SC/D06/J16
MODEL-BASED ANALYSES OF TRENDS OVER TIME IN THE AGE CORRESPONDING TO THE
TRANSITION PHASE FOR ANTARCTIC MINKE WHALES IN THE JARPA RESEARCH AREA
MITSUYO MORI1, DOUGLAS S. BUTTERWORTH2, RYOKO ZENITANI1, HIDEHIRO KATO3
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, South
Africa
3. Tokyo University of Marine Science and Technology, 5-7, Konan 4, Minato-ku, Tokyo, 108-8477, Japan
Contact e-mail: mori@cetacean.jp
ABSTRACTThis study applies a model-based approach similar to that of Thomson et al. (1999) to the transition phase data obtained
from JARPA surveys to examine trends in the age at maturity for the I and P stocks of Antarctic minke whales. The
results, which takes into account various potential biases related to examining trend in transition phase data (i.e.
truncation and fringe effects, differences between readers, and readers learning over time) suggest that the age atmaturity of Antarctic minke whales declined from about 11 years in the late 1940s to 7 years in the late 1960s for bothstocks, and these declining trends are statistically significant at the 5% level. The analyses also suggest that the age atmaturity increased slightly from the late 1960s to the late 1970s and has stabilized thereafter. These trends are
consistent with the results obtained from VPA (Mori et al. 2006), which suggest that for both the I and P stocks,
abundance increased from the 1940s to the late 1960s and thereafter has been stable or declined somewhat. Thisconsistency enhances the confidence to be placed in estimates of parameters (such as natural mortality and MSYR)
from such VPA analyses that may be of value for management purposes. It also serves to demonstrate the utility ofage-at-maturity as an index to monitor stock status, and suggests that continued monitoring of this parameter isdesirable both for this purpose and for contributing to the understanding of the dynamics of the Antarctic ecosystem.
KEYWORDANTARCTIC MINKE WHALES, AGE AT TRANSITION, TRANSITION PHASE, AGE AT MATURITY

INTRODUCTION
It is widely accepted that the position where growth layer spacing changes (the transition phase) in the earplugs ofwhales is related to the age at physical and sexual maturity of the animal (Lockyer 1972, Masaki 1979, Kato 1983).
Lockyer (1972) explains that the ear plug layer formation during the juvenile growth phase is very irregular and thelayers are unevenly spaced; however, after maturity, the new layers become more compact and evenly spaced. Basedon the analysis of the transition phase in the earplug of Antarctic minke whales collected from the commercial whaling
samples, several authors report a decline in age at transition (and hence at sexual maturity) of this species from the1940s cohorts to the late 1960s cohorts (Masaki 1979, Best 1982, Kato 1983, 1985). Zenitani and Kato (2005, 2006)
also report this decline based on the analysis of samples collected from the JARPA surveys.

Whether or not this decline in age at transition of Antarctic minke whales is a real phenomenon has long been under
debate. Details of the discussions are well summarised in Zenitani and Kato (2006), and will thus not be repeated here.
However, in brief, these queries arose essentially because the nominal trend in the mean age at transition may be subject
to some potential biases as a measure of any real trend, such as arise from the truncation and fringe effects, differences
between readers and readers “learning” over time. The truncation bias arises from the under-representation of older
animals in the most recent cohorts sampled, the fringe effect relates to the low proportion of animals recorded as having
an age-at-transition phase close to their age at capture (presumably as lack of contrast between widely and narrowly
spaced rings in such circumstances makes the phase difficult to identify), and “learning” effect is that as readers gain
experience, they tend to detect transition layers in a greater proportion of the sample (Thomson et al. (1999)). The
analyses of Thomson et al. (1999) took all of these effects into account and led to the conclusion that for those animals
that were caught in the IWC Management Area IV, a decline in the average age at transition from roughly 11 for thecohorts of the 1950s to roughly 7 for those of the 1970s had occurred.

At the meeting to review the Japanese Whale Research Program under Special Permit in the Antarctic (JARPA) calledby the Government of Japan in 2005, it was recommended that a model-based transition phase analysis of the kind ofThomson et al. (1999) (as explained above) be updated utilizing further JARPA survey data that have become available
since that time. In response to this recommendation, and further to update results presented to the IWC’s mid-term
JARPA review meeting (IWC 1998), the analyses of Thomson et al. (1999) are updated here. The main difference
between this work and that of Thomson et al. (1999) is that this study conducts the analyses for what seems to be themost plausible stock-hypothesis at this stage: an I-stock (distributed from Area IIIE to Area VW) and P-stock(distributed from Area VE to Area VIW) (Pastene et al. (2005)). Furthermore, given that questions were raised at the
mid-term JARPA review meeting (IWC 1998) about possible differences between ageing of the commercial and
JARPA catches, the analyses here utilize only the data obtained from the JARPA surveys, so that conclusions are
independent of such considerations.

An important aspect of these analyses is that they provide a basis to check trends in abundance estimates for this species
calculated from VPA, which suggest increases in recruitment and abundance from the 1940s to the late 1960s (Mori etal. 2006), by examining whether there are co-incident changes in biological parameters in the directions that might be
expected in such circumstances.

DATA
The data analyzed pertain to Antarctic minke whales sampled by the JARPA surveys that took place in Antarctic
Management Areas IIIE to VIW from 1987 to 2004 1 . Not all the samples could be aged and only those that were aged
were used in the analyses. The analyses are conducted for the two possible stocks identified by Pastene et al. (2005),
which are the I-stock (distributed from Area IIIE to Area VW) and P-stock (distributed from Area VE to Area VIW).
For the I-stock, 3897 Antarctic minke whales were aged from samples taken between 1987 and 2004, and among those,transition phases were identified for 1831 animals. For the P-stock, the corresponding figures are 1891 whales aged
from samples between 1988 and 2004, with transition phases identified for 968. The earplugs taken from 1987-1989
and 1992 were read by H. Kato, and others were read by R. Zenitani. The proportion of earplugs that were aged andwere assigned a transition phase for each year by each reader are shown in Figure 1. Pre-1941 cohorts were excluded
from these analyses for the I-stock, as the associated sample sizes are small. For the P-stock, the corresponding cut-off
date was 1944.

