U.S. Department of the Interior

U.S. Geological Survey

Open-File Report 2015–1029

Prepared in cooperation with the U.S. Fish and Wildlife Service

Resilience and Risk—A Demographic Model to Inform 

Conservation Planning for Polar Bears
Cover: An adult female polar bear and her three yearling cubs on the sea ice of the Chukchi Sea, spring 2010. 

Photograph taken by Eric V. Regehr, U.S. Fish and Wildlife Service.
Resilience and Risk—A Demographic Model to Inform 

Conservation Planning for Polar Bears

By Eric V. Regehr, Ryan R. Wilson, Karyn D. Rode, and Michael C. Runge

Prepared in cooperation with the U.S. Fish and Wildlife Service

Open-File Report 2015–1029

U.S. Department of the Interior

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

SALLY JEWELL, Secretary

U.S. Geological Survey

Suzette M. Kimball, Acting Director

U.S. Geological Survey, Reston, Virginia: 2015 

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Suggested citation:

Regehr, E.V., Wilson, R.R., Rode, K.D., and Runge, M.C., 2015, Resilience and risk—A demographic 

model to inform conservation planning for polar bears: U.S. Geological Survey Open-File Report 2015-

1029, 56 p., http://dx.doi.org/10.3133/ofr20151029. 

ISSN 2331-1258 (online)
iii

Contents

Executive Summary....................................................................................................................................... 1

Introduction.................................................................................................................................................... 2

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

Demographic Model................................................................................................................................... 5

Life Cycle Graph..................................................................................................................................... 5

Vital Rates.............................................................................................................................................. 6

Density Dependence .............................................................................................................................. 6

Relationships between Vital Rates and Density.................................................................................. 8

Maximum Net Productivity Level......................................................................................................... 8

Metabolic Energetic Equivalents....................................................................................................... 10

Environmental Variation ....................................................................................................................... 11

Temporal Variation in Carrying Capacity .......................................................................................... 11

Density-Independent Variation.......................................................................................................... 11

Human-Caused Removals ................................................................................................................... 12

Population Projections.......................................................................................................................... 12

Evaluating ESA and MMPA Criteria......................................................................................................... 13

Proposed Metrics for ESA Criteria........................................................................................................ 13

Proposed Metrics for MMPA Criteria .................................................................................................... 14

Simulations for ESA Criteria................................................................................................................. 15

Simulations for MMPA Criteria ............................................................................................................. 17

Inputs to the State-Dependent Management Framework ................................................................. 17

Robustness of the State-Dependent Management Framework ........................................................ 18

Software................................................................................................................................................... 19

Results......................................................................................................................................................... 19

ESA Criteria ............................................................................................................................................. 23

MMPA Criteria.......................................................................................................................................... 27

Discussion ................................................................................................................................................... 35

Demographic Model................................................................................................................................. 35

ESA Criteria ............................................................................................................................................. 37

MMPA Criteria.......................................................................................................................................... 39

Comparison with Current Polar Bear Harvest Management................................................................. 42

Definition of Sustainability for Polar Bear Harvest Management .......................................................... 42

Acknowledgments ....................................................................................................................................... 43

References Cited......................................................................................................................................... 43

Appendix A. Methods to Adapt Published Vital Rates to the Polar Bear Life Cycle Graph ......................... 52

Appendix B. Methods to Generate Density-Dependent Functions for the Vital Rates ................................ 53

Appendix C. Methods to Generate a Sea Ice-Based Proxy for Carrying Capacity...................................... 55

Appendix D. Methods to Estimate Parameters Using Simulated Population Assessments ........................ 56
iv

Figures 

Figure 1. Polar bear life cycle graph ............................................................................................................6

Figure 2. Graphs showing the model of density dependence for polar bears ..............................................9

Figure 3. Contour plot of the recruitment metric as a function of survival and breeding probability ............ 20

Figure 4. Contour plot of maximum net productivity level (MNPL) as a function of 

survival and recruitment............................................................................................................................... 21

Figure 5. Contour plot of per capita population growth rate at maximum net productivity level (rMNPL) 

as a function of survival and recruitment ..................................................................................................... 22

Figure 6. Contour plot of the combinations of survival and recruitment necessary to achieve a 

90-percent probability of persistence over 100 years for different levels of carrying capacity...................... 25

Figure 7. Graphs showing example projections for a small population with and without human-caused 

removals...................................................................................................................................................... 26

Figure 8. Graph showing probability of population size decreasing to less than NMNPL (P<MNPL) as a 

function of the management factor (FO), for different levels of data precision.............................................. 29

Figure 9. Graph showing relationship between the probability of extirpation (1-Ppersist) and the 

probability of population size decreasing to less than NMNPL (P<MNPL) .......................................................... 30

Figure 10. Graph showing annual yield as a function of time, for different levels of the 

management factor (FO) .............................................................................................................................. 31

Figure 11. Graphs showing example population projections with and without human-caused

removals for stable and declining carrying capacity..................................................................................... 34

Tables

Table 1. Estimated vital rates for polar bear subpopulations.........................................................................7

Table 2. Metabolic energetic equivalents for polar bears ............................................................................ 10

Table 3. Summary of key components in simulations to evaluate Endangered Species Act (ESA) and 

Marine Mammal Protection Act (MMPA) criteria.......................................................................................... 16

Table 4. Levels of data precision used in simulated population assessments............................................. 17

Table 5. Range of survival rates that must be maintained, in conjunction with sufficient recruitment, to 

achieve a 90-percent probability of persistence over 100 years .................................................................. 24

Table 6. Threshold values of the management factor (FO) used to calculate human-caused removals 

under the state-dependent management framework, which meet conservation goals based on a 

placeholder degree of risk tolerance............................................................................................................ 28

Table 7. Increased risk of extirpation compared to no human-caused removals (ΔPextirpation), 

under a sample application of the state-dependent management framework, for different rates of 

declining carrying capacity........................................................................................................................... 33
v

Conversion Factors

International System of Units to Inch/Pound

Multiply By To obtain

Length

kilometer (km) 0.6214 mile (mi)

kilometer (km) 0.5400 mile, nautical (nmi) 

Mass

kilogram (kg) 2.2054 pound avoirdupois (lb)
vi

Abbreviations, Acronyms, and Symbols

Adult Polar bear aged 5 or more years

C0 Cub-of-the-year

C1 Yearling

C2 Two-year-old polar bear still dependent on its mother

CI Confidence interval

CMP Polar Bear Conservation Management Plan

CS Chukchi Sea 

CV Coefficient of variation

ΔPextirpation Increased risk of extirpation

ESA Endangered Species Act

FO Factor that reflects management objectives and risk tolerance when calculating 

sustainable harvest rate

hMNPL Maximum rate of human-caused removals that still maintains populations above the 

equilibrium population size at MNPL

κ Proportional change in the number of ice-covered days

K Carrying capacity 

mee Metabolic energetic equivalent

MMPA Marine Mammal Protection Act

MNPL Maximum net productivity level

N Population size

NMNPL Equilibrium population size at MNPL

OSP Optimum sustainable population

P<MNPL Probability that population size is less than MNPL

PBR Potential biological removal method for assessing take

PBRT Polar Bear Recovery Team

Ppersist Probability of persistence

PVA Population viability analysis

r Per capita population growth rate

rmax Maximum per capita population growth rate at low densities and in the absence of 

human-caused removals

rMNPL Per capita population growth rate at MNPL and in the absence of human-caused 

removals

SB Southern Beaufort Sea 

SR Factor that specifies the male-to-female ratio in human-caused removals

Subadult Independent polar bear aged 2–4 years

Subpopulation One of the 19 polar bear subpopulations recognized by the International Union for 

the Conservation of Nature

t Annual time step in a population projection

TK Traditional knowledge

USFWS U.S. Fish and Wildlife Service

USGS U.S. Geological Survey
1

Resilience and Risk—A Demographic Model to Inform 

Conservation Planning for Polar Bears

By Eric V. Regehr1, Ryan R. Wilson1, Karyn D. Rode2, and Michael C. Runge2

Executive Summary

Climate change is having widespread ecological effects, including loss of Arctic sea ice. This 

has led to listing of the polar bear (Ursus maritimus) and other ice-dependent marine mammals under 

the U.S. Endangered Species Act. Methods are needed to evaluate the effects of climate change on 

population persistence to inform recovery planning for listed species. For polar bears, this includes 

understanding interactions between climate and secondary factors, such as subsistence harvest, which 

provide economic, nutritional, or cultural value to humans. 

