USGS Science Strategy to Support U.S. Fish and Wildlife Service Polar Bear Listing Decision

Polar Bears in the Southern Beaufort Sea II: Demography and Population Growth in Relation to Sea Ice Conditions

By Christine M. Hunter1, Hal Caswell2, Michael C. Runge3, Eric V. Regehr4, Steven C. Amstrup4, and Ian Stirling5

Administrative Report

U.S. Department of the Interior U.S. Geological Survey

ii

U.S. Department of the Interior DIRK KEMPTHORNE, Secretary

U.S. Geological Survey Mark D. Myers, Director

U.S. Geological Survey, Reston, Virginia: 2007

U.S. Geological Survey Administrative Reports are considered to be unpublished and may not be cited or quoted except in follow-up administrative reports to the same federal agency or unless the agency releases the report to the public.

Any use of trade, product, or firm names is for descriptive purposes only and does not imply endorsement by the U.S. Government.

Author affiliations: 1 University of Alaska, Fairbanks 2 Woods Hole Oceanographic Institution, Massachusetts 3 U.S. Geological Survey, Patuxent Wildlife Research Center, Maryland 4 U.S. Geological Survey, Alaska Science Center, Anchorage 5 Canadian Wildlife Service, Edmonton, Alberta

Contents

Abstract 1

1 Introduction 2

2 Demographic model structure 22.1 The life cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 The demographic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Parameter estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.3.1 Fertility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Climate model projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5 Demographic analysis sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Deterministic demography 63.1 Deterministic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.1.1 Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.2 LTRE analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.3 Population projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.4 Correction for harvest mortality . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Results of deterministic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.1 Growth rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2.2 Deterministic population projections . . . . . . . . . . . . . . . . . . . . . . . 93.2.3 Effects of harvest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Stochastic demography 94.1 Stochastic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4.1.1 Stationary stochastic environments . . . . . . . . . . . . . . . . . . . . . . . . 104.1.2 Sensitivity analysis of the stochastic growth rate . . . . . . . . . . . . . . . . 114.1.3 Climate model scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Results of stochastic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.2.1 Stationary stochastic environments . . . . . . . . . . . . . . . . . . . . . . . . 124.2.2 Climate model projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Discussion 145.1 Interpretation of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.2 Coupling GCMs and demographic models . . . . . . . . . . . . . . . . . . . . . . . . 155.3 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

5.3.1 Estimation and inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165.3.2 Model structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.4 Importance of this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.5 Prospects for further demographic modeling . . . . . . . . . . . . . . . . . . . . . . . 18

6 Summary and Conclusions 19

7 Acknowledgments 20

8 Literature cited 20

iii

9 Tables 23

10 Figures 2510.1 Deterministic model figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2610.2 Stochastic model figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

A Appendix figures 39

List of Figures

1 Polar bear life cycle graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252 Bootstrap distribution of growth rates, full model set . . . . . . . . . . . . . . . . . . 263 Stable stage distribution and elasticity of λ . . . . . . . . . . . . . . . . . . . . . . . 274 LTRE analysis for full and non-covariate model sets . . . . . . . . . . . . . . . . . . 285 Population size at t = 45, full, non-covariate, and time-invariant models. . . . . . . . 296 Population growth rate as function of ice . . . . . . . . . . . . . . . . . . . . . . . . 307 Effect of harvest on population growth . . . . . . . . . . . . . . . . . . . . . . . . . . 318 Stochastic growth rate and bad ice years . . . . . . . . . . . . . . . . . . . . . . . . . 329 Frequency of bad ice conditions 1979–2006 . . . . . . . . . . . . . . . . . . . . . . . . 3310 Elasticity of the stochastic growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . 3411 Stochastic projections: distribution of population size . . . . . . . . . . . . . . . . . 3512 Climate model forecasts of ice conditions . . . . . . . . . . . . . . . . . . . . . . . . . 3613 Population projections from climate models . . . . . . . . . . . . . . . . . . . . . . . 3714 Quasi-extinction probability from climate models . . . . . . . . . . . . . . . . . . . . 38A-1 Bootstrap distribution of growth rates, full, non-covariate, and time-invariant model

set comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39A-2 Sensitivity and elasticity of λ for full, non-covariate, and time-invariant models . . . 40A-3 Distribution of population at t = 45: full, non-covariate, and time-invariant models . 41A-4 Distribution of population size t = 45, 75, 100: full model set . . . . . . . . . . . . . . 42A-5 Elasticity of the stochastic growth rate . . . . . . . . . . . . . . . . . . . . . . . . . . 43A-6 Climate model forecasts of frequency of bad ice years . . . . . . . . . . . . . . . . . . 44A-7 Climate model projections: full model set, vital rate samples . . . . . . . . . . . . . 45A-8 Climate model projections: non-covariate model set, vital rate samples . . . . . . . . 46

List of Tables

1 Deterministic population growth rate λ, with 90% confidence intervals, standarderror, and proportion of bootstrap samples < 1, for the time-invariant, full, andnon-covariate model sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2 Ten IPCC AR-4 GCMs used to forecast ice covariates. IPCC model name, countryof origin, approximate grid resolution (degrees), and the number of runs used fordemographic projections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

iv

Abbreviations, Acronyms, and Symbols

Abbreviations, Acronyms, and Symbols

Meaning

AIC Akaike’s Information Criterion

AR-4 IPCC Fourth Assessment Report

cdf Cumulative distribution function

COY Cub-of-the-year

GCM General circulation models

IPCC Intergovernmental Panel on Climate Change

IUCN International Union for Conservation of Nature and Natural Resources

LTRE Life table response experiment

SB Southern Beaufort Sea

SRES Special Report on Emission Scenarios

USFWS U.S. Fish and Wildlife Service

USGS U.S. Geological Survey

Polar bears in the southern Beaufort Sea II: Demography and

population growth in relation to sea ice conditions

by Christine M. Hunter, Hal Caswell, Michael C. Runge, Eric V. Regehr,Steve C. Amstrup, and Ian Stirling

Abstract

This is a demographic analysis of the southern Beaufort (SB) polar bear population. Theanalysis uses a female-dominant stage-classified matrix population model in which individualsare classified by age and breeding status. Parameters were estimated from capture-recapturedata collected between 2001 and 2006. We focused on measures of long-term population growthrate and on projections of population size over the next 100 years. We obtained these resultsfrom both deterministic and stochastic demographic models. Demographic results were relatedto a measure of sea ice condition, ice(t), defined as the number of ice-free days, in year t, inthe region of preferred polar bear habitat. Larger values of ice correspond to lower availabilityof sea ice and longer ice-free periods. Uncertainty in results was quantified using a parametricbootstrap approach that includes both sampling uncertainty and model selection uncertainty.

Deterministic models yielded estimates of population growth rate λ, under low ice conditionsin 2001–2003, ranging from 1.02 to 1.08. Under high ice conditions in 2004–2005, estimates ofλ ranged from 0.77 to 0.90. The overall growth rate estimated from a time-invariant model wasabout 0.997; i.e., a 0.3% decline per year. Population growth rate was most elastic to changesin adult female survival, and an LTRE analysis showed that the decline in λ relative to 2001conditions was primarily due to reduction in adult female survival, with secondary contributionsfrom reduced breeding probability.

Based on demographic responses, we classified environmental conditions into good (2001–2003) and bad (2004–2005) years, and used this classification to construct stochastic models. Inthose models, good and bad years occur independently with specified probabilities. We foundthat the stochastic growth rate declines with an increase in the frequency of bad years. Theobserved frequency of bad years since 1979 would imply a stochastic growth rate of about -1%per year.

Deterministic population projections over the next century predict serious declines unlessconditions typical of 2001–2003 were somehow to be maintained. Stochastic projections predicta high probability of serious declines unless the frequency of bad ice years is less than its recentaverage. To explore future trends in sea ice, we used the output of 10 selected general circulationmodels (GCMs), forced with “business as usual” greenhouse gas emissions, to predict values ofice(t) until the end of the century. We coupled these to the stochastic demographic model toproject population trends under scenarios of future climate change. All GCM models predicta crash in the population within the next century, possibly preceded by a transient populationincrease.

The parameter estimates on which the demographic models are based have high levels ofuncertainty associated with them, but the agreement of results from different statistical modelsets, deterministic and stochastic models, and models with and without climate forcing, speaksfor the robustness of the conclusions.

1

1 Introduction

The US Fish and Wildlife Service (USFWS)proposed listing polar bears (Ursus maritimus)as a threatened species under the EndangeredSpecies Act in January 2007 (US Fish and Wild-life Service 2007). To help inform their final de-cision, they requested that the USGS conductadditional analyses of existing data for polarbears and their sea ice habitat. Part of thiseffort involved demographic analyses to betterunderstand polar bear population status in thesouthern Beaufort Sea (SB), one of 19 IUCN(International Union for the Conservation ofNature) populations of polar bears worldwide(Aars et al. 2006).

This report addresses demography and pro-jections of population growth for the SB polarbear population in relation to current, recent,and projected environmental conditions in thesouthern Beaufort Sea. This is part II of atwo-part report. Part I (Regehr et al. 2007)includes the details of estimation of the vitalrates (survival and breeding probabilities) anddescribes the biological context of the SB popu-lation in relation to the world-wide distributionof polar bears; we will not repeat that informa-tion here.

Our goals are to describe the demographicstatus of polar bears in the SB under currentand recent conditions, to describe the effect ofsea ice conditions on demography, and to projectthe future dynamics of the population under arange of future sea ice scenarios. To do this,we develop and analyze both deterministic andstochastic stage-classified matrix population mod-els. Such models are often used as a mathemat-ical framework for studying populations of con-servation concern (e.g., Fujiwara and Caswell2001, Runge et al. 2004, Hunter and Caswell2005a, Smith et al. 2005, Gervais et al. 2006)because they are flexible enough to be tailoredto the biology of the species and the availabil-ity of data and have a more completely devel-oped mathematical theory than any other typeof population model.

We report measures of potential long-termpopulation growth rate, sensitivity and elastic-

ity of the growth rate to changes in the vitalrates, and other demographic statistics that canhelp provide an understanding of the popula-tion response to environmental conditions. Wealso report the results of short-term populationprojections, focusing on time scales of 45, 75,and 100 years into the future. Our results arebased on estimates of the vital rates obtainedunder a variety of statistical models, and hencehave sampling uncertainty and model selectionuncertainty associated with them. One of ourgoals is to incorporate those sources of uncer-tainty. This uncertainty is essential to evaluat-ing the risks associated with possible manage-ment actions.

2 Demographic modelstructure

We begin by describing the structure of themodel, the estimation of parameters, and therelationship with sea ice conditions.

2.1 The life cycle

Typical of large mammals, polar bears are long-lived, have delayed maturity and low fecundity.Female polar bears in the SB are first avail-able to mate in April-June of their 5th year.1

Implantation of the conceptus is delayed untilSeptember-October when pregnant females en-ter maternal dens (Derocher and Stirling 1992,Amstrup and Gardner 1994). Females give birthin December-January and nurse until their cubsof the year (COYs) are large enough to leave theden the following March–April (Stirling 1988,Amstrup 2003). Cubs that survive remain withtheir mothers until they are weaned as 2-year-olds.

We define the overall model structure witha life cycle graph (Figure 1). The model dis-tinguishes 6 female and 4 male stages based onsex, age and reproductive status. Stage 1 and7 are newly independent 2-year old females andmales, respectively. Stages 2 and 3 are females

1In this regard, the SB population differs from thosein other regions, which typically mate in their 4th year.

2

aged 3 and 4 years, respectively. Stage 4 is fe-males without dependent young and availableto breed (either single or accompanied by 2-year olds). Mothers and their cubs are not in-dependent during the two-plus years of parentalcare, so we modeled the reproductive stages asmother-cub units. Stage 5 is females accom-panied by one or more COYs and stage 6 isfemales accompanied by one or more yearlings.Males are classified by age; stages 7, 8 and 9 areaged 2, 3, and 4 years, respectively, and stage10 is males 5 years of age or older. This lifecycle graph differs from that used in Regehr etal. (2007) only in that it includes reproductionin addition to the transitions of existing indi-viduals.

