Fisheries Research 37 (1998) 77±95
On the effect on cod and herring ®sheries of retuning the Revised
Management Procedure for minke whaling in the greater Barents Sea
Tore Schwedera,*, Gro S. Hagenb, Einar Hatlebakkc
a
Department of Economics, University of Oslo, Box 1095, Blindern, 0317 Oslo, Norway
b
Norwegian Computing Center, 0317 Oslo, Norway
c
Statens Datasentral, 0317 Oslo, Norway
Abstract
In a model with four species ± cod, capelin, herring and minke whales ± the ®sh populations are age and length distributed,
while the minke whale is age and sex distributed. The time step is one month, and there are two areas (The Barents Sea and
parts of the Norwegian Sea). There is a food-web with minke whales as top predators, consuming herring, capelin and cod
according to a non-linear consumption function in available prey abundance. The consumption function for minke whales is
roughly estimated. The opportunistic minke whale may forage on plankton and other ®sh than cod, capelin or herring, and is
thus, modelled as having carrying capacity and demographic parameters independent of the status of the ®sh stocks in the
model. The ®sh-®sheries are managed by ®xed VPA-based ®shing mortalities (cod and herring) and Captool (capelin), while
minke whaling is managed according to the RMP of the IWC. The model is stochastic in ®sh recruitment and in survey indices
for minke whales. The model is simulated over 100-year periods in a number of scenarios spanned by nine experimental
factors. The core of the experimental design is an orthogonal array with 27 points. The primary study variable is the tuning of
the RMP, and the response variables are catches and stock sizes of cod, herring and minke whale. The responses are taken as
yearly means over the last 90 years of the period. When the tuning of the RMP is changed from the current level of targeting
the ®nal stock at 72% of carrying capacity to 60%, the annual catch of whales increases with some 300 animals, while the
annual catch of cod increases with some 0.1 million ton on the average. For herring, no clear main effect was found on catch
or mortality rate. The catch of cod is estimated to increase in annual mean with some 6 ton with a mean reduction in the whale
stock of one animal. The results concerning the effects on the cod and herring ®sheries must be taken as tentative since the
ecosystem model used could be improved, and so could the strategies for managing the ®sheries. The study exempli®es how
scenario experimentation can be used as a tool for investigating the properties of ®shery management regimes. # 1998
Elsevier Science B.V. All rights reserved.
Keywords: Scenario; Experiment; Simulation; Multispecies model
1. Introduction
Sea mammals are important in the northeastern
Atlantic ecosystem. The number of minke whales is
*Corresponding author.
0165-7836/98/$19.00 # 1998 Elsevier Science B.V. All rights reserved.
PII S0165-7836(98)00128-3
estimated to be around 85 000 north of 658N in 1995,
and their total consumption is estimated to be around
1.4 million ton biomass yearly, (Haug et al., 1996
personal communication). Harp seals, Greenland
seals, great baleen whales and toothed whales are also
important top predators in this ecosystem.
78
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
Will the ®shermen be able to take more cod, capelin
and herring if the stocks of sea mammals are reduced?
How much more if whaling is regulated by the management procedure developed by the Scienti®c Committee of the IWC, but tuned to another target than its
present level? We investigate this for minke whales by
running scenario experiments in a multi-species and
multi-¯eet model roughly tailored to the ®sheries and
ecosystem of the Barents and the Norwegian Sea. The
multi-¯eet nature of the ®shery is modelled as in
Hagen et al. (1997).
The conclusions obtained in this study are not
meant as ®nal ®ndings. There is substantial room
for improvement in the ecosystem model used as
the operating model in the simulations. Rather than
settle the debate over whether regulated whaling could
be bene®cial for ®sh-®sheries, our aim is to demonstrate how scenario experimentation can be used to
shed light on this and other management issues, and to
present a tentative analysis.
The use of uncertainty factors and scenario experiments are not without their dif®culties. When data are
unavailable and uncertainty simply is ignorance, it
will be dif®cult to obtain plausibility ranges. In such
cases, the levels of the uncertainty factors will have to
rely on guesswork. It is desirable that the degree of
plausibility is about even across uncertainty factors,
say with the range having `plausibility' 95%. It is also
desirable to have the models parameterized such that
the main effects of the factors on the central output
variables are as linear as possible. This is because we
will use linear statistical models when estimating the
effects of the factors involved, with focus on the
experimental factors. The choice of criterion or output
variable might also be dif®cult to make. In our case,
the long term yield in the cod and herring ®sheries, and
the corresponding stock sizes are of primary interest.
The yield from whaling is also of interest.
1.1. Scenario experimentation
The core of the bio-dynamic model is a system with
cod, capelin and herring. Fisheries on cod and herring
are carried out according to catch limits based on target
®shing mortality and VPA-assessments, while Captool
(Tjelmeland and Bogstad, 1998) is used to determine
®shing mortality for capelin. The minke whale stock is
age and sex speci®c, and it has an autonomous population dynamics speci®ed by the IWC (more precisely
its Scienti®c Committee) for the purpose of performing implementation trials for the Revised Management
Procedure (RMP) for north Atlantic minke whales, see
International Whaling Commission (1993). We use
the program developed by IWC.
Whaling is regulated by the RMP, see International
Whaling Commission (1994), but not necessarily with
the same tuning as currently used by the Norwegian
Government in calculating catch limits for minke
whales (Target level 0.72K, i.e. 72% of carrying
capacity; Internal protection level 0.54K). This procedure takes as input the historical catch series, and
5-yearly abundance estimates (on the summer feeding
grounds) and their measured uncertainty. The output is
a yearly quota that is in force until a new abundance
estimate arrives. We assume that whales are removed
according to the quotas. The RMP is tuned by two
parameters which determine the long-term equilibrium level of the whale stock
Scenario modeling and experimentation are discussed in general terms in Hagen et al. (1997). The
basic idea is to have a dynamic model of the process to
be studied. This model is conditioned to available data
as well as possible, in the sense that parameters are
estimated or matched to data. Understood stochastic
variability in the process, like stochasticity in yearclass strength is modelled in probabilistic terms. This
could also apply to estimated parameters, where the
estimation uncertainty is available in the format of
Bayesian posterior distributions. When uncertainty is
associated with lack of data and knowledge instead of
sampling variability or understood stochasticity, the
parameter in question is treated as an uncertainty
factor in the computer experiments. For these parameters, plausibility ranges with midpoints are sought.
The uncertainty factor is allowed to vary over the three
levels LOW, MID and HIGH, determined by the
plausibility ranges obtained. In the computer experiments, the uncertainty factors are combined with the
factors that specify the management regime governing
the system. These factors are called speci®cation
factors. In the present application, tuning of the catch
limit algorithm for whaling will be the speci®cation
factor.
