NOAA Technical Memorandum NMFS-AFSC-207
 

Population Status of the Eastern 

North Pacific Stock of Gray Whales 

in 2009 

by 

A. E. Punt and P. R. Wade 

U.S. DEPARTMENT OF COMMERCE 

National Oceanic and Atmospheric Administration
 

National Marine Fisheries Service
 

Alaska Fisheries Science Center
 

February 2010 
NOAA Technical Memorandum NMFS 

The National Marine Fisheries Service's Alaska Fisheries Science Center 

uses the NOAA Technical Memorandum series to issue informal scientific and 

technical publications when complete formal review and editorial processing 

are not appropriate or feasible. Documents within this series reflect sound 

professional work and may be referenced in the formal scientific and technical 

literature. 

The NMFS-AFSC Technical Memorandum series of the Alaska Fisheries 

Science Center continues the NMFS-F/NWC series established in 1970 by the 

Northwest Fisheries Center. The NMFS-NWFSC series is currently used by 

the Northwest Fisheries Science Center. 

This document should be cited as follows: 

Punt, A. E., and P. R. Wade. 2010. Population status of the eastern 

North Pacific stock of gray whales in 2009. U.S. Dep. Commer., NOAA 

Tech. Memo. NMFS-AFSC-207, 43 p. 

Reference in this document to trade names does not imply endorsement by 

the National Marine Fisheries Service, NOAA. 
NOAA Technical Memorandum NMFS-AFSC-207
 

Population Status of the Eastern 

North Pacific Stock of Gray Whales 

in 2009 

by 1 2 A. E. Punt and P. R. Wade 

 1School of Aquatic and Fishery Sciences
 

University of Washington
 

Box 355020
 

Seattle, WA 98195-5020
 

2 National Marine Mammal Laboratory
 

Alaska Fisheries Science Center

 7600 Sand Point Way NE
 

Seattle, WA 98115-6489
 

www.afsc.noaa.gov

U.S. DEPARTMENT OF COMMERCE 

Gary F. Locke, Secretary 

National Oceanic and Atmospheric Administration

 Jane Lubchenco, Under Secretary and Administrator 

National Marine Fisheries Service 

James W. Balsiger, Acting Assistant Administrator for Fisheries

February 2010 
This document is available to the public through: 

National Technical Information Service 

U.S. Department of Commerce 

5285 Port Royal Road 

Springfield, VA 22161 

www.ntis.gov 
 

ABSTRACT

An age- and sex-structured population dynamics model is fitted using Bayesian methods to data 

on the catches and abundance estimates for the eastern North Pacific (ENP) stock of gray whales 

(Eschrictius robustus). The prior distributions used for these analyses incorporate revised 

estimates of abundance for ENP gray whales and account explicitly for the drop in abundance 

caused by the 1999-2000 mortality event. A series of analyses are conducted to evaluate the 

sensitivity of the results to different assumptions. The baseline analysis estimates the ENP gray 

whale population to be above the maximum net productivity level (MNPL), because the posterior 

mean for the ratio of 2009 abundance to MNPL, termed the optimal sustainable population ratio, 

is 1.29 (with a posterior median of 1.37 and a 90% probability interval of 0.68-1.51), indicating 

the population is estimated to be well above MNPL. The baseline analysis estimates a probability 

of 0.884 that the population is above its MNPL, which means there is a 0.884 probability that it 

is at its optimum sustainable population size as defined by the U.S. Marine Mammal Protection 

Act. These results are consistent across all the model runs. The baseline model also estimates the 

2009 ENP gray whale population size (posterior mean of 21,911) to be at 85% of its carrying 

capacity (posterior mean of 25,808), and this is also consistent across all the model runs. 

iii
 
 

v

CONTENTS 

Abstract.......................................................................................................................................... iii 

Introduction......................................................................................................................................1 

Methods............................................................................................................................................5 

 Available Data.................................................................................................................5 

 Analysis Methods............................................................................................................5 

 The Population Dynamics Model ......................................................................5 

 Parameter Estimation.........................................................................................9 

 Output Statistics...............................................................................................11 

 Sensitivity Tests...............................................................................................12 

 Prior Distributions............................................................................................13 

Results............................................................................................................................................14 

Discussion......................................................................................................................................16 

Acknowledgments..........................................................................................................................23 

Citations .........................................................................................................................................25 

Tables.............................................................................................................................................32 

Figures............................................................................................................................................39 

Appendix........................................................................................................................................43 

INTRODUCTION

The eastern North Pacific (ENP) gray whale (Eschrichtius robustus) population has been hunted 

extensively by both commercial and aboriginal whalers. Indigenous peoples of both North 

America and Russia have hunted gray whales in some locations for centuries and possibly for 

2,000 years or more (Krupnik 1984, O’Leary 1984). The winter breeding grounds of the ENP 

gray whale (lagoons and adjacent ocean areas in Baja California, Mexico) were discovered by 

Yankee whalers in the early 19th century, and two commercial whaling vessels first hunted gray 

whales (in Magdalena Bay) in the winter of 1845-46 (Henderson 1984). This began a period of 

intense hunting with large catches of ENP gray whales by Yankee whalers from 1846 until 1873 

which decimated the population. Whaling ships and shore-based whalers continued to catch gray 

whales for the next two decades which drove the population to apparent commercial extinction 

by 1893. In the 20th century, modern commercial pelagic whaling of ENP gray whales began in 

1910 and ended in 1946 when gray whales received full protection under the International 

Convention for the Regulation of Whaling (Reeves 1984). Aboriginal catches of ENP gray 

whales along the Chukotka Peninsula of Russia have continued since 1946 until the present. 

Gray whales were listed as endangered under the U.S. Endangered Species Act in 1973 and after 

increasing substantially in population size, they were removed from the endangered species list 

in 1994. 

From 1846 to 1900 recorded commercial kills numbered nearly 9,000 gray whales, and it is 

roughly estimated that about 6,000 gray whales were killed by aboriginal hunters during this 

same period, for a total of more than 15,500 whales caught (Table 1). Since 1900, about 11,500 

additional ENP gray whales have been killed by commercial and aboriginal whalers for a total 

since 1846 of more than 27,000 whales caught (Table 1). The magnitude of the catches, 

particularly for the period of high exploitation during the 1800s, gives some information on the 

likely pre-exploitation population size. For example, Jones et al. (1984) state that “most whaling 

historians and biologists believe the pre-exploitation stock size was between 15,000 and 24,000 

animals”. 

ENP gray whales migrate along the west coast of North America, and the National Marine 

Fisheries Service (NMFS) has taken advantage of this nearshore migration pattern to conduct 
shore-based counts of the population in central California during December-February from 1967-

68 to 2006-07. These survey data have been used to estimate the abundance of the ENP gray 

whale stock over the survey period (Reilly 1981; Buckland et al. 1993; Laake et al. 1994; Hobbs 

et al. 2004; Rugh et al. 2005, 2008a). 

The resulting sequence of abundance estimates has also been used to estimate the population’s 

growth rate (Buckland et al. 1993, Buckland and Breiwick 2002), as well as its status relative to 

the maximum net productivity level (MNPL)1 and carrying capacity (K) (Reilly 1981, Lankester 

and Beddington 1986, Cooke 1986, Wade 2002, Punt and Butterworth 2002). However, 

attempts to model the gray whale population from 1846 until the present, accounting for the 

catch record, assuming that the stock was at its carrying capacity in 1846 (has and this has not 

changed) has run into difficulties because the catch history cannot be reconciled with a 

population that increased at the observed rate from 1967-68 to 1979-80 (Reilly 1981, Cooke 

1986, Lankester and Beddington 1986). The explanation for this is simple; if one assumes a 

relatively low maximum growth rate, the ENP gray whales would not have been able to increase 

between 1967-68 and 1979-80 because of the catches during that time, and if one assumes a high 

maximum growth rate, the population would not be increasing then because it would have 

already returned to carrying capacity. Butterworth et al. (2002) investigated the inability to fit a 

standard population dynamics model to the data for the ENP gray whales extensively and 

concluded that the catch history and the observed rate of increase could be reconciled in one of 

three different ways, which were not mutually exclusive: a) a 2.5× increase in K between 1846 

and 1988, b) a 1.7× increase or more in the commercial catch between 1846 and 1900, and c) a 

3× increase or more in aboriginal catch levels prior to 1846 compared to what was previously 

assumed (Butterworth et al. 2002). 

 

1

The maximum net productivity level is described in the National Marine Fisheries Service's definition of "optimum 

sustainable population" (OSP) (50 CFR 216.3) as the abundance level that results in the greatest net annual 

increment in population numbers or biomass resulting from additions to the population due to reproduction and/or 

growth less losses due to natural mortality. Under the U.S. Marine Mammal Protection Act, populations above 

MNPL are considered to be at OSP; populations below MNPL can be designated as ‘depleted’ and are afforded a 

greater level of protection. 

2
Given these difficulties, recent gray whale population assessments have been conducted by 

modeling the population since 1930 or later, rather than trying to model the population since 

1846 (e.g., Punt and Butterworth 2002, Wade 2002). These analyses differed from the earlier 

assessments by not assuming the population size in 1846 was K. Instead, K is essentially 

estimated by the recent trend in abundance, where a growing population implies that K has likely 

not yet been reached, and a roughly stable population implies the population is at or near K. 

Based on abundance surveys through 1995-96, point estimates of K from these analyses ranged 

from 24,000 to 32,000, but these estimates were relatively imprecise because they had broad 

confidence intervals (Wade 2002, Punt and Butterworth 2002). In particular, the results did not 

exclude the possibility that K could be much larger than this range. However, these analyses did 

suggest that the population was likely close to K and at or above its MNPL. For example, Wade 

(2002) estimated a probability of 0.72 the population was above MNPL in 1996. Punt and 

Butterworth (2002) also conducted analyses projecting the population from the year 1600 under 

various assumptions that historic commercial and aboriginal catches were underestimated (as in 

Butterworth et al. (2002)). Those analyses resulted in point estimates of K that ranged between 

15,000 and 19,000. In those analyses, it was estimated the population was at a very high fraction 

of K in 1996 and had a very high probability of being above MNPL. 