METHODOLOGY
The methodology used here is almost the same as that used in Thomson et al. (1999), which basically fits the data to
two models based on different hypotheses: one model assumes that the transition phase is an artefact, and the otherassumes that the transition phase is a real phenomenon. Details of the models are described below.
Model 1: The transition phase is an artifact
As in Thomson et al. (1999), we first assume that the transition phase is not a real phenomenon. This means that thereader assigns a transition phase in some random fashion in relation to the number of growth layers. To reflect suchmechanism, we assume that the readers assign the transition phase according to a normal distribution which is centered
at a fixed proportion of the length and hence age of the earplug ( μ plug ), with standard deviation σ plug .
The proportion of earplugs in each age group that are assigned a transition phase is described by a vector of points
Pl R where l =5, 7, 9, 11 or 13 2 and refers to the age of the animal at capture, and R=Kato or Zenitani, indicating theperson who read the earplug. The probability of assigning a transition phase in earplugs of other ages is calculated bylinear interpolation between successive points, and is assumed to remain constant after the age of 13. The probabilityof recording an age at transition equal to the age at capture is assumed to be zero.
The values of l given above were selected by examination of model fits, in which a much larger number of values wereused. The subset of values needed to adequately describe the resultant function was then chosen, primarily on the basisof the Akaike Information Criterion (AIC).
Since a “learning” effect amongst the readers was identified in Thomson et al. (1999), we also allow for this possibility,
so that the probability of reader R assigning a transition phase in age group l is assumed to be:(1)
Pl R, y s = Pl R, y min ⋅ 1 + β lR ⋅ ( y s − y min )[]where Pl R, y s is the probability of reader R assigning a transition phase to an earplug aged l, sampled in year y s ,Pl R, y min is the probability of reader R assigning a transition phase to an earplug aged l in his/hers first year
1

Whaling takes place during the austral summer, so that each whaling season spans two years. For convenience of
expression the austral summer of 1987-1988, for example, is referred to as the year 1987.
2
This was taken up to age 15 for the P-stock when Kato was the reader, since this gave better fit to the data in terms ofAIC.
2

y min of reading, andβ lR

is the slope of the straight line that accounts for “learning”.

y min is taken to be 1987 for Kato and 1990 for Zenitani. Pl R, y s is assumed to be less than 1.
Given the assumption that the readers assign the transition phase according to a normal distribution which is centered ata fixed proportion of the age of the earplug ( μ plug ), with standard deviation σ plug , the probability that this transition
phase, observed in an earplug of age a, would be assigned to an age at transition tm can be calculated from:
⎧ tm
*
2
⎪
for 1 ≤ tm ≤ a
ϕ a , tm = ⎨ tm −1 N μ plug , σ plug⎪⎩0otherwise

∫

(

)

(2)

where N * is a normal distribution which has been truncated at the ends of the earplug (i.e. tm=0 and tm=a). The area
under the truncated distribution is scaled upwards so that it is equal to one.
Model 2: The transition phase is a real phenomenon
As in Thomson et al. (1999), in this model it is assumed that the transition phase does reflect a real signal (probably
related to the age at maturity), and that individual whales belonging to a particular cohort ( yc ) mature according to anormal distribution with mean tm y c and standard deviation σ yc .

The variance of this distribution may be higher for cohorts for which the mean age at transition is greater. This istaken into account through the following equation as in Thomson et al. (1999):(

σ y c = σ 1965 + γ ⋅ tm y c − tm1965

)

(3).

The mean age at transition for each cohort ( tm y c ) is assumed to change from T1 to T2 between cohorts Y1 and Y2 andT2 to T3 between cohorts Y2 and Y3 according to:
T1
⎧
1−α
⎪
⎪T − (T − T )⎛⎜ yc − Y1 ⎞⎟
1
2 ⎜
⎟
⎪⎪ 1
⎝ Y2 − Y1 ⎠tm y c = ⎨⎛ yc − Y2 ⎞⎪⎪T2 + (T3 − T2 )⎜⎜ Y − Y ⎟⎟⎝ 3 2⎠⎪⎪⎩
T3yc ≤ Y1
Y1 ≤ yc ≤ Y2
(4)

Y2 ≤ yc ≤ Y3
yc ≥ Y3

where T1 is the mean age at transition in year Y1 ,T2 is the mean age at transition in year Y2 ,T3 is the mean age at transition in year Y3 , andα is a shape parameter such that α =0 corresponds to a straight line.

The probability of recognizing transition phase when one is present, given the number of layers that have accumulated
after the transition phase ( atm ) is given by a vector of points PλR, y s . Choices of λ were made in the same way as
described for the l values of Model 1. In this case, λ values of 1 and 3 were selected for reader Kato, and 1, 3 and 6
forreaderZenitani.
Linearinterpolationisusedbetweensuccessive
points.
Further the probability that reader Kato detects a transition phase is taken to remain constant once at least three layers
have accumulated after the transition phase. For reader Zenitani, such constancy occurs only after six layers.
“Learning” may be taken into account in the same way as described for Model 1.

