ICES Journal of Marine Science, 55: 1061–1070. 1998
Article No. jm980349
A sustainability criterion for the exploitation of North Sea cod
R. M. Cook
Cook, R. M. 1998. A sustainability criterion for the exploitation of North Sea cod. –
ICES Journal of Marine Science, 55: 1061–1070.
The spawning stock biomass of cod, Gadus morhua, in the North Sea has been in
decline for many years. There is evidence that at low stock sizes recruitment is reduced.
Using non-parametric methods, equilibrium spawning stock biomass and yield curves
are constructed, which are consistent with annual observations that show the continuous decline in the spawning stock can be explained by high fishing mortality rates. A
new reference point, Gloss, is defined, which corresponds to the stock replacement line
at the lowest observed spawning stock size. Recent fishing mortality rates yield
replacement lines with a high probability of exceeding Gloss and therefore indicate
there is a high probability of further stock decline and a possibility of collapse. The
replacement line is very sensitive to fishing mortality rate which makes managing the
stock at high rates of exploitation risky. A reduction in fishing mortality rate by
approximately 30–40% is required to ensure the probability of collapse is small.
1998 International Council for the Exploration of the Sea
Key words: North Sea, cod, sustainability.
Received 2 April 1997; accepted 11 December 1997.
R. M. Cook: FRS Marine Laboratory, P.O. Box 101 Victoria Road, Aberdeen AB11
9DB, UK. Correspondence to: tel: +44 1224 876544; Fax: +44 1224 295511; e-mail:
[email protected]
Introduction
North Sea cod, Gadus morhua, have been exploited for
hundreds of years. Most scientific data describing the
dynamics of the stock have been collected during this
century (Daan et al., 1994) with detailed analyses
describing the post World War II fishery (Daan, 1978;
ICES, 1997a). For the years since 1970, the spawning
stock biomass of cod in the North Sea has declined
almost continuously (ICES, 1997b). The decline has
been associated with a continuous rise in fishing mortality rate. The sustained decline in spawning stock
biomass has resulted in the Advisory Committee on
Fishery Management (ACFM) regularly advising that
fishing mortality on the stock should be significantly
reduced. Despite reduced total allowable catches (TACs)
being set in line with lower fishing mortality, there is as
yet little evidence of reduced exploitation rates (ICES,
1997b). Present fishing mortality rates appear to be
approximately double those typical of the first part of
the 20th century (Daan et al., 1994; Pope and Macer,
1996). In the last decade, a number of North-west
Atlantic cod stocks have collapsed (Hutchings and
Myers, 1994; Myers et al., 1996) with high exploitation
rates being implicated as a contributory factor. In these
circumstances it is pertinent to question whether the
1054–3139/98/061061+10 $30.00/0
North Sea stock is also in danger of collapse under the
present exploitation regime.
ACFM conventionally classifies stocks in relation to
‘‘safe biological limits’’. This term has no formal definition but attempts are generally made to measure the
state of the stock against criteria such as the direction of
change of the biomass, the magnitude of fishing mortality rate in relation to Fmed (Sissenwine and Shepherd,
1987), the occurrence of good year classes and current
biomass in relation to MBAL – the ‘‘minimum biologically acceptable level’’ (ICES, 1991). The latter is usually
interpreted as either the spawning stock biomass below
which the probability of poor recruitment increases or
the lowest observed spawning stock biomass if the
stock–recruitment data show no relationship. The North
Sea cod stock has been below the conventional MBAL
of 150 000 t since 1984 (ICES, 1997b). Below this value,
recruitment is considered to be adversely affected. However, the fishing mortality rate for 1995 is estimated by
ICES to be below Fmed and this is often interpreted as
being ‘‘safe’’ because at this level, recruitment should on
average replace the spawning stock. In this case the
estimate Fmed is strongly influenced by the fact that most
of the observations have occurred during a period of
high exploitation and is not therefore a satisfactory
estimate of a safe exploitation level. These apparently
1998 International Council for the Exploration of the Sea
1062
R. M. Cook
b
Gcrash
a
Recruits
Recruits
Gloss
Spawning stock biomass
Figure 1. Theoretical stock–recruitment relationship for a fish
stock. The curve predicts recruitment given stock size, while the
straight lines are the replacement lines. These lines predict
expected spawning stock from recruitment for two levels of
exploitation. At a low exploitation rate (a), the dotted line
indicates the population trajectory expected which cycles
towards an equilibrium point. For high exploitation (b) the
dotted line shows the expected population trajectory collapsing
towards the origin.
conflicting criteria highlight the difficulty of judging the
state of the stock using traditional reference points.
