Not to be quote without Author’s permission

JA/J05/JR22

IMPLICATIONS OF THE UPDATED ADAPT-VPA ASSESSMENTS FOR THE DYNAMICSOF MINKE WHALES IN AREAS IV AND V
D.S. Butterworth and M. Mori
MARAM (Marine Resource Assessment and Management Group)
Department of Mathematics and Applied Mathematics
University of Cape Town, Rondebosch 7701, South Africa

ABSTRACTA stock-recruitment model of the Pella-Tomlinson form is fit to recruitment and adult female abundance estimatesfrom the Base Case ADAPT-VPA assessment of Areas IV+V combined. The trends of these plots require theassumption that minke whale carrying capacity first increased, then later decreased during the 20th Century.

Aninitial attempt at this fit suggests that this carrying capacity increased about five fold from 1930 to the mid-1960s,
and then decreased again by about half. MSYR1+ is estimated at 4.0% for this model.

INTRODUCTION
This paper focuses on the results of the Base Case assessment of Areas IV and V combined that is presented inMori and Butterworth (2005), with a view to improved understanding of the dynamics of the minke whales in thisregion.

Fig. 1a plots the recruitment ( N y ,1 ) time series estimated by this Base Case assessment.

Fig. 1b shows theassociated stock recruitment plot: N y ,1 vs N yf−1 , where the “adult” (reproductive) population is taken to be:
45

N yA =∑N(1)

y ,aa =7

and the number of adult females N yf = 0.5 ⋅ N yA , i.e. an age at first-parturition of 7 is assumed.

This plot

includes a “replacement line”; this is the line where recruitment exactly balances deaths from natural mortality, i.e.
for a point on this line, population size is maintained unchanged for the following year in the absence ofharvesting.

For a conventional stock-recruitment relationship, recruitment levels would lie above the replacement line at low
adult female population sizes, but (in the absence of harvesting) as carrying capacity K was approached,recruitments would cluster around a stable equilibrium point on the replacement line.

This, however, is NOT thebehaviour evident in Fig. 1b. The recruitment series does approach the replacement line (in about 1973), butthereafter drops below this line as both recruitment and the number of reproductive animals decrease.

Such behaviour requires not only the conventional assumption of an increase in the minke whale carrying capacity
K in the mid-20th Century as the then unharvested minkes increased, presumably in response to the depletion ofother krill-feeding Antarctic baleen whale species through overharvesting, but also a subsequent decline in K (aresult also demonstrated earlier by Butterworth and Punt, 1999). Possibly this could be a response to recent
recoveries of these other species.

This paper attempts an initial model fit to the stock-recruitment information in Fig. 1b, to assess to whatquantitative extent changes in K are required.

THE RECRUITMENT MODEL
Recruitment is assumed to follow a Pella-Tomlinson form:  f  z Ny
f 
N y +1,1 = λ ⋅ N y 1 + A1 −  f  K   y  



(2)

whereN y ,1

is the recruitment (1-year-olds) in year y,

λis the combined pregnancy and first year survival rate when the population is at carrying capacity,is the number of adult (past the age of first parturition) females, taken to be given by

N yf

45

∑N0.5

Ay , a = 0.5 N y(i.e. we assume equal numbers of males and females),a =7AK yf

is the resilience parameter (related to MSYR),
is the carrying capacity for adult females, which may change over time, andz

is the degree of compensation parameter, which we set here at 2.39.

