SC/60/IA13

SOME MODIFICATIONS TO THE CURRENT ADAPTVPA MODEL FOR ANTARCTIC MINKE WHALESMITSUYO MORI1, DOUGLAS S. BUTTERWORTH2
1. The Institute of Cetacean Research, 4-5 Toyomi-cho, Chuo-ku, Tokyo, 104-0055, Japan
2. MARAM, Department of Mathematics and Applied Mathematics, University of Cape Town, Rondebosch 7701, SouthAfrica
Contact e-mail: mori@cetacean.jp
ABSTRACTThe current ADAPT-VPA model of Mori et al. (2007) for Antarctic minke whales in Areas IIIE to VW is modified in two respects. One
is the revision of the stock-recruitment relationship, and the other is the simplification of the functional form of the carrying capacity.
AIC indicates that the model which includes the above two modifications is better compared to the “Reference case” scenario of Mori etal. (2007). Thus, future analyses will regard this modified version as a new “Reference case” scenario, which has the incidental advantage
of corresponding more closely to the assumptions made in the SCAA (Statistical Catch at Age) approach of Punt and Polacheck (2006) toanalysis of these data.
KEYWORDS ADAPT-VPA, CATCH-AT-AGE, NATURAL MORTALITY, ANTARCTIC MINKE WHALE

INTRODUCTION
This study is a continuation of the series of studies since Butterworth et al. (1999) (Butterworth et al. 2002, Mori andButterworth 2005, Mori et al. 2006, Mori et al. 2007) and has modified the most recent analyses for Antarctic minke
whales in Areas IIIE to VW (the I-stock) (Mori et al. 2007) in two respects:
1) Improvement of the stock-recruitment relationship; and2) Simplification of the functional form of carrying capacity.
DATA
The data used are exactly the same as in Mori et al. (2007).

METHODOLOGY
The methodology used is exactly the same as in Mori et al. (2007) except in two respects. One is that equation (6) on
Mori et al. (2007) which describes the assumed stock-recruitment relationship as follows:
⎡

⎧

⎢⎣

⎪
⎩

z
⎛ N yf ⎞ ⎫⎪⎤⎥⎟f ⎟ ⎬⎥⎝ K y ⎠ ⎪⎭⎥⎦⎪⎢
N y +1,1 = λ ⋅ N yf ⎢1 + A⎨1 − ⎜⎜is modified as below:

(1)

⎡

⎧

⎢⎣

⎪
⎩z
⎛ N 1y+ ⎞ ⎫⎪⎤⎥⎟
1+ ⎟ ⎬⎥⎝ K y ⎠ ⎪⎭⎥⎦⎪⎢
N y +1,1 = λ ⋅ N yf ⎢1 + A⎨1 − ⎜⎜(2).

The basis for this modification is that traditionally in the IWC Scientific Committee, it is usual to assumeofN yf
K yf

N 1y+
K 1y+

insteadfor the density dependence on recruitment, i.e. that this dependence is on the 1+ rather than the mature femalecomponent of the population. Further, similar density dependence is assumed in the SCAA approach of Punt andPolacheck (2006), and this makes it easier to compare results between their method and this ADAPT-VPA approach.
The second modification concerns the functional form adopted for the change in carrying capacity over time. The
functional form assumed in Mori et al. (2007) is as follows:
⎧ K1f
⎪ff
⎪ K f + K 2 − K1 ( y − y )γ1⎪⎪ 1~
( y2 − y1 )γ
K yf = ⎨ff⎪ K f + K3 − K 2 ( y − y )2⎪ 2( y3 − y 2 )⎪f⎪⎩ K 3()()

y ≤ y1y1 + 1 ≤ y ≤ y2(3)
y 2 + 1 ≤ y ≤ y3
y3 + 1 ≤ ywith the following choices made for the “change” years: y1 = 1930 , y2 = 1960 and y 3 = 2000 . K yf is set to be~εK yf → K yf ⋅ e y(4)where the ε y are estimable parameters which are constrained to change somewhat smoothly over time under theassumption:()

ε y = ε y −1 + η y , where η y ~ N 0, σ 2 .