Furthermore, none of these criteria are applied in a way
which considers the uncertainty in the assessments.
There is therefore a need for better criteria in order to be
able to judge the state of the stock.
In this paper the state of the cod stock is examined by
comparing the present exploitation rate of the stock with
rates that are likely to be sustainable. A new reference
point is developed which considers the probability of
stock decline below observed values. This overcomes
some of the weaknesses of the traditional reference
points used to classify safe biological limits by simultaneously considering both biomass and exploitation
rate. An essential element of the analysis is the assumption that there is a relationship between stock and
recruitment even though this may not be adequately
estimable. Increasingly this is accepted as the appropriate null hypothesis when examining stock–recruitment
data (Rosenberg and Restrepo, 1996).
Theory
The sustainability of harvesting is largely determined by
two factors, the relationship between the size of spawning stock (SSB) and the annual number of offspring (the
recruits) produced, and the subsequent survival of the
recruits on entering the fishery. This is illustrated in
Figure 1 which shows a theoretical stock–recruitment
curve and a recruit survivorship line. Where the two
lines intersect is an equilibrium point to which the
Spawning stock biomass
Figure 2. An example of typical stock–recruitment data where
there is insufficient information to define the recruitment function near the origin (dotted line). Gcrash is the slope of the
stock–recruitment function at the origin and is the replacement
line which would lead to stock collapse. Gloss is the replacement
line which gives an equilibrium at the lowest observed spawning
stock biomass.
population is attracted (Beverton and Holt, 1957). If the
survivorship line lies above the stock recruitment curve
there is no non-zero equilibrium point and the population is attracted to the origin. The slope of the survivorship line is affected by the fishing mortality rate, F. The
more heavily the stock is exploited, the steeper the slope.
This line is also called a replacement line since it defines
the survivorship needed to replace the spawning stock in
the future. It is important to note the distinction
between a replacement line and fishing mortality. A
replacement line (referred to as G), while dependent on
F, is also dependent on a number of biological parameters including growth, maturity and natural mortality. Thus a unique value of F can give a variety of
replacement lines if the biological parameters vary.
With perfect information of the type in Figure 1 it is
easy to define conditions of sustainability and collapse
but this ignores estimation errors and the limitations of
real data. Consider the stock–recruitment data illustrated in Figure 2. This shows the typical problem where
data are scattered and are inadequate to define the left
hand part of the stock–recruitment curve (the broken
line). If we knew the stock–recruitment curve we could
define the slope of the line, Gcrash, the replacement line
for the fishing mortality, which results in stock collapse.
However, the best we can do is to define Gloss, the
replacement line which corresponds to the lowest
observed spawning stock (LOSS). Although this is not
the replacement line we seek, it has certain value
because;
(a) Gloss is a minimum estimate of Gcrash,
(b) any fishing mortality which corresponds to a
replacement line to the right of Gloss should be
A sustainability criterion for the exploitation of North Sea cod
sustainable provided the stock–recruitment relationship is not depensatory at low levels and,
(c) any fishing mortality which corresponds to a
replacement line to the left of Gloss should result in
an equilibrium stock size below the lowest observed
value or stock collapse.
Clearly we wish to establish a fishing mortality rate
which is below Gcrash with some degree of confidence. If
it can be established that F gives a replacement line
below Gloss then this condition is satisfied.
Methods
Distribution of Gloss
The replacement line, Gloss, can be defined as the line
joining the origin of the stock–recruitment plot to the
point given by the expected recruitment value, R
z loss, at
the lowest observed spawning stock biomass, Sloss. The
slope of this line is then simply calculated from:
In order to calculate R
z loss it is necessary to describe a
stock–recruitment relationship in the region of Sloss.