Recently, Rugh et al. (2008b) evaluated the accuracy of various components of the shore-based 

survey method, with a focus on pod size estimation. They found that the correction factors that 

had been used to compensate for bias in pod size estimates were calculated differently for 

different sets of years. In particular, the correction factors estimated by Laake et al. (1994) were 

substantially larger than those estimated by Reilly (1981). Also, the estimates for the surveys 

prior to 1987 in the trend analysis were scaled based on the abundance estimate from 1987-88. 

This meant that the first 16 abundance estimates used one set of correction factors, and the more 

recent 7 abundance estimates used different (and larger) correction factors which would 

influence the estimated trend and population trajectory. In addition, there were other subtle 

differences in the analysis methods used for the sequence of abundance estimates. Thus, a reevaluation of the analysis techniques and a reanalysis of the abundance estimates were warranted 

to apply a more uniform approach throughout the years. Laake et al. (2009) derived a better, 

more consistent, approach to abundance estimation, and incorporated it into an analysis to re3
estimate abundance for all 23 shore-based surveys. These new revised abundance estimates led 

to this re-assessment of the ENP gray whale population. 

Here, we re-assess the population of ENP gray whales by fitting an age- and sex-structured 

population model to these revised abundance estimates, using similar methods as those used by 

Wade (2002) and Punt and Butterworth (2002). Note also that we use recent abundance estimates 

from 1997-98, 2000-01, 2001-02, and 2006-07 that were not available in previous assessments. 

As in Punt and Butterworth (2002), we also perform sensitivity tests to various assumptions or 

modeling decisions. 

These analyses also incorporate new information about the biology of the ENP gray whales 

developed in recent studies. In particular, it is now recognized that the population experienced an 

unusual mortality event in 1999 and 2000. An unusually high number of gray whales were 

stranded along the west coast of North America in those years (Moore et al. 2001, Gulland et al. 

2005). Over 60% of the dead whales were adults, and more adults and subadults stranded in 1999 

and 2000 relative to the years prior to the mortality event (1996-98), when calf strandings were 

more common. Many of the stranded whales were emaciated, and aerial photogrammetry 

documented that migrating gray whales were skinnier in girth in 1999 relative to previous years 

(Perryman and Lynn 2002, W. Perryman, Southwest Fisheries Science Center, SWFSC, pers. 

comm.). In addition, calf production in 1999 and 2000 was less than one third of that in the 

previous years (1996-98). In 2001 and 2002, strandings of gray whales along the coast decreased 

to levels that were below their pre-1999 level (Gulland et al. 2005), and average calf production 

in 2002-2004 returned to the level seen in pre-1999 years (Table 2). A Working Group on 

Marine Mammal Unusual Mortality Events (Gulland et al. 2005) concluded that the emaciated 

condition of many of the stranded whales supported the idea that starvation could have been a 

significant contributing factor to the higher number of strandings in 1999 and 2000. Perryman et 

al. (2002) found a significant positive correlation between an index of the amount of ice-free area 

in gray whale feeding areas in the Bering Sea and their estimates of calf production for the 

following spring for the years 1994 to 2000; the suggested mechanism is that longer periods of 

time in open water provides greater feeding opportunities for gray whales. Whether or not heavy 

ice cover was ultimately the mechanism that caused the 1999-2000 event, it is clear that ENP 

gray whales were substantially affected in those years; whales were on average skinnier, they 

4
had a lower survival rate (particularly of adults), and calf production was dramatically lower. 

Given that this event may have affected the status of the ENP gray whale population relative to 

K, we specified an additional model parameter (“catastrophic mortality”) that allowed for lower 

survival in the years 1999 and 2000 to investigate this effect. 

METHODS

Available Data 

A variety of data sources are available to assess the status of the ENP stock of gray whales. 

These data sources are used when developing the prior distributions for the parameters of the 

population dynamics model, when pre-specifying the values for some of the parameters of this 

model, and when constructing the likelihood function. Table 1 lists the time series of removals. It 

should be noted that the catches for the years prior to 1930 are subject to considerable 

uncertainty, and evaluating these catches remains an active area of research. However, the 

uncertainty associated with these early catches is inconsequential for this report because the 

population projections do not start before 1930. 

The key source of information on the abundance of the ENP gray whales is based on data 

collected from the southbound surveys that have been conducted since 1967-68 near Carmel, 

California (Laake et al. 2009; Table 2). Information on trends in calf numbers are also available 

from surveys of calves during the northbound migration (Perryman et al. 2002, W. Perryman, 

pers. comm., Table 2). The calf abundance data are not included in the baseline analyses, but are 

considered in one of the tests of sensitivity. 

Analysis Methods 

The Population Dynamics Model 

The analyses use an age- and sex-structured population dynamics model which assumes that all 

whaling takes place at the start of the year, and that all animals are ‘recruited’ to the hunted 

population by age 5 (i.e., hunting only occurs on animals age 5 and older) (Punt 1999, Punt and 

Butterworth 2002). The dynamics of the population are assumed to be governed by the 

equations: 

5
M

1 1

1, , 1 , 1 1

, , ,1 ,1 1

0.5

(1 )

(1 ) (1 )

t t

sss

t a ta ta a y

ss s s

tx tx x y tx tx x y

P f

N N F SS

N F SS N F S

+ +

+ − −−

− −−

⎧

⎪⎪ = − ⎨

⎪

⎪ − +− ⎩

%

% %S ,

 (1) 

where is the number of animals of age a and sex s (m / f) at the start of year t, 

a S is the annual survival rate of animals of age a in the absence of catastrophic 

mortality events (assumed to be the same for males and females), 

y S

% is the amount of catastrophic mortality (represented in the form of a survival rate) 

during year y (catastrophic events are assumed to occur at the start of the year 

before mortality due to whaling and natural causes), 

,

s Ft a is the exploitation rate on animals of sex s and age a during year t, 

M Pt is the number of females that have reached the age at first parturition by the start 

of year t, 

M

,

m 1

t

a a

P = += ∑ f xNt a , (2) 

ma is the age-of-maturity, 

t f is pregnancy rate (number of calves of both sexes per ‘mature’ female) during 

year t (note that Equation (1) assumes an equal male : female sex ratio at birth), 

and 

x is the maximum age class, which for convenience is lumped across older age 

classes (i.e., individuals stay in this age class until they die). 

Density dependence on fecundity can be modeled by writing the pregnancy rate, , as follows: 

{ ( ) } 1 1 max 1 1 / ,0 2 2

z

t eq tt f f A SP K + +

− −

⎛ ⎡ = +− ⎜ ⎢ ⎝ ⎠ ⎣

% ⎤ ⎞

⎟ ⎥⎦ , (3) 

6
where is the pregnancy rate at the pre-exploitation equilibrium, 2 Ff = )0( : 

−1

f

1

() 2 ()

m

x

a

a a

fF N F = +

⎧ ⎫ ⎪ ⎪ = ⎨ ⎬ ⎪ ⎪ ⎩ ⎭ ∑ % , (4) 

 is the resilience parameter: 

 , (5) 

 is the maximum (theoretical) pregnancy rate, 

 is the degree of compensation, 

1 Pt+ is number of animals aged 1 and older at the start of year t: 

 1

,

1

x s

t

s a

P +

=

= ∑∑Nt a , (6) 

1 K + is the (current) pre-exploitation equilibrium size (carrying capacity) in terms of 

animals aged 1 and older, 

is the number of animals of sex s and age a when the exploitation rate is fixed at 

F, expressed as a fraction of the number of calves of the same sex s (see 

Appendix 1 of Punt (1999) for details). 

Note that although these equations are written formally as if only the pregnancy rate component 

of ‘fecundity’ as defined here is density-dependent, exactly the same equations follow if some or 

all of this dependence occurs in the infant survival rate (Punt 1999). Catastrophic mortality is 

assumed to occur before density-dependence because many of the deaths in 1999 and 2000 

occurred before mating was likely to have occurred. Note that non-catastrophic natural mortality 

does not appear in Equation 3 because it cancels out. The time-lag in Equation 3 is specified to 

 

2

 The pregnancy rate at the pre-exploitation equilibrium can be considered to be the equilibrium pregnancy rate 

when the exploitation rate, F, is fixed at zero. 

7
match the reproductive cycle of gray whales; mature female gray whales mate and become 

pregnant in early winter, have a gestation period of slightly longer than one year, and give birth 

at the start of the next year (on average in January) (Rice and Wolman 1971, Shelden et al. 

2004). Their body condition at the end of the summer feeding season will help determine their 

probability of becoming pregnant the following winter and producing a calf a year later. 

Therefore, the density-dependent effect on calf production is assumed to be determined by the 

population size during the feeding season two time-steps prior (approximately 1.5 years earlier). 

Following past assessments of the ENP stock of gray whales (e.g., Butterworth et al. 2002, Punt 

and Butterworth 2002, Punt et al. 2004), the catch (by sex) is assumed to be taken uniformly 

from the animals aged 5 and older, that is, 

,

5

/ ,

x s s

ta t ta

a

FC N == ∑ s . (7) 

The population is assumed to have had a stable age-structure at the start of the projection period 

(year ). 

INIT INIT

Tot '

, INIT ' INIT

' '0

( )/ ( ss s

ta t a a

s a

N N NF N F == % % ∑∑ )

x

Tot

 , (8) 

where INIT Nt is the size of the total (0+) component of the population at the start of year . The 

value of FINIT is selected numerically so that: 

INIT

Tot

0 INIT INIT

0

0.5 ( ) ( ) s

t a

s a

N NF NF == ∑∑ % x

 , (9) 

where is the number of calves (of both sexes) at the start of the year when 0 INIT N F( )

F = FINIT : 

1/z 1

INIT

0 INIT 1

eq INIT

1 ( ) ( )1 1 ( )

f F K N F

A f PF

+

+

⎛ ⎞ ⎡ ⎤ =− − ⎜ ⎟ ⎢ ⎥

⎝ ⎠ ⎢ ⎥ ⎣ ⎦ % , (10) 

8
1 P F( ) + % is the size of the 1+ component of the population as a function of F, 

expressed as a fraction of the number of calves (of both sexes). 

Parameter Estimation 

Catastrophic mortality is assumed to be zero (i.e., 1 y S = % ) except for 1999 and 2000 when it is 

assumed to be equal to a parameter . This assumption reflects the large number of dead ENP 

gray whales observed stranded along the coasts of Oregon and Washington during 1999 and 

2000 relative to numbers stranding there annually historically (Gulland et al. 2005, Brownell et 

al. 2007). 

