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SC/O05/BWI4Estimation of the lower bound of MSYR for western North PacificBryde's whale
TAKASHI HAKAMADA, TAKEHARU BANDO AND SEIJI OHSUMIThe Institute of Cetacean Research, 4-5, Toyomi-cho, Chuo-ku, Tokyo, 104-0055, Japan.
Contact e-mail: hakamada@cetacean.jp

ABSTRACTAt the Workshop on the pre-implementation assessment of the western North Pacific Bryde’s whale in March
2005, it was recommended that data on calving intervals and their possible implications for the range for valuesof MSYR(mature) should be summarised and presented to the Scientific Committee no later than the First
Annual Meeting (IWC, 2005). Bando et al. (2005) estimated annual pregnancy rate (i.e. inverse of calvinginterval) as 0.55 and 0.59 using JARPN II data obtained during 2002-2005 and 2000-2001, respectively.
Instantaneous increasing rate using the annual pregnancy rate estimates was calculated using the method inBrandão et al. (2000). We applied inequalities between the estimated instantaneous increasing rate and MSYR,
which were derived by Butterworth and Best (1980), to compare MSYR to the estimated increasing rate. Using
point estimates of annual pregnancy rate estimates in Bando et al. (2005), lower bounds of MSYR are estimatedunder the assumption that population exceeds MSYL. In conclusion, at least 1% should not be used as a
MSYR(mature) in the IST.
KEYWORDS: BRYDE’S WHALES, WESTERN NORTH PACIFIC, MSY RATEINTRODUCTION
At the Workshop on the pre-implementation assessment of the western North Pacific Bryde’s whale, it wasrecommended that data on calving intervals and their possible implications of the range for values ofMSYR(mature) that should be used in Implementation Simulation Trials should be summarised and presented tothe Scientific Committee no later than the First Annual Meeting. This paper is aimed to answer thisrecommendation.

To estimate lower bound of MSYR, we used the method described by Butterworth and Best (1990). They
estimated lower bound of MSYR using increasing rate assuming ‘Strong Convexity’ of sustainable yield rate andapplied to the case of South African right whale. We also consider the case that population level exceeds MSYL
adding to the case without assumption on the population level. From the results of HITTER runs for this stock, itis suggested that population size of this stock is more than MSYL (IWC,1997). In such a condition, lower boundof MSYR would be estimated much lower than real MSYR value.

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MATERIAL AND METHODS
Biological parameters
We used four biological parameters as follow;
Age at the first parturition
It was agreed that the estimate is 9 at the CA of this stock in 1996 (IWC, 1997). Bando et al. (2005) estimatesthis parameter as 7. In this analysis, we use both of the estimates. When we assume this parameter as 9, we
assume that natural mortality coefficient is 0.07. When 7 is assumed as age at the first parturition, we used 0.08
as natural mortality coefficient.
Natural mortality coefficient
It was agreed that the estimate is 0.07 at CA of this stock in 1996 (IWC, 1997). Bando et al. (2005) estimates thisparameter as 0.08. We used both of the estimates as explained in the above paragraph.
Annual pregnancy rate
Bando et al. (2005) estimated 0.55 and 0.59. We use these two estimates.
Juvenile natural mortality coefficient (at 0 age)
Because we don’t have enough information on this parameter, we assume this parameter is more than the naturalmortality coefficient. On the analogy of the case for other large whales (Punt, 1999; Wade, 2002; Punt andButterworth, 2002), we assume this parameter are 0.1, 0.3, and 0.5, respectively.
Estimation of instantaneous increasing rate
From Leslie matrix model applied to mature female component, we can derive equation:e tmδ = e (tm −1)δ S + pq f S j S tm −1(1)where:tm = age at first parturition (adding 1 to age at sexual maturity),δ = instantaneous increasing rate,
S = annual survival rate (=exp(-M), where M is natural mortality coefficient),
P = annual pregnancy rate,
qf = proportion of births that are female（=0.5）
and Sj = juvenile survival rate at age 0 (=exp(-Mj), where Mj is juvenile natural mortality coefficient).

This equation was applied to the case of humpback whales to estimate the maximum increasing rate of thisspecies (Brandão et al., 2000). Given biological parameters except instantaneous increasing rate, by solving
equation (1) with respect to δ, the instantaneous increasing rate is estimated.