3

Parameter estimation
The parameter estimation methodology is again that of Thomson et al. (1999). The parameter values for both models
were estimated by maximizing a log-likelihood based itself on the assumption of multinomial distributions:ln L =⎡ a∑∑ ⎢⎢ ∑ N(

y c , a , tm ⋅ ln ρ y c , a , tm⎛⎞⎤

) + N yimm, a ⋅ ln⎜⎜1 − ∑ ρ y , a,tm ⎟⎟⎥awhere N y c , a , tm

(5)cc
⎝ tm =1⎠⎥⎦⎣tm =1is the number of earplugs sampled and read from cohort yc which have age a and age at transition tm,

N imm
yc ,a

is the number of earplugs sampled and read from cohort yc which have age a that were consideredρ y c , a , tm

not to have a transition phase, andis the model-estimated probability of assigning a transition phase at age tm in an earplug of age aycafrom cohort yc . For Model 1, ρ y c , a , tm = PaR, y s ⋅ ϕ a , tm and a similar equation follows for Model 2.
The number of samples of age a in cohort yc in the model ( Nˆ y c , a ) is taken to be exactly equal to the observed numberof samples N y c , a .

The model then determines how Nˆ y c , a is distributed across the transition phase ages.

The

model-estimated number of earplugs ( Nˆ y c , a , tm ) for cohort yc , age a and age at transition tm, is calculated as:Nˆ y c , a , tm = Nˆ y c , a ⋅ ρ y c , a , tm

(6).

The estimated number of earplugs from cohort yc of age a for which no transition phase was assigned ( Nˆ imm
y c , a ), is thus
given by:
a
⎞
⎛
ˆ
⎜
(7).
Nˆ yimmN1
ρ y c , a ,tm ⎟=⋅−yc , a ⎜c ,a⎟⎠⎝ tm =1
For convenience of expression, these animals are termed “immatures”. Under the assumptions of Model 1, these are
earplugs that were considered not to have a transition phase. However under Model 2, this group includes not only
earplugs that do not have a transition phase but also earplugs whose transition phase was not detected by the earplug
reader.

∑Estimates of precision for the parameters of both Model 1 and Model 2 were obtained based upon the Hessian
approximation calculated using AD Model Builder.

RESULTS
Results for the I-stock (distributed from Area IIIE to Area VW)

Model 1: The transition phase is an artefact
Table 1 shows the estimated parameter values and the associated log-likelihoods ( ln L ) for different forms of the modelin which the transition phase assumed to be an artefact (Model 1) for the I-stock. Based on the AIC, the model with
l=5, 7, 9, 11, 13 for defining Pl R, y s and which assumes no “learning” (for either Zenitani or Kato) was chosen. In
terms of this model, both Zenitani and Kato position the transition phase at about 40% of the total age on average(Figure 2a). Zenitani appears to have been a little more conservative than Kato, recording a lower proportion oftransition phases for younger earplugs (for ages 5 to about 10) (Figure 3a). Both readers recognized a transition phasein roughly 75% of all older earplugs.
The ability of Model 1 to reflect the data can be judged from plots of the observed and model-predicted distribution ofage after transition ( atm ). This model does not achieve a good fit to the distribution of those values in cases where atransition phase was assigned, and there are clear indications of systematic deviations in the residuals (Figure 4a, lowerplot).

Model 2: The transition phase is a real phenomenon
Table 2 shows the estimated parameter values and associated log-likelihoods ( ln L ) for different forms of the model inwhich the transition phase is a real phenomenon (Model 2) for the I-stock. Based on the AIC, the model with λ =1

4

and 3 for PλK, y s and with λ =1, 3 and 6 for PλZ, y s , and which assumes no “learning” (for either Zenitani or Kato) waschosen. Zenitani again appeared to have been a little more conservative than Kato, recording a lower proportion oftransition phases with fewer layers after the transition phase (Figure 5a). Both readers assigned a transition phase insome 70% of the earplugs when there were more than six layers after the transition phase.
Compared to Model 1 (see Figure 4a), Model 2 appears to fit the observed distribution of age after transition very well
(Figure 6a). Both Model 1 and Model 2 fit the number of “immatures” very well. Model 2 indicates that a realdecline in the mean age at transition with cohort occurred for the Antarctic minke whales classified as belonging to the“I-stock” (Figure 7a): roughly from about 11 years for the 1948 cohort to 7 years for the 1967 cohort. This decline isstatistically significant at the 5% level (estimated mean of T2-T1 = -4.50, standard error = 0.20, t = 22.1, P<0.05).
The slight increase estimated in the mean age at transition from the 1968 cohort to the 1981 cohort is again statisticallysignificant at the 5% level (estimated mean of T3-T2 = 0.59, standard error = 0.09, t = 6.8, P<0.05). The drop in theobserved average age at transition from around 1997 is a reflection of the truncation effect impacting results for recent
cohort (Figure 7a).
Results for the P-stock (distributed from Area VE to Area VIW)

Model 1: The transition phase is an artefact
Table 3 shows the estimated parameter values and the associated log-likelihoods ( ln L ) for different forms of Model 1
for the P-stock. Based on the AIC, the model with l=5, 7, 9, 11, 13 for reader Zenitani, and l=5, 7, 9, 11, 13, 15 forreader Kato for defining Pl R, y s , and which assumes no “learning” for either reader was chosen. As for the I-stock, bothZenitani and Kato positioned the transition phase at about 40% of the total age on average (Figure 2b). For this stock,
Kato appeared to have been a little more conservative than Zenitani, recording a lower proportion of transition phases
for younger earplugs (for age 5 to about 10) (Figure 3b); however, there are only two years for which Kato read theearplugs for this stock, and the standard error estimates in Table 3 indicate that the difference between the two readers is
not significant at the 5% level.
As was the case for I-stock, this model does not achieve a good fit to the distribution of age after maturity for instanceswhere a transition phase was assigned, and there are clear indications of systematic deviations in the residuals (Figure
8a, lower plot).