1.2. An overview of the model
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
Our main interest is in the effect of whaling on the
other ®sheries. Our approach is, therefore, to tune the
RMP over a relatively broad spectrum, to see how the
®sheries perform under the various regimes of whaling. With this focus we experiment with three different
strategies for managing the ®sheries on herring, cod
and capelin.
Recruitment in ®sh is a most important natural
factor determining the status of the ®sh stocks. The
parameters of the recruitment functions are hard to
estimate, and are treated as uncertainty factors. In
addition, stochasticity is incorporated in the recruitment process.
For whales, we include uncertainty factors in the
parameters maximum sustainable yield rate (MSYR),
carrying capacity (K), bias in absolute abundance
estimates of whales (BIAS), and in two parameters
regulating the predator±prey relationship between
whales and ®sh.
2. The scenario model
The basic structure of the scenario model we use is
as follows. The ecological system consists of herring,
capelin, cod and minke whales. There are two areas
(the Barents Sea and parts of the Norwegian Sea), and
the time step is one month.
Recruitment is stochastic for the ®sh stocks but
deterministic for minke whales. Except stochasticity
with positive correlation in recruitment between herring and cod, recruitment for each species depends
only on spawning stock of the species. Except for
predation, individual natural mortality is independent
of the state of the system.
In the model, the minke whale prey on herring,
capelin and cod. The cod prey on young cod (cannibalism), herring and capelin, the herring prey on
young capelin while the capelin is at the bottom. The
dynamics of the model is explained in broad terms in
the following. For the ®sh-®sheries model a more
extensive presentation is given in Hagen et al. (1997).
2.1. Stock size and distribution
2.1.1. Minke whales
Since the minke whale is an opportunistic predator,
it may switch to plankton or to food items outside our
79
model if herring, capelin and cod are in short supply.
We, therefore, assume that the population dynamics of
the minke whale is independent of stock status of
herring, capelin and cod. In practical terms this means
that the minke whale is modelled in a separate module,
and its trajectory with respect to population and yield
is simulated before the ®sh model is simulated ± with
mortality caused by simulated ®shermen and minke
whales, in addition to excess natural mortality. In a
more realistic model zoo-plankton and `other ®sh'
would be included in the ecosystem model, and the
dynamics of the minke whale would be dependent on
the abundance of all its prey items. However, with
respect to our main question, the lack of realism due to
minke whale dynamics being independent of the status
of herring, capelin and cod should be a minor concern.
The stock size of the minke whale is determined by
the carrying capacity and the past catches together
with mortality and fertility parameters. These latter
demographic parameters are discussed below. We use
the IWC model, MANA4, as documented in International Whaling Commission (1993). In this model
there are three stocks of minke whales with a total of
10 substocks covering the whole north Atlantic. We
will be concerned only with the eastern stock in the
present study. The substocks do have the same demographic parameters, but they are allowed to differ in
their carrying capacities. The Greater Barents Sea
stock, EB‡ES, has carrying capacities regulated by
an uncertainty parameter, K. K is thus included in the
experimental design, but not as a three level uncertainty factor as most of the other parameters/factors. A
closer description of K in the design is given below
(Section 3).
The minke whales in our model belong to one
substock which is evenly distributed between the
two feeding areas, the Barents Sea and the Norwegian
Sea. The parameters for carrying capacity in the IWC
program we use refer to mature females. The numerical values for the mature female carrying capacities of
the substocks are calculated by assuming that 27% of
the total stock consists of mature females.
The yearly temporal distribution is determined by
the population dynamics of the model. In addition, the
minke whale has a seasonal distribution. The minke
whales migrate to its feeding ground in spring, and
return to their breeding grounds in the autumn. We
assume that the minke whales are present on the
80
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
feeding grounds north of 658N during April±October,
and absent during November±March. The number of
whales in the model at the start of the simulation
period is taken to be the number of whales estimated
north of 658N, based on the July 1995 survey (Schweder et al., 1997): 85 000 minke whales.
2.1.2. Fish
For cod, herring and capelin, the initial stock sizes
are speci®ed as the abundance estimates based on VPA
for the year 1993, as in Hagen et al. (1997).
2.2. Recruitment
2.2.1. Minke whale
The minke whale recruit according to a Pella±
Tomlinson production function with MSYLˆ0.6K,
as de®ned in International Whaling Commission
(1993). The parameter regulating the productivity of
the stock is the maximum sustainable yield rate per
year, MSYR. We use this as an uncertainty parameter.
Parallel to the carrying capacity parameter K, MSYR
is included in the experimental design. As experimental parameters, MSYR and K are linked together, and a
closer description is given in Section 3.
2.2.2. Fish
All three ®sh species have Beverton±Holt recruitment. The recruitment level for cod and herring are
used as experimental factors. The factors, named
RCOD and RHER, are de®ned as multiplicative parameters to the Beverton±Holt function, and the plausibility range is 10% of the reference value for both
parameters.
Recruitment in cod and herring is in¯uenced by
oceanographic and ecological factors that are likely to
produce exceptionally good year classes in the same
years for both species about 10 years apart. This is
roughly modelled by stochastically drawing good
common years for the two species, at least 5 years
apart and with a mean distance of 10 years. More
details of ®sh recruitment are laid out in Appendix A.
2.3. Mortality in excess of predation and fishing
mortality
Minke whales have a piecewise linear natural mortality rate increasing from 0.085 at age 4 to 0.115 at
age 20, with ¯at mortality above and below this
interval.
For cod and herring, we set the excess mortality rate
at 0.05, while for capelin there is no natural mortality
in excess of that caused by minke whales, cod, herring
and man. However, for capelin we have spawning
mortality, which means that a certain fraction of the
mature stock dies immediately after spawning. In the
present study, the spawning mortality for capelin is set
to 100%.
The excess mortality we use might be on the low
side. It was, however, necessary to limit the excess
mortality in our model, since in several scenarios the
minke whale stock grew high and the mortality caused
by these whales was substantial. In Hagen et al.
(1997), the value 0.2 was used for cod, but in that
application cod was the top predator, except for man.
2.4. Predation
Minke whales, cod and herring are predators, while
capelin, herring and small cod act as prey. In the
following, the predation is brie¯y described, while
predation formulas are given in the Appendix A.
2.4.1. Minke whales
Minke whales prey on herring, capelin and cod, and
the whale's diet is determined by availability of the
three ®sh components. Due to migratory patterns and
®sh behavior, we assume that there is a limited overlap
between the prey species and the minke whale. The
predation functions are estimated from data on ®sh
stock abundance and stomach content.
We have chosen the two most important parameters
from the predation model (Appendix A), H and C, as
uncertainty factors. The midpoints are as estimated
from data and the rather arbitrary plausibility range is
the multiplicative interval from half the midpoint to
twice the midpoint for both factors. This corresponds
basically to the switching in the consumption being
half as sharp, or twice as sharp as at the reference level.