Relation of MSYR to instantaneous increasing rate
Butterworth and Best (1990) derived the inequality in the constraint of ‘Strong Convexity’ of increasing rate as afunction of population of the stock. The inequality is:r (0)(2)
MSYR >
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where, r(0) is the maximum increasing rate. For example, it is easy to see that Pella-Tomlinson model with anassumption that MSYL is more than a half of carrying capacity satisfies ‘Strong Convexity’. The instantaneous
increasing rate derived from equation (1). Of course, instantaneous increasing rate δ is less than r(0) (Fig.
1a).Therefore,r ( 0) δ(3)MSYR >>
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As suggested from results of HITTER runs (IWC, 1997), it is possible that population of the stock is more than
MSYL. Assuming that population size is more than MSYL, as illustrated in Fig. 1b,it holds thatMSYR = r ( N MSY ) > r ( N ) = δ(MSYL<N≤K)(4)

because r(N) is strictly decreasing function with respect to N. Inequality (4) would provide better estimate oflower bound of MSYR than inequality (3) under this assumption.

RESULTS
Table 1 and Table 2 show lower bound of MSYR given by inequality (4) assuming that population size is above
MSYL. From these tables lower bound decreases as Mj increases.

DISCUSSIONSAt the CA of the Bryde’s whales, HITTER runs were conducted. The results are shown in Appendix 1. Theysuggest population is above MSYL in 1996 for most of the cases. For MSYR(mature)=1% in the case for (1b),(2a) and (2b), population size in 1996 is less than MSYL but more than 50% of K. For MSYR(mature)=1%, RYis 131-168. These figures are more than double of JARPNII annual catch. This indicates the population havebeen increasing since 1996 and therefore population size is now more than MSYL even if MSYR(mature)=1%.
From this, it is likely that this stock is more than MSYL. Therefore, the instantaneous increasing rate is better
estimate of lower bound of MSYR than a half of it.

Wade (2002) estimates juvenile survival rate for eastern North Pacific grey whales as 0.925. Punt andButterworth (2002) estimates it as 0.607. Punt (1999) estimates that for Bering-Chukchi-Beaufort Seas stock ofbowhead whales 0.905 – 0.929 for several cases, respectively. On the analogy of these cases, estimated juvenile
survival rate ranges 0.929 – 0.607 that corresponds to juvenile natural mortality coefficient of 0.074-0.499.
Therefore, we used 0.1-0.5 as this parameter.

Kitakado et al. (2005) estimates additional variance of abundance estimates for the Bryde’s whale. They also
estimates instantaneous increasing rate as 0.052 assuming that year effect is exponential. This figure is moresimilar to estimated lower bound of MSYR assuming Mj is 0.1 or 0.3 than assuming that Mj =0.5. This may
suggests that the former estimates are better than the latter.

In conclusion, we recommend 3% as the lower bound of MSYR.

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THIS PAPER CAN NOT BE CITED WITHOUT PERMISSION FROM THE AUTHORSACKNOWLEDGEMENTSAuthors would like to appreciate to all the researchers participate JARPNII surveys. We also thank Prof. Doug
Butterworth for suggesting methodology and useful comments to improving manuscript. We also thank Drs.
Hatanaka, Kato, Kawahara, Kitakado, Okamura and Pastene for their valuable comments on this paper.

REFERENCES
Bando, T., and Hakamada, T. and Ohsumi, S. 2005. Estimation of true pregnancy rate of the western NorthPacific. Paper SC/O05/BWI5 presented to this meeting (unpublished). xxpp
Bando, T., Kishiro, T., Ohsumi,.S., Zenitani, R. and Kato, H. 2005 Estimation of some biological parameters ofwestern North Pacific Bryde’s whales by age distributions. Paper SC/O05/BWI7 presented to this meeting(unpublished). xxpp.
Brandão, A., Butterworth, D. S. and Brown, M. R. 2000. Maximum possible Humpback Whale increasing rates
as a function of biological parameter values. J. Cetacean Res. Manage. 2 (suppl.):192-3.
Butterworth, D. S. and Best, P. B. 1990. Implications of the recovery rate of the South African Right Whale
population for baleen whale population dynamics. Rep. Int. Whal. Commn. 40:433-47.
Government of Japan. 2000. Research plan for cetacean studies in the western North Pacific under special permit(JARPN II) (Feasibility study plan for 2000 and 2001). Paper SC/52/O1 presented to the IWC ScientificCommittee, June 2000 (unpublished). 68pp.
Government of Japan. 2002. Research Plan for Cetacean Studies in the Western North Pacific under Special
Permit (JARPN II). Paper SC/54/O2 presented to IWC Scientific Committee, May 2002 (unpublished).
115pp
International Whaling Commission, 1997. Report of the Sub-committee on the North Pacific Bryde’s whales.
Rep. Int. Whal. Commn. 47:163-8.
International Whaling Commission, 2005. Report of the Workshop on the pre-implementation assessment ofwestern North Pacific Bryde’s Whales. Paper SC/57/Rep3 presented Scientific Committee, May 2005
(unpublished). 30pp.
Kitakado, T., Shimada, H., Okamura, H. and Miyashita, T. 2005. Update of additional variance estimate for thewestern North Pacific stock of Bryde's whales. Paper SC/O05/BWI6 submitted to this meeting (unpublished).
xxpp
Ohsumi, S. 1977. Bryde’s whales in the pelagic whaling ground of the North Pacific. Rep. Int. Whal. Commn(special issue 1):140-50.
Punt, A. E. 1999 Results for two selected assessments of the Bering-Chukchi-Beaufort seas stock of bowheadwhales. J. Cetacean Res. Manage. 1 (suppl.):192-4.
Punt, A. E. and Butterworth, D. S. 2002. An examination of certain of the assumptions made in the Bayesian
approach used to assess the eastern North Pacific stock of gray whales (Eschrichtius robustus). J. Cetacean
Res. Manage. 4(1): 99-110.
Wade, P. R. 2002. A Bayesian stock assessment of the eastern Pacific gray whale using abundance and harvest
data from 1967-1996. J. Cetacean Res. Manage. 4(1): 85-98.