(5)The modified functional form is as follows:⎧ K 11+⎪
1+1+⎪ K 1+ + K 2 − K 1 ( y − y )γ1⎪⎪ y( y 2 − y1 )γ
K 1y+ = ⎨1+1+⎪ K 1+ + K 3 − K 2 ( y − y )2⎪ 2( y3 − y 2 )⎪ 1+⎪⎩ K 3()()

y ≤ y1y1 + 1 ≤ y ≤ y 2(6)y 2 + 1 ≤ y ≤ y3y3 + 1 ≤ ywith the following choices made for the “change” years: y1 = 1930 , y 2 = 1960 and y 3 = 1980 . This choice of years
was made because an initial exploration for different values of these change years indicated that the years listed were2better in terms of improved maximum likelihood values. This functional form is much simpler than the previous
functional form shown in equations (3)-(5).

RESULTS
Various output statistics for the I-stock are shown in Table 1, which includes the “Reference case” and the twosensitivity test results. The “Reference case” is the same as in Mori et al. (2007)1. For the two sensitivity test
scenarios, the –lnL values shown are comparable with the “Reference case” since the same catch-at-age overdispersion
parameter of the “Reference case” is used for these scenarios. Various plots for the “Reference case” and the twosensitivity test scenarios are shown in Figure 1.
AIC indicates that the model which includes the two modifications is better compared to the “Reference case”. TheAIC values here are not all exact because no account is taken of the penalty function used to limit K variation over timein two of the models, but the preference for the model with both modifications is nevertheless clear. Thus, future
analyses will regard this modified version as a new “Reference case” scenario. This has the incidental advantage ofcorresponding more closely to the assumptions made in the SCAA approach of Punt and Polacheck (2006) to analysisof these data. Note that the best estimate (scenario “simplify K” in Figure 1) of the recent trend in abundance haschanged with the modifications made from slightly down to slightly up.

REFERENCES
Butterworth, D. S., Punt, A. E., Branch, T. A., Fujise, Y., Zenitani, R. and Kato, H. 2002. Updated ADAPT VPA recruitment and abundance trend
estimates for Southern Hemisphere minke whales in Areas IV and V. International Whaling Commission Document SC/54/IA25: 20 pp.
Butterworth, D. S., Punt, A. E., Geromont, H. F., Kato, H. and Fujise, Y. 1999. Inferences on the dynamics of Southern Hemisphere minke whales
from ADAPT analyses of catch-at-age information. J. Cetacean Res. Manage. 1: 11-32.
Mori, M. and Butterworth, D. S. 2005. Some advances in the application of ADAPT-VPA to minke whales in Areas IV and V. International Whaling
Commission Document SC/57/IA17: 27 pp.
Mori, M., Butterworth, D. S. and Kitakado, T. 2006a. Application of ADAPT-VPA to various stock hypotheses on Antarctic minke whales
distributed through IWC management Areas IIIE to VIW. Paper SC/58/IA8 presented to the IWC Scientific Committee, May 2006 (St. Kitts)(unpublished). 40pp.
Mori, M., Butterworth, D. S. and Kitakado, T. 2007. Further progress on application of ADAPT-VPA to Antarctic minke whales. Paper SC/59/IA13
presented to the IWC Scientific Committee, May 2007 (Alaska) (unpublished). 32pp.
Punt, A. E. and Polacheck, T. 2006. Further statistical catch-at-age analyses for Southern Hemisphere minke whales. International Whaling
Commission Document SC/58/IA2: 40 pp.

1

The results for the “Reference case” differ slightly from Table 4 of Mori et al. (2007) since a coding error was found and has been corrected for thispaper.