Fig. 1 displays partial consumption plots.
2.4.2. Fish
How cod is preying on (young) cod, herring and
capelin is speci®ed in Hagen et al. (1997), where the
predation pattern is estimated from stomach data.
Herring prey on juvenile capelin when present in
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
81
Fig. 1. Partial effects on minke whale's consumption of herring and cod, of varying available cod and herring biomass from 0% to 200% of
median observed level. For the two upper and the lower left panel, H is varied from low to high value. At bottom right C is varied.
the Barents Sea. We model this as impaired recruitment in capelin in years when herring and capelin
overlap, see Appendix A. The strength of this effect is
not well known. Our results are not insensitive to how
the herring capelin interaction is modelled and uncertainty in the herring capelin interaction induces uncertainty in our results concerning the relationship
between whaling and ®shing.
2.5. Observational regime
2.5.1. Minke whales
There are two parameters for the observational
regime for minke whales: the bias in the 5-yearly
abundance estimates, and their coef®cients of variation. The survey noise constants that regulate the
coef®cients of variation are set to values roughly
82
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
matching the estimated coef®cients of variation of the
abundance estimates obtained from the 1995 survey,
see Schweder et al. (1997).
The bias in the absolute abundance estimate is taken
as an uncertainty factor with three levels: We set
BIAS ˆ ÿ40% 0 ‡ 40%
2.5.2. Fish
In the VPA assessments, survey indices are used.
These indices are described in Hagen et al. (1997). In
the present study, we assume that these indices are
linear and unbiased and without error.
2.6. Management regimes
2.6.1. Minke whales
For whaling, quotas will be set by the RMP method
of the IWC, as it is implemented in the IWC program
MANA4, see International Whaling Commission
(1993). For technical reasons, the substocks of
MANA4, ES and EB, are associated with our feeding
areas, the Norwegian Sea and the Barents Sea, respectively. The catch limits are calculated for each of ES
and EB separately (no catch cascading), and whales
are removed according to the catch limits.
The RMP is tuned to three different levels. This is
taken to be the main experimental factor, and is called
TUNING. There are two tuning parameters, internal
protection level (IPL) and the probability of the quantile (PROB) of the posterior distribution of a quantity
related to the replacement yield used as the catch limit.
Both parameters are technical. The values of the
tuning parameters are chosen to meet target values
in a reference scenario with maximum sustainable
yield relative to the mature female component being
1%. At TUNINGˆLOW, IPLˆ54% and the target
stock size after 100 years of managed exploitation is
72% of K. At MEDIUM and HIGH levels, IPLˆ50%
and PROB is chosen to make target stock size 66% and
60%, respectively. The order of this speci®cation is
that TUNING on high level means high catches,
relative to the medium and low levels.
2.6.2. Fish
Cod and herring are managed by quotas calculated
to make ®shing mortality meet ®xed target values. A
VPA routine with Laurec±Shepherd tuning is used to
Table 1
Target fishing mortality rates for cod and herring
Experimental factor SFISH
Cod
Herring
L
M
H
0.45
0.60
0.45
0.2
0.2
0.35
calculate the quotas, see Hagen et al. (1997). The ®sh
management is taken as a factor, F, with three levels.
At each level, the Captool method with a reserved
spawning stock of capelin of 500 000 ton is used to
manage the capelin. For cod and herring, ®xed values
of F are used according to Table 1.
3. Experimental design and levels for uncertainty
factors
In Table 2 the factors of the experiment are speci®ed. Some of the factors are complex, and are further
speci®ed in Tables 1, 3 and 4.
Most of the factors are described in the previous
chapter, and we will now give only a description of the
experimental values for the two parameters MSYR
and K. The parameter choices for these parameters are
a result of ®tting a likelihood function to historical
data (unpublished note: STAT/11/97, Norwegian
Computing Center). It turned out that the log likelihood, as a function of MSYR and K, was a surface
with a maximum point and a banana shaped contour at
the 90% plausibility level. The parameter values for
Table 2
BIAS is relative bias in absolute abundance estimates for minke
whales
Experimental Name
factors
1
2
3
4
5
6
7
8
9
MSYR
H
C
BIAS
K
RCOD
RHER
FISH
TUNING
Levels
Reference
L
M
H
0.593
0.831
0.6
1.186
1.662
1
2.372
3.324
1.4
0.9
0.9
1
1
1.1
1.1
Table 3
Table 3
Table 1
Table 4
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
83
Table 3
Relation between the artificial factors W and L, and the experimental parameters MSYR (TRUE MSY RATE) and the carrying capacity K
Artifcial factors
Exp. parameter
Initial mature female stock
W
L
MSYR
K
ES
EB
MSY
L
L
L
M
M
M
H
H
H
L
M
H
L
M
H
L
M
H
0.0078
0.0155
0.0213
0.0015
0.0163
0.0280
0.0095
0.0171
0.0225
107 000
82 000
69 000
140 000
90 000
65 000
120 000
97 000
81 000
4020
2948
2546
5360
3350
2412
4556
3618
2948
24 656
19 028
15 946
32 169
20 770
15 008
27 604
22 378
18 760
501
763
882
126
880
1092
684
995
1094
MSYR is relative to the 1‡ stock. The unit for K, initial mature female stock and MSYˆ0.6MSYRK is 1 whale.
Table 4
Parameters governing the RMP
PPROB
IPL
L
M
H
0.423
0.54
0.45
0.50
0.51
0.50
the pair (MSYR,K) have been chosen to be at the
maximum point (reference point) and in 8 points on
the 90% contour, see Fig. 2. A simultaneous plausi-
bility range of 90% corresponds roughly to both
parameters having a plausibility range of 95%. When
®tting the likelihood function, only the Greater
Barents Sea stock (substocks ES and EB) was
included, and so the experimental part of the carrying
capacity K is the sum of the values for these two
substocks. The two substocks EN and EC were held at
constant carrying capacities 40 000 and 8000, respectively, and for technical reasons these values will be
used in the present experiment as well, but will not
Fig. 2. Contours of the log-likelihood function used to determine the experimental design in MSYR and K. The inner contour represents a
90% plausibility region, and is approximately spanned by the eight chosen points. The inner point is the maximum likelihood estimate.
84
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
in¯uence the results. For MSYR the parameter value is
the realized maximum increase rate relative to the 1‡
stock without whaling, and it is given as `TRUE MSY
RATE' to the MANA4 program.