S

%

1 The parameters of the population dynamics model are am; S

% ; K + ; the 1+ population size at the 

start of 1968, 1+ 3 P1968 ; MSYL1+ (the maximum sustainable yield level for the 1+ population, which 

is the population size at which maximum sustainable yield (MSY) is achieved when hunting 

takes place uniformly on animals aged 1 and older); MSYR1+ (the ratio of MSY to MSYL1+); 

max f ; and the non-calf survival rate, 1 S + . The analysis does not incorporate a prior distribution 

for the survival rate of calves (S0) explicitly. Instead, following Wade (2002), an implicit prior 

distribution for this parameter is calculated from the priors for the five parameters am, max f , 1 S + , 

MSYR1+ and MSYL1+. For any specific draw from the prior distributions for these five 

parameters, the value for S0 is selected so that the relationships imposed by the population model 

among the six parameters are satisfied. If the resulting value for S0 is less than zero or greater 

than that of , the values for , am, 1+ S S1+ max f , MSYR1+ and MSYL1+ are drawn again4

. Thus, the 

prior for S0 is forced to conform to the intuitive notion that the survival rate of calves must be 

lower than that for older animals (and must be larger than zero) (Caughley 1966).

 

3

 The 1968 population size is taken to be a measure of initial abundance so that the analyses based on different 

starting years are comparable in terms of their prior specifications. 4

 The implications of different treatments of how to handle situations in which the calculated value for S0 is outside 

of plausible bounds is examined by Brandon et al. (2007). 

9
Under the assumption that the logarithms of the estimates of abundance based on the southbound 

surveys are normally distributed, the contribution of these estimates to the negative of the 

logarithm of the likelihood function (ignoring constants independent of the model parameters) is 

obs 1 1 obs 1 , ˆ ˆ n 0.5 n | | 0.5 ( n n )[( ) ] ( n n ) i i ij j

i j

L N P N Pj

+ − + − = +Ω + l l V V ∑∑ l l − +Ω l l − , (11) 

where is the ith estimate of abundance obs Ni

5

, 

ˆ

Pi is the model-estimate corresponding to , obs Ni

V is the variance-covariance matrix for the abundance estimates, and 

Ω is a diagonal matrix with elements (this matrix captures sources of 

uncertainty not captured elsewhere; termed “additional variance” in Wade 

(2002)). 

2 CVadd

A Bayesian approach is used to estimate the ‘free’ parameters of the model based on the prior 

distributions in Table 3 and the sampling/importance resampling (SIR) algorithm (Rubin 1988).

(a) Draw values for the parameters 1 S + , am, max f , MSYR1+ , MSYL1+, 1 K + , 1 P1968

+ , S

% , and 

CVadd from the priors in Table 3. 

(b) Solve the system of equations that relate 0 S , 1 S + , am, max f , MSYR1+, MSYL1+, A and z 

(Punt 1999, Eqs. 18-21) to find values for S0, A, and z, and find the population size in 

year INIT t and the population rate of increase in this year, so that, if the population is 

projected from year INIT t to 1968, the total (1+) population size in 1968 equals the 

generated value for 1 P1968

+ . 

(c) Compute the likelihood for the projection (see Equation 11). 

(d) Repeat steps (a)-(c) a very large number (typically 5 million) of times. 

(e) Select 5,000 parameter vectors randomly from those generated using steps (a)-(d), 

assigning a probability of selecting a particular vector proportional to its likelihood 

 

5

 The abundance estimate for year y/y+1 is assumed to pertain to abundance at the start of year y+1. 

10
The above formulation implies that the year for which a prior on abundance is specified (1968) is 

not necessarily the same as the first year of the population projection (t , baseline value 1930). 

Starting the population projection before the first year for which data on abundance are available 

allows most of the impact of any transient population dynamics caused by the assumption of a 

stable age-structure to be eliminated. Therefore, the model population should mimic the real 

population more closely by allowing the sex- and age-selectivity of the catches to correctly 

influence the sex- and age-distribution of the population once the trajectory reaches years where 

it is compared to the data (i.e., 1967-68 and beyond). 

INIT

Output Statistics 

The results are summarized by the posterior medians, means and 90% credibility intervals for 

MSYR1+, MSYL1+, , , , and K1+ and the following management-related quantities: 1 S + 0 S S

%

(a) 1 1

2009 / + – the depletion level, or the number of 1+ animals at the start of 2009, 

expressed as a percentage of that corresponding to the equilibrium level; 

P K +

(b) 1+ - the OSP ratio, the number of 1+ animals at the start of 2009, expressed as 

a percentage of that at which MSY is achieved; and 

1

2009 P MSYL / +

(c) λmax – the maximum rate of increase (given a stable age-structure and the assumption of 

no maximum age; Breiwick et al. 1984) 

1

2009 P K/ + 1+

1 1

 is termed the depletion level because it provides a measure of how depleted the 

population is relative to the carrying capacity (“K”), as the equilibrium level in a densitydependent model is equivalent to carrying capacity. 2009 P MSYL / + + is referred to as the OSP ratio

because it provides a measure of whether the population is above MNPL (MSYL1+ is essentially 

equivalent to MNPL) and therefore at OSP (e.g, Gerrodette and DeMaster 1990). Because the 

model is age-structured and includes human removals it is parameterized in terms of MSYL1+. 

MSYL0+, which is the population level at which MSY occurs when hunting takes place 

uniformly on animals aged 0 and older, is exactly equivalent to MNPL. The difference between 

MSYL1+ and MSYL0+ is small in a long-lived animal with a relatively slow population growth 

rate. In the case of the ENP gray whale assessment model, MSYL1+ has been found to be nearly 

11
equivalent to MNPL (Punt, unpublished data), so we consider MSYL1+ to be equivalent to 

MNPL here for the calculation of the output statistic OSP ratio. Populations at OSP will have a 

value of 1.0 or higher for OSP ratio. Note that λmax can be equated to rmax (e.g., as in Wade 

(1998)) through the equation rmax =λmax – 1.0. 

Sensitivity Tests 

Our baseline assessment includes the baseline estimates of 1+ abundance (Table 2) and allows 

for a catastrophic mortality event in 1999-2000. The sensitivity of the results of the analyses is 

explored to: 

(a) Vary the first year considered in the population projection (1940, 1950 and 1960); 

(b) Replace the estimates of abundance for the southbound migration by the values used in 

the previous assessment (Table 2, “Unrevised estimates”); 

(c) Replace the abundance estimates with the “Lo” and “Hi” series (Table 2)6

; 

(d) Ignore the catastrophic event in 1999-2000 (abbreviation “No event”); 

(e) Base the analysis on the generalized logistic equation (see Appendix for details) 

(abbreviation “Gen Logist”)7

; 

(f) Split the abundance series after 1987-88 (abbreviation “Split series”), where the first 

abundance series is treated as a relative index of abundance scaled to absolute abundance 

through a constant of proportionality, and the second series is treated as an absolute index 

of abundance; and 

(g) Include the calf counts at Point Piedras Blancas, California (Perryman et al. 2002, 

Perryman, pers. comm.) in the analysis (abbreviation “With calf counts”). 

For the last sensitivity test, the contribution of the data on calf counts to the negative of the 

logarithm of the likelihood function (ignoring constants independent of the model parameters) is 

 

6

 The sequence of gray whale abundance estimates depends in part on the estimates of observer detection probability 

that were measured with the double observer data. Assessment of matches amongst the pods detected by the 

observers depends on the weighting parameters for distance and time measurements (Laake et al. 2009). The 

weighting parameters used for the baseline abundance estimates were selected such that 95% of the observations 

of the same pod would be correctly matched. Sensitivity is explored to matching weighting parameters that gave a 

98% and 90% (Table A2; Laake et al. 2009). 7

 The sensitivity test is provided because the generalized logistic model has been the basis for some previous 

management advice for this stock (e.g., Wade 2002). 

12
based on the assumption that the calf counts are relative indices of the total number of calves and 

are subject to both modeled and unmodeled sources of uncertainty; that is, 

obs m f 2

obs

,0 ,0

2 2

add-2

( nC n( ( ))) 2 2

add-2 + n 0.5 n( + )+0.5 i ii

i

qN N

i CV

i i

L CV

σ

σ

− + − = ∑ ∑ l l l l , (12)

where is the observed number of calves during year i, Ci

q is the constant of proportionality between the calf counts and model estimates of 

the number of calves, 

σi is the standard deviation of the logarithm of , and obs Ci

2 CVadd-2 is the additional variance associated with the calf counts. 

Prior Distributions 

The prior distributions (Table 3) are generally based on those used in recent International 

Whaling Commission (IWC) assessments of ENP gray whales. The prior distributions for S1+,

K1+, ,CVadd, CVadd-2, and were selected to be uniform over a sufficiently wide range so 

that there is effectively no posterior probability outside of that range. 

S

% lnq

The prior for the age-at-maturity differs from that used in previous assessments, Uniform[5,9], 

based on the review by Bradford et al. (unpublished manuscript)8

 who could find no basis for 

that range in the literature. They concluded that the most relevant data set for age-at-maturity 

was that of Rice and Wolman (1971), corrected by Rice (1990) for the underestimation of whale 

ages by one year in the original study, resulting in a median age of 9, and lower and upper 

bounds of 6 and 12. Bradford et al. (unpublished manuscript) note that the only observation of 

the age-at-first-reproduction (AFR) in ENP gray whales (a known whale observed with a calf for 

the first time) was 7 years for a whale first seen as a calf in a lagoon in Mexico. In the western 

Pacific population of gray whales, there have been observations of AFR of 7 and 11 years for the 

 

8

 Bradford, A.L., D.W. Weller, A.R. Lang, A.M. Burdin, and R.L. Brownell, Jr. Unpublished manuscript. 

Comparing age at first reproduction information from western gray whale to age at sexual maturity estimates of 

eastern gray whales. School of Aquatic and Fishery Sciences, Box 355020, University of Washington, Seattle. 

13
only two whales whose first calving has been documented to date (Bradford et al. unpublished 

manuscript). The prior for the maximum birth rate, fmax, was set equal to the prior selected for 

recent assessments (Wade 2002, Punt and Butterworth 2002). This prior implies a minimum 

possible calving interval between 1.67 and 3.33 years. 