Model 2: The transition phase is a real phenomenon
Table 4 shows the estimated parameter values and associated log-likelihoods ( ln L ) for different forms of Model 2 for
the P-stock. Based on the AIC, the model with λ =1 and 3 for PλK, y s and with λ =1, 3 and 6 for PλZ, y s , and whichassumes no “learning” was chosen. The probability of detecting a transition phase after some layers have accumulated
are lower for reader Kato compared to Zenitani, but again this could be a result of low sample sizes for Kato (Figure 5b),for whom this result differs from that for the I-stock. In contrast, results for Zenitani are very similar to hers for the
I-stock.
As was the case for the “I-stock”, when compared to Model 1 (see Figure 8a), Model 2 appears to fit the distribution forage after transition very well (Figure 9a). Both Model 1 and Model 2 fit the number of “immatures” very well.
Model 2 indicates that a real decline in the mean age at transition with cohort occurred for the Antarctic minke whales
classified as belonging to the “P-stock” (Figure 7) from roughly about 11 years for the 1946 cohort to 7 years for the
1966 cohort. This trend is very similar to that of the I-stock, and this decline is again statistically significant at the 5%
level (estimated mean of T2-T1 = -3.79, standard error = 0.22, t = 16.9, P<0.05). The slight increase in the mean age at
transition from the 1969 cohort to the 1977 cohort is again statistically significant at the 5% level (estimated mean of
T3-T2 = 0.56, standard error = 0.111, t = 5.0, P<0.05). As was observed for the I-stock, the impact of the truncation
effects on recent cohorts is evident after about 1997 (Figure 7b). The final steady mean age at transition T3 isestimated to be reached a few years earlier than for the I-stock, but the difference is not statistically significant at the 5%
level.
DISCUSSION
These analyses, which take into account various potential biases that arise in the interpretation of trends in transition
phase data, indicate that the age at transition (and hence likely maturity) of Antarctic minke whales declined from about
11 years at the end of the 1940s to 7 years in the late 1960s for both the I- and P-stocks, and that these declining trends
are statistically significant at the 5% level. This result is similar to that suggested by various authors previously fromexaminations of these data (Masaki 1979, Best 1982, Kato 1983, 1985, Thomson et al. 1999, Zenitani and Kato 2005,

5

2006). These analyses also suggest that the age at maturity of this species increased slightly from the late 1960s to late1970s, and has stabilized thereafter. Again this result is similar to that of Zenitani and Kato (2006).
These trends are also consistent with the results obtained from VPA, which suggest that for both I and P stocks,
abundance increased from the 1940s to the late 1960s, and thereafter has been stable or declined somewhat (Mori et al.
2006). This consistency enhances the confidence to be placed in estimates of parameters (such as natural mortality
and MSYR) from such VPA analyses that may be of value for management purposes.

The reason for the decline in age-at-maturity of minke whales from the 1940s to the 1960s could be a result of the krill
surplus, which likely became available due to the extensive whaling of the large baleen whales (Laws 1977, Kato 1983,
1987). We surmise that during this period, the feeding conditions for minke whales improved due to the krill-surplus
and resulted in their attaining sexual maturity at an earlier age. We also surmise that the constant (perhaps evenslightly increasing) trend in age at maturity since the 1970s suggest that the period of “krill-surplus” came to an end
around this time, so that more recently the species may be more restricted in its feeding. Various recent studies
suggest the recovery of the large baleen whales (e.g. humpback whales, blue whales, fin whales) that were once heavily
exploited (e.g. Matsuoka et al. 2006). These whales share the same prey (krill) with Antarctic minke whales and so
may be playing a role in such enhanced feeding restrictions (see also Mori and Butterworth (2006)).

A trend in age-at-maturity can provide a good indication of the trend in stock abundance, and hence provide a usefulindex for stock monitoring. Continued monitoring of the age-at-maturity of the stock, both to monitor stock status andfor contributing to the understanding of the dynamics of the Antarctic ecosystem is desirable.
ACKNOWLEDGEMENTS
We sincerely thank Ms. Robin Thomson for providing the code for her earlier analyses to assist with this work. We
also appreciate for the useful comments provided by Dr. H. Hatanaka and various other colleagues in improving thismanuscript.