We have previously based our scenario experiments
on design matrices that are orthogonal in the main
effects (concepts and techniques of experimental
design are presented by Box et al. (1978)). In the
present experiment this is not possible, because
MSYR and K are de®ned simultaneously. Instead,
we base the experimental design on an orthogonal
array design when de®ning the scenarios. To do this,
we introduce two arti®cial experimental factors, W and
L, both having three levels (LOW, MED and HIGH),
and when de®ning the scenarios the experimental
parameters MSYR and K are substituted by the two
arti®cial factors. In the simulations (and the analysis),
each point of (W, L) corresponds to a speci®ed set of
values for (MSYR, K). Table 3 displays the relation
between (W, L) and (MSYR, K), while Table 5 shows
the set of scenarios that will be simulated in this
experiment.
The production model has the level of maximum
sustainable yield at MSYLˆ0.6K. The maximum
sustainable yield is thus MSY ˆ0.6KMSYR, which
is given in Table 3 for ease of interpretation.
Each of the scenarios speci®ed in Table 5 was
simulated in two replicates. The experimental design
and the number of replicates is minimal, at least by the
standards of previous scenario experiments related to
the RMP carried out within IWC. However, our
interest is in mean response of ®sheries to increased
whaling and not on the potential for severe depletion.
As will be seen in the Section 4 our minimal design is
suf®cient to allow conclusions to be made ± conditional on the assumptions made.
Table 5
Orthogonal array design for the experiment
No.
W
H
C
BIAS
L
RCOD
RHER
SFISH
TUNING
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
L
L
L
L
L
L
L
L
L
M
M
M
M
M
M
M
M
M
H
H
H
H
H
H
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
L
L
L
M
M
M
H
H
H
M
M
M
H
H
H
L
L
L
H
H
H
L
L
L
M
M
M
L
L
L
M
M
M
H
H
H
H
H
H
L
L
L
M
M
M
M
M
M
H
H
H
L
L
L
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
L
M
H
M
H
L
M
H
L
M
H
L
H
L
M
H
L
M
H
L
M
L
M
H
L
M
H
L
M
H
H
L
M
H
L
M
H
L
M
M
H
L
M
H
L
M
H
L
1
3
3
2
3
1
3
1
2
1
2
3
2
3
1
3
1
2
1
2
3
2
3
1
3
1
2
L
M
H
M
H
L
H
L
M
M
H
L
H
L
M
L
M
H
H
L
M
L
M
H
M
H
L
The levels for SFISH are labeled 1, 2 and 3 not implying any order.
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
4. Results
For each response variable, regression analysis has
been used to summarize the results. As predictors we
use the experimental factors as categorical covariates
and the experimental parameters MSYR (in %) and K
(in unit 1000 whales) are used as numerical covariates.
The descriptive models are ®tted with a constant term.
The categorical covariates are coded as treatment
factors, with their medium level (M or 2) as reference
category. There are, thus, two parameters associated
with the main effect of each of these covariates.
With an orthogonal array design, regression coef®cients for different main effects are uncorrelated. Due
to the inclusion of MSYR and K as numerical covariates `outside' the orthogonal design, we must, however, expect some correlations.
For an orthogonal array design with 27 points
for nine 3-level factors, interactions cannot be
estimated when all the factors are included with main
effects.
4.1. Cod
4.1.1. Mean yearly catches
Fig. 3 shows marginal effects plot for mean catch of
cod over 90 years. For each categorical factor, and for
85
each level of these factors, the mean value over the
simulation period and over scenarios with the factor at
the given level, is shown. In addition, the mean value
for each level of the product MSY is shown to the
right.
For TUNING, the expected order is that at low
level, the whale stock will grow higher and compete
more strongly with cod and also consume more cod,
and so the ordering is expected to be from low to high
level. We observe that the simulation results have the
expected order. This is also true for the factors C and
BIAS. When C is on low level, the whale's preference
for cod (and capelin) is low, and the cod stock will
grow larger, so the expected order of the mean catches
is from high to low level. When BIAS is on low level,
the whale stock is estimated too low, the catches will
be lower and the stock will grow higher. The low
values of MSY occur when MSYR is low. Then the
RMP will keep the whale stock at a reduced level,
while with MSYR higher than 1%, the stock will
approach levels closer to carrying capacity. This is
seen from Fig. 6. The reduction in catches of cod
when MSY increases is thus an effect of an increased
whale abundance.
The ®tted descriptive model includes the factors
TUNING, MSYR, K, BIAS, C and SFISH. In addition
to the main effects, the descriptive model also includes
Fig. 3. Marginal effects plot for mean annual cod catch (in million tons).
86
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
Table 6
Effects on the cod fisherya of tuning the RMP
Intercept TUNING
L
Value 2.711
Std. e 1.079
H
MSYR K
BIAS
L
ÿ0.069 0.041 ÿ1.490 ÿ1.353 0.294
0.026 0.026 15.692 0.800 0.186
SFISH
kC
MSY
H
BIASkC
H
L
H
L
LL
HL
LH
HH
0.256
0.110
0.266
0.110
0.231
0.186
0.030 ÿ0.027 ÿ0.137 ÿ0.519 ÿ0.009 ÿ0.427 ÿ0.406
0.026
0.026 0.029 0.287 0.075 0.270 0.287
a
Regression results: mean yearly cod catches.
For categorical covariates, for MSYR, K and MSY when multiplied by the covariate, the unit is million ton, R2ˆ0.87.
the interaction effect MSY, and that between C and
BIAS, see Table 6.
The main effect of tuning the RMP up from a target
of 72% to a target of 60% is an increase in yearly cod
catches of about 0.1 million ton, see Table 6, where
the estimated contrast in TUNING is 0.11ˆ0.041ÿ
(ÿ0.069). The relative increase will, however, depend
on the values of MSYR and K, because they will
always contribute to the expected value. The main
effects of MSYR and K are not signi®cant, but the
interaction effect MSY is highly signi®cant.
4.1.2. Mean stock size
We would expect the whale factors (TUNING, C,
BIAS) to have similar effect on the mean cod stock
compared to the effect on the cod catches. Fig. 4
shows the marginal effects over all levels of the
categorical covariates, and we observe that the simulated results coincides with our expectations.
The ®tted descriptive model includes the terms
TUNING, MSYR, K, C, BIAS and RCOD, and also
the interaction effect MSY, see Table 7. The main
reason for the relatively modest R2ˆ0.64 is a very low
response value in one of the scenarios (one of the
replicates of scenario number 20, see Table 5), which
the regression model is not capable to re¯ect. The
reason for the low response value in this simulation is a
total collapse of the cod stock after a period of 50
years. The fact that TUNING was at low value in this
scenario and the levels of MSYR and K were such that
the number of whales would be relatively high
throughout the simulation period while the levels of
H and C would make the per whale consumption of
cod relatively high, might hint that the cod stock in the
Fig. 4. Marginal effects plot for mean annual cod stock (in million ton).
