JA/J05/JR6

A Point Estimate of Natural Mortality Coefficient of Southern Minke Whales
Using JARPA data
EIJI TANAKA,1 RYOKO ZENITANI,2 and YOSHIHIRO FUJISE2
1

Tokyo University of Marine Science and Technology, Faculty of Marine Sciences, Konan 4-5-7,

Minato-ku, Tokyo 108-8477, JAPAN2

The Institute of Cetacean Research, Toyomi Shinko Buld. 5F, Toyomi-cho 4-5-7, Chuou-ku, Tokyo

104-0055, JAPANABSTRACTThis paper shows a point estimate of natural mortality coefficient of Southern minke whales using
JARPA data, by modifying the original method proposed by Tanaka (1990). Skew age composition by
mature/immature segregation was corrected using maturity area at age. Net rate of natural increase of
size of stock by a new interpretation of stock boundary was estimated by maximum likelihood. Using
the zero increase rate of stock size from the statistical model selected by the c-AIC, the point
estimates of natural mortality coefficient were 0.0486 (per year) for both the Eastern Indian Ocean
Stock and 0.0490 for Western South Pacific Stock.
INTRODUCTION

Natural mortality coefficient is one of most important biological parameters for both population
dynamics and stock management. The changes in number of recruits and carrying capacity on
Southern minke whales (Sakuramoto and Tanaka, 1985, 1986; Butterworth et al., 1996,1997), which
relates to those in the ecosystems of the Antarctic, have been estimated using the mortality
coefficient.
In the Revised Management Procedure (RMP) which is a current conservative tool of thewhale stock management in IWC, an assumed coefficient of natural mortality was used in the
Implement Simulation of the RMP. In the results of simulation trials, such performance statistics asrealized protection (RP) level was used for one of criteria of stock conservation, and was depending
upon the extent of the natural mortality. The effect of the changes in the carrying capacity , whichcauses those of the recruits, were examined in the past robustness simulation trials for developing the
RMP. The estimate of the extent of changes depends upon the mortality estimate. Thus the natural
mortality coefficient has directly and indirectly been used in the RMP.
JARPA survey a program for investigating the biological aspect of the Southern minke
whales and one of the objectives of the program is to estimate such parameters as the natural
mortality coefficient. Term of the planned survey is so long as sixteen years in order to have a precise
estimate using stock abundances and age compositions (Tanaka,1990, Tanaka et al., 1992; Cooke et al,
1997; Tanaka and Fujise, 1997). Studies on the estimation with such research takes under random1

sampling have been presented (Sakuramoto and Tanaka, 1989; de la Mare, 1990, Nakamura, 1991).
In the JARPA review meeting in 1997, however, information on age segregation wasrecognized, irrespective of random sampling. A portion of young whales does not migrate into the
Antarctic survey areas where is south of 60゜S. The observed frequency in the young age class could
be lower than the true frequency for the whole stock and therefore the estimate of the natural
mortality using directly the data would be biased. Many of the young age classes consist of immaturecomponent of the stock. If the frequency of this component is corrected, we could make unbiased
estimate of age composition.
In addition, Luis et al (2005) proposed a new scenario of stock boundary that is different
from the traditional IWC stock division. In the scenario the boundary is 165°E, There were twostocks in Area III/E, IV, V, and VI/W, and the Eastern Indian Ocean Stock (I-stock) and the WesternSouth Pacific Stock (P-stock). Therefore, new method is necessary for handling and/or analyzing thedata because the JARPA survey was planned and conducted by the traditional division.
This paper proposed an estimation of the natural mortality coefficient introducing a method
for correction of the observed age composition and natural increase rate with segregation parameters.
Issues relating the estimation were discussed.
MATERIALS AND METHODS
Data
Data used for natural mortality coefficient are the annual abundance estimate stratified by area, sex,
mature/immature and age, and estimated curve for maturity rate at age (Bando et al, 2005).
Basic Models for Dynamics of Stock Size
We consider whale population that stock size is changed due to natural mortality andrecruitment, and that stock is consisting of mature and immature components, and that a portion ofeach component is annually migrating into the JARPA survey area. We assume that natural mortality
coefficient is independent of year and that proportion of mature or immature stock is independent ofage.
Let Nt,s,a, Rt,s,a, Ut,s,a , R’t,x,s,a, and U’t,x,s,a be size of stock of sex s(=1,2) at age in yearsa(=1,2,3,…) in year t(=1,3,5,…,T), mature stock size, immature one, mature stock size in Areax(=1,2,3, for Area IV, V/W, V/E), and immature one, respectively. Basic model for the dynamics are
expressed by