The projection interval is from April to April,the time of sampling and when females withCOYs typically leave their maternal dens. Tran-sitions of individuals or mother-cub units amongstages, and creation of new individuals, are in-dicated by arcs in the life cycle graph. Thetransition probabilities of existing individualsare defined in terms of conditional probabilities.The probability that a bear in stage i survivesfrom time t to t + 1 is σi(t). The conditionalprobability that a female in stage i breeds be-tween time t and t + 1, given survival, is βi(t).The probability that at least one member of aCOY litter survives from year t to year t + 1,given survival of the mother, is σL0. Thus, forexample, the transition from stage 4 to stage 5,which requires a female to survive and breed,occurs with probability σ4(t)β4(t). The transi-tion from stage 4 to stage 4, which occurs whena female survives but does not breed, occurswith probability σ4(t)(1− β4(t)).

Fertility in this model appears on the arcfrom stage 6 to stages 1 (female offspring) and7 (male offspring). It depends on three quan-tities: the survival of the mother (σ6), the sur-vival of at least one member of a yearling litter(σL1), and the average number of 2-year olds ina successful litter (f). The calculation of σL1

and f is described in Section 2.3.1.

2.2 The demographic model

The life cycle graph corresponds to a stage-structured matrix population model

n(t + 1) = Atn(t) (1)

where n(t) is a vector giving the number of in-dividuals in each stage, ni(t), and At is a popu-lation projection matrix that projects the pop-ulation from t to t + 1. The entry aij of A isthe coefficient on the arc from stage j to stagei in the life cycle graph. The projection matrixis given by equation (2).

The upper left 6 × 6 corner of the matrixdescribes the production of females by females.The lower right 4× 4 corner describes the sur-vival of males. The lower left corner describesthe production of males by females.

Polar bears are polygynous and only a por-tion of females are available to mate in a givenyear. Therefore, it is unlikely that female fer-tility is limited by availability of males or fluc-tuations in the sex ratio. Thus, we expect thegrowth rate of the population to be determinedby the female portion of the life cycle and baseall subsequent analyses on the female matrix.

2.3 Parameter estimates

Parameters in the population projection ma-trix were estimated from a multi-state mark-recapture analysis based on the life cycle graphin Figure 1 (Fujiwara and Caswell 2002, Caswelland Fujiwara 2004). The data consisted of cap-ture histories for 627 polar bears captured inthe SB study area between 2001 and 2006 (be-tween Wainwright Alaska and Paulatuk, North-west Territories, Canada; see Amstrup et al.1986 and Regehr et al. 2007). We make a fewpertinent comments here but details of the pa-rameter estimation can be found in Part I ofthis report (Regehr et al. 2007).

We used maximum likelihood methods to fita candidate set of statistical models, developedfrom biological considerations. The models dif-fered in the extent of age- and stage-specificityof the vital rates and in the dependence of sur-vival and breeding probabilities on time and/or

3

A =

0 0 0 0 0 σ6σL1f2 0 0 0 0

σ1 0 0 0 0 0 0 0 0 00 σ2 0 0 0 0 0 0 0 00 0 σ3 σ4(1− β4) σ5(1− σL0)(1− β5) σ6 0 0 0 00 0 0 σ4β4 σ5(1− σL0)β5 0 0 0 0 00 0 0 0 σ5σL0 0 0 0 0 00 0 0 0 0 σ6σL1f

2 0 0 0 00 0 0 0 0 0 σ7 0 0 00 0 0 0 0 0 0 σ8 0 00 0 0 0 0 0 0 0 σ9 σ10

(2)

on a covariate describing sea ice conditions (seeSection 2.4). Models were evaluated using Akaike’sInformation Criterion (AIC; Burnham and An-derson 2002). AIC weights were used to pro-duce model-averaged estimates of parametersfor three sets of models. We refer to these asthe full model set, the non-covariate model set,and the time-invariant model set.

The full model set contained all models with∆AIC ≤ 4. Models in this set described breed-ing probabilities (β4, β5) as time-dependent,survival probabilities (σ1–σ6) primarily as co-variate dependent, and COY litter survival (σL0)as either time-invariant, covariate-dependent,or time-dependent.

Covariate dependence was described by alinear function on the logit scale. Because thisdata set contained only 5 years of recaptures,there was a high degree of uncertainty in theslope and intercept parameters for covariate-dependent survival. Moreover, the results ofcovariate-dependent models are influenced bythe specific functional form of the covariate de-pendence. To evaluate variation in the vitalrates without relying on a particular functionalform of covariate-dependence, we used the non-covariate model set, which contained all mod-els with ∆AIC ≤ 4 that did not include thecovariate (with ∆AIC measured relative to thebest model within this set). Comparison of re-sults between the full model set and the non-covariate model set provides a check on robust-ness of the conclusions from the parametric formof the covariate dependence.

Finally, the time-invariant model set included

all models with time-invariant parameters. Thisset of models represents the best single estimatefor the vital rates over the period of the study,although we note at the outset that the datado not support the claim that the rates were,in fact, constant.

Uncertainty in the demographic results arisesfrom sampling variability and model-selectionvariability of the vital rate estimates. To evalu-ate this uncertainty, we used a parametric boot-strap procedure that is described in detail inRegehr et al. (2007). Briefly, the method gen-erates B bootstrap samples. Those samplesare allocated at random among the models inthe model set with probabilities given by theAkaike weights. Within each model, samples ofthe mathematical parameters θ are generatedfrom a multivariate normal distribution usingthe estimated covariance matrix for that model.From each bootstrap set of mathematical pa-rameters, the vital rates π are calculated, whereπ contains the stage-specific survival probabili-ties, breeding probabilities, and fertility. Theseare used to construct projection matrices andcalculate demographic statistics. Unless other-wise noted, we used B = 10, 000. We presentmeasures of variance either by plotting the dis-tribution of the resulting statistic, or as 90%confidence intervals obtained as the 5th and95th percentiles of the bootstrap distributions(Efron and Tibshirani 1993).

2.3.1 Fertility

The fertility parameters, σL1 and f , could notbe directly estimated by the mark-recapture anal-

4

ysis, but instead were calculated from the othervital rates and from an independent value forthe probability distribution of litter size. Thecalculation takes account of both survival of in-dividual cubs and the chance of loss of entirelitters.

The size distribution of a litter (of COYs)in stage 5 was set to

c =(

0.276 0.724)

(3)

where ci is the probability of a litter of size i.These values were obtained from capture datacollected in 2001–2006 in the SB managementunit. We included only one- and two-cub littersbecause triplets are rare for polar bears inhab-iting the Arctic basin (Amstrup 2003). We didnot investigate variation in this parameter.

The probability of loss of a COY litter is

1− σL0 = (1− s)c1 + (1− s)2c2 (4)

where s is a measure of survival of a COY.That is, a COY litter is lost if it consists ofone cub which dies, or two cubs both of whichdie. Given c and an estimate of σL0, we solve(4) for s.

Conditional on not losing the COY litter,the size distribution of a yearling (stage 6) litteris

y1 = [c1s + 2c2s(1− s)] /σL0 (5)y2 = c2s

2/σL0 (6)

The probability of loss of the yearling litter be-tween stage 6 and stage 1 is then

1− σL1 = (1− σ1)y1 + (1− σ1)2y2 (7)

where we assume that the survival of a yearlingcub is the same as that of a 2-year old bear instage 1. Conditional on not losing the yearlinglitter, the size distribution of the litter arrivingat stage 1 is then

z1 = [y1σ1 + 2y2σ1(1− σ1)] /σL1 (8)z2 =

[y2σ

21

]/σL1 (9)

The expected number of new stage 1 bears, con-ditional on not losing the yearling litter, is then

f = z1 + 2z2. (10)

Multiplying this by the survival of the motherand the probability of not losing the litter, anddividing by 2 to account for the sex ratio, givesthe fertility σ6σL1f/2.

2.4 Climate model projections

The SB region is covered with annual sea icefrom approximately October to June, and par-tially or completely ice free from July to Septem-ber, when sea ice retreats northward into theArctic basin (Comiso 2006, Richter-Menge etal. 2006). To represent ice conditions and habi-tat availability for polar bears we derived anindex of sea ice conditions. This index, ice(t),was the number of days during calendar yeart that were ice-free in the region of preferredhabitat for polar bears. Preferred habitat wasdefined as waters within the SB managementunit east of Barrow, AK, with an ocean depthless than 300 m because polar bears in the SBselect strongly for sea ice over the shallow wa-ters of the continental shelf (Durner et al. 2004,2007). A day was considered ice free if the meanice concentration in preferred habitat area wasless than 50%. Mean ice concentration wasthe arithmetic mean of daily ice concentrationvalues for the 139 grid cells (25 × 25 km) inthe preferred habitat area based on passive mi-crowave satellite imagery (from the NationalSnow and Ice Data Center, Boulder, Colorado,USA; ftp://sidads.colorado.edu/pub/).Regehr et al. (2007) found that the ice covari-ate was highly negatively correlated with theavailability of polar bear habitat as describedby resource selection functions (Durner et al.2007)

Larger values of ice correspond to longer ice-free periods and reduced amounts of ice avail-able to polar bears. So we expect the effect onsurvival, breeding, and population growth to benegative. The multistate mark-recapture anal-ysis (Regehr et al. 2007) found evidence of suchnegative effects, using models in which the rela-tionship between survival probabilities and icewas described using a logistic function; i.e.,

logit[σi(t)

]= ai + bi ice(t). (11)

5

2.5 Demographic analysis sequence

A matrix population model (1) can assume sev-eral forms, depending on whether the entriesin A are time-invariant, time-dependent, or de-pendent on an environmental covariate. In whatfollows, we analyze deterministic models, whichassume that environmental conditions remainconstant over time, and stochastic models, whichallow for variation in environmental conditions.By analyzing a sequence of models we are ableto obtain more robust conclusions than wouldbe possible from examining any single model.To orient the reader we give an outline here ofthe sequence of models and analyses. The mod-els we focus on are

1. Deterministic models

(a) time-invariant

(b) year-specific

2. Stochastic models

(a) stationary stochastic environments

(b) climate model scenarios

We begin with deterministic models. Thesimplest of these is time-invariant, in which theprojection matrix is obtained from a single es-timate of the vital rates over the entire pe-riod 2001–2005. Then we consider year-specificmodels, in which a separate projection matrixis obtained for each of the years 2001,. . . ,2005.The calculations from a given year-specific ma-trix give the population dynamics that wouldresult if the conditions in that year were main-tained indefinitely.

Then we incorporate environmental varia-tion by constructing stochastic models. We clas-sify these by the way the environment is de-scribed. The simplest model is obtained by de-scribing the environment as a stationary pro-cess, i.e., one which fluctuates, but whose mean,variance, and other statistical properties do notchange with time. In our models, the stationarystochastic environment is characterized by thefrequency of years with poor sea ice conditions(in a sense to be described precisely below).Rather than assuming a stationary stochastic

environment, our second set of models derivesa stochastic environment from the output ofgeneral circulation models (GCMs) for climateover the next century. These models are non-stationary, because they not only fluctuate, buttheir statistical properties change over time.

The deterministic year-specific calculationsshow the consequences of the conditions in theyears in which the parameters were estimated.The stochastic models use that information toproject the consequences of future environmen-tal fluctuations. Because the stochastic mod-els use the year-specific models as components,it is important to understand the behavior ofthe deterministic models. For both determinis-tic and stochastic calculations we compare theresults of models parameterized from the fullmodel set, the non-covariate model set, and fordeterministic models the time-invariant modelset. For all except the climate model scenarios,we examine both long-term population growthrates and short-term projections of populationsize.

As was emphasized in Part I of this report(Regehr et al. 2007), estimating the vital ratesfrom the available data was subject to large de-grees of sampling and model selection uncer-tainty. Examining multiple models helps doc-ument the robustness, or lack thereof, of theconclusions, as we discuss in Section 5.