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
87
Table 7
Effects on the cod stocka of tuning the RMP
Intercept TUNING
Value
Std. error
2.596
0.751
MSYR
L
H
ÿ0.262
0.101
0.036
0.101
53.035
24.546
K
MSY
0.126
0.547
C
ÿ0.189
0.042
BIAS
RCOD
L
H
L
H
L
H
0.185
0.101
ÿ0.127
0.101
ÿ0.070
0.101
0.252
0.101
ÿ0.208
0.106
ÿ0.005
0.106
a
Regression results: mean stock size (cod).
For categorical covariates, for MSYR, K and MSY when multiplied by the covariate, the unit is million ton, R2ˆ0.64.
model is quite sensitive to the number and behavior of
the minke whales. The sensitivity of the northeast
Arctic cod to the status of the minke whale stock is
a major research issue that is not pursued in the present
paper.
When studying the coef®cient estimates we observe
some obvious unreasonable results. The main effects
of both MSYR and K are estimated positive, which
contradicts our intuition. The interaction effect is,
however, negative, and more than compensates for
the positive main effects for all simulated pairs
(MSYR,K). The reason for this peculiarity is the
strong linkage between those two experimental parameters. Large MSYR-values are simulated only in
combination with small K-values (and vice versa), and
therefore, the main effects will be hidden. Interpreting
the coef®cient estimates further, tuning the RMP to the
most protective level (TUNING on low level) will
result in a mean cod stock that are approximately 0.26
million ton lower than if any of the two other tuning
targets were used. It is no signi®cant difference for the
two higher levels. A positive bias in the abundance
estimates for whale (BIAS on high level) will also
result in a signi®cantly higher mean cod stock. The
interpretation is that increased whale catches as a
result of the high abundance estimates, has a positive
effect on the cod stock.
From Table 7 we observe that the main effect of
MSYR is signi®cant on 5%, but as discussed above we
are convinced that this effect is spurious and due to the
experimental design.
4.2. Herring
4.2.1. Mean yearly catches
Table 8 shows the result of ®tting the preferred
descriptive model with mean yearly catch of herring
as the response variable.
A model consisting of H as the only explanatory
variable has R2ˆ0.74. The preferred regression model
is quite good, having R2ˆ0.88. The effect of TUNING
seems to be somewhat strange, since both the low and
the high level has positive main effects and as such
result in higher yearly herring catches than when
TUNING is on medium level. From the standard errors
we observe, however, that none of the effects are
signi®cant at 5% level, but the effect of TUNING
on high level is signi®cant at 10%. A possible interpretation could then be that when the RMP is tuned
towards the least protective level (TUNING on high
level), the main effect is a slight increase in mean
yearly herring catches, while there is no signi®cant
difference between the two lower TUNING levels.
From the coef®cient table we can also observe that the
Table 8
Effects on the herringa fishery of tuning the RMP
Intercept
Value
Std. error
a
1.024
0.283
TUNING
MSYR
L
H
0.055
0.039
0.072
0.039
13.458
9.156
K
ÿ0.027
0.207
MSY
ÿ0.066
0.015
H
L
H
0.228
0.039
ÿ0.411
0.039
Regression results: mean yearly herring catches.
For categorical covariates, for MSYR, K and MSY when multiplied by the covariate, the unit is million tons, R2ˆ0.88.
88
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
Table 9
Effects on the herring stocka of tuning the RMP
Intercept
Value
Std. error
8.069
3.051
TUNING
MSYR
L
H
0.528
0.404
1.022
0.404
298.037
102.628
K
2.295
2.236
MSY
ÿ0.982
0.172
RHER
H
L
H
L
H
4.86
0.404
ÿ3.982
0.404
ÿ0.043
0.424
1.206
0.418
a
Regression results: mean stock size (herring).
For categorical covariates, for MSYR, K and MSY when multiplied by the covariate, the unit is million tons, R2ˆ0.93.
main effects of MSYR and K are far from being
signi®cant, which once more most probably is due
to the experimental design. The interaction effect
MSY is, however, both signi®cant and it is negative,
as expected.
4.2.2. Mean stock size
Similar to the herring catches, the mean stock size
is mainly affected by the whale's preference for
herring, H. The preferred descriptive model for
mean herring stock is similar to the model describing
the catches, except for the inclusion of the herring
recruitment factor, RHER. When this factor is on
high level, the result is a signi®cant increase in
the mean herring stock size. The coef®cient estimates
for the ®tted regression model is given in Table 9,
and most of the comments concerning the model
for herring catches are also relevant for this model.
The ®tted model has R2ˆ0.93, and ®ts the data
very well.
4.3. Capelin
Capelin stock and catches are mainly affected by
the large ¯uctuations in capelin recruitment. That
capelin also is managed by the Captool approach,
allowing the ®shermen to take whatever is estimated
of the capelin stock when a suf®cient amount is
reserved for the cod to feed on and as spawning stock,
makes the relationship weak between yearly capelin
catches or mean stock size on the one hand and of our
experimental factors on the other. From a marginal
plot one can get the impression that the management
strategy for cod and herring (SFISH) will have large
explanatory effect. We have tried to ®t a model based
on this impression, but unlike the herring responses,
the apparent marginal effect does not result in any
explanatory power. All our alternative regression
models resulted in R2-values less than 0.3, and consequently, none of the models were acceptable for
describing an eventual relation between the capelin
catches (or stock size) and the experimental factors
and parameters.
4.4. Minke whales
For the whale responses, the only relevant covariates are TUNING, BIAS, MSYR and K, and combinations thereof like MSY. The reason is that these are the
only covariates that concern the MANA4-program,
which is used to simulate the whale stock.
4.4.1. Mean yearly catches
In Fig. 5 the marginal effects on the mean yearly
whale catches are shown. TUNING on low level
represents the most protective tuning of the RMP
procedure, and the order of the marginal effects are
from low to high level. Also for the BIAS factor, the
order of the marginal effects are from low to high. This
is as expected, since BIAS on low level means that the
abundance estimates are negatively biased and so the
result is lower quotas (and thus lower catches).
Normally we would expect the order of the marginal
effects of both MSYR and K to be from low to high
level. From the marginal effect plot we observe that
this is not the case in the present experiment. The
reason for the apparent opposite effect of the carrying
capacity factor K is once more the link between the
two factors. When K is high the MSYR is very low,
and the result is that the whale stock decreases until
reaching a minimum before it starts to grow again. The
catches start to decrease when the stock size gets under
a limit (dependent on the tuning of the RMP), and
there is not any increase in the catches during the
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
89
Fig. 5. Marginal effects plot for mean annual whale catch (in number of whales).
simulation period in any of the de®ned scenarios. The
result is that the mean yearly catches over the total
simulation period gets low for this combination of the
MSYR and K parameters.
The four relevant factors are all included in the
®tted regression model, which has R2ˆ0.92. From
Table 10 we observe that the main effects of TUNING
and BIAS are as expected. The low level results in
lower catches and the high level in higher catches, than
does the reference (middle) level. The negative main
effects for MSYR and K are for almost all combinations in the design more than neutralized by a large
and positive interaction effect.