There are many parametric stock–recruitment relationships which can be used to summarize the data (Deriso,
1980; Shepherd, 1982; Schnute, 1985). Although these
are quite flexible in shape the choice of function to use is
usually stock-dependent. To avoid the need to choose a
particular function a non-parametric approach has been
used here. Non-parametric methods have been used
before (Evans and Rice, 1988) and have the advantage
that the data determine the shape of the curve. The
particular method used here is to fit a lowess curve
(Cleveland, 1981) assuming lognormal errors and use
the smoothed value at Sloss as an estimate of R
z loss. The
value of R
z loss used was the fitted log value, backtransformed and corrected for the residual variance in
the conventional way. It was found that the best results
were obtained with the ‘‘stiffest’’ smoother, that is using
all the data points in the local regression estimates. More
flexible smoothers produced curves with multiple inflection points implying a very complex dependence of
recruitment on stock size and were not considered
realistic.
These calculations take no account of the uncertainties in the data. Of particular concern is uncertainty in
the Gloss replacement line. Uncertainty in Gloss can be
considered by calculating a frequency distribution of
the estimate in Equation (1). This can be achieved by
bootstrapping the lowess fit to the stock recruitment
data. It has been done here by re-sampling with
replacement, using a similar approach to Gabriel
1063
(1994). For n observations, n stock recruitment pairs
were drawn at random and the lowess curve fitted. For
repeated realizations, Gloss was calculated using Equation (1). This allowed a distribution for Gloss to be
calculated. In trials, it was found that the estimated
distribution stabilized after 300 realizations. For an
added safety margin, 500 realizations were used in the
analysis.
An alternative procedure might be to make an initial
fit of the lowess curve to obtain a set of residuals and
then bootstrap these. This assumes that the SSB values
are exact. By bootstrapping the data pairs, uncertainty
in the X axis is considered, but at the cost of bias since
in some samples the lowest observed SSB will not be
encountered. However, this bias is in the direction of
underestimating Gloss and is therefore consistent with
the Precautionary Approach (FAO, 1995). Comparative runs of these two alternatives were conducted
and very little difference was shown between the two.
This is not surprising given that for this stock there
are a number of observations at very low SSBs. Bootstrapping the data pairs has the advantage of
reducing the sensitivity of Gloss to single, exceptionally
low values which might be measured with a large
error.
Equilibrium curves
The equilibrium yield, Ye, and equilibrium spawning
stock, Se, can be easily calculated if an adequate
description of the stock–recruitment function is available. Such curves can be useful in understanding the
likely spawning stock and yield associated with a given
exploitation regime. Given the lowess estimated values
of recruitment these equilibrium curves can be
obtained simply by multiplying the fitted recruitment
value, R
z , by the appropriate yield per recruit value,
y(è), or spawning stock biomass per recruit value,
b(è)* i.e.:
Ye =R
z y(è)
(2)
Se =R
z b(è)
(3)
The parameters, è, are the standard vital quantities
of weight at age, w, proportion mature at age, p,
fishing mortality rate, F and natural mortality rate, M.
Distribution of GF
The position of Gloss is determined directly from the
stock–recruitment data. The calculation of replacement
lines for given fishing mortality rates can be made from
1064
R. M. Cook
Table 1. Annual estimates of recruitment at age 1, spawning stock biomass, yield and mean fishing
mortality.