3 Deterministic demography

We begin with the analysis of deterministic ver-sions of the demographic model.

3.1 Deterministic methods

We apply deterministic demographic analysesto the time-invariant model set, the full modelset, and the non-covariate model set. In mostcases, and unless otherwise specified, results forthe full model set and the non-covariate modelset were very similar.

The full model set and the non-covariatemodel set contain year-specific parameter esti-mates. As a result, demographic quantities cal-culated from them are also year-specific, in the

6

following sense. Suppose we take the vital ratesfor 2001, construct the projection matrix A,and calculate a population growth rate. Thatrate describes the long-term population growththat would occur if the conditions in 2001 wereheld constant. Thus the growth rate represents,and summarizes, the conditions present in 2001,by asking what would happen if they were to bemaintained. Such results are called projections,to distinguish them from predictions, or fore-casts, that attempt to say what will actuallyhappen in the future.

We calculate the long-term population growthrate as the dominant eigenvalue λ of A. Thecontinuous-time growth rate r is given by r =log λ. If λ > 1 (i.e., r = log λ > 0) the popu-lation will increase if the conditions remain thesame. If λ < 1 (i.e., r = log λ < 0) the popula-tion will decline to extinction unless conditionschange. Under constant conditions, the popu-lation will eventually converge to a stable stagestructure proportional to the right eigenvectorw corresponding to λ. The left eigenvector vcorresponding to λ gives the distribution of re-productive values among the stages. A popula-tion that is not at the stable stage distributionwill go through a period of transient fluctua-tions in both abundance and structure beforethe asymptotic growth rate is realized. Thesetransient fluctuations can be analyzed in termsof the subdominant eigenvalues and eigenvec-tors of A, or revealed by short-term projections(see Section 3.1.3).

3.1.1 Perturbation analysis

Perturbation analysis measures the effect on pop-ulation growth of changes in parameters. Thesensitivity of population growth rate to a pa-rameter is the absolute change in populationgrowth rate caused by a small absolute changein a parameter. The sensitivity of λ to a pa-rameter π is

∂λ

∂π= vT ∂A

∂πw (12)

where the eigenvectors are scaled so that theirscalar product 〈v, s〉 = 1 (Caswell 1978, 2001)

and the derivative ∂A/∂π is a matrix whose(i, j) entry is ∂aij/∂π. If π = aij , then (12)reduces to

∂λ

∂aij= viwj . (13)

It is often useful to express changes in parame-ters, and their effects, on a proportional scale.The elasticity of population growth rate to aparameter is the proportional change in pop-ulation growth rate caused by a proportionalchange in the parameter, and is given by

π

λ

∂λ

∂π. (14)

3.1.2 LTRE analysis

Differences in population growth rate λ amongyears are the integrated result of all the differ-ences in stage-specific survival, transitions, andreproduction among those years. LTRE (lifetable response experiment) analyses decomposethose differences into contributions from the dif-ference in each of the vital rates (Caswell 1989,2001).

Let πi be one of the demographic parame-ters (survival probability, litter survival, breed-ing, f , or σL1). We choose the conditions in2001 as our reference, and examine the changesin λ relative to this. Let λr be the growth rateunder the reference conditions and λt be thegrowth rate under conditions in year t. Then,to first order,

λt − λr ≈∑

i

(π

(t)i − π

(r)i

) ∂λ

∂πi

∣∣∣∣πi+πr2

. (15)

The sensitivity term is calculated at the meanof the reference and the treatment parameters.The ith term in the summation on the righthand side is the contribution of the differencesin the parameter πi to the difference in λ. Aparameter may make a small contribution if itdoes not differ much, or if λ is not very sensitiveto differences in that parameter.

3.1.3 Population projection

Population growth rate λ gives the long-termrate of population growth. In the short-term,

7

growth may fluctuate because of deviations fromthe stable stage distribution. To evaluate this,we projected population growth for 45, 75, and100 years2 according to (1), starting from aninitial population proportional to

n(0) =(

0.106 0.068 0.106 0.461

0.151 0.108)T

. (16)

This is an estimate of the structure of the SBpopulation, averaged over 2004–2006. It is ob-tained from a Horvitz-Thomson estimator ap-plied to mark-recapture data in the SB popula-tion using recapture probabilities from Regehret al. (2006).

Because the models under consideration hereare linear, the projections produce populationsize measured relative to the initial population,and thus can be directly interpreted in termsof proportional increase or decrease. Bootstrapsamples were used to document the uncertaintyin projected population size for each model set.

3.1.4 Correction for harvest mortality

Polar bears in the SB management unit are har-vested as part of annual regulated hunts by na-tive user groups in the US and Canada (Broweret al. 2002). Regehr et al. (2007) used tag re-turn data to estimate a harvest component h oftotal mortality for female bears in stages 1–4.To find the survival rate in the absence of har-vest, we removed the current level of harvestfrom the estimated survival rates by replacingσi with σi

(1−h) . To evaluate the effect of variousrates of harvest mortality on population growthrate, we then applied a range of harvest rates upto 0.1 to stages 1–4. Harvest mortality was notconsidered for stages 5 and 6 because huntersare discouraged from taking females with de-pendent cubs.

3.2 Results of deterministic analysis

3.2.1 Growth rates

Population growth rates are summarized in Ta-ble 1. The time-invariant model set predicts

2The 45-year projection horizon was specifically re-quested by the FWS.

a growth rate very close to stationarity (λ =0.997, implying a decline of −0.3% per year).The full model set and the non-covariate modelset both predict population growth or stabilityunder the conditions in 2001–2003 and popula-tion decline under the conditions in 2004–2005.The estimates of population growth rate are ac-companied by significant uncertainty (Figure 2).However, in 2004–2005, with a longer ice-freeseason, only a tiny percentage of the bootstrapsamples produced positive population growth.

The stable stage distribution contains moreindividuals in the adult stages than in sub-adultstages. (Figure 3). In years with a longer ice-free season the proportion of individuals in theadult but not reproducing stage increased andthe proportion in all other stages decreased,particularly the reproducing stages (Figure 3).However, the distribution of individuals amongthe six stages was not qualitatively different.Similarly, the reproductive value of reproduc-ing stages increased in years with poor condi-tions but this change was relatively small (notshown).

In long-lived species, the sensitivity and elas-ticity of population growth rate to model pa-rameters is typically greatest for adult survivalrates (e.g., Heppell et al. 2000). This popu-lation was no exception, with λ most sensitiveand most elastic to survival of the adult femalestages (σ4–σ6). See Figure 3 for the full modelset and Figure A-2 for the non-covariate modelset.. Among the other parameters, only thebreeding probability of an adult female withoutcubs (β4) makes a significant contribution topopulation growth rate. Under the conditionsof 2004–2005, the sensitivity and elasticity of λto survival of individuals in stage 4 (adult, notcurrently reproducing) increased. This partlyreflects the increase in the proportion of in-dividuals in this stage under these conditions,which increases its importance to λ.

The LTRE analysis decomposes the differ-ences in λ (measured relative to 2001 condi-tions) into contributions from each of the vitalrates. The decline in λ is mostly due to declinesin adult survival in stages 4–6 (σ4, σ5, and σ6),with secondary contributions from the breeding

8

probability of females in stage 4 (β4) (Figures4).

3.2.2 Deterministic population projec-tions

Short-term projections of the population showthat exponential growth is achieved quickly whenconstant conditions are maintained; with no ap-parent transient effects (Figure 5). This is ex-pected because the initial population vector (16),based on current population estimates, is notfar from the stable stage distribution.

The uncertainty in the short-term projec-tions is shown by the probability distribution,over the bootstrap sample, of the log of total fe-male population size at 45 years relative to thestarting population size, i.e. log

(N(45)N(0)

). Un-

der the conditions experienced in years 2001–2003, a high proportion of the bootstrap sam-ples yield population increase. Under the con-ditions of 2004–2005, almost all the boostrapsamples yield population decline, often to verylow values (Figures 5, A-3, and A-4).

Figure 5 shows the cumulative distributionfunction (cdf) for this relative growth (the cu-mulative distribution at x is the probability thatthe variable in question is less than or equal tox). Most of the mass for the cdf’s in 2001–2003, is gained to the right of the zero line, in-dicating that most of the probability distribu-tion produces positive population change. Thecdf’s for conditions in 2004 and 2005 approach1 to the left of the zero line, indicating thatalmost all of the probability distribution pre-dicts population decline. The distributions inFigure 5 reflect the distribution of populationgrowth rates in Figure 2. Projections to 75 or100 years merely extend the patterns apparentat 45 years (Figure A-4).

The differences among years in the resultsof population projection is partly due to vi-tal rates expressed as functions of the ice co-variate. As noted above and in Regehr et al.(2007), estimating dependence on ice was dif-ficult because the capture-recapture data setspanned only five years, so we also examinednon-covariate models. Figure 6 compares the

population growth rate λ predicted by the fullmodel set as a function of ice with the values ofλ estimated from the non-covariate model set.The similarity is remarkable.

From this figure, it appears that values ofice greater than approximately 125 ice-free daysper year lead to precipitous declines in λ, butthat below this level λ is relatively insensitiveto ice. Of course, the pattern of λ for valuesof ice greater than those observed in the datais an extrapolation that depends completely onthe parametric form (logistic) assumed for thecovariate-dependence. This range of values couldbe explored only through continued capture-recapture studies of the SB population.

3.2.3 Effects of harvest

The effects of the harvest level of females onpopulation growth rate λ are shown in Figure 7;the level of female harvest to which the popula-tion is currently subject has only a small effecton population growth rate.

4 Stochastic demography

Environmental conditions have changed over thecourse of this study, and will no doubt change inthe future. The deterministic analyses (Section3.2) characterize those conditions by project-ing the population growth patterns that wouldresult if the conditions were somehow kept con-stant. Now we turn to the effects of the fluctu-ations themselves, using stochastic models.

A stochastic demographic model contains amodel of the environment, a link from the envi-ronment to the vital rates, and a projection ofthe population using those vital rates. We willpresent results of two stochastic calculations,which differ in the scenarios used to describevariation in the environment.

Based on the patterns in survival (Regehret al. 2007) and in population growth rateλ (Table 1, Figure 6), we classified 2001–2003as “good” years, characterized by high survivaland estimates of population growth rate λ greaterthan 1. We classified 2004–2005 as “bad” years,characterized by reduced survival, and values

9

of λ much less than 1. Our first approach tostochastic models is based on the statistical pat-terns of good and bad years.

Good years are associated with low valuesof ice, and bad years are characterized by highvalues of ice, but our first analysis does notmake any explicit connection to the dynamicsof ice. Instead, it assumes that good and badyears occur independently and with a constantprobability q of a bad year and 1− q of a goodyear. Population growth in such a stochasticenvironment will depend on the value of q (i.e.,on the long-term frequency of bad years), andwe examine this dependence. Although the en-vironment fluctuates, its statistical properties(e.g., the value of q) do not, so we call this astationary stochastic environment.3

Our second approach explicitly links the vi-tal rates to future sea ice conditions. We usedthe output of a set of 10 IPCC (Intergovernmen-tal Panel on Climate Change) climate models(DeWeaver 2007 describes the choice of thesemodels) to forecast scenarios of sea ice condi-tions until the year 2100. We used the outputfrom these climate models to generate stochas-tic sequences of good and bad years, and usedthose sequences to project polar bear popula-tion size through the end of the century.

Stochastic calculations include several dif-ferent kinds of uncertainty. They include uncer-tainty due to environmental stochasticity: evenif the probability of a bad year is fixed, some-times a bad year will occur and sometimes itwill not. They include uncertainty due to thesampling error in the vital rates, which we as-sess using bootstrap samples and by comparingthe full and non-covariate model sets. Finally,they include uncertainty due to the uncertaintyin forecasts of future climate conditions. Weobtain some information on this uncertainty byusing 10 climate models rather than a singlemodel, but it is not possible to examine cli-mate forecast uncertainty within a single cli-mate model because multiple stochastic realiza-

3Mathematically speaking, it is a particularly sim-ple type of stationary environment, in which successivestates of the environment are independently and identi-cally distributed.

tions of climate model output are generally notavailable.