The prior for the population size (in terms of animals aged 1 and older) in 1968 differs from that 

used in previous assessments. Rather than combining a uniform prior on 1968 population size 

with the abundance estimate for 1968 to create an informative prior for 1 P1968

+ as was the case in 

previous assessments, this assessment assumes a broad uniform prior for 1968 population size, 

and includes all of the estimates of abundance in the likelihood function. This is because the 

previous approach cannot be applied because all of the estimates of abundance are correlated 

(Laake et al. 2009). 

The prior for MSYR is bounded below by the minimum possible value and above by a value 

which is above those supported by the data. This prior is broader than those considered in 

previous assessments because those assessments assigned a prior to MSYR when this parameter 

is expressed in terms of removals of mature animals only. The prior for MSYL1+ has been 

assumed to be uniform from 0.4 to 0.8. The central value for this prior reflects the common 

assumption when conducting IWC assessments of whale stocks that maximum productivity 

occurs at about 60% of carrying capacity. The upper and lower bounds reflect values commonly 

used to bound MSYL for whale stocks (e.g., those used in the tests that evaluated the IWC’s 

catch limit algorithm). 

RESULTS 

The baseline assessment estimates that ENP gray whales increased substantially from 1930 until 

1999 when a substantial reduction in population size from close to carrying capacity (in terms of 

median parameter estimates) occurred (Fig. 1). This reduction was associated with an estimated 

decline in non-calf survival from 0.982 to 0.847 (posterior means, where 0.981 × 0.863 = 0.847) 

in each of 1999 and 2000. The population is estimated to have been increasing since 2000. The 

model fits the data well, although, as in previous IWC assessments, the analyses suggest that the 

coefficients of variation for the abundance estimates are underestimated (by 14% median 

14
estimate). The baseline assessment estimates that this stock is currently well above MSYL1+

(posterior mean for of 1.29) (Table 4). The probability that the stock is currently 

greater than MSYL1+ is 0.884. 

1 1 + +

1 1

2009 P MSYL /

The probability that the stock is currently above MSYL1+ is less for the baseline analysis and for 

the analysis in which the original abundance estimates are used (“Unrevised estimates” in Table 

4) than in some earlier assessments. The reasons for this are explored using the analyses in which 

no allowance is made for survival having dropped in 1999-2000 (“No Event” and “Unrevised, 

No event” in Table 4, see also Fig. 2) because the previous assessments did not explicitly 

account for the mortality event. This comparison suggests that allowing for the possibility of a 

catastrophic mortality event in 1999-2000 has reduced the ability to constrain the upper bound 

for carrying capacity because the lower 5% limit for 2009 P MSYL / + + is notably higher for the 

analyses which ignore this event (Table 4). Bayes factors comparing the analyses which include 

a 1999-2000 catastrophic mortality event and those which do not provide support for estimating 

a parameter for the 1999-2000 event; for example, in the baseline analysis the ln(Bayes factor) 

value is 3.00 compared to the “No event” model, which is interpreted as strong but not definitive 

support (Kass and Raftery 1995) for including the catastrophic mortality parameter in the model. 

The results are insensitive to changing the first year of the analysis (Table 4, Fig. 3). The key 

management-related results are also not sensitive to splitting the series in 1987-88, using the calf 

count estimates and using the “Lo” and “Hi” abundance estimates (Fig. 4). The results for the 

generalized logistic model are most comparable with the two “No event” analyses because no 

account is taken of a catastrophic mortality event in 1999-2000 when fitting the generalized 

logistic model (Appendix). While not entirely comparable, the qualitative conclusions from the 

generalized logistic model are identical to those from the age-structured model. 

Figure 5 shows the posterior distributions for the parameters for the baseline analysis. These 

posteriors show that the data update the priors for MSYR1+ and MSYL1+ to a substantial extent. 

The posterior MSYL1+ emphasizes higher values for MSYL1+, which is not unexpected given 

that the rate of increase for the ENP gray whales is assessed to have been high until just before 

this population (almost) reached its current carrying capacity. The posteriors for the age-atmaturity, maximum fecundity, and adult survival place greatest support on low, high, and high 

15
values, respectively. This is consistent with the fairly high growth rates and values for MSYR1+. 

The posterior for the survival multiplier is also updated substantially, with both high (close to 1) 

and low values (below 0.7) assigned low posterior probability. 

The maximum rate of increase, λmax , is well-defined in all of the analyses. The posterior mean 

estimates of this quantity range from 1.057 to 1.068 and are fairly precisely determined 

(Table 4). 

DISCUSSION 

The sensitivity tests were designed to examine the effect of various assumptions on the 

assessment results and to examine the effect of changes in the methods that have occurred, 

particularly in the abundance estimation. Overall, the results are consistent across most of the 

sensitivity tests with some exceptions. In particular, the baseline model fit to the unrevised 

abundance estimates had relatively different results from the other analyses. Leaving aside that 

analysis for the moment, the posterior medians for the parameters of interest were relatively 

consistent. Across all the other analyses, posterior means for K1+ ranged from 21,146 to 27,716, 

for the depletion level ranged from 0.76 to 0.96, and for the OSP ratio ranged from 1.22 to 1.54. 

Therefore, as in previous assessments, the ENP gray whale population is estimated to be above 

MSYL1+ and approaching or close to K. The estimates of depletion level and OSP ratio in Wade 

(2002) and in Punt and Butterworth (2002) are very similar to the results presented here, though 

the current estimates of K are lower. The results in Wade and Perryman (2002) and Brandon 

(2009), which were the only previous assessments to use abundance estimates from the 1997-98 

and subsequent surveys, gave higher and more precise estimates for depletion level and OSP 

ratio than estimated here; however, in common with previous assessments, those results are 

superseded by this new assessment because it uses the revised abundance estimates of Laake 

et al. (2009). 

The posterior means for the life history parameters were very consistent as well, with the 

posterior means for λmax ranging from 1.057 to 1.068, non-calf survival ranging from 0.972 to 

0.983, and calf survival ranging from 0.706 to 0.730. The parameter MSYL1+ was updated to 

strongly emphasize higher values in the baseline analysis. As discussed above, MSYL1+ is nearly 

16
equivalent to MNPL. There are theoretical arguments for why MNPL should be relatively higher 

in marine mammals than, say, marine fishes (Eberhardt and Siniff 1977, Fowler 1981, Taylor 

and DeMaster 1993), but, in general, there has not been empirical data of sufficient quantity and 

quality to estimate this parameter well in marine mammals (Goodman 1988, Ragen 1995, 

Gerrodette and DeMaster 1990). Empirical evidence that is available for large, long-lived 

mammals has shown convex nonlinear density-dependence in life history parameters such as 

age-specific birth and mortality rates (Fowler et al. 1980; Fowler 1987, 1994), which suggest 

MNPL > 0.5K. A relatively long time-series of abundance estimates has documented the 

recovery of harbor seal (Phoca vitulina) populations in Washington state, and Jeffries et al. 

(2003) estimated MNPL to be greater than 0.5K for these populations. In the ENP gray whale 

analysis here, values from 0.40 to 0.54 for MSYL1+ have low probability in the posterior 

distribution (Fig. 5, Table 4) which is consistent with the conclusions of Taylor and Gerrodette 

(1993) that MNPL was likely to be greater than 0.5K. Thus, the posterior distribution for 

MSYL1+ estimated here suggests that the ENP gray whale population experienced a decrease in 

population growth only when it was relatively close to K1+. 

The results did not vary much for a large number of the sensitivity tests, providing assurance that 

the assumptions made for the baseline analysis did not have a substantial influence on the results. 

Changing the initial year from which the model was projected had little effect on the results, 

which is similar to the results seen in Punt and Butterworth (2002) for initial years ranging from 

1930 to 1968, as used here. The results for the ‘Lo’ and ‘Hi’ series of abundance estimates are 

very similar to the baseline results, suggesting that that those assumptions made in calculating 

the abundance estimates do not have a strong influence on the results of the assessment. 

Additionally, splitting the abundance time series in 1987-88 did not have a substantial effect. 

This is particularly reassuring, because some changes in the field methods happened at that time, 

notably the use of a second independent observer during that and subsequent surveys (Laake et 

al. 2009). The generalized logistic model provided similar results to the ‘No-event’ analysis, with 

some small differences. This was similar to results seen in Wade (2002), where the quantitative 

values for some parameters were somewhat different for the generalized logistic, although the 

qualitative results are nearly identical in this case. That the quantitative results differ between the 

generalized logistic and the baseline analyses is to be expected because the analysis based on the 

17
generalized logistic did not account for the dynamics of sex- and age-structure, and also ignored 

time-lags in the dynamics. 

The baseline analysis fits the abundance data better than in the ‘No-event’ analysis because it 

includes the catastrophic mortality event in 1999-2000 (Figs. 1, 2). Furthermore, the Bayes 

factor confirms that there is strong, but not definitive, evidence supporting the use of a model 

including the catastrophic mortality. The model estimates that 15.3% of the non-calf population 

died in each of the years with catastrophic mortality, compared to about 2% in a normal year. In 

that 2-year period, the model estimates the population fell from being at 99% of K1+ in 1998 to 

83% in 1999 and 71% in 2000, before increasing back up to 91% by 2009. In contrast, the ‘Noevent’ analysis estimates the population had reached a level very close to K1+ by ~1995 and has 

remained there since, which clearly does not fit with the evidence regarding the biological effects 

on the population in 1999 and 2000. In the baseline analysis the estimate of the number of 

whales that died in 1999 and 2000 was 3,303 (90% interval 1,235-7,988) and 2,835 (90% 

interval 1,162-6,389), respectively, for a combined total for the two years of 6,138 (90% interval 

2,398-14,377). In comparison, the ‘No-event’ analysis estimates that the number of whales that 

died in 1999 was 587, and in 2000 it was 447. Comparing the number of strandings (from 

Mexico to Alaska) reported in Gulland et al. (2005) in the years around the mortality event to 

these estimates of total deaths from the baseline model indicates that only 3.9-13.0% of all ENP 

gray whales that die in a given year end up stranding and being reported. 