The ®tted model states that TUNING on low level
results in a mean yearly catch (quota) that is 151
animals less than the reference level, while TUNING
on high level has a mean yearly catch that is 169
animals higher than the reference level. The estimated
mean difference in the yearly quota is thus more than
300 animals between the two extreme levels.
4.4.2. Mean stock size
TUNING and BIAS have opposite marginal effects
on the stock size, compared to the catches. The reason
is of course that lower catches result in a larger stock,
and vice versa. Fig. 6 shows the marginal effect plot,
and we can observe that the effects of TUNING and
BIAS are smaller on the stock size than they are on the
catches (compared to the range of simulated values).
Further we would expect MSYR and K to have
similar effects on stock size and catches, and as we can
see from the ®gure the expectation was correct. This
means that even if the experimental design has been
chosen in such a way that the real main effects are
hidden, the apparent main effects are similar for the
two responses.
Table 10
Effects on the whale catchesa of tuning the RMP
Intercept
Value
Std. error
a
206
241
TUNING
MSYR
L
H
ÿ151
33
169
33
ÿ25 326
7 794
K
ÿ75
176
MSY
102
13
BIAS
L
H
ÿ323
33
248
33
Regression results: mean yearly whale catches.
For categorical covariates, for MSYR, K and MSY when multiplied by the covariate, the unit is number of whales, R2ˆ0.92.
90
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
Fig. 6. Marginal effects plot for mean annual whale stock (in number of whales).
The four factors are all included in the ®tted model,
and the model includes both the obvious interaction
effect between MSYR and K, and an interaction effect
between TUNING and BIAS. Table 11 shows the
coef®cient estimates, and the ®tted model has
R2ˆ0.94. We observe that the main effect of managing
the whale stock according to the most protective level
is an increase in mean stock size of approximately
9000 animals, while there is no signi®cant difference
in mean stock size between the two higher TUNING
levels.
4.5. Cod and herring catches versus whale stock
To investigate further the direct effect of the whale
stock on cod and herring ®sheries, we have ®tted linear
models with number of whales in the Barents Sea as an
additional explanatory variable, using unit ton for the
catches of ®sh and unit whale for the whale stock.
4.5.1. Cod catches
The model for cod catches includes the mean
number of individuals in the whale stock over the
period (called WHALE), the cod recruitment (RCOD)
and the whale's preference for cod and capelin (C).
Table 12 displays the regression results.
From this table one can observe that the main effect
on cod catches from one whale is estimated to be a
reduction of about 6.5 ton, and the standard deviation
tells that the effect is signi®cant. The main effects of
the whale's preference for cod and capelin are not
signi®cant, while the interaction effect between C and
WHALE tells that when the whale has a high preference for cod and capelin the cod catches are further
Table 11
Effects on the whale stocka of tuning the RMP
Intercept
Value
87 239
Std. error 13 740
a
TUNING
MSYR
L
H
9164
3320
503
3272
K
ÿ4 079 533 ÿ15 844
472 017
10 285
MSY
14 193
792
BIAS
TUNING BIAS
L
H
LL
HL
LH
HH
14 106
3272
ÿ3670
3320
ÿ12 299
4742
ÿ4596
4610
ÿ1441
4711
ÿ14 004
4743
Regression results: mean stock size (whale).
For categorical covariates, for MSYR, K and MSY when multiplied by the covariate, the unit is number of whales, R2ˆ0.94.
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
91
Table 12
Regression results with number of whales as a covariate (cod catches vs. whale stock)
Intercept
Value
Std. error
1 332 070
92 733
WHALE
ÿ6.57
0.76
RCOD
C
WHALE C
L
H
L
H
WHALEL
WHALEH
ÿ75 087
186 343
233 991
136 174
ÿ718 48
25 922
37 397
26 690
1.46
1.49
ÿ2.39
1.10
Table 13
Regression results with number of whales as a covariate (herring catches vs. whale stock)
Intercept
Value
Std. error
1 205 643
123 884
WHALE
ÿ3.90
0.99
RHER
H
WHALEkH
L
H
L
H
WHALEL
WHALEH
ÿ155 408
151 341
231 602
146 815
ÿ33 722
21766
57 134
22 099
3.12
1.22
ÿ5.21
1.18
reduced. The recruitment covariate is signi®cant on
low level. The model has R2ˆ0.85.
4.5.2. Herring catches
The model for herring catches looks similar to the
model for cod catches, including the whale stock
(WHALE), the whale's preference for herring (H)
and the herring recruitment (RHER). The regression
results are given in Table 13.
This table shows that the estimated main effect on
herring catches from one whale is a reduction of
3.9 ton, and the standard deviation shows that the
effect is signi®cant. The estimated interaction effects
show that when the preference for herring is on low
level, the main effect is nulli®ed while when the
preference is on high level the reduction is increased
by approximately 130%. As for the cod catches, the
main effects of the preference covariate (H) are not
signi®cant. The recruitment covariate is signi®cant on
high level, and the model has R2ˆ0.96.
5. Discussion
The title of this paper is ``On the effect on cod and
herring ®sheries of retuning the revised management
procedure for minke whaling in the Greater Barents
Sea''. So, what are the conclusions of this simulation
experiment? In our model, and in the region of
parameter space that has been explored, the picture
is clear: the higher the whale stock, the stronger
the predation pressure on herring and cod, and the
smaller the long term catches of cod and herring. For
the cod ®shery, it is bene®cial to manage minke
whaling with RMP tuned to the 60% target value.
For herring catches, the effect of tuning RMP was not
signi®cant.
It is possible to imagine situations where whaling no
longer is of bene®t to the cod ®shery. If, say, herring in
the Barents Sea prey harder on capelin larvae than
modelled, a larger stock of whales could reduce the
number of herring in the Barents Sea resulting in a
larger stock of capelin. If the increase in capelin more
than compensates the adverse effect on the cod stock
by the increased predation pressure on cod from minke
whales and also less herring as food for cod, the result
of increased whaling is reduced cod catches. From the
present experiment, this situation seems to require
more non-linearity in the predation pattern, and less
predation of minke whales on cod and capelin than we
have in our model.
Since the design set is limited to the orthogonal
array with only 27 points, there is limited space for
identifying interaction effects in general. This is a
restriction in the process of model ®tting. One exception has been the interaction effect between the two
parameters MSYR and K. These parameters are not
included directly in the orthogonal array design, and
the result is both that the interaction effect can (and
should) be estimated and included in the ®tted models,
92
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
and that the main effects are hidden in the experimental design. We have observed that in most of the
®tted models the main effects of these two parameters
were estimated to contradict our expectations, but also
that the interaction effect more than compensated for
the `wrong' main effects so that the total effect of the
two parameters were `as expected'. Due to the strong
linkage between the two parameters, the main effects
are hidden in the experimental design. One additional
reason for the strange main effect estimates could be
that the interaction effect is not uniquely de®ned. For
each value of MSY there are in®nitely many combinations of MSYR and K that have identical product,
and the estimated main effects could be interpreted as
an adjustment of the total effect according to the actual
values of the two parameters.