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THIS PAPER CAN NOT BE CITED WITHOUT PERMISSION FROM THE AUTHORSTable 1. Estimated lower bound of MSYR using inequality (4) when tm=9and M=0.07 as agreed at CA of theBryde’s whale in 1996 (IWC, 1997).

Mj
p0.55
0.59

0.1
0.037
0.041

0.3
0.026
0.030

0.5
0.016
0.020

Table 2. Estimated lower bound of MSYR using inequality (4) when tm=7and M=0.08 as in Bando et al. (2005)..

Mj
p0.55
0.59

0.1
0.042
0.047

0.3
0.029
0.033

5

0.5
0.016
0.021

THIS PAPER CAN NOT BE CITED WITHOUT PERMISSION FROM THE AUTHORSr
2MSYRr (0)r (N t )MSYRNtMSYLK2MSYLNFig. 1a. Relation between MSYR and instantaneous increasing rate δ without assumption that current population
size is more than MSYL. Bold curve indicates a graph of r=r(N), A dot indicates current population size Nt andcurrent instantaneous increasing rate δ and straight line tangent line of the curve at N=MSYL.

r2MSYR

r (0)

MSYR

r (N t )MSYL

NtK2MSYL

NFig. 1b. Relation between MSYR and instantaneous increasing rate δ under the condition that current population
size is more than MSYL. Bold curve indicates a graph of r= r(N), A dot indicates current population size Nt andcurrent instantaneous increasing rate δ and straight line tangent line of the curve at N=MSYL.

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Appendix 1. Results from HITTER run for western north pacific Bryde’s whales at the 47th IWC/SC meeting(referred from IWC, 1997).

MSYR (mature)0%1%
2%(a1) Hit total population of 25,640 (best estimate)
TotalK
46,03940,32836,220
N(1996)25,83526,56327,091Mature females K
13,14911,518
10,345N(1996)
7,517
7,391
7,294
N/K(%)
57.2
64.2
70.5
RY33151
238(a2) Hit total population of 25,640, 1.2×catch
Total
K
50,105
43,62638,938N(1996)25,874
26,66627,274
Mature females K
14,31012,46011,121
N(1996)
7,556
7,421
7,313
N/K(%)
52.8
59.6
65.8
RY
39168
267
(b1) Hit total population of 18,531 (lower 5th percentile)
Total
K
38,92633,75830,007N(1996)
18,726
19,334
19,817
Mature females K
11,1179,641
8,570
N(1996)
5,487
5,384
5,300
N/K(%)
49.4
55.8
61.8
RY33
131
210
(b2) Hit total population of 18,531, 1.2×catch
Total
K
42,99137,14032,880N(1996)
18,766
19,415
19,952
Mature females K
12,278
10,6079,391
N(1996)
5,526
5,417
5,327
N/K(%)
4551.156.7
RY39144
231

74%

6%

30,824
27,626
8,803
7,162
81.4
336

27,35427,504
7,812
7,092
90.8
341

32,76728,006
9,358
7,156
76.5
395

28,80328,1508,226
7,055
85.8441

25,06820,460
7,160
5,173
72.2
319

21,913
20,722
6,258
5,084
81.2
376

27,276
20,736
7,790
5,185
66.6
361

23,727
21,195
6,777
5,077
74.9
446