N t +1, s , a +1 = N t , s , a exp(− M a )(1)

N t = ∑∑ N t , s ,a(2)

Rt , s , a = β s , a N t , s , a(3)

U t , s , a = (1 − β s ,a )N t , s ,a

(4)

s

a

2

β s ,a =

Here

1
1+ e

(5)

−cs (a − as )N t , s , a = U t , s , a + Rt , s , a(6)

R ' t , x , s ,a = δ t , x Rt , s ,a(7)

U ' t , x , s ,a = γ t , xU t , x ,a(8)γ t , x : rate of size of immature stock in survey area to total immature stock(>0)δ t , x : rate of size of mature stock in survey area to total mature stock(>0)
Nt : stock size in total

β s,a : maturity rate at age
Ma : natural mortality coefficient (per year).

Estimation Procedures for Average Natural Mortality Coefficient
From equation (1), we have

⎛ pt ,a ⎞⎛ N t ,a ⎞⎛ N ⎞⎟ = −2 r + 2 z a⎟ = ln⎜ t ⎟ + ln⎜
M a = ln⎜⎜
⎜N ⎟
⎜p
⎟
⎟
N
⎝ t +2 ⎠⎝ t + 2,a + 2 ⎠⎝ t + 2,a + 2 ⎠

r=

(9)

1 ⎛ N t +2 ⎞⎟ln⎜
2 ⎜⎝ N t ⎟⎠

(10)

pt ,a
1
ln
2 pt + 2,a + 2

(11)

za =

N t ,a = N t p t ,apt ,a =

N a ,t
Nt

=

(12)

∑N

t ,i , a

i

∑∑ N
i

(13)
t ,i , jjHere r(per year) is the net rate of increase of stock size. When age composition is fairly stable, theaverage natural mortality coefficient is expressed by
∞
⎛ N ⎞
⎛
⎞
M = ln⎜⎜ t ⎟⎟ − ln⎜1 − ∑ p a ,t + 2 ⎟ = −2r + 2 z⎝ a ~1⎠⎝ N t +2 ⎠

3

(14)

∞
⎛
⎞
z = − ln⎜1 − ∑ p a ,t + 2 ⎟⎝ a ~1⎠(15)

Therefore those estimates are expressed byMˆ = zˆ − rˆzˆ = −

(16)

T
∞
1
⎛
⎞
ln⎜1 − ∑ pt ,a + k ⎟∑(T − 1) t =3 ⎝ k = 2⎠(17)Correction of Age Composition
Age composition in the survey area was corrected by abundance by school size and age
composition by school size (Fujise et al, 1992). However, if the value of γ t , x is different from that ofδ t , x , the age composition is biased and therefore should be corrected. Assuming that Nt,s,a in some ageclasses are close to each other and using equation (3), (4), (7) and (8), an estimate λt , x is
approximately expressed by

∑∑ R'
s

t , x ,s ,aδ t ,x

a

∑∑ β
s

s ,a

=

a

∑∑ U 'st , x , s ,a∑∑ (1 − β )t ,s ,a

a

∑∑ β

s ,a

s ,a

s

≈

a

∑∑ (1 − β )Nδ t,x ˆ= λt , x
γ t,x

(18)

t ,s ,aa∑∑ (1 − β )
s ,a

s ,a

s

s ,a

s

s

γ t,x

a

∑∑ β Nsaa

Using equation (18), we reduce the following equations:∑ (R't , x , s ,a+λˆt , xU 't , x , s ,a)s∑∑ (R'st , x , s ,a+λˆ U '
t,x

t , x , s ,as) δ ∑∑ (Rt,xt , s ,asa∴ pˆ t ,a ==

δ t , x ∑ (Rt , s ,a + U t , s ,a )∑ (R't , x , s ,a+λˆt , xU 't , x ,s ,a∑∑ (R'st , x,s ,at,x

N t ,aNt= pt ,aa)s+λˆ U '