In a separate analysis (submitted to NAMMCO) we
found that both cod and herring catches are well
approximated by linear functions of the number of
whales in the stock, and that the cod catches are
reduced by some 5 ton and the herring catches by
some 4.5 ton for each extra whale present in the
Barents Sea. The simulations behind those numbers
were performed with whale stock being constant over
the simulation period, all other factors on their medium level and with no replicates, and so there was little
room for uncertainty considerations. To further investigate the effect of whales on the cod and herring
catches we use data from the present uncertainty
experiment. Using the number of whales as an additional explanatory variable when ®tting models to the
data on cod and herring catches, we estimate the cod
catches to decrease with 6.57 ton (s.e. 0.76 ton), and
the herring catches to decrease with 3.90 ton (s.e.
0.99 ton), when the minke whale stock is increased
with one animal. This means that a 95% con®dence
interval for each extra whale's effect on the cod
catches is (ÿ8.06, ÿ5.08) and on the herring catches
the con®dence interval is (ÿ5.84,ÿ1.96). The present
uncertainty experiment thus con®rms the cost on the
herring catches (ÿ4.5 is included in the con®dence
interval), while the cost on the cod catches of each
extra whale perhaps is larger than 5 ton. Note, however, that uncertainty in the herring capelin interaction
has not been taken into account, and that this uncertainty would propagate into uncertainty in the interaction between whale abundance and cod catch. From
Table 12, it is also seen that the effect of whale
abundance on cod catches depends upon the level
of C, which should be no surprise.
As noted in Section 1, the substantive conclusions
obtained from our experiment must be understood as
tentative. There is still too much uncertainty surrounding this study to allow ®rm conclusions. The model
assumed for minke whale predation is based on data.
Observed local abundance of prey was compared to
stomach contents in whales. The local abundance data
are, however, of questionable quality, and our model
for whale predation might, therefore, not be very good.
The predation of herring on capelin larvae and young
capelin is modelled as reduced capelin recruitment in
years with herring in the Barents Sea. The realism of
this important part of the model is also open to
question. This is also the case for a number of other
aspects of the model like assuming the minke whale
dynamics unaffected by the status of the ®sh stocks.
With respect to the main question ± does increased
whaling yield higher catches of cod and herring ± our
qualitative conclusion is likely to stand even if the
density dependence of the minke whale basically is
determined by available amounts of cod, capelin and
herring. Increased whaling will, even in this case, lead
to a decrease in the whale stock ± with decreased
predation pressure on the ®sh stocks, and increased
catches of cod and herring.
In addition to presenting tentative results on the
effect of increased whaling on the ®sheries of herring
and cod, our aim is to present a case of what we call the
method of scenario experimentation. The concepts
and techniques of this method to study the comparative properties of management regimes were presented
and discussed in Hagen et al. (1997). They have been
exempli®ed in the present study.
The extensive simulation studies carried out during
the development of the RMP within the IWC Scienti®c
Committee can be regarded as cases of scenario
experiments, see Kirkwood et al. (1997). The implementation simulation trials required before an actual
implementation of the RMP are de®nitely scenario
experiments, with particular emphasis on uncertainties surrounding the stock in question, see International Whaling Commission (1993) and Kirkwood
et al. (1997).
Simulation studies are not uncommon in the general
®sheries literature. Our impression is, however, that
simulation studies for other ®sheries are usually less
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
where B is the mature biomass of the stock, MAXR is
the expected maximum number of recruits and H is the
stock's half value. The noise is given by a normal
(Gaussian) stochastic variable Z having expectation
and standard deviation . All parameters are, of
course, speci®c for each species. Maximum values and
half values used in the present experiment are given in
Table 14.
Due to environmental factors, recruitment will be
exceptionally high in some years. This is modelled by
having two levels for the maximum value (MAXR),
one `normal' value and one `high' value. The actual
recruitment level is every year (in the simulations) a
result of a stochastic drawing. This stochastic procedure results in years of high recruitment at least 5
years apart and with a mean distance (between 2 years
of high recruitment) of 10 years. Years of high recruitment are common for all species.
When herring is present in the Barents Sea, the
predation on capelin larvae results in reduced capelin
recruitment. How the herring stock affects the capelin
recruitment is modelled as a piecewise linear function.
In the following, the prey stock is de®ned as the
capelin recruits, and the predator stock is de®ned to
be the part of the herring stock present in the Barents
Sea. Roughly the model is as follows:
Let CAPREC be the number of capelin recruits
according to a Beverton±Holt recruitment model, and
HER be the number of herring (0±3 years) present in
the Barents Sea.
If CAPREC < RECRPREYLOW, or if HER <
RECRPREDLOW, then no predation is performed.
If HER > RECRPREDUP, then the number of
capelin recruits, RECR, is set to min(CAPREC,
RECRPREYLOW).
If RECRPREDLOW < HER < RECRPREDUP (and
CAPREC > RECRPREYLOW), then the number of
capelin recruits are reduced according to the function
systematic than those carried out within the IWC, and
those in the present paper and in Hagen et al. (1997).
Comparison of management regimes by simulation is
often done within a model with little regard to uncertainties. The conclusions from the primary comparative analysis might then be put to sensitivity analysis
to investigate their robustness.
Scenario experimentation is, of course, closely
related to sensitivity analysis by way of simulation.
Most cases of sensitivity analysis are, however, characterized by (i) the result that is tested for sensitivity is
arrived at in a given model without regards to the
uncertainty embodied in the sensitivity trials, and (ii)
the experimental design of the sensitivity trials is naive
with one factor perturbed at the time, and possibly
with imbalance between the number of replicates and
the main design. The analysis of the simulation results
is also naive with simply tabulating summaries of the
raw simulation data. This as distinct from scenario
experiments, where a careful experimental design is
chosen, and where the results and the associated
uncertainties are obtained in an integrated analysis.
