SC/J09/JR14

Preliminary assessment of impacts on thesandlance population by consumption ofminke whales oﬀ Sanriku region
Hiroshi Okamura∗1 , Hiroshi Nagashima∗2 , Shiroh Yonezaki∗3*1 National Research Institute of Far Seas Fisheries, Fisheries Research Agency, Kanagawa
236–8648, Japan*2 Miyagi Prefecture Fisheries Research and Development Center 97–5, Watananoha,
Ishinomaki, Miyagi 986–2135, Japan*3 Institute of Cetacean Research, NF bldg. 2F 8-13-11 Nishi-Shinjuku, Shinjuku, Tokyo,
Japan

Abstract
We develop a Bayesian assessment model to investigate the eﬀects of consumption by minke whales on sandlances oﬀ the Sanriku region. The modelallows for various uncertainties making use of time series data historicallycollected by ﬁsheries and researches. The impact of predation was examinedin terms of MSY. When we used the linear functional response curve, theresultant impact was so great that the median value of MSY was increased
by 154%. Whereas using the constant functional response, the impact wasmuch smaller, where we saw only 17% increase of the median value of MSY.
The estimation of functional response forms should therefore be important.

1. Introduction
The role of top predators on ecosystem is one of central interests in ecology and ﬁsheries science. Tamura and Ohsumi (1999) claimed that whales
consumed at least three times as many preys as ﬁsheries in the world. However, Yodzis (2001) stressed that culling top predators does not always leadto an increase of their preys and can even provide a detrimental eﬀect on theﬁshery. The objective of this paper is to quantify the impacts of the predators on the prey species and their ﬁsheries and contribute the information
to the ﬁsheries management of the prey species.
The sandlance oﬀ Sarinku region, which is the northwestern Paciﬁccoastal region of Japan, is an important ﬁshery resource and is preyed bymany predators including marine mammals such as minke whales and northern fur seals. We focus on interaction between sandlances and minke whalesin this paper. Minke whales are transients in spring oﬀ Sanriku region,
however, sandlances are regarded as a closed population (Although Sanriku
sandlances include two stocks, one is much smaller than the other). So, as
the ﬁrst step, we develop a sandlance’s population dynamics model. After the basic model development, we take account of predation by minke
1whales in the ﬁshing season. The devloped model is considered as a kind of“Minimum Realistic Models (MRM)” (Punt and Butterworth 1995).

2. Materials and Methods
Brief History of Sandlance Fishery
Sandlances consist of juveniles and adults, which are categorized by theirbody length (Juveniles: ≤ 10cm, Adults: ≥ 10cm). Juveniles and adults
are caught by diﬀerent ﬁshing gear, so ﬁshermen are basically diﬀerent forjuveniles and adults. Although the amount of catch was about 14,000 ton
on average between 1950s and early 1980s, the oﬀshore trawl-net ﬁshermen
participated in ﬁshery for adults in 1984 so the population size decreased
dramatically (Hashimoto 1991; Nagashima 2004). Consequently, there occurred a serious conﬂict between coastal and oﬀshore ﬁsheries. After discussion, the trawl-net ﬁshery has stopped voluntarily, lift net with light ﬁsherycut ﬁshing period, and the upper limit of total allowable catches was set to
10,000 tons by private ﬁshing regulation (Nagashima 2004).
Data
Basic datasets are as follows:• CPUE series of juvenile sandlances by lift net with light ﬁshery from1994 to 2006 (Top plots of Fig. 1)• CPUE series of adult sandlances by dip nets from 1995 to 2006 (Middle
plots of Fig. 1)• Abundance estimates of adult sandlances by the DeLury method from1984 to 1989 based on the oﬀshore trawl-net ﬁshery data, which aredealt with as relative abundance indices (Bottom plots of Fig. 1)• Ages (Fig. 2), lengths, and weights time series data from 1999 to2005 (Sandlances estivate in autumn after ﬁshing seasons in spring.
Sandlances estivating in sand are sampled and aged. Their lengths
and weights are measured at the same time.)• Catch series by lift net with light ﬁshery, dip nets, and trawl-net ﬁsheryfrom 1901 to 2006 (Fig. 3). Data during the war are missed and data
before the war do not have distinction between juveniles and adults.
Data before the war was classiﬁed into juveniles and adults using theratio after the war. We used a simple linear interpolation for the
missing data during the war.
• Consumption of sandlances by minke whales in 2003 (Tamura et al.
2004).