We developed a matrix-based demographic model for polar bears that can be used for population 

viability analysis and to evaluate the effects of human-caused removals. This model includes densitydependence (the potential for a declining environmental carrying capacity), density-independent 

limitation, and sex- and age-specific harvest vulnerabilities. We estimated values of adult female 

survival (0.93–0.96), recruitment (number of yearling cubs per adult female; 0.1–0.3), and carrying 

capacity (>250 animals) that must be maintained for a hypothetical population to achieve a 90-percent

probability of persistence over 100 years. 

We also developed a state-dependent management framework, based on harvest theory and the 

potential biological removal method, by linking the demographic model to simulated population 

assessments. This framework can be used to estimate the maximum sustainable rate of human-caused 

removals, including subsistence harvest, which maintains a population at its maximum net productivity 

level. The framework also can be used to calculate a recommended sustainable harvest rate, which 

generally is lower than the maximum sustainable rate and depends on management objectives, the 

precision and frequency of population data, and risk tolerance. The historical standard 4.5-percent 

harvest rate for polar bears, at a 2:1 male-to-female ratio, is reasonable under many biological and 

management conditions, although lower or higher rates may be appropriate in some cases. 

1

U.S. Fish and Wildlife Service. 2

U.S. Geological Survey. 
2

Our modeling results suggest that harvest of polar bears is unlikely to accelerate population 

declines that result from declining carrying capacity caused by sea-ice loss, provided that several 

conditions are met: (1) the sustainable harvest rate reflects the population’s intrinsic growth rate, and the 

corresponding harvest level is obtained by applying this rate to an estimate of population size; (2) the 

sustainable harvest rate reflects the quality of population data (e.g., lower harvest when data are poor); 

and (3) the level of human-caused removals can be adjusted. Finally, our results suggest that stopgap 

measures (e.g., further reduction or cessation of harvest when the population size is less than a critical 

threshold) may be necessary to minimize the incremental risk associated with harvest, if environmental 

conditions are deteriorating rapidly. We suggest that the demographic model and approaches presented 

here can serve as a template for conservation planning for polar bears and other species facing similar 

challenges. 

Introduction

Climate change is expected to be a primary driver of ecological change and biodiversity loss in 

the foreseeable future (Groffman and others, 2014). This may be especially true in the Arctic (Post and 

others, 2009; Convey and others, 2012) where the rate of warming has been approximately twice the 

global mean (Intergovernmental Panel on Climate Change, 2013). In the Arctic marine environment, 

warming temperatures and changing circulation patterns have led to declines in the extent, temporal 

availability, and thickness of sea ice (Maslanik and others, 2007; Stroeve and others, 2012). Sea-ice loss 

is expected to continue throughout the 21st century with some models predicting a nearly ice-free Arctic 

in the summer within 20 years (Overland and Wang, 2013). Arctic marine mammals depend on sea ice 

for many aspects of their life history (Laidre and others, 2008; Kovacs and others, 2011). In recent 

years, several species of Arctic marine mammals have been listed under the U.S. Endangered Species 

Act (ESA) prior to the occurrence of range-wide population declines, based on climate change concerns 

(Schliebe and others, 2006; Cameron and others, 2010; Kelly and others, 2010). These listings present 

new challenges, including the fact that the government agencies primarily responsible for implementing 

recovery plans do not have the regulatory authority to address global greenhouse gas emissions (e.g., 

U.S. Fish and Wildlife Service, 2013), the primary cause of anthropogenic climate change. Near-term 

recovery actions likely will focus on addressing secondary factors or threats, with the goal of increasing 

the chances of persistence until global action leads to a stabilized climate system (Ruhl, 2008; Seney 

and others, 2013). Demographic models can be used to evaluate the effects of climate change on 

persistence, including interactions between climate and secondary factors. Such models also can 

establish links between conservation goals and specific metrics for monitoring, provide an objective 

assessment of the benefits of recovery actions, and inform decisions about the tradeoffs that will 

necessarily be part of recovery planning. 

Polar bears (Ursus maritimus) exemplify the challenges discussed here. The species was listed 

as “threatened” under the ESA in 2008 based on observed and forecasted population declines owing to 

sea-ice loss (U.S. Fish and Wildlife Service, 2008). Of the 19 recognized polar bear subpopulations 

(Obbard and others, 2010), multiple lines of evidence suggest that 2 subpopulations have experienced 

sea ice-related declines to date (Stirling and others 1999; Regehr and others, 2007, 2010; Bromaghin 

and others, in press). Several subpopulations apparently are either productive or stable despite sea-ice 

loss (Obbard and others, 2007; Stirling and others, 2011; Stapleton and others, 2012; Peacock and 

others, 2013; Rode and others, 2014), and many subpopulations are data deficient (Obbard and others, 

2010). Although the current status of polar bear subpopulations is variable, all polar bears depend on sea 

ice for fundamental aspects of their life history (Amstrup, 2003), including access to their primary prey, 
3

ringed seals (Phoca hispida) and bearded seals (Erignathus barbatus). The long-term ecological effects 

of continued climate change are expected to be negative for most subpopulations (Stirling and Derocher, 

2012). Effective conservation planning will require consideration of regional and temporal variation in 

the manifestation of such effects. 

Polar bears represent an important cultural, nutritional, and in some regions economic resource 

for native people throughout the Arctic (Dowsley, 2009; Born and others, 2011; Voorhees and others, 

2014). Fifteen polar bear subpopulations, including the two subpopulations that occur partially within 

United States territory, currently support a legal subsistence harvest (Laidre and others, 2015). When 

polar bears were listed under the ESA, the species was classified as “depleted” under the U.S. Marine 

Mammal Protection Act (MMPA; 16 USC §1361 et seq.). Both the ESA and MMPA prohibit taking 

(i.e., to harass, hunt, capture, or kill) protected species, with several exemptions including take by 

coastal-dwelling Alaskan Natives for subsistence purposes including making and selling handicrafts. 

Concurrent with the ESA listing, the U.S. Fish and Wildlife Service (USFWS, 2013) published a 

Section 4(d) “Special Rule,” establishing that take regulations for polar bears would continue as 

determined by the MMPA prior to listing. Although the ESA listing concluded that overutilization (i.e., 

excessive take) does not currently threaten polar bears, future interactions between climate change and 

human-caused removals are less clear. For example, habitat loss could increase vulnerability to 

overutilization due to smaller population sizes or reduced intrinsic population growth rates. Climate 

change also could lead to increased human-bear conflicts (Dyck, 2006) if longer ice-free seasons result 

in more polar bears on shore for longer periods (Schliebe and others, 2008; Towns and others, 2009; 

U.S. Geological Survey, unpub. data, 2014). In addition to the biological considerations listed here, how 

polar bears are managed likely will be subject to increased international attention and politicization in 

coming years (Peacock and others, 2011). Thus, although loss of Arctic sea ice is the primary long-term 

threat to polar bears, subsistence harvest likely will represent a key secondary factor in future 

discussions about the species. 