+ U t ,s ,a )=t , x , s ,a

(19)

)

a

4

Estimation of r
Suppose thatδ t ,1 + δ t , 2 = 1(20)

then the stock size in Area IV or V/W is expressed by()

N t , x = ∑∑ R't , x , s ,a +λˆt , xU 't , x , s ,a = δ t , x ∑∑ (Rt , s ,a + U t , s ,a ) = δ t , x N t
s

a

s

(21)

a

The mean of proportion of stock size in Area IV to that in Areas IV+V/W is denoted by δ .For the
I-stock, r is estimated by maximizing the following log-likelihood:

LLW = LL1 + LL2

(22)

{

( )

}

2
ln (N t ,1 ) − ln (δ ) − ln Nˆ 1 − rˆ(t − 1) ⎤
1 ⎡
2
2
⎥
LL1 = − ∑ ⎢ln{2π (σ t ,1 + υ1 )}+2 t ⎢σ t2,1 + υ12
⎥
⎣
⎦

{

( )

(23)

}

2
ln(N t , 2 ) − ln(1 − δ ) − ln Nˆ 1 − rˆ(t − 2) ⎤
1 ⎡
2
2
⎥
LL2 = − ∑ ⎢ln{2π (σ t , 2 + υ 2 )}+2 t ⎢σ t2, 2 + υ 22
⎥
⎣
⎦

(24)

For the P-stock, assuming δ =1, r is estimated by maximizing the following log-likelihood:

{

( )

}

2
ln(N t ,3 ) − ln Nˆ 1 − rˆ(t − 1) ⎤
1 ⎡
2
2
⎥
LLE = − ∑ ⎢ln{2π (σ t ,3 + υ 3 )}+2 t ⎢σ t2,3 + υ 32
⎥
⎣
⎦

where

(25)

υ x2 : additional variance by stockσˆ t2, x : measurement error of abundance estimate.

Results
Tables 1-3 summarize the results of estimations and Figures 1 and 2 illustrate the average of thecorrected age compositions. Comparing the compositions before and after the correction in Figure 1,
the corrections seemed to be effective in the Area V, but not in Area IV. From the c-AIC values inTable 2, the models using r=0 were selected and the point estimates of natural mortality coefficientwere 0.0486 (per year) for both the Eastern Indian Ocean Stock and 0.0490 for Western South Pacific
Stock.

Discussions
The corrected age composition will be useful for estimating the selectivity of research takes by the
JARPA comparing the corrected abundance at age and the catch at age. Also the composition will beavailable for monitoring changes in number of recruits, which relates to stock dynamics.
The performance of the presented method for correcting age compositions depends upon
precision of the age composition and maturity rate at age. The age composition in survey area was5

estimated by 1) abundance estimation by sub-area and school size and 2) age composition by sub-areaand school size (Kato et al, 1991, Kishino et al., 1991).
The abundance estimate used in the paper was computed using the only data from samplingand scouting vessels (SSV) because the bootstrap samples for CV computation are made by pair of
the abundance and age composition from SSV and the data size for scouting vessel (SV) is too small
to estimate abundance by sub-area and school size. However, abundance estimate adopted in the
JARPA survey was made from the combined data because the estimate is expected to modify by using
the SV data. The age composition will be modified introducing the SV data.
The maturity rate at age (Bando et al, 2005) was estimated by the raw data and might have
uncertainty due to segregation. However, the estimate was much revised than that from thecommercial catch data and will be reliable.
In the estimated value of r, we note that the r estimate may reflect changes in δ t . The
discrimination, in particular for the P-stock, was difficult because the only data in Area V/E wasavailable and such comparison of the trend for the same stock as trend in Area IV and V/W was
impossible. In that sense, statistical criterion is not applicable and the comparable data are necessary
for the discrimination.
The estimate of M is close to that by the ADAPT VPA (Butterworth et al, 2002) and that for
Area V (Kitakado et al., 2005). The CV was not yet estimated but that can be computed usingbootstrap samples (Efron, 1979). This is an issue in the near future.