2

Population Dynamics Model
We use a hierarchical Bayesian delay-diﬀerence model (Hilborn and Walters 1992; Meyer and Miller 1999a). The process model is given by:• Juveniles (Number)：Nt+1,J

2 ]∼ LN[f Nt+1,A {1 + r(1 − Nt+1,A /KA )}, σA,1

• Adults (Number)：
0.5 N , σ 2 ]Nt+1,A ∼ LN[(1 − FA,p )SA Nt,A + (1 − FJ,p )θSAt,J
A,2

• Juveniles (Biomass):
Bt+1,J = w0 Nt+1,J• Adults (Biomass)：
0.5
Bt+1,A = (αNt,A + ρBt,A )SA (1 − FA,p ) + w1 Nt,J θSA(1 − FJ,p )where：
NA,t : Adult abundance in year t
BA,t : Adult biomass in year t
NJ,t : Juvenile abundance in year t
BJ,t : Juvenile biomass in year t
SA ： Adult survival rate
θ： Juvenile survival rate (early half)
w0 : body weight in age 0
w1 : body weight in age 1
f ：fecundity in equilibriumr：density-dependence eﬀect of recruitment
KA : carrying capacity of adult abundanceα, ρ: Ford-Brody growth model parameters
σA,i : process errors
FA,p : ﬁshing rate for adult ﬁsh in period p
FJ,p : ﬁshing rate for juvenile ﬁsh in period p
Fishing rate is estimated with ﬁve periods according to transition ofﬁsheries:
Period 1: 1901–1936 (before the World War II)
Period 2: 1937–1964 (among and after the war)
Period 3: 1965–1983 (addition of ﬁsheries by Fukushima prefecture)
Period 4: 1984–1989 (expansion of ﬁshing by trawl-net ﬁshery)3Period 5: 1990–2006 (modern ﬁshing operation)
In equilibrium,
NJ,0 = KJ = f KA
and
0.5 .
KA (1 − SA ) = KJ θSA

Therefore,
0.5
f = (1 − SA )/(θSA).

We assumed the population was in equilibrium without any ﬁshing before
1901.
With relation to MSY, given YA = FA BA (Catch of adult sandlances)and YJ = FJ BJ (Catch of juvenile sandlances), F̃ = FMSY is given bysolving the equations,¯

dYA ¯¯= 0,
dFA ¯FA =F̃A¯

dYJ ¯¯= 0.
dFJ ¯FJ =F̃J
Equilibrium under constant ﬁshing mortality F is given by
0.5[1 − (1 − FA )SA ]NA (FA , FJ ) = (1 − FJ )NJ (FA , FJ )θSA,

NJ (FA , FJ ) = f NA (FA , FJ )[1 + r{1 − NA (FA , FJ )/KA }].
Since the above equations cannot be solved analytically, we calculate
FMSY and MSY using a numerical optimization and the posterior samplesof basic parameters f, r, θ, etc.
The observation models (likelihood function) are composed of three partson CPUE, catch, and age composition. The CPUE models are given by:
Adult:

X

1{log(IA,t ) − log(qA,k BA,t )}2
2CVO 2where IA,t is adult abundance index (CPUE or abundance estimate froma DeLury method). The ﬁshing eﬃciencies qA,k are diﬀerent parameters
between CPUEs and DeLury abundance estimates.