4.1 Stochastic methods

In this section we describe the methods used forstochastic calculations.

4.1.1 Stationary stochastic environments

Let A(1), . . .A(5) denote the projection matri-ces, for females, estimated in years 2001–2005,and let q be the probability of occurrence of abad year. In a bad year, the population experi-ences a matrix chosen from the set {A(4),A(5)}with probability 1/2. In a good year, the popu-lation experiences a matrix chosen from the set{A(1),A(2),A(3)} with probability 1/3. Thusthe population grows according to

n(t + 1) = Atn(t) (17)

where the matrix At is

At =

A(1) with probability (1− q)/3A(2) with probability (1− q)/3A(3) with probability (1− q)/3A(4) with probability q/2A(5) with probability q/2

(18)This algorithm uses the variability within thegood years, and within the bad years, as es-timates — admittedly crude but better thannothing — of the variation in the vital rateswithin these categories (see Caswell and Kaye2001 for another example).

Because this environmental process is sta-tionary and ergodic, and because the projectionmatrices form an ergodic set (e.g., Tuljapurkar1990), the population has an expected per-yeargrowth rate given by

log λs = limT→∞

1T

log ‖AT−1 · · ·A0n0‖ (19)

where ‖ · ‖ is the 1-norm and n0 is an arbitraryinitial population vector.

This growth rate is the stochastic analog ofthe population growth rate λ in a time-invariantenvironment, and is thus the relevant measure

10

of long-term population growth in a stochasticenvironment. If log λs ≤ 0, the population willeventually decline to extinction (unless some-thing happens to change the environment or thevital rates). If log λs > 0 the population willeventually grow, but even so, stochastic fluctu-ations may carry it to sufficiently low densitiesto cause extinction.

We estimated log λs by applying (19) withT = 5000. The effects of parameter uncertaintywere evaluated by repeating these calculationsfor 2000 samples from the parametric bootstrapdistribution of the vital rates.

The stochastic growth rate is an asymptotic(long-term) index of population growth, as is λin the deterministic model. Thus we also car-ried out short-term projections, starting withan initial population with the observed popula-tion structure (16), and calculated the propor-tional increase or decrease in total populationsize at times t = 45, 75, and 100 years.

4.1.2 Sensitivity analysis of the stochas-tic growth rate

To calculate the sensitivity of log λs and theelasticity of λs to changes in parameters, weused the perturbation analysis of Tuljapurkar(1990) and Caswell (2005, 2007).

Following Regehr et al. (2007), let π be thevector of demographic parameters

π =(

σ1 · · · σ6 σL0 β4 β5 f)T (20)

and let

d log λs

dπT=

(d log λs

dπi

)(21)

be the 1×10 vector of the sensitivities of log λs.To calculate these sensitivities, we generate arandom sequence of matrices At for t = 0, . . . , T−1. We use this sequence to generate a sequenceof stage distribution vectors

w(t + 1) =Atw(t)‖Atw(t)‖ , t = 0, . . . , T − 1,

(22)one-step growth rates

Rt = ‖Atw(t)‖, (23)

and reproductive value vectors

vT(t− 1) =vT(t)At−1

‖vT(t)At−1‖ t = 1, . . . , T.

(24)In these calculations, the starting vectors w(0)and v(T ) are arbitrary non-zero, non-negativevectors with ‖w(0)‖ = ‖v(T )‖ = 1. Then thesensitivities are calculated as

d log λs

dπT= lim

T→∞1T

T−1∑

i=0

[wT(i)⊗ vT(i + 1)

]

RivT(i + 1)w(i + 1)

×(

dvecAi

dπT

). (25)

The elasticity of λs is

limT→∞

1T

T−1∑

i=0

[wT(i)⊗ vT(i + 1)

]

RivT(i + 1)w(i + 1)

×(

dvecAi

dπT

)diag (πi). (26)

In these expressions, ⊗ denotes the Kroneckerproduct, and dvecAi/dπT is the matrix (di-mension 36 × 10) of derivatives of the entriesof Ai with respect to the parameters in π. Wecalculated all sensitivities and elasticities usingT = 5000.

4.1.3 Climate model scenarios

To project population growth under future cli-mate change scenarios, we extracted forecastsof the availability of sea ice for polar bears inthe SB region, using monthly forecasts of sea iceconcentration from 10 IPCC Fourth AssessmentReport (AR-4) fully-coupled general circulationmodels (GCMs; Table 2; see DeWeaver 2007 fordetails of the selection process and properties ofthe models). We used forecasts for the 21st cen-tury based on a “business as usual” greenhousegas forcing scenario (Special Report on Emis-sion Scenarios SRES-A1B). The 10 GCMs wereselected based on concordance between their20th century simulations (20c3m) of sea ice ex-tent and the observational record of ice extentduring 1953–1995 (DeWeaver 2007).

11

Because GCMs do not provide suitable fore-casts for areas as small as the SB, we used seaice concentration for a larger area composed of5 IUCN polar bear management units (Aars etal. 2006) with ice dynamics similar to the SBmanagement unit (Barents Sea, Beaufort Sea,Chukchi Sea, Kara Sea and Laptev Sea; seeRigor and Wallace 2004, Durner et al. 2007).We assumed that the general trend in sea iceavailability in these 5 units was representativeof the general trend in the Southern Beaufortregion.

From each GCM output we calculated thenumber of ice-free months during the calendaryear in the region of preferred habitat for po-lar bears within the 5 IUCN units. Preferredhabitat was defined as waters over the conti-nental shelf and less than 300m deep (Interna-tional Bathymetric Chart of the Arctic Ocean;http://www.ngdc.noaa.gov/mgg/bathymetry/arctic/arctic.html). A month was consideredice-free if the mean ice concentration in this re-gion was less than 50%. Mean ice concentra-tion was the arithmetic mean ice concentrationfor all grid cells in the region of preferred habi-tat. Among the 10 GCMs analyzed, the seaice grids had model-specific spatial resolutionsranging from 1× 1 to 3× 4 degrees of latitudeand longitude (DeWeaver 2007).

We transformed the ice-free months to ice-free days by multiplying by 30. Then the out-put of each GCM was shifted so that the pro-jected 2005 value coincided with the mean (114.4days) of the ice-free days observed in the SBduring 2000–2005. This calibrated all the mod-els and produced sea ice projections scaled rel-ative to the observed current (or recent past)conditions. Each year in the shifted trajec-tory was then classified as good or bad depend-ing on whether the number of ice-free days wasgreater than or less than 127 (a value midwaybetween the values observed during the goodyears 2001–2003 and the bad years 2004–2005).To extract trends in the frequency of bad yearsfrom this binary series, we applied a Gaussiankernel smoother as described by Copas (1983;see Smith et al. 2005 for an ecological applica-tion similar to this one). The kernel standard

deviation (2.5 years) was chosen, as suggestedby Copas (1983), as a subjective compromisebetween smoothing and variation.

Because these are short-term projections andbecause the stochastic properties of the envi-ronment change, there is no single measure ofpopulation growth. Instead, we projected thepopulation from 2005 to 2100, with n(0) givenby (16), and with a matrix At chosen randomlyat each year with probabilities (18), with q asthe probability of a bad year in year t. Theproportional increase or decrease in total pop-ulation size was recorded for each year.

4.2 Results of stochastic analysis

4.2.1 Stationary stochasticenvironments

Population growth rates. The stochasticgrowth rate declines with increases in the fre-quency of bad years, shown in Figure 8. Theresults for the full model set have been par-titioned into two subsets because of an unex-pected (and pathological) behavior of the thesemodels with ice-covariate dependence. Becauseof the high variance in, and strong negativecorrelation between, the estimates of the slopeand intercept of this response, a fraction (about11%) of the bootstrap samples showed a posi-tive response to bad ice years. Because of thebehavior of the logistic function, these led to ex-tremely negative population growth rates underconditions of good ice years (log λs as low −20,which corresponds to a population half-life of12 days). Neither a positive response to ice-freeconditions nor a population half-life measuredin days are biologically realistic, so we have par-titioned the bootstrap samples of the full modelset into a subset conditional on a non-increasingresponse and an (unrealistic) subset conditionalon an increasing response, and we focus on thenon-increasing subset. However, this partitionobviously affects our conclusions about trends,so we rely on comparison with the non-covariateset of models, which is not partitioned in anyway.

In both sets of models, the stochastic growth

12

rate decreases nearly linearly with increases inthe frequency q of bad years. The slope impliesthat an increase of 0.1 in q would reduce thestochastic growth rate log λs by 0.02y−1 (non-covariate model set) or 0.03y−1 (full model set).There is a critical threshold frequency abovewhich log λs < 0 and the population will de-cline; this is q ≈ 0.175 (non-covariate modelset) or q ≈ 0.165 (full model set). These esti-mates of the critical frequency should be com-pared to the average frequency of bad ice yearsobserved since 1979 (6/28 = 0.21), with an ap-parent increasing trend and decadal oscillations(Figure 9). This observed frequency implies astochastic growth rate on the order of −0.014for the full model set and −0.007 for the non-covariate model set.

The elasticity of the stochastic growth rateλs to changes in parameters is shown in Fig-ure 10 for the full model set and Figure A-5for the non-covariate model set, for a range offrequencies (q) of bad years. Regardless of q,the stochastic growth rate is most affected bychanges in adult female survival, in a patternsimilar to that of the deterministic growth rate(Figure 3).

Population projection. The distributions ofrelative population size at t = 45, 75, and 100are shown in Figure 11. When bad years arevery rare (q = 0.01), the probability of de-cline is approximately 0.25 for the non-covariatemodel set and 0.20 for the full model set. Whenq = 0.25 (similar to the observed average fre-quency of bad ice years since 1979), the prob-ability of decline at 45 years is approximately0.6 for the non-covariate set and 0.9 for the fullmodel set. Decline to a population only 1% ofits current size might be considered a severe de-cline; it would certainly put the population atrisk of extinction. With q = 0.25, the probabil-ity of such a decline within 45 years is approx-imately 0.2 for either model set.

Extending the time horizon increases therisk of extreme decline. With q = 0.25, as t in-crease from 45 to 75 to 100 years, probability ofextreme decline (to 1% of initial size) increases

from about 0.2 to 0.4 to 0.55 for the full modelset. The corresponding probabilities from thenon-covariate model set are about 0.15, 0.2, and0.25 (Figure 11).

4.2.2 Climate model projections

Figure 12 (top panel) shows the coarse-grainedoutputs of the climate models (integer values ofice-free months). The 10 models differ in theirtrends and intercepts, but they all predict in-creases over the next century. Figure 12 showsthe results of rescaling these projections (mid-dle panel) and transforming the trends into thepredicted probability of bad ice conditions (bot-tom panel). Figure A-6 displays these trends foreach model individually. All 10 of the climatemodels agree in forecasting an increase in theincidence of bad years, reaching an incidence of1 within the next century; indeed, all but oneof the models makes this prediction within thenext 50 years. Two GCMs (miub echo g andmpi echam5) differ from the rest in predictinga long delay (20–40 years) before the probabil-ity of a bad ice year rises above zero.

Figure 13 shows population trajectories sam-pled from the bootstrap distribution of matricesand the 10 climate models. Most trajectoriesdecline to very low values well before the end ofthe century. A few outliers increase greatly be-fore declining. Closer examination (Figures A-7 and A-8) reveals that these are associatedwith the two GCMs that predict no bad yearsfor the next several decades.

The declines shown in Figure 13 obviouslycarry some risk of extinction. However, themodels we use here can project population sizeas declining exponentially towards zero, but cannever actually reach zero. Thus any interpre-tation in terms of extinction must be accom-panied by a definition of an extinction thresh-old.4 If the current SB population numbersabout 1500 individuals (Regehr et al. 2006),

4Such thresholds are often called quasi-extinctionthresholds, to distinguish them from true extinction.They may represent values believed to result in real ex-tinction, or simply thresholds felt to be of managementconcern.