REFERENCES
Best, P.B. 1982. Seasonal abundance, feeding, reproduction, age and growth in minke whales off Durban (with incidental observation from theAntarctic). Rep. int. Whal. Commn 32: 759-786.
International Whaling Commission. 1998. Report of the Intersessional working group to review data and results from special permit research onminke whales in the Antarctic, Tokyo, 12-16 May 1997. Rep. Int. Whal. Commn. 48. 377-412.
Kato, H. 1983. Some consideration on the decline in age at sexual maturity of the Antarctic minke whale. Rep. int. Whal. Commn 33: 393-399.
Kato, H. 1985. Further examination of the age at sexual maturity of the Antarctic minke whale as determined from earplug studies. Rep. int. Whal.
Commn 35: 273-277.
Kato, H. 1987. Density dependent changes in growth parameters of the southern minke whale. Sci. Rep. Whales Res. Inst., Tokyo 38: 47-73.
Laws, R. M. 1977. Seals and whales of the Southern Ocean. Phil. Trans. R. Soc. Lon. B, Biol. Sci. 279: 81-96.
Lockyer, C. 1972. The age at sexual maturity of the southern fin whale (Balaenoptera physalus) using annual layer counts in the ear plug. J. Cons. Int.
Explor. Mer. 34 (2): 276-294.
Masaki, Y. 1979. Yearly change of the biological parameters for the Antarctic minke whale. Rep. int. Whal. Commn 29: 375-396.
Matsuoka, K., Hakamada, T. and Nishiwaki, S. 2006. Distribution and abundance of humpback, fin and blue whales in the Antarctic Areas IIIE, IV, V
and VIW (35±E-145±W). JARPA review document SC/D06/J7
Mori, M. and Butterworth, D. S. 2006. A first step towards modelling the krill-predator dynamics of the Antarctic ecosystem. CCAMLR Science 13.
217-277.
Mori, M., Butterworth, D, S. and Kitakado, T. 2006. Application of ADAPT-VPA to Antarctic minke whales in the JARPA research area. Document
prepared for the JARPA-review meeting.
Pastene, L.A., Goto, M., Kanda, N., Bando, T., Zenitani, R., Hakamada, T., Otani, S. and Fujise, Y. 2005. A new interpretation of the stock identity in
the Antarctic minke whale (Balaenoptera bonaerensis) based on analyses of genetics and non-genetics markers. Paper JA/J05/JR3 presented to the
Review Meeting of the Japanese Whale Research Program under Special Permit in the Antarctic (JARPA) called by the Government of Japan,
January 2005 (unpublished). 30pp.
Thomson, R. B., Butterworth, D. S. and Kato, H. 1999. Has the age at transition of Southern Hemisphere minke whales declined over recent decades?
Mar. Mamm. Sci 15: 661-682.
Zenitani, R. and Kato, H. 2005. Long- term trend of age at sexual maturity of Antarctic minke whales by counting transition phase in earplugs. Paper
JA/J05/JR7 presented to the JARPA Review Meeting called by the Government of Japan, January 2005. 12 pp.
Zenitani, R. and Kato, H. 2006. Temporal trend of age at sexual maturity of Antarctic minke whales based on transition phase in earplugs obtained
under JARPA surveys from 1987/88-2004/05. JARPA review document SC/D06/J15

6

Table 1. Parameter estimates and log-likelihoods ( ln L ) for different forms of the model in which the transition phase isassumed to be an artefact (Model 1) for the I-stock. The figures shown in parenthesis are standard deviations provided
by the Hessian approximation. Results are shown for various choices for the l values of equation (1) and for cases where
the probability that an earplug reader will assign a transition phase to an earplug either does not (no “learning”), or does(with “learning”) increase over time. For the results with “learning”, the values quoted for Pl refer to the reader’s firstyear of sampling ( y min ) – 1987 for Kato and 1990 for Zenitani. The “best” model form is selected on the basis of AIC.
BEST MODELμ plugσ plug

P5
P7
P9
P 11
P 13
P 15
P 17
P 19
P 21
P 23
P 25
P 27
P 29
P 31
P 33
P 35
β5
β7
β9
β 11
β 13
ｌｎL
AIC

l= 5,7,9…35

l= 5,7,9…15

No "learning"

No "learning"

Kato
0.397
0.151
0.000
0.219
0.400
0.558
0.759
0.660
0.685
0.745
0.703
0.540
0.647
0.603
0.823
0.698
0.804
0.792

Kato
0.397
0.151
0.000
0.219
0.400
0.559
0.750
0.694

Zenitani
0.363
0.159
0.000
0.025
0.224
0.551
0.707
0.763
0.668
0.733
0.651
0.760
0.803
0.743
0.707
0.831
0.779
0.787

-6415.75
6487.75

l= 5,7,9…13
No "learning"

Zenitani
Kato
Zenitani
0.363 0.397 (0.008) 0.363 (0.004)
0.159 0.151 (0.006) 0.159 (0.003)
0.000 0.000 (0.000) 0.000 (0.000)
0.025 0.219 (0.053) 0.025 (0.010)
0.224 0.398 (0.083) 0.225 (0.032)
0.551 0.568 (0.094) 0.544 (0.042)
0.709 0.698 (0.023) 0.746 (0.011)
0.750-6428.18
6460.18-6428.88
6456.887

l= 5,7,9…11
No "learning"

l= 5,7,…9
No "learning"

l= 5,7,9…13
With "learning"

Kato
0.397
0.151
0.000
0.221
0.380
0.688

Kato
0.397
0.151
0.000
0.197
0.662

Kato
0.397
0.151
0.000
0.219
0.398
0.568
0.698

Zenitani
0.363
0.159
0.000
0.025
0.205
0.729

-6441.50
6465.50

Zenitani
0.363
0.159
0.000
0.021
0.682

-6539.23
6559.23

Zenitani
0.363
0.159
0.000
0.022
0.178
0.545
0.695

0.006 0.000
0.000 0.021
0.000 0.037
0.000 0.000
0.000 0.010-6424.46
6472.46

Table 2. Parameter estimates and log-likelihoods (lnL) for different forms of the model in which the transition phase isa real phenomenon (Model 2) for the I-stock. The figures shown in parenthesis are standard deviations provided by the
Hessian approximation. Results are shown for various choices for the λ values, and both with and without a “learning”
effect. For the results with “learning”, the values quoted for Pλ refer to the reader’s first year of sampling ( y min ).
a)
Lamda=1-6 Lamda=1-7 Lamda=1-10 Lamda=1,3,6 Lamda=1,3,6 Lamda=1-6