Year
Recruits
age 1 (millions)
SSB
(’000 t)
Yield
(’000 t)
Mean F
(2–8)
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
195
374
415
507
489
195
209
782
911
174
320
264
486
247
839
488
525
899
314
618
324
592
152
706
277
194
277
134
169
312
159
399
248
151.5
166.1
205.4
230.7
250.0
258.3
256.0
276.9
277.3
231.1
209.2
231.0
211.9
182.6
159.9
159.8
164.6
182.0
195.5
189.4
153.5
131.0
122.3
110.4
100.7
95.2
89.7
77.1
70.2
67.5
63.1
63.3
74.7
116.5
126.0
181.0
221.3
253.0
288.4
200.8
226.1
328.1
354.0
239.1
214.3
105.2
234.2
209.2
297.0
270.0
293.6
335.5
303.3
259.3
228.3
212.9
196.1
209.6
176.4
139.9
125.3
102.5
114.0
121.9
110.6
138.8
0.473
0.493
0.546
0.515
0.613
0.616
0.574
0.551
0.660
0.824
0.691
0.658
0.707
0.703
0.707
0.824
0.681
0.800
0.761
0.898
0.943
0.848
0.811
0.905
0.891
0.866
0.933
0.790
0.930
0.876
0.974
0.944
0.809
the standard ‘‘per recruit’’ formulae*. The slope, GF, of
the replacement line for a particular value of fishing
mortality rate, F, is simply 1/b(è) i.e.:
The parameters, è, are generally measured either with
error or are by their nature variable quantities. Growth,
*Standard ‘‘per recruit’’ formulae referred to in Equations (2)
and (3) of the main text used in the calculation of the
replacement line slope, equilibrium SSB and equilibrium yield.
Z is the total mortality and is the sum of F and M.
for example would be expected to vary from year-toyear leading to different annual mean weights at age.
These sources of error need to be considered when
calculating a frequency distribution of GF. The calculation of such a frequency distribution has been achieved
here by simulation. A mean and coefficient of variation
(c.v.) for each parameter was specified with an associated distribution. The quantity GF was then repeatedly
calculated by drawing parameter values at random from
the specified distributions. The methods for estimating
the parameters, è, and their c.v.s are given below.
Probability that GF >Gloss
Given the estimated distributions of the replacement
lines it is simple to calculate the probability that the
present fishing mortality rate, F, has a replacement line
above Gloss. This probability is given by considering the
distribution of the ratio Gloss/GF. The ratio will be
centred on one if Gloss =GF. If Gloss <GF, then the ratio
will be less than one. Hence the probability we seek is
A sustainability criterion for the exploitation of North Sea cod
simply the probability that this ratio is less than or equal
to one, i.e.:
It can be calculated by drawing at random value of
Gloss and GF as described above, forming the ratio and
then accumulating the proportion of the total sample
which is less than or equal to one.
Floss distribution
For a unique value of Gloss and a unique set of parameters, è, it is possible to calculate a multiplier, floss, on
the exploitation pattern, s, which satisfies the equation:
This multiplier leads to the fishing mortality rate,
Floss, above which the stock would be expected to
decline to an equilibrium spawning stock below the
lowest observed value. A distribution of Floss can be
obtained by combining the procedures described earlier
to find the distribution of Gloss and GF. For each
bootstrapped value of Gloss, a set of parameters è is
selected at random from their given distributions and
Equation (6) is solved. This gives a distribution of
fishing mortality rates which are likely to lead to stock
decline below the lowest observed spawning stock.
Parameter estimates and c.v.s
The input values required to estimate GF are fishing
mortality rate at age, natural mortality rate at age,
weight at age, and maturity at age. In order to obtain a
distribution of GF, it is necessary to estimate these
values and their variances. Nominal values for these
quantities have been obtained from standard ICES
assessments (ICES, 1997a) and the required parameter
values and c.v.s were calculated as follows:
Fishing mortality
The required estimate of fishing mortality is a value that
quantifies the typical exploitation rate of over a period
of years, not an estimate for the single most recent year.
This is because it is of no great significance if GF exceeds
Gloss in any one year. What is important is how frequently GF exceeds Gloss. In the example presented in
this paper, fishing mortality is estimated from XSA
(Darby and Flatman, 1994) which gives annual estimates by age. It is assumed that fishing mortality can be
decomposed into an age specific selectivity effect, sa, and
a year effect fy:
Fay =safy
(7)
1065
Values of fy were estimated as the mean F over a
standard age range (2–8) in each year. The sample
variance of the fys over a number of years was taken as
the required variance for calculating the parameter c.v.
This variance expresses the annual year on year variability of F caused by both process error and measurement error in the catch at age data. It also implicitly
assumes the exploitation rate is in an equilibrium state
over a number of years. It is likely to underestimate the
variances if there are large correlations in the values of
F. However, where fishing mortality is large the error in
individual estimates of F is dominated by the error in the
catch in the same age/year stratum (Bailey and Kunzlik,
1989) so that correlations due to the sequential
calculations in the VPA equations are reduced.