13

then a decline to 0.001 of current size corre-sponds to a population of 1.5 bears, almost cer-tainly incompatible with persistence. A declineto 0.01 of current size corresponds to 15 bears,which would certainly be at a high risk of ex-tinction. Figure 14 shows the risk of declin-ing to the thresholds of 0.01 and 0.001, treat-ing the sampled set of population trajectoriesas a probability distribution measuring our un-certainty about future dynamics. The risk ofquasi-extinction, under either threshold and fromeither model set, exceeds 75% by the end of thecentury.

5 Discussion

We developed models for the current and fu-ture demography of polar bears in the southernBeaufort Sea that link sea ice conditions to pop-ulation dynamics, embedding the parameter es-timation directly in the modeling effort and tak-ing full account of uncertainty that derives froma number of sources. We were able to makeuse of state-of-the-art demographic methods,coupling them with results from GCM models.This work is an important new analysis of thestatus of the polar bear population in this re-gion, and is also one of the first efforts to inte-grate demographic and climate models. Below,we discuss the results, their importance, andconsiderations in the interpretation of our con-clusions.

5.1 Interpretation of results

In this study, we were challenged to provide de-mographic analyses of a long-lived species livingin a highly dynamic environment which is un-dergoing dramatic changes, and to do so on thebasis of a relatively short-term set of estimatesof the vital rates. We have also been chargedwith going beyond characterizing current con-ditions to projecting the results of future sce-narios.

Neither of these are easy tasks, which ex-plains our continual invocation of results frommultiple sets of statistical models, from both

deterministic and stochastic demographic mod-els, from stochastic models that do and that donot rely on GCMs for future sea ice scenarios,and from 10 different GCM projections. This,unfortunately, may leave the reader lost in ablizzard of detail. So, here we discuss and in-terpret the overall results.

The reduction in survival and breeding prob-abilities documented in Part I of this report(Regehr et al. 2007) translate into declines inthe population growth rate λ from 2001 to 2005.An LTRE analysis, which decomposes the dif-ference in λ into contributions from the vitalrates, shows that the decline in λ is due pri-marily to reductions in adult female survival,and secondarily to reductions in breeding prob-ability.

Projecting population size using the year-specific vital rates gives increases under the con-ditions in 2001–2003, but decreases under theconditions of 2004–2005. The interpretation ofthese results is clear: the SB environment de-clined in quality (as far as polar bear popula-tions are concerned) over the study period to apoint where, if nothing changes, the populationwill decline rapidly to extinction.

Of course, things do change, and our stochas-tic analyses address these changes. We phrasethese results in terms of good and bad years.It is important to understand these categories.There was a qualitative difference in the poten-tial for population growth between the condi-tions of 2001–2003 and of 2004–2005. This dif-ference was associated with an approximately50% increase in the duration of the ice-free pe-riod (as measured by the ecologically-motivatedcovariate ice). Our division of the years 2001–2005 into good and bad is not based on a pri-ori ideas of the effects of ice, but on the basisof the population response to those conditions.It is an admittedly crude binary classificationof a much more complicated response. Contin-uation of capture-recapture studies in the SBwould make it possible to refine this descrip-tion of the effects of the environment.

Even so, at this level of resolution it is clearthat the frequency of bad years in the future willdetermine the fate of the SB polar bear popula-

14

tion. A frequency greater than about 0.17 (onebad year in 6) will produce a negative stochas-tic growth rate and (unless something changes)lead to eventual extinction. The observed fre-quency of bad years appears to have been in-creasing since records began in 1979 (Figure 9),5

but even taking the average observed frequency(0.21) predicts a population decline of about 1%per year.

This prediction is not far from the deter-ministic growth rate calculated from a time-invariant model. It would be useful to haveindependent data on recent trends in size ofthe SB population to corroborate these results,but such information is not available. How-ever, Amstrup et al. (1986) estimated an ap-proximate population size of N ≈ 1800 in themid-1980s and Regehr et al. (2006) estimated apopulation size of approximately 1500 in 2006.A decrease from N = 1800 to N = 1500 be-tween 1986 and 2006 represents an average de-cline of 0.9% per year. Anecdotal evidence sug-gests that the population may have increasedslightly from its mid-1980s level into the 1990s(Amstrup et al. 2001). If the decline from 1800to 1500 took place over 15 years, it would rep-resent a rate of 1.2% per year, over 10 years, adecline at a rate of 1.8% per year. Such com-parisons are difficult because of the scant in-dependent historical data and a high degree ofuncertainty around historical population esti-mates. Nevertheless, the comparison supportsour estimates, to the extent that such limiteddata can provide support.

No one, of course, predicts that Arctic seaice conditions will be stationary (e.g., IPCC2007, Overland and Wang 2007, Stroeve et al.2007). Our projections of future populationdynamics based on sea ice forecasts from 10GCMs all show polar bear populations crash-ing within the next century. The probabilityof quasi-extinction, with thresholds of 0.01 or

5As we write this in August of 2007, a month prior tothe end of the sea ice melt season, the decline in Arcticsea ice extent has set a new record for the available timeseries from1979-2006 (National Snow and Ice Data Cen-ter, http://nsidc.org/news/press/2007 seaiceminimum/20070810 index.html).

0.001 (1% or 0.1% of current size) exceeds 0.7.The details, but not the outcome, are sensitiveto the choice of GCM (see especially Figures A-7 and A-8).

The links between loss of sea ice and sur-vival, reproduction, and population growth couldbe simple or complex. Sea ice retreat forc-ing polar bears to swim longer distances, thusincreasing the risk of drowning or starvation,would be a fairly simple connection betweenincreased open water and survival. However,a reduction in the duration of ice cover overwater depths less than 300m will also affectmany ice-dependent species such as ringed seals(Phoca hispida), bearded seals (Erignathus bar-batus), and arctic cod (Boreogadus saida); seee.g. Gaston et al. (2005). Many food web linksmay be involved in these effects. Intuitively,reduction in the availability of ringed seals, aprimary prey species of polar bears, whose pre-ferred habitat is ice over waters less than 300 mdepth (Stirling et al. 1982), must play a role.

The formal spatial scope of inference for thiswork is the southern Beaufort Sea population,but to the extent that the SB represents otherpopulations in the divergent polar basin (Am-strup et al. 2007), inference could be applied tothis larger zone. The ice covariate was calcu-lated from GCMs over this larger zone (whichencompasses the Barents, Beaufort, Chukchi,Kara, and Laptev Seas; DeWeaver 2007), sothe climate predictions do apply over this largerarea of inference. The demographic informa-tion is derived from just the SB (Regehr et al.2007), but the life-history and behavior of polarbears in these regions is similar (Amstrup et al.2007).

5.2 Coupling GCMs and demographicmodels

To our knowledge, there are few, if any, exam-ples of demographic models coupled to GCMs,thus, this work represents a substantial break-through in the integration of two major classesof quantitative environmental analysis. Thereare substantial challenges to such a coupling:including (1) finding a way to link models that

15

operate at substantially different spatial scalesand over different time intervals, and (2) discov-ering a link between some environmental fac-tor predicted by the climate models and a de-mographic response that can be represented inthe population model. Future examples of suchcoupled models will require these same issuesto be resolved for each new application.

Several aspects of our use of these climatemodels deserve comment. First, we focusedon the IPCC A1B forcing scenario. This sce-nario projects future global CO2 levels basedon a business-as-usual assumption about futureglobal economies and energy use. Several otherforcing scenarios have been explored by IPCC,based on different assumption about patternsof CO2 production and uptake, and these sce-narios project different trends in global temper-atures and other climatic variables. Thus, ourresults do not capture uncertainty in populationprojections derived from uncertainty about fu-ture patterns of atmospheric CO2. The A1Bscenario, however, enjoys support as a middle-of-the-road forecast, thus in this context, it isneither precautionary nor optimistic.

Second, we chose 10 of the global circula-tion models to include in our analysis, ratherthan the whole suite of available models. Therationale for this choice is described in a com-panion report (DeWeaver 2007) but was basedon how well models match observed ice con-ditions from 1953-1995. We believe these tobe the most appropriate models for forecastingarctic sea ice dynamics. Some excluded modelsdo a very poor job of predicting sea ice, pre-dicting either no change over the next centuryor complete loss of sea ice. These models wouldnot provide useful projections.

Finally, we examined sea ice dynamics overa larger area than the southern Beaufort Sea,because of the spatial limitations of the globalcirculation models. To the extent that this largerarea does not represent the patterns that will beseen in the SB, our projections might change ifmore detailed information were available. How-ever, the use of the larger spatial area in theclimate models does make us more comfortabledrawing tentative inference about the popula-

tion dynamics in that part of the polar basin.

5.3 Caveats

Evaluation of the strength of our conclusionsrequires consideration of a number of factorsthat went into the calculations. Below we dis-cuss several sets of issues, examining assump-tions and exploring issues that could affect ourconclusions.

5.3.1 Estimation and inference

The conclusions of this report depend, in part,on the parameter estimation methods used tocalculate inputs for the demographic model. PartI of this report (Regehr et al. 2007) discussesthose methods in detail. However, we feel itpertinent to mention that one concern that hasarisen in short-term mark-recapture studies ofanimals with high survival rates is the potentialfor negative bias in survival rates at the end of atime series, due to non-random temporary emi-gration. Regehr et al. (2007) carefully considerthis potential source of bias, concluding that itis unlikely, but the possibility cannot be com-pletely eliminated from any study.

Our evaluation of the effect of the frequencyof bad years on population growth rate dependson using the periods 2001-2003 and 2004-2005to characterize good and bad years, respectively.These sample sizes are small (3 and 2 years,respectively) and may not be representative ofthe classes of conditions we wish to encapsu-late, and there is considerable uncertainty inthe estimation of the growth rates for each ofthose years. Nevertheless, our projections takefull account of this uncertainty, thus consideringthe possibility that the difference between thegood and bad years is not as great as the meansimply, or is perhaps greater. The strength ofthe patterns in the projections, even consider-ing the uncertainty, is evidence of the signifi-cance of the differences among these years.

Our projections of future population trend,from the linked GCM-demographic model, arepredicated on a causal relationship between seaice extent and polar bear survival. No obser-

16

vational (as opposed to experimental, which isclearly impossible in this context) study canever provide iron-clad evidence of causation. How-ever,the fact that we only considered one ex-planatory variable, which was identified a pri-ori, lends support to our argument. More im-portantly, our understanding of the ecology ofthe species clearly indicates that sea ice is anenvironmental factor with effects on polar bearsurvival and reproduction.

Finally, the models that included ice as a co-variate were quite sensitive to uncertainty, giv-ing rise in some cases to results that defied intu-ition, such as a small fraction of replicates thatindicated current growth rates of 0. While thisraises some concern, it is understandable as astatistical result of fitting a parametric model toa short time series. We used the non-covariatemodel set to explore the projections in the ab-sence of this sensitivity. That both model setsprovide the same qualitative predictions, andagree quantitatively in many instances, leadsus to believe that this sensitivity does not un-dermine our conclusions.

5.3.2 Model structure

The conclusions of this report also depend, inpart, on the structure of the demographic model.Several elements of the model structure deservecomment.

The demographic model contains no density-dependent mechanisms (either positive or neg-ative). It is tempting to speculate that at lowpopulation sizes some compensatory mechanismmight make growth rates increase because of re-duced competition and provide some resiliencethat our model does not capture. However, adensity-dependent analysis of the SB polar bearpopulation could not be as simple as that. Itwould have to account for the prospect of dra-matic declines in carrying capacity due to cli-mate change. As a very crude calculation, ifthe carrying capacity varies with overall Arcticsea ice, it would have been declining at a rateof about 1.8% per year since 1995 (Stroeve etal. 2007). In the best of all density-dependentworlds, the population would have been track-

ing that decline, leading to λ ≈ 0.98 withoutany additional effects (e.g., mortality due todrowning) of sea ice decline. To the extent thatpolar bears rely on sea ice on the continentalshelf, near the southern edge of the BeaufortSea, future declines would have even more dra-matic impact on carrying capacity. Alternately,and perhaps more likely, if there is a depen-satory mechanism (such that growth rates de-crease even more because of Allee effects), thenthe demise of the population could be acceler-ated. In any event, the effects of density de-pendence can only be speculation, pending thedevelopment of models linking the carrying ca-pacity and changes in the environment.