In this report, we develop a demographic model for polar bears that can be used to perform

population viability analysis (PVA; Beissinger and McCullough, 2002) and to evaluate the effects of 

human-caused removals (i.e., the combination of subsistence harvest, animals killed in conflicts, and 

other sources of human-caused mortality). The model extends previous work for polar bears (Hunter 

and others, 2010) by (1) including density-dependent functions to link vital rates (e.g., survival and 

recruitment) to changes in population size and environmental carrying capacity; (2) including sex- and 

age-specific harvest vulnerabilities; and (3) linking the demographic model to simulated population 

assessments, so that management strategies can be evaluated in light of uncertain information and time 

lags (e.g., Milner-Gulland, 2011). 

A primary motivation for developing this demographic model was to inform conservation 

planning for polar bears currently underway within the United States. In 2013, the USFWS convened a 

Polar Bear Recovery Team (PBRT) to fulfill statutory requirements of the ESA and MMPA. The team 

was tasked with developing a plan that includes “objective, measurable” criteria for the recovery of 

polar bears under the ESA, as well as measures to conserve and restore the species to its optimum 

sustainable population (OSP) under the MMPA. Optimal sustainable population is defined as being 

greater than the maximum net productivity level (MNPL), which is the number of animals that results in 

maximum sustainable yield (Wade, 1998). Development of criteria that meet these ESA and MMPA 

requirements, and strategies for achieving the criteria, are being combined into a single plan referred to 

as the Polar Bear Conservation Management Plan (CMP). An initial draft of the CMP is being 

developed by the PBRT and is scheduled to be released for public comment in 2015. 
4

The PBRT has approached conservation planning for polar bears using a three-tier framework. 

Fundamental goals are at the highest tier and reflect overarching desires for the species, including 

maintaining population persistence, diversity (e.g., genetic, behavioral, ecological), and opportunities 

for subsistence use. Within the draft CMP, these fundamental goals are expressed as recovery criteria, 

which define conditions under which protection of the ESA is no longer needed, and conservation 

criteria, which reflect management goals under the MMPA. One tier below the fundamental goals and 

associated ESA and MMPA criteria are quantitative demographic criteria. The demographic criteria are 

more proximate, measurable, and specific expressions of the ESA and MMPA criteria. The 

demographic criteria can serve as proxies for achieving the fundamental goals, and can be used to guide 

recovery actions. A third and final tier consists of specific threats-based criteria that must be addressed 

to achieve recovery. 

As members of the Science and Traditional Ecological Knowledge Working Group of the PBRT, 

the authors of this report were asked to develop a demographic model for polar bears, and to 

demonstrate how to estimate demographic criteria that correspond to higher level ESA and MMPA 

criteria. To do this, we first estimated combinations of vital rates that could serve as proxies for 

population persistence, a potential metric for expressing fundamental goals under the ESA. These vital 

rates are easier to measure than the probability of persistence and can be used as an initial indicator of 

whether, if conditions are maintained, a specified target for persistence is likely to be met. They also can 

be used to evaluate tradeoffs and efficiencies when planning recovery actions—for example, whether to 

focus on improving recruitment or survival (Runge and others, 2004). Second, we estimated the 

sustainable rate of human-caused removals that could maintain populations greater than the MNPL, as 

required by the MMPA. We expected this rate to vary as a function of population status, uncertainty in 

the information available for management, time lags in the management system, and risk tolerance (i.e., 

how certain that managers want to be of population size remaining greater than the MNPL). Therefore, 

rather than estimating a unique sustainable rate, we developed a state-dependent management 

framework that can be used to estimate the sustainable rate of human-caused removals under different 

conditions. State-dependent management refers to using updated information on the state of a system to 

inform decisions about management of the system (Lyons and others, 2008). 

In this report, we present the polar bear demographic model, proposed metrics for demographic 

criteria that correspond to candidate ESA and MMPA criteria, and estimates of threshold values for 

these metrics. Our intention was not to predict the future status of polar bears; we projected hypothetical

polar bear populations forward in time under different conditions, for the purpose of evaluating ESA 

criteria, but did not use the model for inference about the future of specific subpopulations (e.g., Hunter 

and others, 2010) or of the species (e.g., Amstrup and others, 2008). Furthermore, our intention was not 

to prescribe a specific management approach. We, therefore, estimated metrics related to human-caused 

removals based on a placeholder degree of risk tolerance. Several structural elements of the 

demographic model were informed by data for the Chukchi Sea (CS) and Southern Beaufort Sea (SB) 

polar bear subpopulations, because these subpopulations occur within U.S. territory (Obbard and others, 

2010) and are a focus of the draft CMP. We intend for the structure and application of the demographic 

model to be refined in an adaptive manner as information on polar bears improves and recovery actions 

are implemented. 
5

Methods

We first describe the demographic model and how it can be used to project hypothetical polar 

bear populations forward in time. We then propose demographic metrics to serve as proximate measures 

of ESA and MMPA criteria, and finally describe simulations consisting of multiple population 

projections with different combinations of vital rates, environmental conditions, and management 

approaches, which were used to estimate threshold values for these metrics. 

Demographic Model

Life Cycle Graph

We constructed a matrix-based population model (Caswell, 2001), based on a life cycle graph 

(fig. 1) developed previously for a multistate capture-recapture analysis (Regehr and others, 2010) and 

demographic assessment (Hunter and others, 2007, 2010) of the SB polar bear subpopulation. The life 

cycle graph is based on a post-breeding census from the spring of year t to the spring of year t + 1. It 

includes six female stages representing age and reproductive status, and four male stages representing 

age. Transitions between stages are defined by the following vital rates: 

• σi is the probability that an individual in stage i survives from time t to t + 1; 

• σL0 and σL1 are the probabilities that at least one member of a cub-of-the-year (C0) or

yearling (C1) litter, respectively, survives from time t to t + 1; 

• f is the expected size of a C1 litter that survives to 2 years; and

• βi is the conditional probability, given survival, of an individual in stage i breeding and 

thereby producing a C0 litter with at least one member surviving until the following spring. 

All survival rates in the life cycle graph represent natural (i.e., unharvested) survival. 

For the current analysis, we modified the life cycle graph from Hunter and others (2007, 2010) 

by adding a transition between stages 3 and 5, to allow 4-year-old females to breed and produce a C0 

litter in the following year. Although uncommon in the SB, this occurs in some other subpopulations 

(e.g., Lunn and others, 2014). 
6

Figure 1. Polar bear life cycle graph (modified from Hunter and others, 2010, fig. 2). Stages 1–6 are females and 

stages 7–10 are males. σi is the probability of survival of an individual in stage i, σL0 and σL1 are the probabilities of 

at least one member of a cub-of-the-year (C0) or yearling (C1) litter surviving to the following spring, f is the 

expected size of C1 litters that survive to 2 years, and βi is the conditional probability, given survival, of an 

individual in stage i breeding, thereby producing a C0 litter with at least one member surviving until the following 

spring.

Vital Rates

We obtained time-invariant estimates of vital rates for polar bears (table 1) from the scientific 

literature and from proceedings of the Polar Bear Specialist Group of the International Union for the 

Conservation of Nature (Obbard and others, 2010). These vital rates were used to inform some aspects 

of the demographic model and to define the range of conditions under which hypothetical polar bear 

populations were projected, as described throughout section, “Methods.” We adapted the published vital 

rates, which were derived from capture-recapture models with different structures, to the life cycle 

graph using conventions described in appendix A. 

Density Dependence

It is necessary to include density dependence in a demographic model, if the model will be used 

to evaluate persistence relative to changing carrying capacity or human-caused removals (Guthery and 

Shaw, 2013). We developed density-dependent functions based on polar bear biology, population 

dynamics theory, and the nutritional requirements of polar bears. 
7

Table 1. Estimated vital rates for polar bear subpopulations.

[Survival rates are estimates of natural (i.e., unharvested) survival unless otherwise noted. Age classes are cub-of-the-year (C0), yearling (C1), subadult (2–4 yr) and adult (≥5 yr).