4

Juvenile:X

1{log(IJ,t ) − log(qJ BJ,t )}2
2CVO 2where IJ,t is juvenile abundance index (CPUE).
The catch models are given by:
Adult:

X

1
2
2 {log(YA,t ) − log(ŶA,t )}2CVYwhere YA,t is the observed catch of adult ﬁsh and ŶA,t = FA,t BA,t .
Juvenile:X

1{log(YJ,t ) − log(ŶJ,t )}2
2CVY 2where YJ,t is the observed catch of juvenile ﬁsh and ŶJ,t = FJ,t BJ,t .
The age composition models are given by:
nt,a ∼ Multinom(nt , pt,0 , . . . , pt,2+ )wherept,0 ∝ (1 − FJ,t )NJ,t θ
pt,1 ∝ (1 − FA,t−1 )(1 − FA,t )NJ,t−1 SA θ
1.5
pt,2+ ∝ (1 − FA,t−1 )(1 − FA,t )NA,t−1 SA
and

P2+

a=0 pt,a = 1 and nt =P2+

a=0 nt,a .

The prior distributions on the parameters are given by:
SA ∼ U[0, 1]
θ ∼ U[0, 1]
r ∼ U[0.1, 5.0]
KA ∼ U[103 , 105 ]− log(q·,k ) ∼ U[0, 9]
FA,p ∼ U[0, 1]
FJ,p ∼ U[0, 1]
2 ∼ Ga[0.01, 0.01]
1/σA,·
2
1/σO ∼ Ga[0.01, 0.01]
1/σY2 ∼ Ga[0.01, 0.01]

5w0 , w1 , α, ρ were ﬁrst estimated outside the Bayesian analysis and incorporated into the Bayesian analysis using the estimated values and their precisions.
The parameters and state variables in the models were inferred with theirposterior distributions. However, the posterior distributions do not have
explicit formulae for our state-space model. Thus, a numerical algorithm
such as Markov chain Monte Carlo and Sequential Monte Carlo is necessary
to assess the posterior distributions (Meyer and Miller 1999b; Newman et al.
2006). We used the software program WinBUGS (Bayesian Analysis Using
Gibbs Sampler Windows version: Spiegelhalter et al. 2003) to generate
the posterior samples. We generated two independent MCMC sequences of
30,000 iterations with diﬀerent initial parameter values. We discard the ﬁrst
15,000 iterations from each sequence as the burn-in samples, and thinned
each sequence by keeping every 6th simulation draw. All inferences were
derived using the remaining 5,000 samples obtained from concatenating thetwo chains.
Long–term Prediction
We examined the impacts of predation (i.e., reduction of natural mortality) on MSY to examine the long-term eﬀect of culling predators. For
simplicity, we assumed FA = FJ and maximized YA + YJ in this exercise.
The following 3 scenarios were examined:
Scenario 1 (No predation eﬀect): No change from 2006.
Scenario 2 (linear functional response): the ratio that corresponds
to the amount of predation by minke whales is added to SA . The′ = S + R where R = 3, 885/B̃
modiﬁed survival rate is then SA
A
A,2003 .
The value 3,885 is the annual consumption of sandlances by minke
whales estimated in 2004 in Tamura et al. (2004) and B̃A,2003 is themedian value of estimated adult biomass in 2003.
Scenario 3 (constant functional response): The adult dynamics equation was replaced by
0.5 N
Nt+1,A = (1−FA ){SA (Nt,A −RN2003,A )+RN2003,A }+(1−FJ )θSAt,J
where R is the same as that deﬁned in Scenario 2 and the biomass
equation is accordingly modiﬁed. Thus, the abundance in equilibrium
is obtained by solving the quadratic equation. Because the constantamount, RN2003,A , is always excluded from the natural mortality andindependent from the prey abundance, NA,t , the functional response
for predators becomes constant to changes in prey abundance.

6

Scenario 2 corresponds to a so-called Type I functional response (Krebs
2001; Gotelli 2001). Since Type 2 and Type 3 functional response curves
reach saturation as the prey density increases, Scenario 3 is considered as
the most extreme functional form. The realistic functional form may be
between Scenarios 2 and 3.

