the management reference point, ignoring uncertainty of Fcur
resulted in underestimation of risk in all scenarios (see
the circles in Fig. 3).
4. Discussion
This study suggests that incorporating uncertainty of
the indicator reference point and the management reference point into risk assessment of fisheries is important,
and may influence the estimation of overexploitation
risk and determination of stock status. Ignoring uncertainty may lead to incorrect conclusion about the status
of the stock.
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
This study also suggests that the magnitude of the
uncertainty in the parameters of the YPR model could
influence the estimation of P(Fcur > F0.1 ), thus affecting the conclusion as to whether the fishery is overfished. Higher uncertainty in the model parameters resulted in higher uncertainty in the estimate of F0.1 .
By comparing the results of the medium-, high-, and
low-variation scenarios considered in this study, we
found that large uncertainty of parameters would reduce P(Fcur > F0.1 ) when F̄cur > F̄0.1 , and thus is less
likely to make a conclusion that the stock was overfished (the case for 1999). We also found that large uncertainty of parameters would increase P(Fcur > F0.1 )
when F̄cur < F̄0.1 , and thus is more likely to make a
conclusion that the stock was overfished (the case for
2000). The uncertainty associated with current fishing
mortality Fcur is important in determining the status of a
fish stock. It can be readily predicted that P(Fcur > F0.1 )
decreases with increasing uncertainty in Fcur when
F̄cur > F̄0.1 and that P(Fcur > F0.1 ) increases with increasing uncertainty in Fcur when F̄cur < F̄0.1 . For example, if we increase variations associated with Fcur
from current CV of 25–30% to CV = 40%, for Scenario
1, P(Fcur > F0.1 ) changed from 0.97 to 0.94 for 1999,
and 0.38 to 0.41 for 2000, respectively (Table 2). To
verify the above conclusion, risk of F̄cur > F̄0.1 was
calculated when both F0.1 and Fcur followed normal
distribution and had different CVs varying from 20 to
181
Table 2
Estimated probability of current fishing mortality (Fcur ) larger than
F0.1 , P(Fcur > F0.1 ) when increased uncertainty with Fcur was used
for the medium-variation scenario (i.e., Scenario 1, Table 1)
Year
F̄cur (year−1 )
σ
P(Fcur > F0.1 )
1999
0.30
σ 1 = 0.08
σ 2 = 0.12
0.97
0.94
2000
0.12
σ 1 = 0.03
σ 2 = 0.048
0.38
0.41
σ 1 is the standard deviation of Fcur with lower uncertainty, which
is the same as used in estimating Figs. 2 and 3. σ 2 is the standard
deviation of Fcur with higher uncertainty.
40% (Fig. 4). The results were consistent with those
derived for the cod fishery. Because Fcur is often overestimated due to retrospective problems (Bishop and
Shelton, 1997; Shelton and Rice, 2002), the consideration of this uncertainty should be the focus of future
studies.
The probability distribution of F0.1 for the high natural mortality variation scenario was different from
that for Scenario 1 but similar to that for Scenario 2
(Fig. 1), suggesting that the uncertainty in natural mortality was the main contributor to the estimated uncertainty for F0.1 and was most important in determining
the uncertainty of F0.1 . The consistency of Scenarios
4, 5 and 6 with the medium-variation scenario suggests that the uncertainty in selectivity, growth, and
Fig. 4. Risk of Fcur > F0.1 when both F0.1 and Fcur follow normal distribution and have different CVs ranging from 20 to 40%. Abscissa is the
mean of F0.1 divided by the mean of Fcur .
182
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
discarding parameters is not the main contributor to the
estimated uncertainty for F0.1 . Thus the estimate of F0.1
is less sensitive to uncertainty in these parameters. This
result is consistent with previous studies (Chen and
Wilson, 2002). Good estimation of natural mortality
and its associated uncertainty is thus critical in assessing the status of a fishery. Natural mortality perhaps is
one of the most difficult parameters to be estimated, and
is likely to vary greatly with the change in the ecosystem, indicating natural mortality is likely to have large
uncertainty. This is also the case for northern cod (G.
morhua) (Myers and Doyle, 1983). Recent variations
in the cod life history and environmental changes increased the uncertainty of natural mortality estimation
(Dutil and Lambert, 2000; Hutchings, 2004). Thus, incorporation of uncertainty in natural mortality in the estimation of indicator and management reference points
should become mandatory in stock assessment and
management.