Litter production rate is defined in Obbard and others (2010). yr, year; ≥, greater than or equal to] 

Survival Reproduction

Subpopulation Female Male Litter production rate C0 litter 

size

Reference

C0 C1 Subadult Adult C0 C1 Subadult Adult 5 yr ≥6 yr

Baffin Bay 0.62 0.94 0.94 0.95 0.57 0.94 0.94 0.95 0.88 1.00 1.59 Taylor and others, 

2005

Davis Strait-Central1 0.92 0.94 0.94 0.96 0.92 0.94 0.94 0.97 0.54 0.44 1.49 Peacock and others, 

2013

Davis StraitNorthern10.89 0.92 0.92 0.95 0.89 0.92 0.90 0.94 0.54 0.44 1.49 Peacock and others, 

2013

Davis StraitSouthern10.90 0.93 0.92 0.96 0.90 0.93 0.91 0.94 0.54 0.44 1.49 Peacock and others, 

2013

Gulf of Boothia 0.89 0.90 0.90 0.96 0.89 0.90 0.90 0.96 0.19 0.97 1.65 Taylor and others, 

2009

Kane Basin 0.41 0.76 0.76 1.00 0.35 0.66 0.66 1.00 0.00 0.98 1.67 Taylor and others, 

2008a

Lancaster Sound 0.75 0.90 0.90 0.95 0.63 0.84 0.84 0.97 0.11 0.95 1.69 Taylor and others, 

2008b

McClintock Channel 0.62 0.98 0.98 0.98 0.62 0.98 0.98 0.92 0.11 0.93 1.70 Taylor and others, 

2006; Obbard 

and others, 2010

Northern Beaufort 

Sea2

0.52 0.33 0.91 0.91 0.52 0.32 0.83 0.83 0.28 0.88 1.76 Stirling and others, 

2011

Norwegian Bay 0.75 0.90 0.90 0.95 0.63 0.84 0.84 0.97 0.54 0.54 1.71 Taylor and others, 

2008b

Southern Beaufort 

Sea

0.34 0.93 0.93 0.97 0.34 0.90 0.90 0.96 0.44 0.44 1.72 Obbard and others, 

2010; Regehr 

and others, 2010

Southern Hudson 

Bay2

0.73 0.72 0.92 0.91 0.59 0.58 0.86 0.86 0.97 0.97 1.58 Obbard and others, 

2007, 2010

Viscount Melville 0.69 0.96 0.96 0.96 0.45 0.92 0.92 0.92 0.62 0.87 1.64 Taylor and others, 

2002

Western Hudson Bay 0.73 0.73 0.92 0.93 0.71 0.71 0.94 0.94 0.26 0.79 1.54 Regehr and others, 

2007; Obbard 

and others, 2010

1

Peacock and others (2013) estimated vital rates for three regions within the Davis Strait subpopulation.

2

Estimates of total survival not corrected for harvest mortality.
8

Relationships between Vital Rates and Density

For long-lived species, relationships between vital rates and density are decreasing, likely 

convex, and have been represented in various ways including the theta-logistic and Ricker equations 

(Morris and Doak, 2002). We defined these relationships using a logistic equation as described in 

appendix B. Sample density-dependent curves for several vital rates are shown in figure 2a. In the 

absence of empirical data for polar bears (Taylor, 1994), we used insights from population and 

evolutionary ecology to develop the functions. For large mammals, density-dependent effects typically 

appear first in subadult survival rates, then in breeding rates and juvenile survival, and finally in adult 

survival (Fowler, 1987). The relative positions of the inflection points of the curves for each vital rate 

were determined by the order in which polar bear life-history events are affected by density (e.g., σL0

typically decreases before β4). Studies in evolutionary ecology show that there is a negative correlation 

between the variance of a vital rate and its importance to population growth (“demographic buffering”; 

Pfister, 1998). Thus, the sensitivity of each vital rate to density was inversely proportional to its 

elasticity, as calculated using a matrix population model based on the life cycle graph (Caswell, 2001). 

To evaluate the sensitivity of the results to the form of density dependence, we developed an 

alternative linear density-dependent function (sample curves in fig. 2d). This function represented the 

classic logistic growth model for populations (Ross, 2009), which is unrealistic for polar bears (Taylor, 

1994) but was useful for two reasons. First, the classic logistic growth model has been widely used in 

wildlife management (Caughley and Sinclair, 1994) and thus provided a benchmark for comparison 

with our density-dependent functions. Second, we expected the model to provide a more conservative 

assessment of human-caused removals because linear relationships between vital rates and density 

would result in lower MNPL and weaker density-dependent changes in vital rates at population sizes 

close to MNPL. 

Maximum Net Productivity Level

The density-dependent functions of the vital rates (figs. 2a and 2d) underlie several key 

demographic behaviors. For each set of curves, there is a unique combination of vital rates that produces 

a per capita population growth rate (r) equal to 0. By definition, this unique combination of vital rates 

corresponded to a location on the x-axis representing the equilibrium population size (N) at carrying 

capacity (K), at which the dimensionless ratio N/K = 1.0 (figs. 2a and 2d). We used the densitydependent functions to derive yield curves, which describe the relationship between population density 

and sustainable removals. Maximum net productivity level is defined as the ratio N/K at which the 

annual sustainable removals were maximized (figs. 2b and 2e). When performing population 

projections, equilibrium population size at MNPL was calculated as NMNPL(t) = MNPL × K(t); where K

is carrying capacity, expressed as a number of animals, and referenced to time (t) because projections 

included temporal variation in K, as described in subsequent sections. The per capita growth rates rmax

and rMNPL provide measures of a population’s capacity for growth, at small population sizes and at 

NMNPL, respectively (figs. 2c and 2f). In this report, we use rMNPL as a measure of a population’s 

resilience, defined as its capacity to grow (in the absence of human-caused removals) following 

reduction to a population size of less than K. Reported estimates of rMNPL were calculated assuming a 

stable stage distribution and asymptotic population dynamics (Caswell, 2001); potential departures from 

these assumptions are discussed where they occur. 
9

Figure 2. Graphs showing the model of density dependence for polar bears. Sample results are shown for a 

population with medium resilience; the per capita growth rate at a population density corresponding to maximum 

net productivity level (rMNPL) was 0.043. Density is expressed as the ratio of population size (N) to carrying capacity 

(K). σ4 is the probability of survival of an individual in stage 4, σL0 and σL1 are the probability of at least one 

member of a cub-of-the-year (C0) or yearling (C1) litter surviving to the following spring, and β4 is the conditional 

probability, given survival, of an individual in stage 4 breeding, thereby producing a C0 litter with at least one 

member surviving until the following spring. The parameter r is the asymptotic per capita population growth rate. 

Vital rates were nonlinear (left column of panels) or linear (right column of panels) functions of density. Panels (a) 

and (d) are sample density-dependent curves of the vital rates. The dashed vertical lines correspond to N/K = 1 at 

carrying capacity. Panels (b) and (e) are yield curves, which show normalized annual sustainable removals as a 

function of density. The vertical lines correspond to N/K = 0.69 at maximum net productivity level. Panels (c) and (f) 

show per capita growth rate as a function of density.
10

Metabolic Energetic Equivalents

We hypothesized that density dependence in polar bears is largely the result of direct (e.g., 

contest) and indirect (e.g., scramble) competition for nutritional resources (Taylor, 1994; Rode and 

others, 2012). However, individual polar bears vary greatly in body size and nutritional requirements 

(Stirling and Oritsland, 1995), and, therefore, can be expected to vary in their contributions to 

competitive effects. To account for this variation, we calculated metabolic energetic equivalent (mee) 

values as the mean body mass for each sex and age class, raised to the 2/3 power to convert from actual 

body mass to metabolic body mass (White and Seymour, 2003; Capellini and others, 2010), and 

standardized to give a mee value of 1.0 for solitary adult females. We used data on polar bear mass from 

the CS and SB subpopulations (Rode and others, 2014; U.S. Fish and Wildlife Service and U.S. 