3. Results and Discussion
The goodness-of-ﬁt plots for the model were shown in Figs. 1–3. The
ﬁtting was good in general, however, some of recently observed abundanceindices were outside 95% credible intervals of the predicted values. The
density plots of posterior samples for the parameters were shown in Figs.
4–5. Some parameters show a bimodal distribution, so the convergence
was incomplete. In fact, R̂ (Spigelhalter et al. 2003) for some parameters
was much larger than 2. When we conducted more MCMC iterations, theresult had not been improved. Since the parameter α had highest R̂ and weestimated the α outside the Bayesian analysis, the cause of unconvergence
may be in the handling of growth parameters in our analysis. In addition,
the bimodality of α was likely to be propagated to other parameters. We
will modify the way to handle the growth equation in the future analysis.
The summary statistics (median, and the 2.5th and 97.5th percentiles
which deﬁne the 95% Bayesian posterior credible interval) of the marginal
posterior distributions for each parameter were given in Table 1. Median
adult survival rate was 74%. This value is similar to previous estimate, 80%(Nagashima et al. 1995). Fishing rates were somewhat larger than estimatesin Nagashima et al. (1995).
The predicted values of state variables, biomass and ﬁshing rate, were
given in Fig. 6. Biomass ﬂuctuated greatly over time. The predictedbiomass trajectories reproduced the rapid decline by participation of trawlnet ﬁshery between 1984 and 1989. Although the patterns of changes inﬁshing rates was similar between adults and juveniles, the rise at period
4 for adults was keener than juveniles. This should be due to the direct
impacts of participation to adult ﬁshery by the oﬀshore trawl-net ﬁshermen.
The FMSY s estimated by long–term predictions were 0.068 for Scenario
1, 0.112 for Scenario 2, and 0.080 for Scenario 3 in median. The corresponding MSYs were 7483 ton for Scenario 1, 19585 ton for Scenario 2, and 8889
ton for Scenario 3 (Table 2). Since the current observed yield for sandlances
was 11533, the population could be overexploited under the current ﬁshingrate. FMSY was low than we expected. When we preliminarily ﬁtted themodel without process errors to the data, the resultant FMSY was muchhigher. When there are measurement errors of stock and recruitment, optimum harvest rate will be overestimated (Hilborn and Walters 1992). In
addition, Cooke (2007) showed using a simulation model that environmental
variability can have a great impact on estimation of MSYR and MSYL. As

7

another example, Meyer and Miller (1999b) ﬁtted three models, observationerror model, process-error model, state-space model, to the tuna data. Fromtheir results, the state-space model estimated the lowest intrinsic rate of increase among three models. The diﬀerence was not large compared to our
result. However, the estimated process error in Meyer and Miller (1999b)
was much smaller compared to our process error estimates. Possibly, ourresult might be the implication for ﬁsheries management same as or similar
to Cooke (2007), that is, when we ignore process errors, the MSYR would
be overestimated so that the management implication might become too
optimistic and dangerous, although MSY might be more stable since MSY
is a function of product of increasing rate and carrying capacity, which is
negatively correlated with increasing rate.
Comparing the results for Scenarios 2 and 3, Scenario 2 showed more
explicit impacts of predation on the sandlance population dynamics. This
result can be predicted from the deterministic models. Using the models
used in Scenarios 2 and 3 with ﬁxed parameter values, the predicted carrying capacities were shown in Fig. 8. The direct change of survival rate hasa greater impact on equilibrium population size. The more realistic functional curve is between Scenarios 2 and 3 and should be Type 2 or Type
3 responses (Krebs 2001). When we have time-series data in abundanceand consumption for predators, we would be able to estimate the functional
response curve including Type 2 and 3 responses. JARPNII data can contribute toward providing necessary information to such trials in the near
future (Tamura et al. 2009).
The results presented here are not meant to be a deﬁnitive assessment.
Some of the assumptions used in the model were not ideal. In addition,
there remains much to be done. In particular, the functional response weused is being oversimpliﬁed in terms of contemporary modeling such as
Ecosim (Walters and Martell 2004; Mori et al. 2009). The model and theresults in this paper should be considered as the ﬁrst step in development.
In the future, the more reﬁned model will be used for sandlance ﬁshery
management taking the predator eﬀect into consideration.