3

Table 1. Results of various statistics for the previous “Reference case” and sensitivity tests for the I-stock.
Average proportions in eachmanagement areaIDI-stock

1

Reference case (m =30)

2

N1+/K1+

3

N1+/K1+ , simplify K

M (CV)

IIIE

IV

VW

9.187

0.056 ( 0.163)

0.205

0.425

0.371

0.71

0.643

4.468

0.050 (0.139)

0.225

0.417

0.358

0.576

5.007

0.046 (0.087)

0.204

0.428

0.368

b rec,1945-68

b rec,1968-88

b rec,1988-lastyr

N tot,1945-68

N tot,1968-88

N tot,1988-lastyr

N lstyr-5,1 /N 1968,1

K 19300.052-0.032-0.0100.060-0.020-0.007

0.374

17047

0.626

0.060

-0.025-0.0040.062-0.012-0.006

0.38256183

0.062-0.024

0.005

0.062

-0.015

0.003

0.473

51783

* Excludes K variation penalty function contribution for ID1 and ID2.

4K 2000 / K 1960 K 1960 / K 1930

Survey q MSYR (1+)-lnL *AIC *No.Est.par0.06

319.561045.12

203

0.62

0.07

319.871045.75

203

0.71

0.06

318.07894.137

129

Total(1+) population

N f (num ber of m ature fem ales)

70000

Ref

S im plify K

50000
Nf

150000100000

40000
30000

R ef
N 1+/K 1+
S im plify K

0.8

15000

10000

20000

50000

N 1+/K 1+R ef
S im plify K

20000

R ef

60000

Sim plify K

R ecruitm ent

200000N 1/N f1

N 1+/K 1+

N 1/N f

N 1+/K 1+

Total(1+) population

R ecruitm ent
25000

80000

250000

0.6
0.4
0.2

5000

10000
1970
Year

1980

1990

0
1930

2000

1950

function of K

250000

70000

Kf

50000

150000

40000

K 1+

200000

60000

10000030000
Ref
Sim plify K
N 1+/K 1+

20000
10000

Total(1+) population

80000

0
1930

50000
0

1940

1950

1960

1970

1980

1990

200000180000160000140000120000100000
80000
60000
40000
20000
0
1930

R ecruitm ent (1 year old)R ecruitm ent (1 year old)

1980

1990

2000

15000
10000
5000
0
10000

20000

30000

40000

1940

1940

1950

1960

1970
Year

50000

60000

70000

80000

1960

1970

1980

1990

2000

1980

1990

2000

200000180000160000140000120000100000
80000
60000
40000
20000
0
1930

10000
5000

ID C RJA R P A
R ef
N 1+/K 1+
S im plify K

1940

1950

1960

1970
Year

1980

1990

2000

10000

20000

30000
40000 50000
N um ber of adult fem ale

1960

60000

70000

80000

15000
10000
5000

010000

1980

1990

2000

1990

2000

ID C RJA R P A
S im plify K
R ef
N 1+/K 1+

1940

1950

1960

1970
Year

1980

Residuals of the recruitm ent trend for the N 1+/K 1+. sim plify K case

VPA
S-R m odel

20000

1970
Year

3000

00

1950

Total(1+) population size for V W
200000180000160000140000120000100000
80000
60000
40000
20000
0
1930

Stock-recruitm ent plot for the N 1+/K 1+, sim plify K case
25000

15000

N um ber of adult fem ale

1940

Total(1+) population size for IV

VPA
S-R m odel

20000

1950

Y ear

00

1930

Stock-recruitm ent plot for the N 1+/K 1+ case
25000

VPA
S-R m odel

20000

1970
YearID C R
JA R P A
R ef
N 1+/K 1+
S im plify K2000

Stock-recruitm ent plot for the R eference case
25000

1960

Total(1+) population size for IIIE
300000

90000

01930

01940

Total(1+) population

1960

R esiduals(R ec.of V P A - R ec.of S-R m odel)

1950

Total(1+) population

1940

R ecruitm ent (1 year old)

0
1930

20000

30000
40000
50000
N um ber of adult fem ale

60000

70000

2000
1000
0-1000-2000-3000
1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 1990 1995 2000
Year

Figure 1. Plots of total population size by stock and by Area, number of mature females, recruitment, recruitment rate and carrying capacity for the I-stock for various
N 1y+
N yf
sensitivity test runs, such as 1) replacing 1+ instead offor the following stock recruitment relationship and 2) together with simplifying the carrying capacity function
KyK yfas detailed in the text.

5