Values of sa were calculated by dividing the age
specific Fs by the fys each year and then taking a mean
across a standard range of years. The sample variance
for each sa was then used to calculate the appropriate
c.v. for selectivity. This will approximate the variability
in selectivity when year effects are removed.
Natural mortality
A similar approach to that for fishing mortality was
adopted. The natural mortality, M, was decomposed
into an age effect, ma, and year effect ky such that:
May =maky
(8)
The values for m were taken as the conventional values
of M used in the assessment. An approximate value for
the c.v. was obtained by taking a 10 year mean of the
predation mortalities estimated by the multispecies
working group (ICES, 1994). These values are approximately 0.2–0.3. For the year effect, k, a nominal value of
1.0 was used with an arbitrary c.v. of 0.1.
Weight at age
This quantity was taken as the mean over a range of
years. The sample variance was used as an estimate of
the variability in weight. This variance will not
adequately describe longer term systematic changes
in growth rate but should serve as an estimate of
cohort specific growth rate changes assuming an overall
stationary mean over time.
Maturity
For the stock considered there is little information about
changes in maturity and a ‘‘conventional’’ maturity
ogive is used. The c.v. for maturity was taken to be 0.1
for those age classes that were partially mature.
Inspection of the frequency distribution of the raw
fishing mortality rates over the 10 year time period and
for fully selected ages (age 3 and older) indicated a more
or less symmetrical distribution. The distribution was
therefore assumed to be normal. For mortality, the
1066
R. M. Cook
Table 2. Input data used in spawning stock per recruit and yield per recruit calculations. s=selectivity,
w=weight at age, m=natural mortality, and p=proportion mature. Figures in parentheses are the c.v.s
used.
Age
1
2
3
4
5
6
7
8
9
10
11
s (1989–1993)
0.107
0.757
0.895
0.865
0.747
0.758
0.831
0.813
0.958
0.720
0.720
(0.38)
(0.15)
(0.09)
(0.07)
(0.06)
(0.14)
(0.10)
(0.13)
(0.75)
(0.18)
(0.18)
s (1993–1995)
0.063 (0.53)
0.583 (0.14)
0.827 (0.05)
0.805 (0.11)
0.765 (0.01)
0.829 (0.04)
0.870 (0.08)
0.987 (0.19)
0.771 (0.21)
0.808 (0.19)
0.808 (0.19)
Year effects
1989–1993
1993–1995
All years
parameter distribution was assumed to be normal in
the absence of adequate real observations. For weight,
the distribution was assumed lognormal which is a
conventional assumption (Schnute and Fournier, 1980).
In the case of proportion mature, this was taken to be
normally distributed after a logit transformation.
Sensitivity analysis
The intersection of the replacement line with the stock–
recruitment curve determines the equilibrium spawning
stock biomass. The stock–recruitment curve is a natural
phenomenon beyond the control of fishery managers.
The slope of the replacement line, however, is potentially
influenced by managers by regulating the overall fishing
mortality rate or the selectivity by age (sa). It is of some
interest to know how much this slope can be controlled
by these factors since growth, natural mortality and
maturity also play a part. In order to investigate this a
sensitivity analysis was performed on the replacement
line slope by calculating the conventional elasticities, ä,
(Prager and MacCall, 1988):
The elasticities, or sensitivity coefficients, quantify the
degree to which the slope, G, depends on the magnitude
of each parameter, èi.