References
Cooke, J. G. (2007) The inﬂuence of environmental variability on
baleen whale sustainabe yield curves. SC/59/RMP10.
Gotelli, N. J. (2001) A Primer of Ecology. Sinauer Assosciates, Inc.
Hashimoto, H. (1991) A study of Japanese sandlance populations.
Hiroshima University Seibutsu Seisangakubu kiyou 30:135–192 (inJapanese).

8Hilborn, R. and Walters, C. J. (1992) Quantitative Fisheries Stock
Assessment. Chapman and Hall.
Krebs, C. J. (2001) Ecology, 5th edn. Benjamin Cummings, San
Francisco, Calfornia.
Meyer, R., and Millar, R. B. (1999a) Bayesian stock assessment using astate-space implementation of the delay difference model. Canadian
Journal of Fisheries and Aquatic Sciences 56: 37–52.
Meyer, R., and Millar, R. B. (1999b) BUGS in Bayesian stock
assessments. Canadian Journal of Fisheries and Aquatic Sciences 56:
1078–1086.
Mori, M., Watanabe, H., Hakamada, T., Tamura, T., Konishi, K.,
Murase, H. and Matsuoka, K. (2009) Development of an ecosystemmodel of the western North Pacific. Paper SC/J09/JR21 presented to
the JARPNII review Workshop, Tokyo, January 2009(unpublished).
49pp.
Nagashima H., Kobayashi, T., Izumi, Y., and Kodama, J. (1995)
Simulation model of sandlance population dynmaics in Sendai Bay.
Miyagi Suisan Center Kenpo 14:51–53 (in Japanese).
Nagashima, H.（2004）Population dynamics of sandlances in Sendai
Bay. Miyagi Suisan Kenpo 4:51–60 (in Japanese).
Newman, K. B., Buckland, S. T., Lindley, S. T., Thomas, L., andFern´andez, C. (2006) Hidden process models for animal population
dynamics. Ecological Applications 16: 74–86.
Punt, A. E. and Butterworth, D. S. (1995) The effects of future
consumption by the Cape fur seal on catches and catch rates of theCape hakes. 4. Modelling the biological interaction between cape fur
seals Arctocephalus pusillus pusillus and the cape hakes Merluccius
capensis and M. paradoxus. South African Journal of Marine Science16, 255–285.

9

Spiegelhalter, D., A. Thomas, N. Best, and D. Lunn. 2003. WinBUGSVersion 1.4 User Manual. MRC Biostatistics Unit. Institute of Public
Health, London.
Tamura, T. and Ohsumi, S. (1999) Estimation of total foodconsumption by cetaceans in the world’s oceans. The Institute of
Cetacean Research, Tokyo. 16pp.
Tamura, T., Konishi, K., Miyashita, T., Yoshida, H., Kato, H., andKawahara, S. (2004) Preliminary analyses of interaction between
common minke whales and sand lance fisheries off Sanriku region.
Paper SC/56/O17 presented to the 56th IWC Scientific Committee
Meeting, 9pp.
Tamura, T., Konisihi, K., Goto, M., Bando, T., Kishiro, T., Yoshida, H.,
Okamoto, R. and Kato, H. (2009) Prey consumption and feeding habitsof Common minke whales in coastal areas off Sanriku and Kushiro.
Paper SC/J09/JR9 presented to the JARPNII review Workshop, Tokyo,
January 2009(unpublished). 18pp.
Walters, C. J. and Martell, S. J. D. (2004) Fisheries Ecology andManagement. Princeton University Press: Princeton, New Jersey.
Yodzis, P. (2001) Must top predators be culled for the sake of fisheries?
Trends in Ecology and Evolution 16:78–84.