Geological Survey, unpub. data, 2014). We further modified mee values to reflect sex- and age-based 

variation in polar bear diet. Nutritional ecology studies suggest that 10–20 percent of the diet of adult 

male polar bears consists of large prey items—for example, bearded seals and beluga whales 

(Delphinapterus leucas), which are less accessible to females and younger bears (Thiemann and others, 

2008; Cherry and others, 2011). Therefore, we reduced the mee value for adult males by 15 percent to 

reflect the portion of male diet that does not overlap with, or cause competition with, polar bears in 

other sex and age classes. The resulting mee values (table 2) represent the energetic requirements of 

each animal, relative to the requirements of solitary adult females. 

Carrying capacity at the beginning of each population projection [i.e., K(t=1)] was specified as a 

number of animals, and subsequently was transformed into energetic equivalents based on the starting 

stage distribution (see section, “Population Projections”) and the sex- and age-specific mee values (table 

2). To accommodate the different energetic demands of individual polar bears, population density at 

each time step t = 1,2... k was based on the sum of mee values for the individuals in the population 

(rather than the number of individuals) divided by K(t) expressed as energetic equivalents. Thus, K

within the demographic model can be understood as the total amount of energy accessible to polar bears 

at any given point in time, and population density can be understood as the fraction of that energy being 

used by polar bears. 

Table 2. Metabolic energetic equivalents for polar bears.

[Values estimated using body mass data for the Chukchi Sea and Southern Beaufort Sea polar bear subpopulations. Age classes are cub-ofthe-year (C0), yearling (C1), 2-year-old (C2), subadult (2–4 yr) and adult (≥5 yr). yr, year; ≥, greater than or equal to] 

C0 C1 C2 Subadult Adult

both sexes both sexes female male female male female male

0.2 0.6 0.7 0.9 0.8 1.0 1.0 1.3
11

Environmental Variation

The Arctic marine environment is highly variable (Walsh, 2008), which can influence 

persistence even when population sizes are large (Boyce and others, 2006). Given the primary threat to 

polar bears of climate change, methods to evaluate persistence should incorporate relationships between 

environmental variation and vital rates. We represented environmental variation as the additive effects 

of two components: (1) temporal variation in K, which operated on vital rates through the densitydependent functions; and (2) density-independent temporal variation in vital rates.

Temporal Variation in Carrying Capacity

We included temporal variation in K in a manner that was biologically relevant to polar bears. 

As described in appendix C, we used satellite imagery for the CS and SB regions to calculate the 

number of days per year (1979–2013) that sea ice covered greater than (>) 50 percent of the mean sea 

ice area in March, when maximum ice extent usually occurs. We then projected the proportional change 

in the number of ice-covered days, relative to a baseline period. The resulting dimensionless metric (κ) 

captures the variance and trend in the duration of the period that polar bears likely have greatest access 

to their prey, and is similar to metrics used to link sea ice to polar bear demography and nutritional 

status (Regehr and others, 2010; Peacock and others, 2013; Rode and others, 2014; Lunn and others, 

2014). 

When performing population projections, carrying capacity at any point in time was calculated 

as K(t) = K(t=1) × κ(t). The resulting values of K(t) represent a proxy for true environmental carrying 

capacity, which is unknown, and were used only for the purpose of exploring the demographic 

consequences of variation in the Arctic environment. All population projections included interannual 

variation in κ. As described subsequently, population projections included several multiples of the 

temporal trend in κ, ranging from 0 to 2 times the observed sea-ice trend within the CS and SB 

subpopulation boundaries, depending on whether the goal of the projection was to evaluate populations 

under conditions of stable or decreasing K. We did not use the demographic model to infer actual 

probabilities of persistence for the CS and SB subpopulations because that would require the additional 

assumption that values of K(t) were accurate predictors of future environmental carrying capacity. 

Density-Independent Variation

Density-independent variation in vital rates for polar bears can arise from weather fluctuations 

(Stirling and Smith, 2004), changes in prey populations (Stirling and Lunn, 1997), or other factors. We 

assumed that such variation constitutes 25 percent of total uncertainty (i.e., temporal variation plus 

sampling uncertainty) in estimated vital rates (table 1), following the example of Taylor and others 

(2002). The resulting estimate of the temporal coefficient of variation for σ4, the most important vital 

rate, was 0.014. Because population persistence is influenced by both the magnitude and correlation 

structure of variation (Doak and others, 2005), we derived a correlation matrix from annual estimates of 

vital rates for the SB subpopulation (Regehr and others, 2010), and used this correlation matrix for all 

population projections. We determined that there was nearly complete correlation among survival rates 

(e.g., a mean correlation coefficient of 0.99 between σ4 and other survival rates) and positive correlation 

between survival and reproductive rates (e.g., 0.67 between σ4 and β4). 
12

We used correlated non-normal distributions to capture density-independent temporal variation

in vital rates. To simulate these distributions, we used methods described in Morris and Doak (2002, p. 

282–287) to transform correlated standard normal deviates into correlated values, from either a 

multivariate beta distribution (for all vital rates in the life cycle graph except f) or a stretched beta 

distribution (for C0 litter size, which was used to calculate f). For the vital rates β3 and β4, shape 

parameters for the beta distribution were occasionally inestimable due to high variances. In such cases, 

these vital rates were randomly selected from a uniform distribution with a range [0, 0.5] or [0.5, 1.0] 

depending on whether the standard normal deviate was less than or greater than 0.5, respectively. 

Human-Caused Removals

Human-caused removals were implemented annually at a specified rate (i.e., fraction of 

population size) or level (i.e., number of animals) depending on the management scenario included in a 

given projection. The reproductive value of polar bears varies with sex and age (Hunter and others, 

2007). Furthermore, subsistence harvests generally are sex-selective, and young polar bears are more 

likely than older bears to be killed by humans due to behavioral and nutritional factors (Derocher and 

others, 1997; Dyck, 2006). To reflect this variation, the demographic model included stage-specific 

harvest vulnerabilities (e.g., Taylor and others, 2005) so that removals could be allocated accurately 

over life cycle stages. We estimated harvest vulnerability for subadults compared to adults using the 

stage distribution for the SB subpopulation from 2001 to 2006 (Regehr and others, 2010) and hunterreported age classes of harvested bears during the same period (U.S. Fish and Wildlife Service, unpub. 

data 2010). This suggested that subadults of both sexes (stages 1–3 and 7–10) were twice as likely to be 

killed by humans, relative to their stage distribution, compared to adults (stages 6 and 10). Similar 

harvest vulnerabilities have been estimated using 28 years of data for the western Hudson Bay 

subpopulation (Derocher and others, 1997). We set harvest vulnerability to 0.1 (relative to single adult 

females in stage 4) for stages 5 and 6, because taking adult females with dependent young is restricted 

and occurs rarely in most subpopulations (Brower and others, 2002). 

Population Projections

For a given population projection, inputs included maximum values of vital rates (i.e., survival 

and reproductive rates at low density), the starting value of K expressed as a number of animals, and the 

rate of human-caused removals. Populations started with a stable stage distribution at a population size 

equal to NMNPL, which was calculated from the starting value of K [e.g., NMNPL ≈ 0.69 × K(t=1), as 

described in section, “Results”]. Density-dependent curves for the vital rates were generated as 

described in section, “Density Dependence”. At each time step t = 2,3,...k, the following operations 

were performed. 