When N f = K f , the recruitment must equal the number of 1+ whales dying annually as a result of natural
mortality, i.e:(λ ⋅ K yf = K 1+ 1 − e − M)(3)(ignoring for simplicity the small correction that accounts for an assumed infinite natural morality from age 45).
Further, expressions for unexploited equilibrium numbers at age values yield:
45

K 1+Kf=∑e− M ⋅aa =145

0.5 ⋅

∑e=µ(4)− M ⋅aa =7

where µ can be computed given the value of M (0.068 for the assessment to be considered here).
Thus equation (2) can be rewritten:(N y +1,1 = µ 1 − e−M) N yf f K y 
N yf 1 + A1 − 2.39 

The unknown parameters of this model are A and the parameters describing K and its temporal variation.
2(5)

These

are then estimated by minimizing:SS =∑ (ln(N )− ln(Nobsy ,12modely ,1))

(6)

ywhereN yobs
,1is the “observed” recruitment for year y from the Base Case assessment for Areas IV+V combined,as shown in Fig. 1a, andl
N ymodeisthe recruitment for year y predicted by the model of equation (5).
,1

RESULTS
Thus far, the following functional form for K yf has been considered (see Fig. 2):

 K 1f

K 2f − K 1f
 f
( y − y1 )γ+
K
 1γ( y 2 − y1 )K yf = K 3f − K 2f
 f(y − y2 )+K 2(y3 − y 2 ) f K 3()()

y ≤ y1y1 + 1 ≤ y ≤ y 2(7)y 2 + 1 ≤ y ≤ y3
y3 + 1 ≤ ywith the following choices made for the “change “ years: y1 = 1930 , y 2 = 1966 and y 3 = 2000 .

The results of the fitting process yield: A=1.28, γ = 1.530 , K 1f = 36639 , K 2f = 151455 and K 3f = 69089 ,suggesting that carrying capacity originally increased about five-fold, but has since approximately halved.

The model fit itself is shown for recruitment vs year in Fig. 3a, and for the stock recruitment plot in Fig. 3b.

Theestimated trends in carrying capacity are shown in Fig. 4.

DISCUSSION
The fit of the model to the recruitment and stock-recruit plots in Fig. 3, while showing qualitative consistency,
clearly manifests some systematic deviations. Further work will seek to modify the functional form assumed forK yf to achieve a better fit.

The estimation of the value of resilience A in turn allows an estimate of MSYR (=FMSY) to be effected.

The resultis:
MSYR1+ = 0.040(8)with a corresponding adult-female population size in terms of the corresponding carrying capacity:
MSYLmat = N f (FMSY ) K f = 0.548(9)3

The model of equation (5) could be used to project the abundance trajectories estimated in Mori and Butterworth(2005) forward in time under alternative future catch levels.

Indeed, the stock-recruitment model here could beincorporated in the ADAPT-VPA assessment process itself, then also providing an improved approach to theprocess used by Mori and Butterworth (2005) to “adjust” earlier estimates of abundance for “missing” cohorts.

REFERENCES
Butterworth, D. S. and Punt, A. E. 1999. An initial examination of possible inferences concerning MSYR ofSouthern Hemisphere minke whales from recruitment trends estimated in catch-at-age analyses. Journal ofCetacean Research and Management 1: 33-39.

Mori, M. and Butterworth, D. S. 2005. Progress on application of ADAPT-VPA to minke whales in Areas IV andV given updated information from IDCR/SOWER and JARPA surveys. Document prepared for this pre-JARPA
review meeting. 27pp.

a)b)4

Stock-recruitment plot for the"Base Case" Area combined assessment

Recruitment trend for the"Base Case" Area combined assessment
45000

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Number of adult femalesYearFigure 1. The “observed” recruitment time trend and stock-recruit plot for the Base Case Areas IV+V combined
ADAPT-VPA assessment of Mori and Butterworth (2005).

K 2f

K 3f
K1f
Yeary1y2y3

Figure 2. The functional form assumed for the change over time in carrying capacity of adult females.

a)

b)
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Stock-recruitment plot for the"Base Case" Area combined assessment

Recruitment trend for the"Base Case" Area combined assessment
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Number of adult females

YearFigure 3. Fit of the model of equation (5) to the recruitment trend and stock-recruit plot for the Base Case Areas
IV+V combined ADAPT-VPA assessment.

160000140000120000100000
80000
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01920 1930 1940 1950 1960 1970 1980 1990 2000
YearFigure 4. The estimated trend in carrying capacity of adult females over time.

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