Data
Input data were taken from the Report of the Working
Group on the Assessment of Demersal stocks in the
North Sea and Skagerrak (ICES, 1997a). Table 1 gives
w
0.664
1.013
2.116
3.918
6.237
8.174
9.827
11.048
12.697
13.925
14.526
(0.08)
(0.09)
(0.12)
(0.12)
(0.10)
(0.04)
(0.03)
(0.05)
(0.05)
(0.07)
(0.08)
m
0.80
0.35
0.25
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
(0.20)
p
0.01
0.05
0.23
0.62
0.86
1.00
1.00
1.00
1.00
1.00
1.00
(0.10)
(0.10)
(0.10)
(0.10)
(0.10)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
(0.00)
f=1.113 (0.08)
f=1.000 (0.10)
k=1.000 (0.10)
the recruitment, spawning stock, yield, and mean fishing
mortality rate estimates since 1963. Table 2 gives the
parameters è. Weight at age is calculated over the period
1986–1995. For the selectivities, s, 2-year ranges were
used. The first range was for the 5-year period 1989–
1993. The period should be typical of recent exploitation
yet insensitive to estimation errors in the terminal values
in the catch at age analysis (Pope, 1972). The second
range use was for the period 1993–1995 which corresponds to the most recent years in the assessment and is
the standard reference period used by the Working
Group to estimate the exploitation pattern.
Results
The difference between the two exploitation patterns
investigated is shown in Figure 3. This shows the ratio of
the old to the new selectivity pattern. It shows that the
older pattern has higher selection on the younger age
groups, particularly on age 1 and 2 but also on ages 3
and 4. This difference is important because it leads to
rather different conclusions about the state of exploitation of the stock even though the selectivity for most age
groups is similar.
Figure 4 shows the stock and recruitment plot with
the fitted lowess curve. Mean recruitment appears to
increase with increasing spawning stock biomass and
reaches a plateau at about 150 000 t; the conventional
MBAL for the stock. The fitted curve at the lowest
spawning stock biomass has been used to calculate the
Gloss distribution. Figure 5 shows the cumulative probability distributions for the Gloss/GF ratio [Equation (5)]
for each of the two selectivity patterns. For the old
pattern this plot shows that the probability that GF
A sustainability criterion for the exploitation of North Sea cod
1.8
Probability ratio < x
1
1.6
Ratio sold/s new
1067
1.4
1.2
0.8
0.6
0.4
0.2
1
0
0.5
0.8
1
2
3
4
5
6
7
Age (years)
8
9
Probability F > Floss
70
79
Recruitment at age 1
3
3.5
4
1
76
800
69
85
600
0
2
2.5
Ratio Gloss /GF
Figure 5. The cumulative probability of the ratio Gloss/GF. The
solid line is for the old selectivity pattern and the dashed line for
the new pattern. For values of the ratio below one, GF exceeds
Gloss. For the old pattern there is a high probability GF exceeds
Gloss.
1000
200
1.5
10 11
Figure 3. The ratio of the selectivity pattern 1989–1993 to the
pattern 1993–1995. The selectivity patterns have been scaled to
the mean over ages 2–8 in each case. If the patterns were the
same, the observed points would lie around a value of one.
400
1
83
78
77
93
83
91
94
50
65
72
73
75
84
66
74
64
8082
88 86
87
9290 89
81
0.6
0.4
0.2
6768 71
100
150
200
250
Spawning stock biomass
0.8
300
0
0.6
0.8
1
1.2
1.4
1.6
F
Figure 4. North Sea cod stock–recruitment data. Recruitment
values are identified with the year class. The fitted line was
drawn using a lowess algorithm.
Figure 6. The cumulative probability distribution of Floss for
the old selectivity pattern () and the new pattern (). For
both patterns, recent historical values are well within the Floss
distribution.
exceeds Gloss is approximately 0.5, while for the new
pattern it is approximately 0.1. It means that for the old
pattern, fishing mortality rate had a high probability of
causing the stock to reach lower equilibrium values than
the lowest observed value.
The cumulative probability distribution of fishing
mortality rates corresponding to Gloss is shown in Figure
6. Regardless of the pattern used, the values of Floss are
in the range of recent historical values (Table 1) indicating that the stock has been fished at levels expected to
produce declining stock sizes. To ensure that any
replacement line lies below Gloss, F needs to be reduced
to approximately 0.6–0.7.
Equilibrium yields and SSB are shown in Figure 7.
Super-imposed on the equilibrium lines are the annual
estimates of SSB and yield. The equilibria show steady
declines as F increases and this is consistent with the
historical data, notably for the older exploitation pattern. The trajectory of equilibrium SSB suggests that the
stock has been exploited historically at levels close to
those which would cause collapse. The steepness of the
slope of the equilibrium SSB and yield curves indicates
that small increases in F lead to large reductions in SSB
and yield. The newer exploitation pattern produces
equilibrium curves to the right of those for the earlier
period. It implies that the changed exploitation pattern
would allow the stock to withstand a slightly higher level
of exploitation.