The baseline analysis is more conservative regarding status relative to K1+ than the ‘No-event’ 

analysis. On the other hand, it can be argued that the ‘No-event’ analysis provides a more 

accurate estimation of current average K1+. In other words, the baseline analysis does a better job 

of modeling the actual time-course of the population by including the mortality event, but it 

might provide an overestimate of the average recent K1+ by essentially considering high 

abundance estimates to be near K1+, but lower abundance estimates to be lower than K1+. The 

different interpretations hinge on whether K1+ is viewed as relatively fixed, with the 1999-2000 

mortality event considered to be unrelated to density-dependence (and therefore K1+), or whether 

K1+ is viewed as something that can vary from year to year, with the 1999-2000 years viewed as 

an event when K1+ itself was low. As populations increase in density, the impact of densityindependent factors on population dynamics probably become more pronounced (Durant et al. 

18
2005, Wilcox and Eldred 2003). The actual carrying capacity of the environment, in terms of 

prey available for the ENP gray whale population, is likely to vary from year to year to a greater 

or lesser extent due to oceanographic conditions affecting primarily benthic production. In terms 

of the model, the parameter K1+ that is being estimated is interpreted as the average carrying 

capacity in recent years. In the baseline analysis, the estimated K1+ is approximately (though not 

exactly) the average recent K1+ for the years before the mortality event, whereas in the ‘Noevent’ analysis, the estimate of average recent K1+ includes all the recent years, including 1999-

2000, and is lower. This is clear from the results, where the baseline estimate of K1+ is 25,808 

(90% interval 19,752- 49,639), whereas the ‘No-event’ estimate of K1+ is substantially lower, 

21,640 (90% interval 18,301-25,762). Further work that could be conducted would be to model 

K1+ with a distribution defined by two parameters, a mean and a variance, to explicitly allow K1+ 

to vary from one year to the next. If such a model included density-dependence in survival, the 

1999-2000 event could be modeled as a substantial decline in K1+ in those years rather than as a 

single event affecting mortality. Such an analysis would provide an estimate of average recent 

K1+ that included the low recent years, 1999-2000. 

The analysis using the original unrevised estimates is not a sensitivity test in the usual sense. 

Those results are provided simply to aid in interpretation of the results of the other analyses 

relative to past results using the unrevised estimates. For example, no previous analyses other 

than Brandon (2009) had used the 2006-07 abundance estimate, so this sensitivity test provides a 

comparison in which both analyses use that estimate. In the ‘No-event’ model, the analyses using 

the original and revised abundance estimates are nearly identical for estimates of depletion level

and OSP ratio. K was estimated to be higher in the analysis that used the original abundance 

estimates, but even though K is lower using the revised abundance estimates, overall the entire 

time-series is shifted such that the estimates of status relative to K are unchanged. 

In contrast, in the baseline model, the original abundance estimates give a fairly different result 

from any other analysis. From the discussion of how correction factors for the abundance 

estimates were calculated in different years in Laake et al. (2009), it is clear that the revised 

abundance estimates should be more accurate, and there were shifts of certain sequences of 

abundance estimates relative to one another that influence the results. For example, the three 

estimates from 1993-94 to 1997-98 are the three highest estimates in the original time-series, 

19
whereas the three estimates from 1984-85 to 1987-88 are the three highest estimates in the 

revised time-series. This has an effect on the baseline analysis results because the model is trying 

to fit the drop in abundance that occurred after the 1997-98 abundance estimate. That drop is 

substantially larger in the unrevised data set than it is in the revised data set, and therefore the 

results for the baseline model differ somewhat between the revised and unrevised data sets. 

The only previous assessment that modeled the 1999-2000 mortality event was that of Brandon 

(2009), whose point estimates of total natural mortality in those years ranged from 1,300 to 

5,200, depending upon a variety of assumptions he explored, lower than the 6,138 estimated here 

in the baseline model. The difference presumably arises because Brandon (2009) modeled 

mortality as a function of a sea-ice index for the Bering Sea, following the relationship found 

between calf production and sea-ice (Perryman et al. 2002). This constrains the dynamics of the 

mortality in Brandon (2009) to reflect the dynamics of the index to some extent. In contrast, the 

1999-2000 mortality was unconstrained in the baseline analysis here and is essentially estimated 

by what value fit the drop in abundance estimates best. Brandon (2009) noted this difficulty in 

his analysis, stating it was not possible in his analysis to fit the strandings data for the 1999-2000 

mortality event without allowing for some additional process error in the survival rates during 

those years. 

λmax is estimated to be 1.062 (90% interval 1.032-1.088) in the baseline analysis. This is similar 

to, but a little lower than, the estimate from Wade (2002) of 1.072 (90% interval 1.039-1.126) 

and the estimates from Wade and Perryman (2002). The posterior for λmax from the ‘No-event’ 

analysis is very similar to this, as is that from the ‘No-event’ analysis using the unrevised 

abundance estimates, indicating the lower estimates of λmax seen here are not due entirely to the 

revision of the abundance estimates but are instead partly due to the additional four abundance 

estimates used here (1997-98 to 2006-07) that were not available at the time the Wade (2002) 

analysis was conducted. To get an estimate of λmax of 1.062, the posterior distribution favored a 

low age-of-maturity, a high maximum fecundity, and a high adult survival. λmax appears to be 

well-defined, as the posterior medians from most of the sensitivity tests are very similar. It 

should be noted that these are theoretical estimates of the population growth rate at a very low 

population size, based upon the density-dependent assumptions of the population model; the 

20
ENP gray whale has not been observed to actually grow this rapidly because the population was 

estimated to be approaching K by the time its growth rate was monitored; consequently, the 

observed population growth rate was less than its theoretical maximum. 

The small and endangered western North Pacific population of gray whales has been estimated to 

have an annual population increase that is between 2.5% and 3.2% per year, but there is concern 

that this growth rate is low because of possible Allee effects and from ongoing human-caused 

mortality (Bradford et al. 2008). Best (1993) summarized the growth rates of eight severely 

depleted baleen whale populations (other than gray whales) and the values ranged from 3.1% to 

14.4%. Some of these estimates were not very precise, and Zerbini et al. (in press) has pointed 

out that the higher rates are implausible given life-history constraints for (at least) humpback 

whales (Megaptera novaeangliae). In more recent studies of other species, a number of estimates 

of trend have been similar to the estimates of λmax reported here. In a simulation study based on 

empirical estimates of life history parameters for humpback whales, Zerbini et al. (in press) 

estimated maximum rates of increase of 7.5%/year (95% CI = 5.1-9.8%) using one approach and 

8.7%/year (95% CI = 6.1-11.0%) using a second approach. Calambokidis et al. (2008) 

calculated point estimates of 4.9% to 6.7% for the North Pacific humpback whale population 

using data from a recently completed North Pacific study of humpback whale abundance. Zerbini 

et al. (2006) used line transect data from sequential surveys to estimate an annual rate of increase 

for humpback whales in shelf waters of the northern Gulf of Alaska from 1987 to 2003 of 6.6% 

per year (95% CI: 5.2-8.6%), and for fin whales of 4.8% (95% CI 4.1-5.4%). On the other hand, 

Mizroch et al. (2004) estimated a rate of increase for North Pacific humpback whales in Hawaii 

using mark-recapture methods for the years 1980-1996 of 10% per year, but the confidence 

limits were wide (95% C.I. of 3-16%). Other unpublished estimates are available spanning 

essentially a similar range as originally reported by Best (1993) (i.e., see IWC SC/61/REP6). In 

summary, the estimates of λmax reported here are similar to trend estimates seen in other species, 

but there are also lower and higher values that have been recorded. 

As noted above, one of the primary management goals of the U.S. Marine Mammal Protection 

Act is to maintain populations at OSP, which is considered to be a level above their MNPL. The 

baseline analysis described here estimates the ENP gray whale population to be at a level above 

their MNPL (i.e., MSYL1+) because the posterior mean for the OSP ratio is 1.29 (with a 

21
posterior median of 1.37 and a 90% interval of 0.68-1.51), meaning the population is estimated 

to be at 129% of MNPL. The baseline model estimates a probability of 0.884 that the population 

is above MNPL (i.e., the probability that the value of the OSP ratio is >1.0). These results are 

consistent across all our model runs. The baseline model also estimates the ENP gray whale 

population to be close to K, and this is also consistent across all the other model runs. 

22
ACKNOWLEDGMENTS 

We thank Jeff Laake, Phillip Clapham, Gary Duker, and James Lee for helpful reviews of drafts 

of this paper. We also thank the many observers who have collected the gray whale abundance 

data over the last 40 years. 

23

CITATIONS

Best, P.B. 1993. Increase rates in severely depleted stocks of baleen whales. ICES J. Mar. Sci. 

50: 169-186. 

Bradford, A., D.W. Weller, P.R. Wade, A.M. Burdin, and R.L. Brownell, Jr. 2008. Population 

abundance and growth rate of western gray whales Eschrichtius robustus. Endang. 

Species Res. 6:1-14. 

Brandon, J.R. 2009. Quantifying uncertainty and incorporating environmental stochasticity in 

stock assessments of marine mammals. Ph.D. diss., Univ. Washington, Seattle, WA. 

161 p. 

Brandon, J.R., J.M. Breiwick, A.E. Punt, and P.R. Wade. 2007. Constructing a coherent joint 

prior while respecting biological bounds: Application to marine mammal stock 

assessments. ICES J. Mar. Sci. 64:1085-1100. 

Breiwick, J.M., L.L. Eberhardt, and H.W. Braham. 1984. Population dynamics of western Arctic 

bowhead whales (Balaena mysticetus). Can. J. Fish. Aquat. Sci. 41:484-496. 

Brownell Jr., R.L., C.A.F. Makeyev, and T.K. Rowles. 2007. Stranding trends for eastern gray 

whales, Eschrichtius robustus:1975-2006. SC/59/BRG40 presented to the IWC Scientific 

Committee, May 2007, Anchorage, U.S.A. (unpublished). 

Buckland, S.T., and J.M. Breiwick. 2002. Estimated trends in abundance of eastern Pacific gray 

whales from shore counts, 1967/68 to 1995/96. J. Cetacean Res. Manage. 4:41-48. 

Buckland, S.T., J.M. Breiwick, K.L. Cattanach, and J.L. Laake. 1993. Estimated population size 

of the California gray whale. Mar. Mammal Sci. 9:235-249. 

Butterworth, D.S., J.L. Korrubel, and A.E. Punt. 2002. What is needed to make a simple densitydependent response population model consistent with data for eastern North Pacific gray 

whales? J. Cetacean Res. Manage. 4:63-76. 