Y1
Y2
Y3
T1
T2
T3
α

σ 1965
γ

P1
P2
P3
P4
P5
P6
P7
P8
P9
P 10
lnL
AIC

Lamda=1-3 Lamda=1,3

Lamda=1,3,6

Kato only

Kato only

Kato only

Zenitani only

1948.1
1965.5
1981.0
11.0
7.2
6.4
0.010
1.311
0.054
0.284
0.447
0.689

1948.4
1965.4
1981.0
10.9
7.2
6.4
0.010
1.311
0.054
0.273

1948.51969.31977.9
11.1
6.5
7.2
0.010
0.960
1.16E-05
0.125

0.688

0.559

0.689

1947.7
1965.5
1981.0
11.0
7.2
6.4
0.010
1.311
0.054
0.284
0.448
0.723
0.543
0.868
0.685

-896.886
920.886-895.981
925.981-896.928
920.928

ReaderspooledReaderspooledReaderspooledReaderspooledKato only

1948.41968.6
1978.0
11.1
6.6
7.2
0.010
1.055
6.35E-03
0.152
0.426
0.551
0.621
0.719
0.733

1948.41968.6
1978.0
11.1
6.6
7.2
0.010
1.055
6.36E-03
0.152
0.427
0.551
0.613
0.691
0.833
0.729

1948.41968.6
1978.0
11.1
6.6
7.2
0.010
1.055
6.33E-03
0.162

1946.01967.01985.1
11.3
7.0
6.3
0.058
1.320
0.050
0.274

0.590

0.685

0.734

-4376.19
4406.19-4375.03
4407.03

1948.41968.6
1978.0
11.1
6.6
7.2
0.010
1.055
6.08E-03
0.152
0.427
0.550
0.613
0.695
0.850
0.663
0.795
0.616
0.736
-4372.87
4410.87-4376.99
4400.99b)
BEST MODEL
No "learning"
Kato

Y1
Y2
Y3
T1
T2
T3
α

σ 1965
γ

P1
P3
P6
β1
β3
β6
ｌｎL
AIC

Zenitani1948.4 (1.029)
1968.3 (0.435)
1981.2 (2.208)
11.1 (0.191)
6.7 (0.072)
7.2 (0.050)
0.010 (0.002)
1.057 (0.019)
0.005 (0.019)
0.356 (0.099) 0.122 (0.032)
0.698 (0.022) 0.558 (0.042)
0.743 (0.011)-4370.64
4398.64

With "learning"
Kato

Zenitani1948.41968.6
1978
11.1
6.6
7.2
0.010
1.055
0.006
0.352
0.111
0.699
0.481
0.696
0.000
0.014
0.000
0.021
0.009
-4365.01
4403.018

0.744-896.969
918.969-3417.68
3439.68

Table 3. Parameter estimates and log-likelihoods ( ln L ) for different forms of the model in which the transition phase isassumed to be an artefact (Model 1) for the P-stock. The figures shown in parenthesis are standard deviations provided
by the Hessian approximation. Results are shown for various choices for the l values of equation (1) and for cases where
the probability that an earplug reader will assign a transition phase to an earplug either does not (no “learning”), or does(with “learning”) increase over time. Because there were only two years for which Kato read earplugs for this stock, andbecause he had already acquired considerable experience by that time, no allowance was made for possible “learning”
in his case here. For the results with “learning” the values quoted for Pl refer to the reader’s (Zenitani’s) first year ofsampling ( y min ).

μ plugσ plug

P5
P7
P9
P 11
P 13
P 15
P 17
P 19
P 21
P 23
P 25
P 27
P 29
P 31
P 33
P 35
β5
β7
β9
β 11
β 13
ｌｎL
AIC

l= 5,7,9…35
No "learning"

l= 5,7,9…15
No "learning"

Kato
0.380
0.128
0.000
0.000
0.088
0.372
0.386
0.631
0.554
0.787
0.446
0.545
0.768
0.531
0.739
0.800
0.986
0.596

Kato
0.380
0.128
0.000
0.000
0.088
0.372
0.384
0.645

Zenitani
0.360
0.148
0.000
0.033
0.203
0.587
0.768
0.758
0.712
0.708
0.756
0.771
0.807
0.698
0.694
0.902
0.773
0.750

-3282.95
3354.95

BEST MODEL
l= 5,7,9…13 (15)No "learning"

Zenitani
Kato
Zenitani
0.360 0.380 (0.012) 0.360 (0.005)
0.148 0.128 (0.009) 0.148 (0.004)
0.000 0.000 (0.000) 0.000 (0.000)
0.033 0.000 (0.001) 0.033 (0.022)
0.203 0.088 (0.060) 0.203 (0.049)
0.587 0.372 (0.116) 0.591 (0.064)
0.769 0.384 (0.110) 0.754 (0.014)
0.753 0.645 (0.039)-3292.18
3324.18-3292.23
3322.239

l= 5,7,9…13
With "learning"
Kato
0.380
0.128
0.000
0.000
0.088
0.372
0.384
0.645

Zenitani
0.360
0.148
0.000
0.033
0.147
0.506
0.681

0.001
0.000
0.045
0.019
0.012
-3287.97
3327.97

Table 4. Parameter estimates and log-likelihoods (lnL) for different forms of the model in which the transition phase isa real phenomenon (model 2) for the P-stock. The figures shown in parenthesis are standard deviations provided by the
Hessian approximation. Results are shown for various choices for the λ values, and both with and without a “learning”
effect. For the results with “learning”, the values quoted for Pλ refer to the reader’s first year of sampling ( y min ).
a)
Lamda=1-6 Lamda=1-7 Lamda=1-10 Lamda=1,3,6 Lamda=1,3,6