Figure 8 shows the sensitivity analysis of the slope of
the current replacement line. The sensitivity coefficients
are plotted in rank order for those parameters with
coefficients greater than 0.1. The slope is most sensitive
to the overall exploitation rate and the selectivities at
ages 2–4 years, the juvenile stage. There is also some
1068
R. M. Cook
300
71
70
(a)
Spawning stock biomass ('000t)
69
250
68
67 74
66
65
72
73
200
75
81
82
76
79
64
63
150
80
78
77
85
83
84
88
100
86
87
89
90 95
91
92
50
0
0.4
0.5
0.6
94
93
0.9
0.7
0.8
Fishing mortality
1
1.1
1
1.1
400
(b)
72
350
81
71
82
80 78
68
300
Yield ('000t)
79
250
66 70
67
69
200
63
74
76
84
85
77
64
87
86
75
65
150
83
73
88
90
89
95
100
93
92
91
94
50
0
0.4
0.5
0.6
0.7
0.8
Fishing mortality
0.9
Figure 7. (a) The equilibrium spawning stock biomass estimated from the stock–recruitment function fitted in Figure 4 as a
function of total fishing mortality. The plotted points are the observed values from the most recent assessment joined as a time
series. (b) The equilibrium yield estimated in the same way as (a). The plotted points are the observed catches plotted as a time
series. Both figures show the observed SSB and catch declining in line with the expected equilibrium function. The heavy line is for
the old selectivity pattern and the thin line for the new pattern.
sensitivity to natural mortality. Overall the analysis
shows that relatively small changes in the exploitation
rate and selectivity pattern can have a large effect on the
replacement line.
Discussion
The sensitivity analysis in Figure 8 reveals much about
the problem of the state of the North Sea cod stock. It
shows that the slope of the replacement line at present
exploitation rates is driven largely by the level of fishing
mortality and the selectivity pattern. The latter is of
particular interest due to the differing results obtained
from the two patterns investigated. It is worth noting
that the effects of the selectivities are cumulative because
the sign of the sensitivity coefficient is the same for all
age groups. Although the value of the coefficient for any
selectivity parameter is much lower than for the year
effect, f, the cumulative effect of increasing selectivities
on a number of the younger age groups can be large. It
means there is an increased cumulative mortality on
juveniles before they reach the age of maturity and hence
A sustainability criterion for the exploitation of North Sea cod
2.5
Sensitivity coefficient, δ
2
1.5
1
0.5
0
–0.5
f
k m1 s2 s3 s4 m2 m4 w4 p5 w5 m3 p3 w3 s5 p6 w6 s1
Parameter, θ
Figure 8. Sensitivity analysis of the replacement line slope to
input parameters. The sensitivity coefficients are plotted in rank
order with values less than 0.1 omitted. f=fishing mortality year
effect, k=natural mortality year effect, s=selectivity, m=
natural mortality selectivity, p=proportion mature, w=weight
at age. The numerals refer to the age groups.
a reduction in the equilibrium spawning stock biomass.
This is why the apparently small change in selectivity
pattern between the periods 1989–1993 and 1993–1995
leads to a large change in the equilibrium conditions.
The dependence of the replacement line on fishing
mortality also illustrates an interesting problem for
management. The steepness of the equilibrium SSB
curve (Fig. 7) is testimony to the fact that small increases
in mortality lead to substantial declines in expected SSB.
The danger of this property is that estimation errors in
the assessment will make it difficult to manage the stock
safely at high exploitation rates. A small underestimation of the true fishing mortality rate could be the
difference between stock collapse and sustainability. The
effect of fishing mortality rate reductions, however, are
quite encouraging. Because so much of the stock dynamics are driven by man made mortalities, reducing the
exploitation rate should rapidly improve the fortunes of
the fishery. A reduction in fishing mortality of approximately 10%, for example, would be expected to more or
less double the spawning stock biomass. Thus the stock
should be highly responsive to effective management
measures.