25
Calambokidis, J., E.A. Falcone, T.J. Quinn, A.M. Burdin, P.J. Clapham, J.K.B. Ford, C.M. 

Gabriele, R. LeDuc, D. Mattila, L. Rojas-Bracho, J.M. Straley, B.L. Taylor, J. Urbán, D. 

Weller, B.H. Witteveen, M. Yamaguchi, A. Bendlin, D. Camacho, K. Flynn, A. Havron, 

J. Huggins, and N. Maloney. 2008. SPLASH: Structure of populations, levels of 

abundance and status of humpback whales in the North Pacific. Final report for Contract 

AB133F-03-RP-00078 prepared by Cascadia Research for the U.S. Deptartment of 

Commerce. 

Caughley, G. 1966. Mortality patterns in mammals. Ecology 47:906-918. 

Cooke, J.G. 1986. On the net recruitment rate of gray whales with reference to inter-specific 

comparisons. Rep. int. Whal. Commn. 36:363-366. 

Durant J.M., D.Ø. Hjermann, T. Anker-Nilssen, G. Beaugrand, A. Mysterud, N. Pettorelli, and 

N.C. Stenseth. 2005. Timing and abundance as key mechanisms affecting trophic 

interactions in variable environments. Ecol. Lett. 8:952-58. 

Eberhardt, L.L. and D.B. Siniff. 1977. Population dynamics and marine mammal management 

policies. J. Fish. Res. Board Can. 34:183-190. 

Fowler, C.W. 1981. Density dependence as related to life history strategy. Ecology 62:602-610. 

Fowler, C.W. 1987. A review of density dependence in populations of large mammals, p. 401-

441. In H.H. Genoways (editor), Current mammalogy, Volume 1. Plenum Publishing 

Corporation, New York, NY. 

Fowler, C.W. 1994. Further consideration of nonlinearity in density dependence among large 

mammals. Rep. Int. Whal. Commn. 44:385-391. 

Fowler, C.W., W.T. Bunderson, M.B. Cherry, R.J. Ryel, and B.B. Steele. 1980. Comparative 

population dynamics of large mammals: a search for management criteria. U.S. Dep. 

Commer., Nat. Tech. Infor. Serv. PB80-178627. 

Gerrodette, T., and D.P. DeMaster. 1990. Quantitative determination of optimum sustainable 

population level. Mar. Mammal Sci. 6: 1-16. 

26
Goodman, D. 1988. Dynamic response analysis I. Qualitative estimation of stock status relative 

to maximum net productivity level from observed dynamics. Mar. Mammal Sci. 4:183-

195. 

Gulland, F.M.D., H. Pérez-Cortés, M.J. Urbán, R.L. Rojas-Bracho, G. Ylitalo, J. Weir, S.A. 

Norman, M.M. Muto, D.J. Rugh, C. Kreuder, and T. Rowles. 2005. Eastern North Pacific 

gray whale (Eschrichtius robustus) unusual mortality event, 1999-2000. U.S. Dep. 

Commer., NOAA Tech. Memo. NMFS-AFSC-150, 33 p. 

Henderson, D.A. 1984. Nineteenth century gray whaling: Grounds, catches and kills, practices 

and depletion of the whale population, p. 159-186. In M.L. Jones, S.L. Swartz, and S. 

Leatherwood (editors), The Gray Whale. Academic Press, New York. 

Hobbs, R.C., D. Rugh, J. Waite, J.M. Breiwick, and D.P. DeMaster. 2004. Abundance of eastern 

North Pacific gray whales on the 1995/96 southbound migration. J. Cetacean Res. 

Manage. 6:115-120. 

Jeffries S.J., H.R. Huber, J. Calambokidis, and J. Laake. 2003. Trends and status of harbor seals 

in Washington State: 1978-1999. J. Wild. Manage. 67:208-219. 

Jones, M.L., S.L. Swartz, and S. Leatherwood (editors) 1984. The Gray Whale. Academic Press, 

New York. 

Kass, R.E., and A.E. Raftery. 1995. Bayes factors. J. Am. Statist. Assoc. 90:773-795. 

Krupnik, I.I. 1984. Gray whales and the aborigines of the Pacific Northwest: the history of 

aboriginal whaling. pages 103-120. In M.L. Jones, S.L. Swartz, and S. Leatherwood 

(editors). The Gray Whale. Academic Press, New York. 

Laake, J.L., D.J. Rugh, J.A. Lerczak, and S.T. Buckland. 1994. Preliminary estimates of 

population size of gray whales from the 1992/93 and 1993/94 shore-based surveys. 

Unpubl. Doc. SC/46/AS7, Int. Whal. Comm. Sci. Commn. [Paper available from 

http://www.iwcoffice.org/]. 

27
Laake, J., A. Punt, R. Hobbs, M. Ferguson, D. Rugh, and J. Breiwick. 2009. Re-analysis of gray 

whale southbound migration surveys, 1967-2006. U.S. Dep. Commer. NOAA Tech. 

Memo. NMFS-AFSC-203, 56 p. 

Lankester, K., and J.R. Beddington, 1986. An age structured population model applied to the 

gray whale (Eschrichtius robustus). Rep. Int. Whal. Commn 36: 353-358. 

Mizroch, S.A., L.M. Herman, J.M. Straley, D. A. Glockner-Ferrari, C. Jurasz, J. Darling, S. 

Cerchio, C.M. Gabriele, D.R. Salden, and O. von Ziegesar. 2004. Estimating the adult 

survival rate of central North Pacific humpback whales (Megaptera novaenagliae). J. 

Mammal. 85:963-972. 

Moore, S.E., R.J. Urbán, W.L. Perryman, F. Gulland, M.H. Perez-Cortes P.R. Wade, L. Rojas 

Bracho, and T. Rowles. 2001. Are gray whales hitting ‘K’ hard? Mar. Mammal Sci. 17: 

954-958. 

O’Leary, B.L. 1984. Aboriginal whaling from the Aleutian Islands to Washington State, p. 79- 

102. In M.L. Jones, S.L. Swartz, and S. Leatherwood (editors), The Gray Whale. 

Academic Press, New York. 

Perryman, W.L., and M.S. Lynn. 2002. Evaluation of nutritive condition and reproductive status 

of migrating gray whales (Eschrichtius robustus) based on analysis of photogrammetric 

data. J. Cetacean Res. Manage. 4:155-64. 

Perryman, W.L., M.A. Donahue, P.C. Perkins, and S.B. Reilly. 2002. Gray whale calf production 

1994-2000: Are observed fluctuations related to changes in seasonal ice cover? Mar. 

Mammal Sci. 18:121-44. 

Punt, A.E. 1999. A full description of the standard Baleen II model and some variants thereof. J. 

Cetacean Res. Manage. 1(Suppl.):267-276. 

Punt, A.E., and D.S. Butterworth. 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. 4:99-110. 

28
Punt, A.E., C. Allison, and G. Fay. 2004. An examination of assessment models for the eastern 

North Pacific gray whale based on inertial dynamics. J. Cetacean Res. Manage. 6:121-

132. 

Ragen, T. 1995. Maximum net productivity level estimation for the northern fur seal 

(Callorhinus ursinus) population of St. Paul Island, Alaska. Mar. Mammal Sci. 11:275 

300. 

Rice, D.W. 1990. Life history parameters of the gray whale: a review of published estimates. 

Paper SC/A90/G22 presented to the IWC Scientific Committee Special Meeting on the 

Assessment of Gray Whales, April 1990 (unpublished). [Paper available from 

http://www.iwcoffice.org/]. 

Rice, D.W., and A.A. Wolman. 1971. The Life History and Ecology of the Gray Whale 

(Eschrichtius robustus). Special Publication No. 3, The American Society of 

Mammalogists, 142 p. 

Reeves, R.R. 1984. Modern commercial pelagic whaling for gray whales, p. 187-202. In M.L. 

Jones, S.L. Swartz, and S. Leatherwood (editors), The Gray Whale. Academic Press, 

New York. 

Reilly, S.B. 1981. Population assessment and population dynamics of the California gray whale 

(Eschrichtius robustus). Ph.D. diss., Univ. Washington, Seattle, WA. 265 p. 

Rubin, D.B. 1988. Using the SIR algorithm to simulate posterior distributions, p. 395-402. In

J.M. Bernardo, M.H. DeGroot, D.V. Lindley and A.F.M. Smith (editors), Bayesian 

Statistics 3: Proceedings of the Third Valencia International Meeting, June 1-5, 1987. 

Clarendon Press, Oxford. 

Rugh, D., R.C. Hobbs, J.A. Lerczak, and J.M. Breiwick. 2005. Estimates of abundance of the 

eastern North Pacific stock of gray whales (Eschrictius robustus) 1997-2002. J. Cetacean 

Res. Manage. 7:1-12. 

29
Rugh, D., J. Breiwick, M. M. Muto, R. Hobbs, K. Shelden, C. D'Vincent, I. M. Laursen, S. Reif, 

S. Maher, and S. Nilson. 2008a. Report of the 2006-2007 census of the eastern north 

Pacific stock of gray whales. AFSC Processed Rep. 2008-03, 157 p. Alaska Fish. Sci. 

Cent., NOAA, Natl. Mar. Fish. Serv., 7600 Sand Point Way NE, Seattle WA 98115. 

Rugh, D.J., M.M. Muto, R.C. Hobbs, and J.A. Lerczak. 2008b. An assessment of shore-based 

counts of gray whales. Mar. Mammal Sci. 24:864-880. 

Shelden, K.E.W., D.J. Rugh, and A. Schulman-Janiger. 2004. Gray whales born north of 

Mexico: indicator of recovery or consequence of regime shift? Ecol. App.14:1789-1805. 

Taylor, B.L., and T. Gerrodette. 1993. The uses of statistical power in conservation biology: The 

vaquita and northern spotted owl. Conserv. Biol. 7:489-500. 

Taylor, B.L., and D.P. DeMaster. 1993. Implications of non-linear density dependence. Mar. 

Mammal Sci. 9:360-371. 

Wade, P.R. 1998. Calculating limits to the allowable human-caused mortality of cetaceans and 

pinnipeds. Mar. Mammal Sci. 14:1-37. 

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. 4:85-98. 