Y1
Y2
Y3
T1
T2
T3

α
σ 1965
γ

P1
P2
P3
P4
P5
P6
P7
P8
P9
P 10
ｌｎL
AIC

ReaderspooledKato only

Kato only

Kato only

Kato only

Zenitani only

1946.01969.31975.1
11.5
6.6
7.2
0.028
0.978
1.10E-02
0.053
0.352
0.444
0.673
0.608
0.734

1946.01969.21975.1
11.4
6.6
7.2
0.013
0.978
9.56E-03
0.053
0.353
0.445
0.670
0.600
0.765
0.733

1946.01969.21975.1
11.4
6.6
7.2
0.013
0.978
9.67E-03
0.060

1946.01971.2
1979.0
12.4
6.8
7.3
0.473
0.976
0.000
0.138

1948.01969.1
1975.0
11.2
6.5
7.2
0.010
0.974
0.021
0.064

0.175

1946.01972.6
1986.9
12.5
6.7
8.0
0.496
0.982
0.000
0.000
0.397
0.580

1946.01971.7
1985.4
12.5
6.8
7.7
0.482
0.981
0.000
0.000

0.526

0.584

0.615

0.733

0.632

1946.01971.51977.0
12.4
6.8
7.2
0.474
0.977
0.000
0.000
0.443
0.000
0.381
0.413
0.635

-2226.92256.9-2226.83
2258.83

1946.01969.21975.1
11.4
6.6
7.2
0.011
0.978
9.62E-03
0.053
0.353
0.444
0.668
0.601
0.780
0.728
0.634
0.801
0.734
-2226.03
2264.03-2228.09
2252.09-318.211
342.211-312.555
342.555-322.513
346.513

Kato

AIC

Lamda=1,3,6

ReaderspooledNo "learning"

P1
P3
P6
β1
β3
β6
ｌｎL

Lamda=1,3

ReaderspooledBEST MODELασ 1965γLamda=1-3

Readerspooled

b)

Y1
Y2
Y3
T1
T2
T3

Lamda=1-6

Zenitani

1946.0 (0.004)
1969.0 (0.005)
1976.5 (1.639)
11.4 (0.308)
6.6 (0.090)
7.2 (0.050)
0.014 (0.109)
0.979 (0.024)
0.012 (0.034)
0.000 (0.001) 0.064 (0.036)
0.581 (0.035) 0.617 (0.061)
0.751 (0.014)-2221.56
2249.56

With "learning"
Kato

Zenitani

1946.01969.01976.5
11.4
6.6
7.2
0.017
0.979
1.20E-02
0.000
0.065
0.581
0.371
0.690
0.000
0.077
0.010
-2216.65
2254.65100.751-322.77344.77-1894.42
1918.42Proportion assigned TP (P-stock)

Proportion assigned TP (I-stock)

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

1993

1992

1991

1990

1989

Z
K

1988

Proportion mature(No.TP/No.Aged)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

2004

2003

2002

2001

2000

1999

1998

1997

1996

1995

1994

1993

1992

1991

1990

1989

1988

Z
K

1987

Proportion mature(No.TP/No.Aged)

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Year of sampling (y s )Year of sampling (y s )Figure 1. Proportion of earplugs sampled and read in year ys, for which a transition phase was detected. “Z” refers toreader Zenitani, and “K” refers to reader Kato.

a)b)
I-stock

P-stock
3.5

3Reader K
Reader Z

Reader K
Reader Z

3
Probability density

Probability density

2.5
2
1.5
1
0.5

2.5
2
1.5
1
0.5

0

0
0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

Proportion of age (tm / a )

0.6

0.8

1

Proportion of age (tm / a )Figure 2. Probability density functions that Zenitani (or that Kato) identified a transition phase at certain proportion of
the total extent of an earplug (expressed as tm/a where tm is age at transition and a is age at capture) for Model 1
without “learning”. The left hand side plot shows results for the I-stock, and the right hand side plot for the P-stock.

a)b)
P-stock
Probability of detecting atransition phase

Probability of detecting atransition phase

I-stock
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Reader K
Reader Z
0

5

10

15

20

25

30

35

40

45

50

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Reader K
Reader Z
0

55

5

10

15

20

25

30

35

40

45

50

55

Age at capture (a )Age at capture (a )Figure 3. Probability that Zenitani (or that Kato) identified transition phase in earplug as function of age at capture (a)reading for that earplug for Model 1 without “learning”. The left hand side plot shows results for the I-stock and theright hand side plot for the P-stock.

11

I-stock (Model 1 – the transition phase is an artefact)
a 140

b 140
Observed
Model

100

120

Number of Immatures

Number of Matures

120

80
60
40

100

Observed
Model

80
60
40
20

20
0
0

5

10

15

20

25

30

35

40

45

50

19
41
19
45
19
49
19
53
19
57
19
61
19
65
19
69
19
73
19
77
19
81
19
85
19
89
19
93
19
97
20
01

0
55

Age after Transition (a tm )Cohort (y c )60

Observed-Model

Observed-Model

40
20
0
-20
-40
-60
0

5

10

15

20

25

30

35

40

45

50

55

Age after Transition (a tm )

10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
1941 1946 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001
Cohort (y c )Number of Immaturesc 300
Observed
Model

250
200
150
100
50
0
0

5

10

15

20

25

30

35

40

45

50

55

40

45

50

55

Observed-ModelAge at capture(a )
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
0

5

10

15

20

25

30

35

Age at capture (a )Figure 4. Observed and model-predicted distributions of age after transition ( atm ) for the complete dataset (upper plot),and residuals for this fit (lower plot) for Model 1 without “learning” are shown in (a). Similar representations for thenumber of “immatures” against cohort and against age are shown in (b) and in (c), respectively [results for the I-stock].

12

a)

b)
P-stock

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Reader K
Reader Z
0

5

10

15

20

25

30

35

40

45

50

Probability of detecting atransition phase

Probability of detecting atransition phase

I-stock

55

1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0

Reader K
Reader Z
0

Age after transition (a tm )

5

10

15

20

25

30

35

40

45

50

55

Age after transition ( a tm )Figure 5. Probability that Zenitani (and that Kato) detected a transition phase in an earplug in which atm layers haveaccumulated since onset of transition for Model 2 without “learning”. The left hand side plot shows results for the
I-stock and the right hand side plot shows those for the P-stock.