Both selectivity patterns give estimates of Floss, which
are close to F=1. Given the number of stock–
recruitment observations that are near the origin (Fig.
4), this value might be expected to be close to an
estimate of the mortality rate, which would cause the
stock to collapse. Recent fishing mortality rates on cod
are estimated to be in the range 0.79–0.97 (Table 1)
which falls well within the range of the Floss distribution
(Fig. 6). This indicates that the recent exploitation of the
1069
stock has probably been at dangerous levels close to
conditions producing a collapse. The fishing mortality
estimate for the most recent year in the assessment
(1995) is towards the lower end of this range and
comprises a more benign selectivity pattern which leads
to a probability of exceeding Gloss of only 10%. Much of
the improvement in this probability is due to the change
in selection pattern on the youngest age groups. These
values, however, are estimated with the lowest precision
in conventional catch at age analyses and it remains to
be seen if the apparent improvement in the selectivity
pattern is real or the result of estimation error.
The form of the equilibrium curve depends to a large
degree on the shape of the stock–recruitment function
since this determines expected recruitment for a particular stock size. Clearly the scatter of data points in the
stock–recruitment plot (Fig. 4) is such that the functional relationship is not well determined and a variety
of equally plausible equilibrium curves could be generated dependent on the curve drawn through the data. In
this paper a non-parametric method has been used to
describe the recruitment function and the resulting
analysis suggests fishing mortality rates near unity are
risky. This is consistent with the results of Cook et al.
(1997) who fitted a variety of different parametric
recruitment curves to North Sea cod data and found the
fishing mortality producing collapse would be in the
range 0.9–1.13. This suggests that the estimation of
fishing mortality rates, which are likely to produce stock
collapse, is not sensitive to the form of the assumed
stock recruitment relationship and is determined largely
by the data.
Much of the analysis presented here makes the
assumption that recruitment is dependent on stock size.
Since the observed stock–recruitment data are a time
series, which show a more or less continual downward
trend, it might be argued that environmental factors
could explain the relationship. This remains a possibility. Cushing (1984) argued that there was a ‘‘gadoid
outburst’’ during the 1960s when recruitment was higher
than expected and probably related to an environmental
effect. However, examining the stock–recruitment plot
in Figure 4 suggests that only the 1969 and 1970 year
classes were higher than expected with most year classes
in the 1960s being comparable to other decades. It is also
worth noting that the productivity of the stock, as
measured by the slope of the stock–recruitment curve,
has been higher in recent years (Fig. 4) when stock sizes
are lower. If this increased productivity is environmentally driven, then the stock is declining despite any
beneficial environmental change. It would mean that the
Gloss replacement line estimated here is over-optimistic
and that there is, in reality, a higher probability of
further stock decline.
The tendency for stocks to increase or decrease will
depend upon the size of the existing stock in relation to
1070
R. M. Cook
the expected equilibrium. The expected equilibrium
value is, in turn, conditional on the growth rate, maturity ogive, and natural mortality, which determine the
slope of the replacement line. These may all change quite
independently of the fishing mortality rate. This means
that there is no unique fishing mortality rate that can be
associated with a unique equilibrium spawning stock
biomass. Furthermore, as this analysis shows, a change
in selectivity pattern can significantly alter what may be
regarded as a ‘‘safe’’ fishing mortality. It indicates that
the state of a stock should not be judged on the basis of
an exploitation rate alone. It is necessary to consider all
the biological factors which influence the replacement
line.
Some care is needed in the choice of the estimate of
fishing mortality rate and its variance when performing
the analysis. The choice of value will depend on the
question being posed. In this analysis the problem being
considered is whether recent historical fishing levels have
been non-sustainable and hence the estimates of F are
based on those calculated for a period of years. However, it may be of interest to consider whether the
most recent fishing mortality, if propagated into the
future, will also lead to problems of sustainability. In
this case the recipe for obtaining the variance of F used
here may not be appropriate. Where F is estimated
using a maximum likelihood method (Methot, 1990;
Gudmundsson, 1994) it may be possible to obtain the
required variances directly as part of the model fitting
procedure. Alternatively the stock assessment can be
bootstrapped.
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