Quantification of uncertainty in life history and fishery parameters is an essential step in estimating uncertainty of F0.1 and Fcur . Uncertainties can arise from
the variation in the statistical estimation of the parameters and natural variability in the parameters between
cohorts and geographic areas (Ferson and Ginzburg,
1996). However, in practice it is unlikely that “correct”
values can be identified to define the uncertainty for
the parameters. A sensitivity analysis through a Monte
Carlo simulation can help overcome this problem. In
the simulation, different levels of uncertainty are considered for each parameter. The ranges of uncertainty
for each parameter evaluated in the simulation were
rather large, reflecting our efforts to cover all possible ranges of uncertainty within them. The comparison
of results derived from different levels of uncertainty
would help us identify the sensitivity of the input parameters to the output reference point distribution and
provide us with enough evidence in identifying impacts
of uncertainty in reference points on the determination
of the stock status and its possible management implications in decision making. In Fig. 4, when a risk of 5%
is used in a precautionary approach and when F0.1 is
used as the management reference point, the possible
fishing mortality used as the management target (i.e.,
future Fcur ) would differ when CVs of Fcur and F0.1
were different. When CVs of both Fcur and F0.1 were
between 20 and 40%, the Fcur should be between (1/3
and 1/1.62) of F0.1 .
The composite risk assessment method assumes
the independence between f(x) and g(y) in Eqs.
(6) and (7). Because of the difficulty in estimating
the covariance structure between f(x) and g(y) and
complex interactions between parameters in the
two functions, the covariance is often ignored in
risk assessment. The formulation of composite risk
assessment cannot be extended from Eqs. (6) and (7) to
incorporate the covariance between f(x) and g(y). One
method of considering the covariance currently used
in risk assessment in other fields is the Monte Carlo
method, which often requires unjustified assumptions
(Bratley et al., 1983; Haas, 1999). Dependency bounds
suggested to be used in risk assessment (Ferson and
Ginzburg, 1996) might be applied in fisheries in
the future when comparing the indicator reference
point and the management reference point. However,
the dependency bounds method has limitations in
uncertainty estimation. For example, the uncertainty
estimation of the F0.1 depend on many assumptions,
such as independency of the parameters in the YPR
model (Eq. (11)) in the Monte Carlo simulation.
We suggest using the composite risk analysis
method to assess the risk resulting from uncertainty
in developing precautionary approaches and in managing the fisheries in the current stage. Further studies that
can incorporate correlations in risk analysis are needed.
A precautionary approach considering uncertainty in
both the management reference point and the indicator
reference point is needed to prevent overexploitation
and to promote the recovery of the overexploited fisheries. Because of lack of considerations of correlations
in risk analysis, a conservative acceptable risk level is
suggested in fisheries decision making though composite risk assessment has considered uncertainty in both
management reference point and in indicator reference
point.
Acknowledgements
Y. Jiao was supported by a Canadian International
Development Agency/Memorial University of Newfoundland Marine Science Scholarship. Partial financial assistance was provided by the “Coasts Under
Stress” (CUS) research project. CUS is funded by
the Social Sciences and Humanities Research Council of Canada (SSHRC), and the Natural Sciences and
Y. Jiao et al. / Fisheries Research 72 (2005) 173–183
Engineering Research Council of Canada (NSERC),
through the SSHRC Major Collaborative Research Initiatives (MCRI) Program. Support from the Maine Sea
Grant and Maine Department of Marine Resources to
Y. Chen is appreciated. We would like to thank Nichole
Stenberg for editing the paper. We gratefully acknowledge two anonymous reviewers for providing constructive comments, which improved the manuscript.
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