First, populations were projected forward 1 year using a stage-structured matrix model: n(t+1) = 

A(t) × n(t), where n(t) is a stage distribution vector representing the number of animals in each life cycle 

stage at time step t, and A(t) is a 10×10 matrix that projects the population from time t to t + 1 (Caswell, 

2001). Entries in A(t) were defined in terms of vital rates in the life cycle graph (fig. 1). Second, humancaused removals were allocated among stages using a multinomial distribution with the probability for 

each stage calculated as the product of its proportional stage distribution and harvest vulnerability. If the 

target removal level for a particular stage exceeded the number of bears in that stage, excess removals 

were allocated to the more populous stages 4 or 10, for females or males, respectively. If the male 

segment of the population was exhausted, which happened occasionally for male-biased removals at 

high rates, excess male removals were allocated to stage 4 females. Third, population density, N/K, was 
13

calculated by summing mee values across animals in the population, then dividing by the total mee

values available at carrying capacity. The survival and reproductive rates corresponding to this density 

were determined from the density-dependent curves. Fourth, these vital rates were subject to densityindependent variation, as described previously. Finally, the resulting vital rates were used to construct a 

projection matrix for the next time step A(t+1). 

For each set of conditions under which a population was projected, we performed 1,000 

iterations to assess variability. For each iteration, we recorded (1) demographic outputs including the 

time-constant values of MNPL, rMNPL, and rmax; n(t) and a vector of stage-specific removals at each time 

step; and (2) whether the population persisted or the time step at which it was extirpated. We defined 

persistence as maintaining a population size greater than a quasi-extinction threshold of 15 percent of 

starting population size or 100 animals, whichever was larger. We used a proportional quasi-extinction 

threshold because of potential Allee effects caused by reduced breeding encounters and mating success 

at low population densities (Molnár and others, 2008). Low population densities could result if N

declines while the amount of habitat during the breeding season remains stable. This could occur for 

polar bears because they mate in April and May (Amstrup, 2003), and projected sea-ice loss at that time 

of the year is relatively small (Stroeve and others, 2012). We used the lower quasi-extinction threshold 

of 100 animals because demographic stochasticity, which our model did not include, and other smallpopulation effects can become important below this level (Wieglus and others, 2001; Morris and Doak, 

2002). 

Evaluating ESA and MMPA Criteria

In this section, we propose metrics for demographic criteria that correspond to higher-level ESA 

and MMPA criteria. We then demonstrate how the demographic model can be used to estimate 

threshold values for these metrics.

Proposed Metrics for ESA Criteria 

We consider the following candidate criterion for recovery under the ESA: “the probability of 

persistence in each ecoregion is at least 90 percent over 100 years.” Ecoregions refer to the four 

grouping of polar bears defined by Amstrup and others (2008) based on spatial and temporal dynamics 

of sea ice, which are consistent with broad patterns of life history and genetic diversity for polar bears

(Peacock and others, 2015). The following three metrics were defined to serve as proxies for 

persistence, and, taken together, can be used to infer whether the goal for persistence is met: 

1. Natural (i.e., unharvested) survival of adult females. Adult female survival is the most important 

determinant of population growth for polar bears (Eberhardt, 1990) and is often a focus of 

demographic studies (Amstrup and Durner, 1995). 

2. Recruitment, measured as the ratio of C1s to adult females in the population. This metric 

integrates breeding success and C0 survival (Rode and others, 2014). Populations need 

recruitment to persist. Furthermore, recruitment may vary more than adult survival for long-lived 

animals and thus drive most of the observed changes in population growth (Mitchell and others, 

2009). 

3. Carrying capacity (K) expressed as a number of animals. This represents an approximate lower 

limit on the number of animals that the environment must be able to support, for a subpopulation 

with sufficient survival and recruitment (as defined by the first two metrics) to persist in the face 

of environmental variation and other chance events (Flather and others, 2011). 
14

We chose these three metrics in light of several considerations. First, climate change is leading 

to unprecedented ecological changes in the Arctic (Hinzman and others, 2013), including the potential 

for declining polar bear populations (Stirling and Derocher, 2012). In this report, we estimate threshold 

values for these metrics, which are values that must be met for populations to persist in accordance with 

the candidate ESA recovery criterion. Additional work is necessary to evaluate whether, and for how 

long, specific subpopulations are likely to meet these threshold values. Second, recruitment and survival 

are functions of population density for large mammals, including bears (Taylor, 1994; Zedrosser and 

others, 2006; Czetwertynski and others, 2007). To be meaningful, threshold values for these metrics 

must be associated with a particular population size relative to carrying capacity. We reference survival 

and recruitment to population size at MNPL (i.e., NMNPL) because this population size is relevant to 

harvest theory (Runge and others, 2009) and requirements under the MMPA (Wade, 1998). Third, for 

populations subject to human-caused removals, total survival is a function of natural (i.e., unharvested) 

survival and direct human-caused mortality (e.g., Peacock and others, 2013). The metric proposed here 

is natural survival, which—in conjunction with recruitment—determines a population’s capacity for 

growth. For natural survival to be a meaningful predictor of persistence, human-caused removals must 

be conducted at a sustainable rate that does not drive populations far below NMNPL. 

Proposed Metrics for MMPA Criteria 

We consider the following candidate criterion for conservation under the MMPA: “each 

subpopulation is managed so that its population size is at least at the maximum net productivity level 

relative to carrying capacity, both currently and as forecast over the next 50 years.” We define the 

metric hMNPL as the maximum rate of human-caused removals that still achieves this goal. Our 

framework for estimating hMNPL extends the formula for allowable take in Runge and others (2009), 

which was based on the potential biological removal (PBR) method (Wade, 1998) and harvest theory 

(Caughley and Sinclair, 1994). We calculated human-caused removal levels for females and males as 

follows: 

Hffffff(t) = FO × r̃MMMM(t) × 0.5 × N�(t), and (1) 

H (t) = Hffffff(t) × SS, (2) 

where H female is the number of females that can be removed annually while meeting the 

conservation goal;

FO is a factor that reflects management objectives and risk tolerance;

r̃MMMM is an estimate of the per capita population growth rate referenced to population 

density at MNPL (i.e., rMNPL), selected from its sampling distribution to represent risk 

tolerance; 

0.5 is a factor to calculate female removals assuming an equal sex ratio in the population; 

N� is an estimate of N selected from its sampling distribution to represent risk tolerance; 

H male is the number of males that can be removed annually; and 

SR is a factor that specifies the male-to-female ratio in removals. 

The notation for time (t) indicates that parameters change annually or as new information becomes 

available. Equations 1 and 2 are written in terms of removal levels (i.e., numbers of animals) for 

convenience. The removal rate hMNPL for females is the right side of equation 1 before multiplying by 

population size [i.e., hMMMM

ffffff = FO × r̃MMMM(t)]. 
15

The management factor FO directly adjusts the removal rate, and is conceptually similar to the 

“recovery factor” in the PBR method (Wade, 1998). Under ideal conditions, populations will stabilize at 

approximately NMNPL when FO = 1. Increasing FO will result in an equilibrium population size of less 

than NMNPL, until an upper limit on FO is reached. Beyond the upper limit, the removal rate will exceed 

the maximum per capita growth rate (rmax) and is no longer sustainable. This upper limit is FO = 2 for 

the classic logistic model of population growth (Runge and others, 2009), and in reality is expected to

vary among species based on life history (Williams, 2013). The practical use of FO is to direct a 

population toward a target size, and to specify risk tolerance with respect to missing that target or 

achieving some other undesired outcome, such as extirpation. Such risks can result from stochasticity, 

uncertain information about population status, time lags in management actions, and other imperfections 

in the modeling or management framework. 

The parameter N� makes this approach state-dependent with respect to population size. This 

ensures that a constant removal rate will be maintained in the face of density-dependent effects—for 

example, declining carrying capacity due to habitat loss. The parameter r̃MMMM makes this approach 

state-dependent with respect to density-independent effects that could limit a population’s capacity for 

growth. It also bases the approach on a parameter that is more relevant and readily measured for 

populations that are managed for maximum sustainable yield, compared to previous formulations that 

included a time-constant value of rmax (Runge and others, 2009). Both rMNPL and N must be estimated 

and, therefore, will include sampling uncertainty. Furthermore, if a direct estimate of r is available from 

a population assessment, it may need to be adjusted to approximate rMNPL if population density is not 

close to MNPL. Selection of r̃MMMMand N� from within their sampling distributions is a decision 

reflecting the risk tolerance of managers, which will affect the removal level and thus the probability of 

meeting conservation goals.