References
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whale based on materials collected by the JARPA survey in 1987/88 to 2003/04. Paper
JA/J05/PJR5 presented to this meeting.
Butterworth, D.S., Punt, A.E., Geromont, H.F., Kato, H. and Miyashita, T. 1996. An ADAPTapproach to the analysis of catch-at-age data information for Southern Hemisphere minke
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Butterworth, D.S., Punt, A.E., Geromont, H.F., Kato, H. and Fujise, Y. 1997. Further ADAPT
analysis of catch-at-age information for Southern Hemisphere minke whales in Areas
Ⅳ and Ⅴ. Paper SC/M97/6 presented to the IWC Scientific Committee.
Cooke, J., Fujise, Y., Leaper, R., Ohsumi, S., and Tanaka, S. 1997. A exploratory analysis of the age
distribution of minke whales collected during JARPA expeditions 1987/88 through 1995/96.
Paper SC/M97/21 presented to the IWC Scientific Committee.
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distribution and segregation of the Japanese researches. Paper SC/44/SHB10 presented to6the IWC Scientific Committee, June, 1992.
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Antarctic Areas IV and V as revealed by a logistic regression model. Paper SC/M97/13
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Kishino, H., Kato, H., Kasamatsu, F., and Fuijise, Y. 1991.
Detection of heterogeneity and estimation of population characteristics from the field survey
data:1987/88 Japanese feasibility study of the Southern Hemisphere minke Whales.
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Nakamura, T. 1991. A new look at a Bayesian cohort model for time-series data obtained fromresearch takes of whale. Rep. int. Whal. Commn

41:345-48.

Nishiwaki, S., Matsuoka, K., and Kawasaki, M. 1997. Review of the sighting surveys in the JARPA.
Paper SC/M97/1 presented to the IWC Scientific Committee, May , 1997.
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A new interpretation of the stock identity in the Antarctic minke whale based on analyses ofgenetics and non-genetics markers. Paper JA/J05/PJR3 presented to this meeting.
Punt, A. E., Cooke, J. G., and Borchers, D.L.. 1996. Estimating the extent of process error forSouthern Hemisphere minke whales from the results of the IWC/IDCR cruises.
Rep. int. Whal. Commn 46:113.
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Hemisphere minke whales using multi-cohort method. Rep. int. Whal. Commn 36:207-12.
Sakuramoto , K.. and Tanaka , S.

1989. On the estimation of age-development natural mortality.

Rep. int. Whal. Commn 39:371-73.
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whales using JARPA data. Paper SC/M97/11 presented to the IWC Scientific Committee.
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Scientific Committee, June, 1995.
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7

Table 1. Results of estimation of λt , x .
a) Area IV
Area / Season

1989/90 1991/92

1993/94 1995/96 1997/98 1999/00 2001/02 2003/04

0.894

2.046

IV(x=1)

3.125

0.531

0.224

0.879

0.987

0.929

b) Areas V/W and V/E
Area / Season

1990/91 1992/93

1994/95 1996/97 1998/99 2000/01 2002/03 2004/05

V/W(x=2)

1.027

4.331

1.207

0.225

0.702

0.042

5.778

-

V/E(x=3)

0.507

3.001

0.981

0.804

4.026

3.983

2.388

-

Table 2. Results of estimation of r , δ x and υ , and N1.
Stock

Model

c-AIC

LL

n

p

r

N1

υ1

υ2

υ3

δ

I-stock

Full

349.8

-166.515

50.027245,245

0.413

0.703

-

0.553

r=0

345.7

-166.915

4-54,103

0.417

0.726

-

0.548

Full

168.5

-77.2

7

3

0.047054,380

-

-

0.000

-

r=0

168.2

-80.2

7

2

-

73,556

-

-

0.241

-

P-stock

Table 3. Results of point estimates of average M
Stock

I-stock

P-stock

M

0.04860.0490

8

0.1

pa

Area IV
0.05

0
0

10

20

30

40

50

60

50

60

50

60

Age in years : a

pa

0.1

Area V/E

0.05

0
0

10

20

30

40

Age in years : a

pa

0.1

Area V/W

0.05

0
0

10

20

30

40

Age in years : a

Figure 1. Comparison of uncorrected age compositions (open circles) and corrected one (solid
curves).

9

0

10

20

30

40

50

60

ln(p at )

0

-5
IV
V/W
V/E

-10
Age in years : a

Figure 2. Comparison of the age composition in three areas.

10