10Table 1. Summary statistics of the estimated posterior distributions foreach parameter.
Parameter
w0
w1
α
ρ
θ
SA
f
qA,1
qA,2
qJ
r
KA
KJ
σA1
σA2
σO
σY
FA,1
FJ,1
FA,2
FJ,2
FA,3
FJ,3
FA,4
FJ,4
FA,5
FJ,5

Notationweight at age 0
Bycatch rate of adults
Ford-Brody intercept
Ford-Brody coeﬃcient
Juvenile survival rate
Adult survival rate
fecundity
ﬁshing eﬃciency of adult index 1
ﬁshing eﬃciency of adult index 2
ﬁshing eﬃciency of juvenile index
density-depend. eﬀect
carrying capacity of adult abund. (×106 )carrying capacity of juvenile abund. (×106 )process error on fecundity
process error on adult dynamics
observation error of abund. index
observation error of catch
Adult ﬁshing rate in period 1
Juvenile ﬁshing rate in period 1
Adult ﬁshing rate in period 2Juvenile ﬁshing rate in period 2
Adult ﬁshing rate in period 3
Juvenile ﬁshing rate in period 3
Adult ﬁshing rate in period 4Juvenile ﬁshing rate in period 4
Adult ﬁshing rate in period 5Juvenile ﬁshing rate in period 5

11Median3.264
10.330
9.519
0.234
0.245
0.741
1.263
0.016
0.255
0.067
0.601
12470
14925
0.575
0.280
0.511
0.640
0.00160.00270.02110.03390.04010.18850.32630.38000.11530.2324

95% CI[1.619, 9.504][6.833, 14.430][6.539, 12.180][–0.218, 0.806][0.120, 0.534][0.636, 0.851][0.575, 1.794][0.004, 0.052][0.086, 0.599][0.028, 0.139][0.174, 1.616][5434, 25391][3681, 35441][0.432, 0.744][0.161, 0.412][0.382, 0.708][0.562, 0.729][0.0008, 0.0034][0.0012, 0.0054][0.009, 0.0483][0.0146, 0.0702][0.0163, 0.0951][0.0877, 0.3512][0.1270, 0.5862][0.1694, 0.7468][0.0238, 0.2983][0.1095, 0.3966]

Table 2. Summary statistics of the estimated posterior distributions for
FMSY and MSY.
Parameter
FMSY,SC1
MSYSC1FMSY,SC2MSYSC2
FMSY,SC3MSYSC3

Notation
FMSY at Scenario 1
MSY at Scenario 1
FMSY at Scenario 2
MSY at Scenario 2
FMSY at Scenario 3
MSY at Scenario 3

Median0.06797483
0.111919585
0.07988889

12

95% CI[0.0200, 0.1248][1080, 18474][0.0645, 0.1658][10444, 36837][0.0310, 0.1329][1721, 20236]

100004000
0Biomass index

Juvenile (Kounago)

1994

1996

1998

2000

2002

2004

2006

2004

2006

Year

1500
500

Biomass index

Adult (Mero−do)

1996

1998

2000

2002

Year

80000
20000

Biomass index

Adult (Mero−do) 2

1984

1985

1986

1987

1988

1989

Year

Fig. 1. Goodness of ﬁt for abundance index data. The solid lines denote themedian posterior predicted values and the broken lines denote 95% credibleintervals. Open circles denote the observed values.

13

2

0

1

2

0

1
Age

03

04

2

0

Age

2

0 80

Age

1

2

0 50

02
Number

01

0 100

0

1
Age

Number00 80

Number
1
Age

0 100

Number0

Number00

0 100

Number

99

1

2

Age

120
0

Number

05

0

1

2

Age

Fig. 2. Goodness of ﬁt for age composition data. The histograms denote theobserved values and the triangle marks denote the median predicted values.

14200000

Yield

Juvenile

1900

1920

1940

1960

1980

2000

1980

2000

Year

8e+04
0e+00

Yield

Adult

1900

1920

1940

1960
Year

Fig. 3. Goodness of ﬁt for catch statistics. The solid lines denote themedian posterior predicted values and the broken lines denote 95% credibleintervals. Open circles denote the observed values.