The parameter SR can be used to implement sex-specific removals, which is practical for polar 

bears because most subsistence harvests are designed for a male-to-female removal ratio greater than 1 

(Taylor and others, 2008c). The term 0.5 in equation 1 could be replaced with an estimate of the 

proportion of females in the population. We have included 0.5 as a conservative management rule 

because most polar bear harvests select for males and the resulting proportion of females in populations 

tends to be greater than 0.5 

Simulations for ESA Criteria

We estimated threshold values for the survival, recruitment, and carrying capacity metrics by 

projecting populations under the conditions described below. For each projection, the key outcome was 

the probability of persistence (Ppersist), calculated as the proportion of iterations that did not cross below 

the quasi-extinction threshold between the start of the projection and the final time step t = 100. 

Projections were performed for all combinations of the following inputs: 

• 400 sets of input vital rates. This parameter space was intended to broadly represent the range of 

vital rates that polar bears could exhibit. It was constructed by combining 20 equal-increment 

intervals in natural survival, from 3 percent less than the minimum values to the maximum 

values observed in case studies (table 1); with 20 equal-increment intervals in breeding 

probability and C0 litter size, from 30 percent less than the minimum values to the maximum 

values observed in case studies (table 1). 

• 6 values for carrying capacity, K(t=1) = 4,000; 2,000; 1,000; 500; 250; and 150 animals; with 

populations starting at a stable stage distribution and a population size corresponding to NMNPL

relative to K. 
16

• Two approaches to human-caused removals: (1) Removals at a constant rate rMNPL, using SR = 1 

and with removals implemented in direct proportion to stage distribution (i.e., without databased harvest vulnerability), and (2) No removals. 

These input combinations resulted in 400 × 6 × 2 = 4,800 projections. Threshold values for the metrics 

were the values of survival, recruitment, and K that resulted in Ppersist >0.90. Inference for these metrics 

was based on projections with human-caused removals at rMNPL. Including human-caused removals

referenced survival and recruitment to NMNPL, making the estimated threshold values for these metrics 

broadly relevant to subpopulations that are harvested. It also ensured that estimates of Ppersist reflected 

low equilibrium population sizes that result from harvest. Comparison of threshold values for the 

metrics, estimated from projections with and without human-caused removals, served as an initial 

evaluation of how removals at a sustainable rate affect population persistence. Key components of this 

and subsequent simulations are summarized in table 3. 

Table 3. Summary of key components in simulations to evaluate Endangered Species Act (ESA) and Marine 

Mammal Protection Act (MMPA) criteria.

Simulation

Component ESA Criteria

MMPA Criteria

(Inputs to the state-dependent 

management framework)

MMPA Criteria

(Robustness of the state-dependent 

management framework)

Population projections:

Parameter space 400 sets of vital rates 3 sets of vital rates corresponding to 

high, medium, and low resilience

400 sets of vital rates

Density dependence density-dependent functions 

using logistic equation (e.g., fig. 2a, 

b, and c)

density-dependent functions using 

logistic equation (e.g., fig. 2a, b,

and c) 

density-dependent functions using 

logistic equation (e.g., fig. 2a, b, 

and c)

Starting carrying capacity [K(t=1)] 4000, 2000, 1000, 5000, 205, 

and 150

1000 1000

Interannual variation in K yes yes yes

Temporal trend in K no no 3 trends

Human-caused removals:

Management factor (FO) 1 31 values from 0.5 to 1.25 0.75

Data precision in simulated 

population assessments (table 4)

true levels 1 to 4 level 3

Sex ratio (SR) 1 2 2

Data-based harvest vulnerability no yes yes

Other:

Key outcome probability of persistence (Ppersist) probability that population size is 

less than maximum net productivity 

level (P<MNPL) 

increased risk of extirpation compared 

to no human-caused removals 

(ΔPextirpation) 

 
17

Simulations for MMPA Criteria

We explored the framework for the candidate MMPA criterion by performing two simulations. 

First, we demonstrated how to calculate inputs to the state-dependent management framework necessary 

to meet conservation goals in the presence of uncertain information on population status and time lags in 

management. Second, we demonstrated how to evaluate the robustness of the state-dependent 

management framework by using a single set of inputs to calculate harvest levels that were applied to

populations with different vital rates and under conditions of stable and declining K. 

Inputs to the State-Dependent Management Framework

We calculated inputs to the state-dependent management framework by projecting populations 

under the conditions described below. For each projection, the key outcome was the probability of 

violating the candidate MMPA criterion, calculated as the proportion of iterations for which N <NMNPL

at the final time step t = 50 (i.e., P<MNPL). Projections were performed for all combinations of the 

following inputs: 

• Three levels of population resilience corresponding to rMNPL = 0.085, 0.043, and 0.015; 

subsequently referred to as high, medium, and low resilience, respectively. These values are 

within the range of r values estimated using the vital rates in table 1, and, therefore, are 

representative of observed per capita growth rates for polar bears. 

• Five levels of data precision. During population projections, N� and the vital rates used to 

estimate r̃MMMM were derived from simulated population assessments that included sampling 

uncertainty (table 4). The first level of data precision corresponded to true values of N and the 

vital rates, updated on an annual basis, to evaluate the effects of human-caused removals under 

perfect sampling and management. Other levels of data precision were based on the amount of 

sampling uncertainty in case studies for polar bears, using the methods described in appendix D. 

These provided a realistic assessment of the effects of imperfect information on polar bear 

management. 

Table 4. Levels of data precision used in simulated population assessments. 

[Coefficients of variation (CV) for adult female survival (σ4) 

and population size (N), representing the amount of sampling

uncertainty in simulated population assessments.]

Data 

precision 

level

Adult female 

survival

CV(σ4)

Population size

CV(N) 

true1 0.000 0.00

1 0.003 0.04

2 0.008 0.08

3 0.018 0.15

4 0.089 0.25

1

Simulated population assessments used the true

values of all population parameters, updated annually

instead of on a 10-year management interval.
18

• Thirty-one equal-increment values of FO from 0.5 to 1.25. We expected that the lower limit of 

0.5 would result in conservative removal rates because the conceptually related recovery factor 

in the PBR method often is set to 0.5 for threatened stocks (Wade and Angliss, 1997). We 

expected that the upper limit of 1.25 would encompass most sustainable removal rates. 

• Three methods to select N�, corresponding to the lower 5th, lower 15th, and 50th percentiles of 

the sampling distribution for N from simulated population assessments. Using a lower percentile 

of N to calculate removal levels, as opposed to using the mean value, protects against 

overestimates when uncertainty is large (Wade, 1998). 

These input combinations resulted in 3 × 5 × 31 × 3 = 1,395 projections. Threshold values for the 

candidate MMPA criterion were the values of FO, and the method to select N� that achieved the 

conservation goal described below. Simulated population assessments were performed at 10-year 

management intervals, and the annual number of human-caused removals remained constant between 

management intervals. Removals were implemented using SR = 2 and data-based harvest vulnerability. 

All projections used starting carrying capacity K(t=1) = 1,000 animals. The parameter r̃MMMM was 

selected as the 50th percentile of the sampling distribution for rMNPL. We included a management rule 

constraining r̃MMMM ≤0.10, to protect against excessive removal rates when per capita growth rates were 

high and data precision was low. 

In the face of uncertainty, any management framework carries a risk of violating the candidate 

MMPA criterion. Thus, to be fully specified, the MMPA criterion also needs to include a risk tolerance. 