Wade, P.R., and W. Perryman. 2002. An assessment of the eastern gray whale population in 

2002. Paper SC/54/BRG7 presented to the IWC Scientific Committee, April 2002, 

Shimonoseki, Japan (unpublished). 16 p. 

Wilcox, C., and B. Eldred. 2003. The effect of density-dependent catastrophes on population 

persistence time. J. Appl. Ecol. 40: 859-71. 

Zerbini, A.N, P.J. Clapham, and P.R. Wade. In press. Plausible maximum rates of increase in 

humpback whales. Mar. Biol. 

30
Zerbini, A.N., J.M. Waite, and P.R. Wade. 2006. Abundance and distribution of fin, humpback 

and minke whales from the Kenai Fjords to the central Aleutian Islands, Alaska: Summer 

2001-2003. Deep-Sea Res. I 53:1772-1790. 

31
Table 1a. -- Historical (pre-1944) aboriginal catches from the Eastern North Pacific stock of gray 

 whales (C. Allison, IWC Secretariat, pers. commn.). 

Years Annual kill 

1600-1675 182 

1676-1750 183 

1751-1840 197.5 

1841-1846 193.5 

1847-1850 192.5 

1851-1860 187 

1861-1875 111 

1876-1880 110 

1881-1890 108 

1891-1900 62 

1901-1904 61 

1905-1915 57 

1916-1928 52 

1929-1930 47 

1931-1939 10 

1940-1943 20 

32
Table 1b. -- Commercial and recent aboriginal (post-1943) catches from the Eastern North 

 Pacific stock of gray whales (C. Allison, IWC Secretariat, pers. commn). 

Year Male Female Year Male Female Year Male Female Year Male Female

1846 23 45 1889 7 13 1932 3 7 1975 58 113 

1847 23 45 1890 7 13 1933 36 69 1976 69 96 

1848 23 45 1891 7 13 1934 64 92 1977 86 101 

1849 23 45 1892 7 13 1935 48 96 1978 94 90 

1850 23 45 1893 0 0 1936 74 114 1979 57 126 

1851 23 45 1894 0 0 1937 5 9 1980 53 129 

1852 23 45 1895 0 0 1938 18 36 1981 36 100 

1853 23 45 1896 0 0 1939 10 19 1982 56 112 

1854 23 45 1897 0 0 1940 39 66 1983 46 125 

1855 162 324 1898 0 0 1941 19 38 1984 59 110 

1856 162 324 1899 0 0 1942 34 67 1985 55 115 

1857 162 324 1900 0 0 1943 33 66 1986 46 125 

1858 162 324 1901 0 0 1944 0 0 1987 47 112 

1859 162 324 1902 0 0 1945 10 20 1988 43 108 

1860 162 324 1903 0 0 1946 7 15 1989 61 119 

1861 162 324 1904 0 0 1947 0 1 1990 67 95 

1862 162 324 1905 0 0 1948 6 13 1991 69 100 

1863 162 324 1906 0 0 1949 9 17 1992 0 0 

1864 162 324 1907 0 0 1950 4 7 1993 0 0 

1865 162 324 1908 0 0 1951 5 9 1994 21 23 

1866 79 159 1909 0 0 1952 15 29 1995 48 44 

1867 79 159 1910 0 1 1953 19 29 1996 18 25 

1868 79 159 1911 0 1 1954 13 26 1997 48 31 

1869 79 159 1912 0 0 1955 20 39 1998 64 61 

1870 79 159 1913 0 1 1956 41 81 1999 69 55 

1871 79 159 1914 6 13 1957 32 64 2000 63 52 

1872 79 159 1915 0 0 1958 49 99 2001 62 50 

1873 79 159 1916 0 0 1959 66 130 2002 80 51 

1874 79 159 1917 0 0 1960 52 104 2003 71 57 

1875 17 33 1918 0 0 1961 69 139 2004 43 68 

1876 17 33 1919 0 0 1962 53 98 2005 49 75 

1877 17 33 1920 1 1 1963 60 120 2006 57 77 

1878 17 33 1921 13 25 1964 81 138 2007 50 82 

1879 21 42 1922 6 4 1965 71 110 2008 64 66 

1880 17 34 1923 0 0 1966 100 120 

1881 17 33 1924 1 0 1967 151 223 

1882 17 33 1925 70 64 1968 92 109 

1883 19 39 1926 25 17 1969 93 121 

1884 23 45 1927 7 25 1970 70 81 

1885 21 41 1928 4 8 1971 62 91 

1886 17 33 1929 0 3 1972 66 116 

1887 7 13 1930 0 0 1973 98 80 

1888 7 13 1931 0 0 1974 94 90 

33
Table 2. -- Baseline estimates of 1+ abundance (and associated standard errors of the logs) from 

southbound surveys (Laake et al. 2009), the estimates of 1+ abundance used in 

previous assessments, two alternative series of abundance estimates, and estimates of 

calf numbers from northbound surveys (W. Perryman, SWFSC, pers. commn). 

1+ abundance Calf counts 

Laake et al. (2009) Unrevised estimates 

Year Estimate CV Estimate CV Year Estimate SE 

1967/68 13426 0.094 13776 0.078 1994 945 68.2 

1968/69 14548 0.080 12869 0.055 1995 619 67.2 

1969/70 14553 0.083 13431 0.056 1996 1146 70.7 

1970/71 12771 0.081 11416 0.052 1997 1431 82.0 

1971/72 11079 0.092 10406 0.059 1998 1388 92.0 

1972/73 17365 0.079 16098 0.052 1999 427 41.1 

1973/74 17375 0.082 15960 0.055 2000 279 34.8 

1974/75 15290 0.084 13812 0.056 2001 256 28.6 

1975/76 17564 0.086 15481 0.060 2002 842 78.6 

1976/77 18377 0.080 16317 0.050 2003 774 73.6 

1977/78 19538 0.088 17996 0.069 2004 1528 96.0 

1978/79 15384 0.080 13971 0.054 2005 945 86.9 

1979/80 19763 0.083 17447 0.056 2006 1020 103.3 

1984/85 23499 0.089 22862 0.060 2007 404 51.2 

1985/86 22921 0.081 21444 0.052 2008 553 53.0 

1987/88 26916 0.058 22250 0.050 2009 312 41.9 

1992/93 15762 0.067 18844 0.063 

1993/94 20103 0.055 24638 0.060 

1995/96 20944 0.061 24065 0.058 

1997/98 21135 0.068 29758 0.105 

2000/01 16369 0.061 19448 0.097 

2001/02 16033 0.069 18178 0.098 

2006/07 19126 0.071 20110 0.088 

34
Table 2. -- Continued. 

1 + abundance 

Lo series Hi series 

Year Estimate SE Estimate SE 

1967/68 12961 0.094 14298 0.095 

1968/69 14043 0.080 15493 0.081 

1969/70 14049 0.082 15498 0.084 

1970/71 12328 0.081 13601 0.082 

1971/72 10695 0.092 11799 0.093 

1972/73 16763 0.079 18493 0.080 

1973/74 16772 0.081 18503 0.083 

1974/75 14760 0.084 16283 0.085 

1975/76 16955 0.086 18705 0.087 

1976/77 17739 0.079 19570 0.081 

1977/78 18860 0.088 20806 0.089 

1978/79 14850 0.080 16383 0.081 

1979/80 19077 0.082 21046 0.083 

1984/85 22684 0.089 25025 0.090 

1985/86 22126 0.081 24409 0.082 

1987/88 25661 0.057 28692 0.056 

1992/93 14785 0.065 17879 0.072 

1993/94 19468 0.057 21124 0.056 

1995/96 20636 0.063 22314 0.063 

1997/98 20426 0.063 22378 0.065 

2000/01 16051 0.063 17145 0.062 

2001/02 15162 0.066 16883 0.067 

2006/07 18775 0.071 20129 0.072 

35
Table 3. -- The parameters and their assumed prior distributions. 

 Parameter Prior distribution 

Non-calf survival rate, S1+ U[0.950, 0.999] a

Age-at-maturity, am U[6,12]b

Maximum birth rate, fmax U[0.3, 0.6]a

Carrying capacity, K1+ U[10 000, 70 000] c

Population size in 1968, 1 P1968

+ U[5 000, 20 000]c

Maximum Sustainable Yield Level, MSYL1+ U[0.4, 0.8]a

Maximum Sustainable Yield Rate, MSYR1+ U[0, 0.1]a

Extra mortality, S

% U[0.2, 1.0]c

Additional variance, 1+ abundance estimates, CVadd U[0, 0.35]a,c

Additional variance, calf counts, CVadd-2 U[0.2, 0.8]c,d

Constant of proportionality, lnq U[-∞, ∞]

d,e

a. Equal to the prior distribution used in the most recent assessments (Punt et al. 2004). 

b. Bradford et al. (unpublished manuscript). 

c. Preliminary analyses provided no evidence of posterior support for values outside this range. 

d. Not used in the baseline analysis. 

e. The non-informative prior for a scale parameter (Butterworth and Punt 1996). 

36
Table 4. -- Posterior distributions for the key model outputs (posterior mean, posterior median [in square parenth

90% intervals) for the baseline analysis and the sensitivity tests. 

Baseline 

INIT t =1940 INIT t =1950 INIT t =1960 Unrevised 

Estimates 

No Event Gen Log

1 K + 25808 [22756] 

(19752 49639) 

 25450 [22506] (19537 49109) 

 24681 [22282] (19454 43887) 

 24396 [222047] (19212 43307) 

41046 [37889] (24214 66564) 

21640 [20683] (18301 25762) 

21146 [2(18229 2

MSYR1+ 0.046 [0.048] 

(0.022 0.064) 

0.047 [0.048] (0.022 0.067) 

0.049 [0.049] (0.024 0.068) 

0.048 [0.049] (0.024 0.070) 

0.035 [0.034] (0.025 0.050) 

0.052 [0.053] (0.026 0.068) 

 0.065 [0(0.034 0.

MSYL1+ 0.656 [0.669] 

(0.532 0.725) 

0.664 [0.677] (0.535 0.741 

0.677 [0.689] (0.541 0.762) 

0.691 [0.702] (0.545 0.786) 

0.611 [0.611] (0.506 0.706) 

0.672 [0.684] (0.577 0.730) 

 0.630 [0(0.441 0.