15

Density of w[2]Density of alpha

0

2

4

6

8

12

0.15
0.00

0.00

0.00

0.15

0.10

0.30

Density of w[1]

6 8

12

16

6

8

10

12

14

N = 5000 Bandwidth = 0.4067

N = 5000 Bandwidth = 0.424

Density of rho

Density of theta

Density of sM

−0.5

0.0

0.5

1.0

4
2
0

0

0.0

2

0.6

4

6

6

1.2

8

N = 5000 Bandwidth = 0.3952

0.1

0.3

0.5

0.60

0.70

0.80

0.90

N = 5000 Bandwidth = 0.01304

Density of f

Density of qA1

Density of qA2

0.5

1.0

1.5

2.0

N = 5000 Bandwidth = 0.07072

0.0

0.0

0 10

0.6

1.5

30

3.0

N = 5000 Bandwidth = 0.01069

1.2

N = 5000 Bandwidth = 0.05315

0.00

0.04

0.08

N = 5000 Bandwidth = 0.002679

0.0

0.8

N = 5000 Bandwidth = 0.02521

Fig. 4. Posterior probability densities of hyper parameters.

16

0.4

Density of r

Density of KA

0e+00

0.0

0

0.10

0.20

0.0

1.0

2.0

5000

20000

35000

N = 5000 Bandwidth = 0.07688

yDensity of KJ

Density of sigmaA[1]Density of sigmaA[2]

0

20000

50000

6
4
2
0

3e−05

0 1 2 3 4 5

N = 5000 Bandwidth = 0.005495

0e+00

0.3

0.5

0.7

0.9

0.1

0.3

0.5

N = 5000 Bandwidth = 0.01178

Density of sigmaO

Density of sigmaY

Density of deviance

0.4

0.6

0.8

1.0

N = 5000 Bandwidth = 0.01599

0.000

0.010

0 2 4 6 8

0.020

N = 5000 Bandwidth = 0.01531

0 1 2 3 4 5yDensityDensity

0.00

6e−05

Density

5

1.0

10

15

Density of qJ

0.5

0.6

0.7

0.8

N = 5000 Bandwidth = 0.008296

Fig. 5. Posterior probability densities of hyper parameters.

17

4050

4100
y

4150

0e+00

3e+05

Biomass

250000

6e+05

Adult

0 100000

BiomassJuvenile

1940

1980

1900

1940

1980

Year

Year

Juvenile

Adult

0.4
0.0

0.2

Fishing Rate

0.6
0.4
0.2
0.0

Fishing Rate

0.6

1900

1

2

3

4

5

1

Period23

45PeriodFig. 6. Posterior probability densities of state variables. The solid lines
denote the median posterior predicted values and the broken lines denote
95% credible intervals.

18

MSY: SC1 (no change)

0.00

0.05

0.10

0.15

0.20

0.25

0.00000

05150.00008

F_MSY: SC1 (no change)

0

10000 20000 30000 40000 50000
N = 5000 Bandwidth = 789.5

F_MSY: SC2 (prop. pred. effect)

MSY: SC2 (prop. pred. effect)

0e+00

0 5

15

4e−05

N = 5000 Bandwidth = 0.004698

0.00

0.05

0.10

0.15

0.20

0.25

0

10000 20000 30000 40000 50000
N = 5000 Bandwidth = 1177

F_MSY: SC3 (const. pred. effect)

MSY: SC3 (const. pred. effect)

0e+00

0 5

15

6e−05

N = 5000 Bandwidth = 0.004378

0.00

0.05

0.10

0.15

0.20

0.25

0

N = 5000 Bandwidth = 0.004256

10000 20000 30000 40000 50000
N = 5000 Bandwidth = 816

Fig. 7. Long–term predictions (FMSY and MSY).

19

1.4
1.0

1.2

Carrying Capacity

1.6

Scenario 2
Scenario 3

0.00

0.02

0.04

0.06

0.08

0.10

R

Fig. 8. Predicted carrying capacity to changes in R using the mathematicalequations in Scenarios 2 and 3. We set SA = 0.75, θ = 0.2, f = (1 −0.5 ), r = 0.6, and NSA )/(θSA
2003,A /KA = 0.3.

20