Appendix A
A.1 Fish recruitment
For all three species recruitment is basically modelled as a Beverton±Holt function, subjected to some
noise. The noise can be additive or multiplicative, both
possibilities are present in the simulation model. In
this simulation experiment we have used the additive
alternative, because we are most experienced with
that. The general ®sh recruitment model in this experiment is then given by the following formula:
Number of recruits ˆ MAXR 93
B
‡ Z…; †
B‡H
Table 14
Parameter values in fish recruitment
Species
Capelin
Cod
Herring
Maximum value, MAXR (in billion)
Normal years
Extreme years
1800
1.33
26.52
1800
2.22
280.63
Half value, H (in million tons)
0.2
0.21
1.29
94
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
RECRˆ RECRPREYLOW ‡ …CAPREC
The combined potential capelin and cod consump0
tion, Ccc
, is divided between capelin and cod according to the formula
acap Bcap
0
0
ˆ Ccc
:
Ccap
acap Bcap ‡ …1 ÿ acap †Bcod
ÿRECRPREYLOW† F
with F being the fraction
RECRPREDUP ÿ HER
:
RECRPREDUP ÿ RECRPREDLOW
The
three
parameters
RECRPREYLOW,
RECRPREDLOW and RECRPREDUP are parameters to the simulation program, and in the present
experiment the following numbers are used:
CAPRECLOW
HERLOW
HERUP
ˆ
ˆ
ˆ
20 109
2:0 109
3:0 109
A.2 Predation formulas
A.2.1. Whale-fish predation
We use the following model discussed and roughly
estimated elsewhere (submitted to NAMMCO). The
estimated model must be regarded as tentative.
Let Bher, Bcod and Bcap denote the total herring, cod
and capelin biomass respectively, and let Bcc ˆ Bcod
‡Bcap denote the total biomass of cod and capelin.
Further, we de®ne Cmax to be the maximum daily consumption of ®sh from one minke whale. From energetic
calculations, one has found that Cmaxˆ0.09 ton (Haug et
al., 1996).
The potential daily consumption of herring and
0
0
and Ccc
respectively, are
cod‡capelin, denoted Cher
given as
0
Cher
0
Ccc
ˆ
ˆ
Cmax ch …Bher ; Bcc †
Cmax ÿ Cher †ccc …Bcc †
(1)
The proportions of herring and cod‡capelin in the
daily consumption, ch(Bher, Bcc) and ccc(Bcc), are
modelled as modi®ed logistic functions:
ch …Bher ; Bcc † ˆ
exp… ‡ H Bher ‡ C Bcc †
1 ‡ exp… ‡ H Bher ‡ C Bcc †
‰1 ÿ exp…ÿH Bher †Š
exp… ‡ C Bcc †
ccc …Bcc † ˆ
‰1 ÿ exp…ÿC Bcc †Š
1 ‡ exp… ‡ C Bcc †
(2)
With H>0, C<0 and C>0 the consumption function
will have a reasonable structure.
We will assume 0.5acap1, since capelin is a fatter
®sh than cod, and it is presumable no less attractive to
the minke whale.
The time step in the model is one month. To prevent
the predation of the minke whales to empty, or even to
create negative stocks, we assume that the actual consumption during one month, Cher etc., is at the most
half the biomass of the food item, generically Cher ˆ
0
; Bher =2†, while for cod a further restriction is
min…Cher
imposed, Ccod ˆ min…20; Ccod ; Bcod =2† (in tons).
Point estimates for the parameters of this predation
model were obtained by at least squares ®t of the
estimated consumed quantities calculated from minke
whale stomach samples and estimated prey abundance
in regions of the Barents Sea. The estimates are given
in Table 15.
A.2.2. Cod predation
We de®ne the following quantities:
H
CAP1
CAP2
CAP3
HER
COD1
COD2
COD3
CY
Indicator for period. Hˆ1 for second half
year.
Number of immature capelin <10 cm.
Number of immature capelin 10 cm.
Number of mature capelin.
Number of herring in Barents Sea.
Number of 1-year-old cod.
Number of 2-year-old cod.
Number of cod 3 years and older.
Indicator for 2-year-old cod.
The cod predation is modelled in two steps. First we
model the total stomach content as a linear function of
number of ®sh on log-scale. The response is the ratio
between the weight of the stomach content and the
total weight of the ®sh. The model for young cod (1±2
year old) and older cod differs, and the stomach
content models are
R1 ˆ 0 ‡ 1 log…CAP1 † ‡ 2 CYlog…CAP2 †
‡ 3 CY…1 ÿ H†log…CAP3 † ‡ 4 CYlog…HER†
‡ 5 log…COD1 † ‡ 6 log…COD2 †
‡ 7 log…COD3 †
(3)
T. Schweder et al. / Fisheries Research 37 (1998) 77±95
95
Table 15
Rough estimates of predation parameters in the whale predation model
Cmax
H
C
C
acap
0.09 ton
ÿ2.75
1.18
ÿ0.24
ÿ0.27
1.66
0.50
Cmax is estimated from energetic considerations (Haug et al., 1996).
for 1 and 2-year-old cod, and for cod of age 3‡,
R3 ˆ 0 ‡ 1 log…CAP1 † ‡ 2 log…CAP2 †
‡ 3 …1 ÿ H†log…CAP3 † ‡ 4 log…HER†
‡ …5 ‡ 6 …1 ÿ H††log…COD1 † ‡ 7 log…COD2 †
‡ 8 log…COD3 †
(4)
These formulae tell that 1-year-old cods feed on
small immature capelin, competing with the older
cod. 2-year-old cods feed on capelin and herring,
competing with both younger and older cods. The
older cods feed on both capelin, herring and young
cod, competing with the rest of the cod stock. The
mature capelin is only preyed upon in ®rst half year,
while for the old cod the eventual cannibalism has a
different pattern in the ®rst and second half year. We
distinguish between the two half years due to the
spawning migrations.
The above models are computed for each age group
of cod, and the next step is to model the distribution of
the different prey species. This is done separately for
each group of prey species (immature capelin, mature
capelin, herring, 1 and 2 year old cod), and the
modelled value is the percentage of the actual prey
specie for each predator group. The linear model is
parallel to the stomach content models:
P ˆ 0 ‡ 1 log…CAP1 † ‡ 2 log…CAP2 †
‡ 3 …1 ÿ H†log…CAP3 † ‡ 4 log…HER†
‡ …5 ‡ 6 …1 ÿ H††log…COD1 †
‡ 7 log…COD2 † ‡ 8 log…COD3 †
(5)
The left-hand quantity (P) corresponds to the percentage of the actual prey species, e.g. CAP1, and the
model is for each predator group estimated for every
prey group. The model describes a mixture of availability (the prey species) and competition (the other
predator groups).
All models are estimated separately (unpublished
note: STAT/02/1995, Norwegian Computing Center),
and one is not assured that the percentages for one
predator group will sum to 100. One is not even sure
that the model will give only non-negative percentages. It is therefore necessary to do an adjustment.
After computing the model percentages, all negative
numbers are substituted with zeros and the remaining
percentages are scaled to sum to 100.
A.2.3. Herring predation
The herring predation is not modelled explicitly,
and the reason is that the only relevant predation is
from herring on 0-group capelin. This model is
described in Section A.1.
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