1 1

2009 P K/ + + 0.849 [0.919] 

(0.393 1.006) 

0.865 [0.933] (0.403 1.016) 

0.885 [0.946] (0.451 1.022) 

0.899 [0.959] (0.453 1.043) 

0.615 [0.598] (0.334 0.948) 

0.956 [0.977] (0.872 0.987) 

 0.964 [0(0.922 0.

1

2009 P MSYL / + 1+ 1.288 [1.366] 

(0.681 1.508) 

1.295 [1.362] (0.701 1.522) 

1.302 [1.355] (0.775 1.516) 

1.296 [1.343] (0.786 1.513) 

1.002 [0.992] (0.580 1.459) 

1.423 [1.424] (1.303 1.583) 

1.541 [1.(1.252 2.

λmax 1.062 [1.063] 

(1.032 1.088) 

1.063 [1.063] (1.033 1.094) 

1.063 [1.062] (1.035 1.094) 

1.062 [1.060] (1.035 1.092) 

1.054 [1.052] (1.036 1.081) 

1.068 [1.069] (1.038 1.091) 

 0.107 [0(0.042 0.

1 S + 0.981 [0.982] 

(0.957 0.997) 

0.981 [0.982] (0.957 0.997) 

0.980 [0.982] (0.957 0.997) 

0.980 [0.982] (0.957 0.997) 

0.978 [0.980] (0.956 0.997) 

0.983 [0.985] (0.960 0.998) 

N/A 

0 S 0.711 [0.732] 

(0.423 0.950) 

0.716 [0.734] (0.426 0.949) 

0.713 [0.727] (0.426 0.952) 

0.706 [0.720] (0.425 0.949) 

0.662 [0.666] (0.400 0.926) 

0.730 [0.747] (0.437 0.955) 

N/A 

S

% 0.863 [0.865] 

(0.772 0.951) 

0.866 [0.867] (0.778 0.951) 

0.868 [0.870] (0.779 0.960) 

0.870 [0.870] (0.781 0.961) 

0.814 [0.809] (0.725 0.915) 

1 N/A 

* r rather λmax . 

37
Baseline Split series Lo series Hi series Unrevised No event Calf co

No eve

1 K + 25808 [22756] (19752 49639) 274891 [22870] (19640 55929) 25826 [22030] (19129 52878) 26902 [24181] (21043 48118) 24162 [23044] (20946 29554) 21501 [2

(18439 2

MSYR1+ 0.046 [0.048] (0.022 0.064) 0.046 [0.047] (0.024 0.062) 0.046 [0.048] (0.021 0.064) 0.046 [0.048] (0.023 0.063) 0.047 [0.048] (0.032 0.061) 0.049 [0

(0.028 0

MSYL1+ 0.656 [0.669] (0.532 0.725) 0.648 [0.663] (0.529 0.721) 0.654 [0.670] (0.520 0.725) 0.654 [0.664] (0.537 0.725) 0.663 [0.673] (0.568 0.722) 0.668 [0

(0.577 0

1 1

2009 P K/ + + 0.849 [0.919] (0.393 1.006) 0.819 [0.908] (0.358 1.003) 0.837 [0.917] (0.355 1.008) 0.855 [0.913] (0.428 1.005) 0.957 [0.975] (0.881 0.985) 0.958 [0

(0.906 0

1

2009 P MSY / + 1+ 1.288 [1.366] L (0.681 1.508) 1.253 [1.357] (0.642 1.502) 1.270 [1.361] (0.632 1.504) 1.301 [1.366] (0.748 1.512) 1.446 [1.442] (1.344 1.608) 1.438 [1

(1.314 1

λmax 1.062 [1.063] (1.032 1.088) 1.063 [1.064] (1.037 1.088) 1.062 [1.063] (1.032 1.088) 1.063 [1.064] (1.034 1.089) 1.063 [1.062] (1.043 1.087) 1.065 [1

(1.037 1

1 S + 0.981 [0.982] (0.957 0.997) 0.981 [0.982] (0.957 0.997) 0.980 [0.982] (0.957 0.997) 0.981 [0.982] (0.957 0.998) 0.982 [0.984] (0.959 0.997) 0.980 [0

(0.958 0

0 S 0.711 [0.732] (0.423 0.950) 0.711 [0.729] (0.420 0.949) 0.710 [0.728] (0.420 0.949) 0.708 [0.725] (0.425 0.949) 0.705 [0.716] (0.420 0.950) 0.720 [0

(0.426 0

S

% 0.863 [0.865] 

(0.772 0.951) 

0.860 [0.862] (0.763 0.958) 

0.862 [0.862] (0.775 0.950) 

0.855 [0.857] (0.772 0.939) 

1 1 

Table 4. -- Continued. 

38
Figure 1. -- Posterior distributions (medians and 90% credibility intervals) for the timetrajectories of 1+ population size (left panels) and 1+ population size expressed 

relative to (current) carrying capacity (right panel) for the baseline analysis. 

Figure 2. -- Posterior distributions (medians and 90% credibility intervals) for the timetrajectories of 1+ population size (left panels) and 1+ population size expressed 

relative to (current) carrying capacity (right panel) for the “No event” analysis. 

39
Figure 3. -- Posterior median time-trajectories of 1+ population size (left panel) and 1+ 

population size expressed relative to (current) carrying capacity (right panel) for the 

baseline analysis and the sensitivity tests which vary the value for tINIT.

Figure 4. -- Posterior median time-trajectories of 1+ population size (left panel) and 1+ 

population size expressed relative to (current) carrying capacity (right panel) for the 

baseline analysis and a subset of the sensitivity tests.

40
Figure 5. -- Posterior distributions for the parameters of the baseline analysis. 

41

y

APPENDIX 

Analyses Based on the Generalized Logistic Equation 

The dynamics of the population are assumed to be governed by the generalized logistic model: 

1 (1 ( / ) )z N N rN N K C y yy y + =+ − − , (App.1) 

where is the number of animals at the start of year y, Ny

r is the intrinsic rate of growth, 

z is the extent of compensation, 

K is the (current) carrying capacity, and 

C y is the catch (in numbers) during year y. 

The parameters of Equation 1 are r, z, K, and while the data available to estimate these 

parameters are the estimates of abundance and their associated variance-covariance matrix. The 

analysis based on the same likelihood function (Eq. 11 of the main text) and priors as the 

baseline analysis using the age- and sex-structured model. 

N1968

43

RECENT TECHNICAL MEMORANDUMS
 

Copies of this and other NOAA Technical Memorandums are available from the 

National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22167 

(web site: www.ntis.gov). Paper and electronic (.pdf) copies vary in price. 

AFSC206	 ALLEN, B. M., and R. P. ANGLISS. 2010. Alaska marine mammal stock assessments, 2009, 276 p. 

NTIS number pending. 

205	 CAHALAN, J., J. MONDRAGON, and J. GASPER. 2010. Catch sampling and estimation in the 

Federal groundfish fisheries off Alaska, 42 p. NTIS number pending. 

204	 LAUTH, R. R. 2010. Results of the 2009 eastern Bering Sea continental shelf bottom trawl survey of 

groundfish and invertebrate resources, 228 p. NTIS number pending. 

203	 LAAKE, J., A. PUNT, R. HOBBS, M. FERGUSON, D. RUGH, and J. BREIWICK. 2009. Re-analysis of 

gray whale southbound migration surveys 1967-2006, 55 p. NTIS No. PB2010-103939. 

202	 FOWLER, C. W., and L. HOBBS. 2009. Are we asking the right questions in science and 

management?, 59 p. NTIS number pending. 

201	 CHILTON, E. A., C. E. ARMISTEAD, and R. J. FOY. 2009. The 2009 eastern Bering Sea continental 

shelf bottom trawl survey: Results for commercial crab species, 101 p. NTIS No. PB2010-103938. 

200	 BOVENG, P. L., J. L. BENGTSON, T. W. BUCKLEY, M. F. CAMERON, S. P. DAHLE, B. P. KELLY, 

B. A. MEGREY, J. E. OVERLAND, and N. J. WILLIAMSON. 2009. Status review of the spotted seal 

(Phoca largha), 153 p. NTIS No. PB2010-101436. 

199	 CONNERS, M. E., J. CAHALAN, S. GAICHAS, W. A. KARP, T. LOOMIS, and J. WATSON. 2009. 

Sampling for estimation of catch composition in Bering Sea trawl fisheries, 77 p. NTIS No. PB2010­

100895. 

198	 BARBEAUX, S. J., and D. FRASER. 2009. Aleutian Islands cooperative acoustic survey study for 

2006, 90 p. NTIS No. PB2010-100894. 

197	 HOFF, G. R., and L. L. BRITT. 2009. Results of the 2008 eastern Bering Sea upper continental slope 

survey of groundfish and invertebrate resources, 294 p. NTIS No. PB2010-100893. 

196	 BUCKLEY, T. W., A. GREIG, and J. L. BOLDT. 2009. Describing summer pelagic habitat over the 

continental shelf in the eastern Bering Sea, 1982-2006, 49 p. NTIS No. PB2009-115399. 

195	 LAUTH, R. R., and E. ACUNA. 2009. Results of the 2008 eastern Bering Sea continental shelf bottom 

trawl survey of groundfish and invertebrate resources, 219 p. NTIS No. PB2009-113760. 

194	 HONKALEHTO, T., D. JONES, A. MCCARTHY, D. MCKELVEY, M.GUTTORMSEN, 

K. WILLIAMS, and N. WILLIAMSON. 2009. Results of the echo integration-trawl survey of walleye 

pollock (Theragra chalcogramma) on the U.S. and Russian Bering Sea shelf in June and July 2008, 

56 p. NTIS No. PB2009-110982. 

193	 ANGLISS, R. P., and B. M. ALLEN. 2009. Alaska marine mammal stock assessments, 2008, 

258 p. NTIS No. PB2009-109548. 

192	 FOWLER, C. W.,T. E. JEWELL and M. V. LEE. 2009. Harvesting young-of-the-year from large 

mammal populations: An application of systemic management, 65 p. NTIS No. PB2009105146. 

191 	 BOVENG, P. L., J. L. BENGTSON, T. W. BUCKLEY, M. F. CAMERON, S. P. DAHLE. A. MEGREY, 

J. E. OVERLAND, and N. J. WILLIAMSON. 2008. Status review of the ribbon seal (Histriophoca 

fasciata), 115 p. NTIS No. PB2006-104582.