We have not included emigration or immi-gration in the demographic model. If thereis permanent emigration from the SB popula-tion, then the estimated survival rates reflectthe apparent survival rates, not the true sur-vival rates. In this case, the population growthrate would be accurate as far as the SB popula-tion is concerned, but negatively biased relativeto the contribution the SB population makes tothe rangewide population (Runge et al. 2006).Likewise, if there is net immigration to the SBpopulation from neighboring areas, which wehave not included, then the current and futuregrowth rates would not be as low as we haveestimated. From the standpoint of persistenceof the SB population, however, we might viewemigration as equivalent to death, because theanimals no longer remain in this region. Fur-ther, immigration, even if it is occurring now, islikely to become less important in the future, ifthe SB region does not contain suitable habitat,and if the neighboring regions begin to experi-ence population declines. The effects of immi-gration and emigration could only be evaluatedby developing a spatial model for polar bearpopulations.

We have not included demographic stochas-ticity in this model. Such stochasticity will ac-celerate extinction once the population reachesvery small (10s of animals) absolute populationsize (Caswell 2001). If the population becomessmall enough for demographic stochasticity tobe important, it is already at risk simply due to

17

its small size. Thus this omission does not un-dermine any of the major conclusions. The pri-mary results of this analysis are driven by thedeterministic loss of sea ice and the concomi-tant reduction in survival; stochastic effects addvariance to this pattern, but do not change thegeneral trends.

Finally, of course, no model constructed inthe present can predict how polar bears mightrespond behaviorally to substantial changes intheir environment in the future. As summerice in the SB region retreats beyond the con-tinental shelf, one might speculate that polarbears could develop a new strategy for seekingfood (e.g., by becoming more land-based), theycould emigrate out of the region, or they couldcontinue to remain in the area. Our projec-tions can only assume that they remain in thearea (and face reduced survival) or emigrate outof the area (and be effectively dead as far asthe SB management unit is concerned). But,as polar bears are one of the most mobile ofquadrupeds, and have evolved in a highly dy-namic environment, one might speculate theycould adapt in some manner. However, all stud-ies to date of annual movement patterns of po-lar bears indicate a very high degree of fidelityto their natal populations, including in SB andwestern Hudson Bay, where they are alreadybeing significantly and negatively affected bychanges in ice conditions. Further, the poten-tial nutritive benefits of terrestrial feeding arecalorically insignificant and unlikely to providea panacea.

5.4 Importance of this work

This work was motivated by the need for a sta-tus assessment to inform a regulatory decision.It was facilitated by the existence of intensiveon-going field work in, and a long-term studyof, the SB polar bear population, and an in-tensive although short-term capture-recapturestudy from 2001–2006.

Our work provides decision-makers with thescientific information needed to evaluate the sig-nificance of the threat of the decline in the ex-tent of sea ice to the SB polar bear population.

The results quantify the risk of decline over thenext 4 to 10 decades based on current knowl-edge, and taking into account uncertainty thatarises from parameter estimation, a short time-series of capture-recapture data, the form ofthe population model, environmental variation,and climate projections. The results are prob-abilistic, in order to represent the uncertaintythat arises from all of these sources. Decision-makers will need to grapple with how that un-certainty affects the conclusion they make. In arisk analytic framework, decision-makers coulduse this information by identifying the event ofconcern, deciding what level of risk (what prob-ability) would trigger a particular decision, andinterpreting this analysis to see if that level ofrisk was reached.

5.5 Prospects for further demographicmodeling

One of the advantages of matrix population mod-els is that their construction lays bare the pro-cesses which are included, and those which areexcluded, from any particular model. We notehere that further demographic modeling wouldlead to further understanding of the dynam-ics of this population. Spatial models (usingthe multiregional matrix model formulation ofHunter and Caswell 2005b) and subsidized mod-els (e.g., Caswell 2007) could consider immi-gration and emigration. An extension of thecapture-recapture study in the SB would even-tually permit quantitative rather than qualita-tive (“good” vs. “bad”) links to climate (anexample of one way to approach this is Lawleret al. 2007). A connection to resource andhabitat models could help to develop a density-dependent analysis.

The polar bear is a top predator in a rapidlychanging ecosystem. Further analysis of po-lar bear populations will not only be importantfor management of the species, but will play acritical role in understanding the entire Arcticecosystem.

18

6 Summary and Conclusions

Demographic analysis links the condition of theenvironment to the fates of individuals and thefates of individuals to the dynamics of popula-tions. Reflecting these linkages, we can describeour results in terms of a particular set of goals:

Goal 1. To evaluate the status of the south-ern Beaufort Sea polar bear populationin terms of its current potential for pop-ulation growth in both deterministic andstochastic environments.

1. If conditions over the period 2001–2005 remained constant they wouldlead to a nearly stationary popula-tion with a decline of about 0.3% peryear.

2. The population would be capable ofincreasing if conditions experiencedin 2001–2003 were maintained. Thepopulation would decline dramaticallyif conditions in 2004–2005 persisted.

3. A frequency of years with conditionssimilar to 2004–2005 (“bad” years)greater than a critical value of about0.17 would lead to population decline.The observed frequency of bad yearssince 1979 is approximately 0.21, andappears to have been increasing.

4. A stochastic environment in whichgood (2001–2003) and bad (2004–2005)years occurred at the frequency ob-served since 1979 would lead to pop-ulation decline at a rate of about 1%per year.

5. Population growth rate (both deter-ministic and stochastic) is most elas-tic to adult female survival and, to alesser degree, breeding probability.

Goal 2. To quantify the response of popula-tion growth to the number of ice free daysover the continental shelf.

1. The deterministic population growthrate showed a marked decrease above

a threshold number of ice-free days(≈ 125) over the continental shelf. Asimilar decrease in population growthrate in the last two years, which hadabove normal ice-free periods, wasobserved from non-covariate models.

2. The stochastic growth rate declineswith increasing frequency of occur-rence of years with above normal ice-free days. An increase of 0.1 in thefrequency of such years reduces thepopulation growth rate by 2–3 per-centage points per year.

Goal 3. To project future population growthon a time scale of decades to a century, inconstant environments, in stochastic en-vironments, and in environments forecastby global climate models.

1. If conditions were to remain similarto 2001-2003, the population wouldincrease over the next 45-100 years.If conditions remained similar to 2004-2005, the population would declineprecipitously within 45 years.

2. In a stochastic environment with afrequency of bad years equal to theobserved value over the period 1979-2006, the population would probablydecline to between 1% and 10% ofits current size before the end of thecentury.

3. A stochastic environment describedby forecasts of sea ice conditions froma suite of 10 GCMs all lead to crashesof the population within the next 100years, usually within the next 50 years.

4. The current population structure isclose to the stable stage distribution,so asymptotic population growth ratesprovide accurate descriptions of rel-atively short-term (45 year) projec-tions.

Goal 4. To evaluate the effects of uncertaintyin parameter estimates and model selec-

19

tion on population growth rate and pop-ulation projections.

1. Parametric bootstrap approaches per-mit us to document the probabilityof outcomes based on the samplinguncertainty. In no case does the un-certainty change the conclusions ofthe analysis.

The intent of this analysis was to integrateclimate projections indicative of polar bear habi-tat with demographic models of polar bear pop-ulation dynamics and to evaluate the severity ofthe threat faced by polar bears in the southernBeaufort Sea due to changes in sea ice. We fo-cused on the increasing number of ice-free daysover water 300m in depth or less. This rep-resents the areas where ringed seals in the SBare most abundant (Stirling et al. 1982). Ouranalysis suggests that polar bears in the SBare likely to experience an important declineby mid-century as the Arctic ocean becomes in-creasingly ice free in the summer.

7 Acknowledgments

C. Hunter and H. Caswell acknowledge fund-ing for model development and analysis fromthe U.S. Geological Survey and U.S. Fish andWildlife Service, the National Science Founda-tion, NOAA, the Ocean Life Institute at WHOI,and the Institute of Arctic Biology at the Uni-versity of Alaska Fairbanks. Funding for thecapture-recapture effort in 2001-2006 was pro-vided by the U.S. Geological Survey, the Cana-dian Wildlife Service, the Department of Envi-ronment and Natural Resources of the Govern-ment of the Northwest Territories, and the Po-lar Continental Shelf Project, Ottawa, Canada.We thank G.S. York, K.S. Simac, M. Lockhart,C. Kirk, K. Knott, E. Richardson, A.E. De-rocher, and D. Andriashek for assistance in thefield. D. Douglas and G. Durner assisted withanalyses of radiotelemetry and remote sensingsea ice data. J. Nichols, and M. Udevitz pro-vided advice with data analysis. We thank E.

Cooch, J-D. Lebreton, and J. Nichols for help-ful comments and discussions. Funding for thisanalysis was provided by the U.S. GeologicalSurvey and U.S. Fish and Wildlife Service.

8 Literature cited

Aars, J., N.J. Lunn, and A.E. Derocher. 2006.Polar Bears: Proceedings of the 14th Work-ing Meeting of the IUCN/SSC Polar BearSpecialists Group. International Union forConservation of Nature and Natural Re-sources, Gland, Switzerland and Cambridge,United Kingdom.

Amstrup, S.C. 2003. Polar bear, Ursus mar-itimus. pp. 587–610 in G.A. Feldhamer,B.C. Thompson, and J.A. Chapman, ed-itors. Mammals of North America: biol-ogy, management, and conservation. JohnsHopkins University Press, Baltimore, Mary-land, USA.

Amstrup, S.C., and C. Gardner. 1994. Po-lar bear maternity denning in the BeaufortSea. Journal of Wildlife Management 58:1-10.

Amstrup, S.C., B.G. Marcot, and D.C. Dou-glas. 2007. Forecasting the rangewide sta-tus of polar bears at selected times in the21st century. USGS Alaska Science Center,Anchorage, Administrative Report.

Amstrup, S.C., T.L. McDonald, and I. Stirling.2001. Polar bears in the Beaufort Sea: A30 year mark-recapture case history. Jour-nal of Agricultural, Biological, and Envi-ronmental Statistics 6:221–234.

Amstrup, S.C., I. Stirling, and J.W. Lentfer.1986. Past and present status of polar bearsin Alaska. Wildlife Society Bulletin 14:241–254.

Brower, C.D., A. Carpenter, M.L. Branigan,W. Calvert, T. Evans, A.S. Fischbach, J.A.Nagy, S. Schliebe, and I. Stirling. 2002.The polar bear management agreement forthe southern Beaufort Sea: An evaluationof the first ten years of a unique conserva-tion agreement. Arctic 55:362–372.

Burnham, K. P., and D. R. Anderson. 2002.

20

Model Selection and Multimodel Inference:A Practical Information-Theoretic Approach.Second edition. Springer-Verlag, New York,New York, USA. 488 pp.

Caswell, H. 1978. A general formula for the sen-sitivity of population growth rate to changesin life history parameters. Theoretical Pop-ulation Biology 14:215–230.

Caswell, H. 1989. The analysis of life table re-sponse experiments. I. Decomposition ofeffects on population growth rate. Ecolog-ical Modelling 46:221–237.

Caswell H. 2001. Matrix population models.Second edition. Sinauer, Sunderland, Mas-sachusetts, USA

Caswell H. 2005. Sensitivity analysis of thestochastic growth rate: Three extensions.Australian and New Zealand Journal of Statis-tics 47:75–85.

Caswell H. 2007. Sensitivity analysis of tran-sient population dynamics. Ecology Let-ters 10:1–15.

Caswell H, Fujiwara M (2004) Beyond survivalestimation: markrecapture, matrix popula-tion models, and population dynamics. An-imal Biodiversity and Conservation 27:471–488.

Caswell, H., and T. Kaye. 2001. Stochasticdemography and conservation of an endan-gered plant (Lomatium bradshawii) in a dy-namic fire regime. Advances in EcologicalResearch 32:1–51.