13

I-stock (Model 2 – the transition phase is a real phenomenon)
a 120

b 120

Number of Matures

100

Number of Immatures

Observed
Model

100
80
60
40

80

Observed
Model

60
40
20

20

0
0

5

10

15

20

25

30

35

40

45

50

19
41
19
45
19
49
19
53
19
57
19
61
19
65
19
69
19
73
19
77
19
81
19
85
19
89
19
93
19
97
20
01

0
55

Cohort (yc )

60
50
40
30
20
10
0
-10
-20
-30
-40
-50

Observed-Model

Observed-ModelAge after Transition (a tm )

0

5

10

15

20

25

30

35

40

45

50

55

Age after Transition (a tm )

10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
1941 1946 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001
Cohort (y c )Number of Immaturesc 300
Observed
Model

250
200
150
100
50
0
0

5

10

15

20

25

30

35

40

45

50

55

40

45

50

55

Observed-ModelAge at capture (a )
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
0

5

10

15

20

25

30

35

Age at capture (a )Figure 6. Observed and model-predicted distributions of age after transition ( atm ) for the complete dataset (upper plot),and residuals for this fit (lower plot) for Model 2 without “learning” are shown in (a). Similar representations for thenumber of “immatures” against cohort and against age are shown in (b) and in (c), respectively [results for the I-stock].

14

Data
Observed
Model

Average age at transition (tm )

14
13
12

200
180
160
140
120
100
80
60
40
20
0

11
10
9
8
7
6
19
41
19
45
19
49
19
53
19
57
19
61
19
65
19
69
19
73
19
77
19
81
19
85
19
89
19
93
19
97
20
01

5

Number of samplesa) I-stock

Cohort (y c )Data
Observed
Model

70
60
50
40
30
20
10

00

96

20

92

19

88

19

84

19

80

19

76

19

72

19

68

19

64

19

60

19

56

19

52

19

19

48

0

19

19

80
Number of samples

14
13
12
11
10
9
8
7
6
5

44

Average age at transition(tm )b) P-stock

Cohort (y c )Figure 7. Observed mean age at transition tm against cohort compared to “true trend” as predicted by Model 2 without“learning”. A histogram of number of whales sampled (and aged) in each cohort is also plotted. Probability intervals(mean ≤2SE) are also shown for the observed and model estimated average age at transition. The upper plot (a) is for
the I-stock, and the lower plot (b) for the P-stock. Note that the fringe effect causes observations to be above true values
for most cohorts, but for the most recent cohorts this effect is more than compensated by the truncation effect working
in the opposite direction.

15

P-stock (Model 1 – the transition phase is an artefact)
a

b
80

80
70

Observed
Model

60

Number of Immatures

Number of Matures

70

50
40
30
20

60
50
40
30
20
10

10

0
0

5

10

15

20

25

30

35

40

45

50

19
44
19
48
19
52
19
56
19
60
19
64
19
68
19
72
19
76
19
80
19
84
19
88
19
92
19
96
20
00
20
01

0
55

Age after Transition (a tm )Cohort (y c )30

10Observed-Model

Observed-Model

20

0
-10
-20
-30
-40
0

5

10

15

20

25

30

35

40

45

50

55

Age after Transition (a tm )

c

10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
1944 1949 1954 1959 1964 1969 1974 1979 1984 1989 1994 1999 2001
Cohort (y c )

140
120

Number of Immatures

Observed
Model

Observed
Model

100
80
60
40
20
0
0

5

10

15

20

25

30

35

40

45

50

55

40

45

50

55

Observed-ModelAge at capture(a )
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
0

5

10

15

20

25

30

35

Age at capture (a )Figure 8. Observed and model-predicted distributions of age after transition ( atm ) for the complete dataset (upper plot),and residuals for this fit (lower plot) for Model 1 without “learning” are shown in (a). Similar representations for thenumber of “immatures” against cohort and against age are shown in (b) and in (c), respectively [results are for the
P-stock].

16

P-stock (Model 2 – the transition phase is a real phenomenon)
a

b
80

80
70

Observed
Model

60

Number of Immatures

Number of Matures

70

50
40
30
20

60
50
40
30
20
10

10

0
0

5

10

15

20

25

30

35

40

45

50

19
41
19
45
19
49
19
53
19
57
19
61
19
65
19
69
19
73
19
77
19
81
19
85
19
89
19
93
19
97
20
01

0
55

Age after Transition (a tm )Cohort (yc )

30
20
10

Observed-Model

Observed-Model

Observed
Model

0
-10
-20
-30
-40
0

5

10

15

20

25

30

35

40

45

50

55

Age after Transition (a tm )

10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
1941 1946 1951 1956 1961 1966 1971 1976 1981 1986 1991 1996 2001
Cohort (y c )c140
Number of Immatures

120

Observed
Model

100
80
60
40
20
0
0

5

10

15

20

25

30

35

40

45

50

55

40

45

50

55

Observed-ModelAge at capture (a )
10
8
6
4
2
0
-2
-4
-6
-8
-10
-12
0

5

10

15

20

25

30

35

Age at capture (a )Figure 9. Observed and model-predicted distributions of age after transition ( atm ) for the complete dataset (upper plot),and residuals for this fit (lower plot) for Model 2 without “learning” are shown in (a). Similar representations for thenumber of “immatures” against cohort and against age are shown in (b) and in (c), respectively [results are for the
P-stock].

17