Comiso, J.C. 2006. Arctic warming signals fromsatellite observations. Weather. 61(3):70–76.

Copas, J.B. 1983. Plotting p against x. AppliedStatistics 32:25–31.

Derocher, A.E., and I. Stirling. 1992. The pop-ulation dynamics of polar bears in west-ern Hudson Bay. Pages 1150–1159 in D.R. McCullough and R. H. Barrett, editors.Wildlife 2001: Populations. Elsevier Ap-plied Science.

DeWeaver, E. 2007. Uncertainty in climate modelprojections of Arctic sea ice decline: Anevaluation relevant to polar bears. USGSAlaska Science Center, Anchorage, Admin-istrative Report.

Durner, G.M., S.C. Amstrup, R.M. Nielson, andT.L. McDonald. 2004. The use of sea icehabitat by female polar bears in the Beau-fort Sea. Minerals Management Service OCSStudy 2004–14.

Durner, G.M., D.C. Douglas, R.M. Nielson, S.C.Amstrup, and T.L. McDonald. 2007. Pre-dicting the future distribution of polar bearhabitat in the Polar Basin from resource se-lection functions applied to 21st cdenturygeneral circulation model projections of seaice. USGS Alaska Science Center, Anchor-age, Administrative Report.

Efron, B., and R.J. Tibshirani. 1993. An In-troduction to the Bootstrap. Chapman andHall.

Fujiwara, M. and H. Caswell. 2001. Demogra-phy of the endangered North Atlantic rightwhale. Nature 414:537-541.

Fujiwara M, and H. Caswell. 2002. Estimatingpopulation projection matrices from multi-stage mark-recapture data. Ecology 83:3257–3265.

Gaston, A.J., H.G. Gilchrist, and J.M. Hipfner.2005. Climate change, ice conditions, andreproduction in an Arctic nesting marinebird: Brunnich’s guillemot (Uria lomvia L.).Journal of Animal Ecology 74:832–841.

Gervais, J. A., C. M. Hunter, and R. G. An-thony. 2006. Interactive effects of preyand p,p’ DDE on burrowing owl populationdynamics. Ecological Applications 16:666-677.

Heppell, S.S., H. Caswell, and L.B. Crowder.2000. Life histories and elasticity patterns:Perturbation analysis for species with min-imal demographic data. Ecology 81:654–665.

Hunter, C.M. and H. Caswell. 2005a. Selec-tive harvest of sooty shearwater chicks: ef-fects on population dynamics and sensitiv-ity. Journal of Animal Ecology 74:589-600.

Hunter, C.M. and H. Caswell. 2005b. Theuse of the vec-permutation matrix in spa-tial matrix population models. EcologicalModelling 188:15-21.

IPCC, 2007: Summary for Policymakers. In:Climate Change 2007: The Physical Sci-

21

ence Basis. Contribution of Working GroupI to the Fourth Assessment Report of theIntergovernmental Panel on Climate Change[Solomon, S., D. Qin, M. Manning, Z. Chen,M. Marquis, K.B. Averyt, M.Tignor andH.L. Miller (eds.)]. Cambridge UniversityPress, New York, NY, USA

Lawler, R.L., H. Caswell, A.F. Richard, J. Rat-sirarson, R.E. Dewar, and M. Schwartz. 2007.Demographic analysis of a protected lemurpopulation in southwest Madagascar. Sub-mitted.

Overland, J.E. and M. Wang. 2007. Future re-gional Arctic sea ice declines. GeophysicalResearch Letters 34: In press.

Regehr, E. V., S. C. Amstrup, and I. Stirling.2006. Polar bear population status in thesouthern Beaufort Sea. U.S. Geological Sur-vey Open-File Report 2006-1337.

Regehr, E.V., C.M. Hunter, H. Caswell, I. Stir-ling, and S.C. Amstrup . 2007. Polar bearsin the southern Beaufort Sea I: survival andbreeding in relation to sea ice conditions,2001–2006. USGS Alaska Science Center,Anchorage, Administrative Report.

Richter-Menge, J., J. Overland, A. Proshutin-sky, V. Romanovsky, L. Bengtsson, L. Brig-ham, M. Dyurgerov, J.C. Gascard, S. Ger-land, R. Graversen, C. Haas, M. Karcher,P. Kuhry, J. Maslanik, H. Melling, W. Mas-lowski, J. Morison, D. Perovich, R. Przyby-lak, V. Rachold, I. Rigor, A. Shiklomanov,J. Stroeve, D. Walker, and J. Walsh. 2006.State of the Arctic Report. NOAA OARSpecial Report, NOAA/OAR/PMEL, Seat-tle, WA, 36 pp.

Rigor, I.G., and J.M. Wallace. 2004. Variationsin the age of Arctic sea-ice and summer sea-ice extent. Geophysical Research Letters.31:L09401.

Runge, J.P., M.C. Runge, and J.D. Nichols.2006. The role of local populations withina landscape context: defining and classify-ing sources and sinks. American Naturalist167:925–938.

Runge, M. C., C. A. Langtimm and W. L. Kendall.2004. A stage-based model of manatee pop-ulation dynamics. Marine Mammal Science

20:361-385.Smith M, H. Caswell and P. Mettler-Cherry

2005. Stochastic flood and precipitationregimes and the population dynamics of athreatened floodplain plant. Ecological Ap-plications 15:1036–1052.

Stirling, I. 1988. Polar bears. University ofMichigan Press, Ann Arbor, Michigan, USA.

Stirling, I., M. Kingsley, and Calvert, W. 1982.The distribution and abundance of seals inthe eastern Beaufort Sea, 1974-79. Cana-dian Wildlife Service Occasional Paper 47.23 pp.

Stirling, I., T.L. McDonald, E.S. Richardson,and E.V. Regehr. 2007. Polar bear popu-lation status in the northern Beaufort Sea.USGS Alaska Science Center, Anchorage,Administrative Report.

Stroeve, J.C., M.M. Holland, W. Meier, T. Scam-bos, and M. Serreze. 2007. Arctic sea icedecline: Faster than forecast. GeophysicalResearch Letters 34(9):L09501, doi:10.1029/2007GL029703.

Tuljapurkar, S. 1990. Population dynamics invariable environments. Springer-Verlag, NewYork, New York, USA.

U.S. Fish and Wildlife Service. 2007. Endan-gered and threatened wildlife and plants:12-month petition finding and proposed ruleto list the polar bear (Ursus maritimus) asthreatened throughout its range. FederalRegister 72(5):1064–1099.

22

9 Tables

Table 1: Deterministic population growth rate λ, with 90% confidence intervals, standard error,and proportion of bootstrap samples < 1, for the time-invariant, full, and non-covariate model sets.

Model set Year λ lower CI upper CI SE proportion < 1time-invariant 0.997 0.755 1.053 0.105 0.57full 2001 1.059 0.083 1.093 0.269 0.24

2002 1.061 0.109 1.094 0.265 0.242003 1.036 0.476 1.107 0.207 0.412004 0.765 0.541 0.932 0.120 1.002005 0.799 0.577 0.959 0.122 0.99

non-cov 2001 1.017 0.810 1.088 0.092 0.432002 1.022 0.836 1.088 0.084 0.402003 1.075 0.903 1.129 0.077 0.192004 0.801 0.549 1.000 0.135 0.952005 0.895 0.446 1.020 0.185 0.88

23

Table 2: Ten IPCC AR-4 GCMs used to forecast ice covariates. IPCC model name, countryof origin, approximate grid resolution (degrees), and the number of runs used for demographicprojections.

IPCC model name Country Grid resolution Runsncar ccsm3 0 USA 1.0× 1.0 8cccma cgcm3 1 Canada 3.8× 3.8 1cnrm cm3 France 1.0× 2.0 1gfdl cm2 0 USA 0.9× 1.0 1giss aom USA 3.0× 4.0 1ukmo hadgem1 UK 0.8× 1.0 1ipsl cm4 France 1.0× 2.0 1miroc3 2 medres Japan 1.0× 1.4 1miub echo g Germany/Korea 1.5× 2.8 1mpi echam5 Germany 1.0× 1.0 1

24

10 Figures

2(3 yr)

1(2 yr)

3(4 yr)

Females

10(5+ yr

Adult)

8(3 yr)

7(2 yr)

9(4 yr)

Males

4(5+ Adult)

6(adult w/ yrlg)

5(adult w/ coy)

1

4(1- 4)

32

4 4

5(1-L0)(1-5)

6

5 L0

5(1- L0) 5

7

10

( 6 L1f)/2

( 6 L1f)/2

98

Figure 1: The polar bear life cycle graph; stages 1–6 are females and stages 7–10 are males. σi isthe probability of survival of an individual in stage i, σL0 and σL1 are the probability of at leastone member of a cub-of-the-year (COY) or yearling litter surviving to the following spring, f isthe expected size of yearling litters that survive to 2 years, and βi is the conditional probability,given survival, of an individual in stage i breeding, thereby producing a COY litter with at leastone member surviving until the following spring.

25

10.1 Deterministic model figures

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1000

2000

3000

Fre

quen

cy2001

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1000

2000

30002002

Fre

quen

cy

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1000

2000

3000

Fre

quen

cy

2003

0 0.2 0.4 0.6 0.8 1 1.2 1.40

250

500

750

10002004

Fre

quen

cy

0 0.2 0.4 0.6 0.8 1 1.2 1.40

250

500

750

1000

Population growth rate, λ

Fre

quen

cy

2005

−4 −2 0 20

500

1000

15002001

−4 −2 0 20

500

1000

15002002

−4 −2 0 20

500

1000

15002003

−4 −2 0 20

500

1000

15002004

−4 −2 0 20

500

1000

15002005

Log10

proportional population change

Figure 2: Bootstrap distribution of deterministic population growth rates (left column) and log ofpopulation size at 45 years relative to starting population size (right column), from the full modelset for conditions in 2001–2005. Bar at left of histogram (right column) gives all instances belowaxis minimum. Bootstrap sample size B = 10, 000.

26

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Pro

po

rtio

n

2001

1 2 3 4 5 60

0.2

0.4

0.6

0.8

12002

Pro

po

rtio

n

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Pro

po

rtio

n

2003

1 2 3 4 5 60

0.2

0.4

0.6

0.8

12004

Pro

po

rtio

n

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

Pro

po

rtio

n

Stage

2005

0

0.2

0.4

0.6

0.8

1

Ela

sticity

2001

0

0.2

0.4

0.6

0.8

12002

Ela

sticity

0

0.2

0.4

0.6

0.8

1

Ela

sticity

2003

0

0.2

0.4

0.6

0.8

12004

Ela

sticity

0

0.2

0.4

0.6

0.8

1

Ela

sticity

Parameter

2005

1 2 3 4 5 6 L0 4 5 f

Figure 3: Stable stage distribution (left column) and elasticity of λ to model-averaged parameterestimates (right column) from the full model set for conditions in 2001–2005.

27

-0.1

-0.05

0

2002

Co

ntr

ibu

tio

n

-0.1

-0.05

0

Co

ntr

ibu

tio

n

2003

-0.1

-0.05

0

2004

Co

ntr

ibu

tio

n

-0.1

-0.05

0

Co

ntr

ibu

tio

n

Parameter

2005

-0.1

-0.05

0

2002

Co

ntr

ibu

tio

n

-0.1

-0.05

0C

on

trib

utio

n

2003

-0.1

-0.05

0

2004

Co

ntr

ibu

tio

n

-0.1

-0.05

0

Co

ntr

ibu

tio

n

Parameter

2005

1 2 3 4 5 6 L0 4 5 f L1

Full model set Non-covariate model set

1 2 3 4 5 6 L0 4 5 f L1

Figure 4: LTRE decomposition of the contributions to differences in λ of parameters from thefull model (left column) and the non-covariate model set for 2002–2005. Differences are measuredrelative to conditions in 2001.

28

0 10 20 30 400

1

2

3

4

5

Pop

ulat

ion

size

Full model set

20012002200320042005

0 10 20 30 400

2

4

6

8

10

12

Pop

ulat

ion

size

Non−covariate model set

0 10 20 30 400

0.5

1

1.5

Pop

ulat

ion

size

Time

Time−invariant model set

10−3

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion Full model set

20012002200320042005

10−3

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1Non−covariate model set

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion

10−3

10−2

10−1

100

101

102

0

0.2

0.4

0.6

0.8

1Time−invariant model set

Proportional population change

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion

Figure 5: 45-year population projections (left column) for parameter estimates from the full, non-covariate and time-invariant model sets. Cumulative bootstrap probability distribution (right col-umn) of the log of population size at 45 years relative to the starting population size, from the full,non-covariate and time-invariant model sets.

29

60 80 100 120 140 160 180 2000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Number of ice−free days

Pop

ulat

ion

grow

th r

ate

Figure 6: Solid line: population growth rate λ as a function of the ice covariate estimated from thefull model set. Diamonds: population growth rate estimated for 2001–2005 from the non-covariatemodel set.

30

0 0.02 0.04 0.06 0.08 0.10.5

0.6

0.7

0.8

0.9

1

1.1

Pop

ulat

ion

grow

th r

ate,

λ

Harvest rate

2001, 2002

2003

Time−invariant

2004

2005

Figure 7: Population growth rate, λ, as a function of harvest rate for model-averaged parameterestimates from the full model set for 2001–2005 and for time-invariant estimates. Harvest is assumedto affect only stages 1–4. The vertical dashed line is the estimated harvest rate of females withoutdependent young.

31

10.2 Stochastic model figures

−1

−0.5

0

0.5Full model set, non−increasing

−30

−20

−10

0

10

Sto

chas

tic g

row

th r

ate

Full model set, increasing

−0.6

−0.4

−0.2

0

0.2Full model set, non−increasing

−15

−10

−5

0

5Full model set, increasing

0 0.5 1−1.5

−1

−0.5

0

0.5Non−covariate models

Frequency (q ) of bad years0 0.5 1

−1

−0.5

0

0.5Non−covariate models

Frequency (q ) of bad years

Figure 8: The stochastic growth rate log λs as a function of the frequency q of years with greaterthan 127 ice-free days. Multiple lines show a sample of 100 models from the bootstrap distributionof (from top to bottom) the full model set conditional on non-increasing response, the full modelset conditional on increasing response, and the non-covariate model set. The right panels show theestimates of log λs and 90% parametric bootstrap confidence intervals.

32

1975 1980 1985 1990 1995 2000 2005 20100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Year

Pro

babi

lity

of b

ad ic

e ye

ar

u=1.2523

Figure 9: Frequency of bad ice conditions from 1979–2006, smoothed with a Gaussian kernelsmoother with standard deviation u = 1.5, 2, 3 (Copas 1983). Data from Figure 3 in Part I of thisreport (Regehr et al. 2007).

33

1 2 3 4 5 6 L0 4 5 f L1

Parameter

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Full model set

Ela

sticity o

f sto

chastic g

row

th r

ate

q=0.01

0.5

0.99

Figure 10: Elasticity of the stochastic growth rate λs for three values of the probability q of badice conditions for parameter estimates from the full model set.

34

10−4

10−2

100

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportional population change

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion

Full model set, 45 year projection

q=0.010.250.50.750.99

10−4

10−2

100

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportional population change

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion

Full model set, q = 0.25

t=4575100

10−4

10−2

100

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportional population change

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion

Non−covariate models, 45 year projection

q=0.010.250.50.750.99

10−4

10−2

100

102

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Proportional population change

Cum

ulat

ive

prob

abili

ty d

istr

ibut

ion

Non−covariate models, q = 0.25

t=4575100

Figure 11: Cumulative distribution of population change from the full and non-covariate modelsets. Left panels show the cdf at t = 45 years as a function of the frequency q of bad years. Rightpanels show the cdf at 45, 75, and 100 years for q = 0.25. Bootstrap sample of size B = 2000.

35

0 20 40 60 80 1000

100

200

300P

roje

cted

ice−

free

day

s

0 20 40 60 80 1000

100

200

300

400

Pro

ject

ed s

cale

d ic

e−fr

ee d

ays

0 20 40 60 80 1000

0.5

1

Years

Pro

babi

lity

of b

ad ic

e ye

ar

Figure 12: Upper panel: projected ice-free days from each of 10 climate models, from 2005–2100.Middle panel: projected ice free days rescaled to agree with the observed average ice-free days from2000–2005. Lower panel: the projected probability of a bad ice year, from 2005–2100, from all 10climate models.

36

0 20 40 60 80 1000

10

20

30

40

50

60

Year

Pro

port

iona

l pop

ulat

ion

size

Full model set

0 20 40 60 80 1000

5

10

15

20

25

30

35

Year

Pro

port

iona

l pop

ulat

ion

size

Non−covariate model set

Figure 13: Stochastic projections based on climate models. Plotted are 200 bootstrap samplestaken over both vital rate variability and all 10 climate models.

37

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

Year

Ext

inct

ion

prob

abili

ty

Full model set

Extinction = 0.0010.01

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Year

Ext

inct

ion

prob

abili

ty

Non−covariate model set

Extinction = 0.0010.01

Figure 14: Probability of quasi-extinction, defined as a decline to a fraction 0.001 or 0.01 of initialpopulation size, obtained from 1000 bootstrap samples combining all 10 climate models.

38

A Appendix figures

0 0.2 0.4 0.6 0.8 1 1.2 1.40

1000

2000

3000

Fre

quen

cy

Full model set, 2001

0 0.2 0.4 0.6 0.8 1 1.2 1.40

500

1000Full model set, 2005

0 0.2 0.4 0.6 0.8 1 1.2 1.40

500

1000

1500

Fre

quen

cy

Non−covariate model set, 2001

0 0.2 0.4 0.6 0.8 1 1.2 1.40

500

1000

1500Non−covariate model set, 2005

Population growth rate, λ

0 0.2 0.4 0.6 0.8 1 1.2 1.40

500

1000

1500

Population growth rate, λ

Fre

quen

cy

Time−invariant model set

Figure A-1: Bootstrap distributions of deterministic population growth rates, λ, from the full,non-covariate and time-invariant model sets for conditions in 2001 and 2005. Bootstrap sample sizeB = 10, 000.

39

0

0.2

0.4

0.6

0.8

1

Se

nsitiv

ity

Full model set

0

0.2

0.4

0.6

0.8

1

Ela

sticity

Full model set

0

0.2

0.4

0.6

0.8

1

Se

nsitiv

ity

Non-covariate model set

0

0.2

0.4

0.6

0.8

1

Ela

sticity

Non-covariate model set

0

0.2

0.4

0.6

0.8

1

Parameter

Se

nsitiv

ity

Time-invariant model set

0

0.2

0.4

0.6

0.8

1

Parameter

Ela

sticity

Time-invariant model set

1 2 3 4 5 6 L0 4 5 f1 2 3 4 5 6 L0 4 5 f

Figure A-2: Sensitivity (left column) and elasticity (right column) of population growth rate toparameter estimates from the full, non-covariate and time-invariant model sets for conditions in2001.

40

−4 −2 0 20

500

1000

1500F

requ

ency

Full model set, 2001

−4 −2 0 20

500

1000

1500Full model set, 2005

−4 −2 0 20

500

1000

1500

Fre

quen

cy

Non−covariate model set, 2001

−4 −2 0 20

500

1000

1500Non−covariate model set, 2005

Log10

proportional population change

−4 −2 0 20

500

1000

1500

Fre

quen

cy

Time−invariant model set

Log10

proportional population change

Figure A-3: Distribution of the log of population size at 45 years relative to the starting populationsize for parameter estimates from the full and non-covariate model sets in 2001 and 2005.

41

−4 −2 0 20

500

1000

1500

Fre

quen

cy

45 year projection, 2001

−4 −2 0 20

500

1000

1500

Fre

quen

cy

75 year projection, 2001

−4 −2 0 20

500

1000

1500

Fre

quen

cy

Log10

proportional population change

100 year projection, 2001

−4 −2 0 20

500

1000

150045 year projection, 2005

−4 −2 0 20

500

1000

150075 year projection, 2005

−4 −2 0 20

500

1000

1500

Log10

proportional population change

100 year projection, 2005

Figure A-4: Distribution of the log of population size at 45, 75 and 100 years relative to the startingpopulation size for parameter estimates from the full model set in 2001 and 2005.

42

s1 s2 s3 s4 s5 s6 sL0 b4 b5 f sL10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Non-covariate model set

Parameter

Ela

sticity o

f sto

chastic g

row

th r

ate

q=0.01

0.5

0.99

1 2 3 4 5 6 L0 4 5 f L1

Parameter

Figure A-5: Elasticity of the stochastic growth rate λs for three values of the probability q of badice conditions for parameter estimates from the non-covariate model set.

43

050

100

0

0.2

0.4

0.6

0.81

ccsm

3 04

0

050

100

0

0.2

0.4

0.6

0.81

cgcm

3 t4

7

050

100

0

0.2

0.4

0.6

0.81

cnrm

cm

3

050

100

0.5

0.6

0.7

0.8

0.91

gfdl

cm

2

050

100

0

0.2

0.4

0.6

0.81

giss

aom

050

100

0

0.2

0.4

0.6

0.81

hadg

em1

Probability of bad ice year

050

100

0.4

0.5

0.6

0.7

0.8

0.91

ipsl

cm

4

050

100

0

0.2

0.4

0.6

0.81

miro

c32

med

res

050

100

0

0.2

0.4

0.6

0.81

miu

b ec

ho

050

100

0

0.2

0.4

0.6

0.81

mpi

ech

am5

Yea

rs

Fig

ure

A-6

:P

roje

cted

prob

abili

tyof

bad

ice

year

s,fr

om20

05–2

100,

from

each

of10

GC

Mcl

imat

em

odel

s.

44

050

100

0

0.51

1.52

2.53

ccsm

3 04

0

050

100

0

0.51

1.52

cgcm

3 t4

7

050

100

012345cn

rm c

m3

050

100

0

0.51

1.5

gfdl

cm

2

050

100

0

0.51

1.52

2.5

giss

aom

050

100

012345ha

dgem

1

050

100

0

0.2

0.4

0.6

0.81

1.2

1.4

ipsl

cm

4

050

100

0

0.51

1.52

2.53

miro

c32

med

res

050

100

0102030405060m

iub

echo

050

100

024681012m

pi e

cham

5

Yea

rs

Proportional population size

Fig

ure

A-7

:St

ocha

stic

proj

ecti

onof

tota

lpop

ulat

ion

size

base

don

sea

ice

proj

ecti

ons

from

10G

CM

clim

ate

mod

els

(see

text

for

deta

ils).

The

mul

tipl

elin

esin

each

pane

lar

ea

set

of50

boot

stra

psa

mpl

esfr

omth

ew

eigh

ted

mod

els

for

the

vita

lra

tes

(ful

lm

odel

set)

,ap

plie

dto

asi

ngle

real

izat

ion

ofth

een

viro

nmen

t.

45

050

100

0123456cc

sm3

040

050

100

0246810cg

cm3

t47

050

100

0246810cn

rm c

m3

050

100

0123456gf

dl c

m2

050

100

0246810gi

ss a

om

050

100

02468ha

dgem

1

050

100

012345ip

sl c

m4

050

100

01234567m

iroc3

2 m

edre

s

050

100

05101520253035m

iub

echo

050

100

024681012m

pi e

cham

5

Yea

rs

Proportional population size

Fig

ure

A-8

:St

ocha

stic

proj

ecti

onof

tota

lpop

ulat

ion

size

base

don

sea

ice

proj

ecti

ons

from

10G

CM

clim

ate

mod

els

(see

text

for

deta

ils).

The

mul

tipl

elin

esin

each

pane

lare

ase

tof

50bo

otst

rap

sam

ples

from

the

wei

ghte

dm

odel

sfo

rth

evi

talr

ates

(non

-cov

aria

tem

odel

set)

,ap

plie

dto

asi

ngle

real

izat

ion

ofth

een

viro

nmen

t.

46