That is, a complete criterion would identify a target value for P<MNPL that would have allowed us to 

calculate corresponding inputs to the state-dependent management framework. Because such guidance 

does not yet exist, we used a placeholder degree of risk tolerance by inferring an upper limit on P<MNPL

based on the candidate ESA criterion. Specifically, this upper limit corresponds to human-caused 

removals that, absent all other threats, would decrease the probability of persistence to less than 90 

percent. We estimated this limit by plotting (1 - Ppersist) as a function of P<MNPL, selecting a polynomial 

fit using Akaike’s Information Criterion and standard regression methods (Zar, 2010), and identifying 

the value of P<MNPL for which the upper 95-percent prediction interval corresponded to (1 – Ppersist) = 

0.10. We present this placeholder degree of risk tolerance to illustrate how the demographic model can 

be used to calculate inputs to the state-dependent management framework that achieve a specified 

conservation goal. 

 Robustness of the State-Dependent Management Framework 

The goal of the second simulation was to assess the robustness of the state-dependent 

management framework, using a single set of inputs to equations 1 and 2, under a wide range of 

conditions including declining K. The key outcome was the incremental change in the probability of 

extirpation (ΔPextirpation), calculated as the difference in Ppersist for identical projections with, and without, 

human-caused removals. Projections were performed for all combinations of the following inputs: 

• 400 sets of vital rates, as defined by the parameter space. 

• Three rates of change in the proxy for K, corresponding to (1) twice the mean temporal trend in 

κ, as estimated from sea-ice data for the CS and SB regions; (2) the mean trend in κ; and (3) no 

temporal trend (i.e., stable carrying capacity).

• Two approaches to human-caused removals: (1) State-dependent management using FO = 0.75, 

SR = 2, and N� selected as the lower 15th percentile of the sampling distribution for N, with a 

data precision level of 3; and (2) No removals.
19

These combinations of inputs resulted in 400 × 3 × 2 = 2,400 projections. Other inputs were identical to 

the first simulation for the candidate MMPA criterion. 

Software

The demographic model was built using the R computing language (version R 3.1.0; The R 

Project for Statistical Computing; http://www.r-project.org). The package “popbio” was used to analyze 

matrix models (Stubben and others, 2008). Computations were performed on the Amazon Elastic 

Compute Cloud (http://aws.amazon.com/ec2/), where we ran R using an Amazon Machine Image for 

RStudio Server (RStudio, 2014) developed by L. Aslett (http://www.louisaslett.com/RStudio_AMI/). 

Results

Across the 400 sets of vital rates that made up the parameter space, the recruitment metric, 

defined as the number of C1s per adult female, ranged from 0.09 to 0.46. The recruitment metric was 

positively correlated with the natural survival rate for adult females and with breeding probability (fig. 

3). The mean estimate of MNPL was 0.69 (95-percent confidence interval [CI] = 0.63–0.74). 

Specifically, hypothetical populations produced maximum sustainable yield at an equilibrium 

population size that was 69 percent of the population size at carrying capacity. Maximum net 

productivity level was negatively correlated with survival and recruitment (fig. 4). That is, populations 

with higher vital rates (and thus higher resilience) produced maximum sustainable yield at smaller 

equilibrium sizes, compared to populations with low resilience. Values of rMNPL ranged from -0.09 to 

0.12. The mean value of the ratio rMNPL/rmax was 0.82 (95-percent CI = 0.79–0.84). As expected, rMNPL

was positively correlated with survival and recruitment (fig. 5). Observed survival and recruitment rates 

for polar bear subpopulations (double boxplot in fig. 5) provide an orientation to the most relevant 

portions of the contour plot. Under the assumption that vital rates in table 1 were for harvested 

subpopulations at densities near MNPL, the corresponding median value of rMNPL would be 

approximately 0.04 based on the location of the crosshairs in the double boxplot. For comparison, the 

actual mean estimate of natural (i.e., unharvested) r for the case studies was 0.05 (95-percent CI = 0.02–

0.09), as shown by the asterisk. The estimates reported in this paragraph are conditional on several 

assumptions of the demographic model, particularly the form of our density-dependent functions. 

The linear density-dependent functions of the vital rates (fig. 2d), which are not realistic for 

polar bears and were included for sensitivity analysis, resulted in a mean estimate of MNPL of 0.49 (95-

percent CI = 0.48–50). The mean value of the ratio rMNPL/rmax also was 0.49 (95-percent CI = 0.48–

0.50). This corresponded closely to the expected behavior of the classic logistic growth model for 

populations, for which MNPL = rMNPL/rmax = 0.50. 
20

Figure 3. Contour plot of the recruitment metric as a function of survival and breeding probability. Recruitment is 

the number of yearlings per adult female. Survival is the natural adult female survival rate, averaged over the three 

adult female stages (4, 5, and 6) in the life cycle graph. Breeding probability is the parameter β4, the conditional 

probability, given survival, of an individual in stage 4 breeding, thereby producing a cub-of-the-year litter with at 

least one member surviving until the following spring. Vital rates are referenced to population density at maximum 

net productivity level. 
21

Figure 4. Contour plot of maximum net productivity level (MNPL) as a function of survival and recruitment. Survival 

is the natural adult female survival rate, averaged over the three adult female stages (4, 5, and 6) in the life cycle 

graph. Recruitment is the number of yearlings per adult female. Vital rates are referenced to population density at 

maximum net productivity level. The dashed region encompasses the combinations of survival and recruitment that 

are feasible under the demographic model given the life cycle of polar bears (fig. 1), density-dependent functions 

for the vital rates (fig. 2), and parameter space. 
22

Figure 5. Contour plot of per capita population growth rate at maximum net productivity level (rMNPL) as a function 

of survival and recruitment. Survival is the natural adult female survival rate, averaged over the three adult female 

stages (4, 5, and 6) in the life cycle graph. Recruitment is the number of yearlings per adult female. Vital rates are 

referenced to population density at maximum net productivity level. The double boxplot shows the location of 

survival and recruitment values calculated using vital rates from case studies (table 1). The asterisk is the mean 

estimated per capita growth rate for case studies. Values for the Northern Beaufort Sea and Southern Hudson Bay 

subpopulations are not shown because estimates of natural survival were not available. 
23

ESA Criteria

We report threshold values of the survival, recruitment, and carrying capacity metrics necessary 

to achieve a probability of persistence (Ppersist) ≥90 percent over 100 years, which was used as an 

example of a candidate ESA recovery criterion. For populations subject to human-caused removals at a 

rate equal to rMNPL, natural adult female survival rates from 0.93 to 0.96, depending on recruitment, are 

necessary for persistence when K ≥1,000 animals (table 5a and fig. 6). As K decreases to less than

approximately 1,000, the required range of vital rates shifts higher, because the quasi-extinction 

threshold switched from a proportional value, calculated as 15 percent of starting population size, to a 

fixed value of 100 animals. Thus, the relative distance between K and the quasi-extinction threshold was 

reduced. As K decreases to less than approximately 250, the effects of environmental variation become 

dominant. At such small population sizes, even populations with the capacity for positive growth (i.e., 

rMNPL >0) risk crossing below the quasi-extinction threshold due to stochasticity. We note that, for 

K = 250, the upper limit on survival required for persistence was approximately 0.97 for most sets of 

vital rates. For projections with K = 150, there were no vital rates that resulted in persistence. This is 

because the equilibrium size of populations subject to human-caused removals was approximately 

NMNPL = 150 × 0.69 = 104, which was so close to the quasi-extinction threshold that all populations 

crossed below it due to stochasticity. 

Threshold values for the proposed demographic metrics were identical for populations with and 

without human-caused removals when K ≥500 (table 5). Survival rates necessary for persistence started 

to differ as K decreased to less than approximately 250. This is not because human-caused removals 

were unsustainable in the sense of causing a negative population trend; all removals were implemented 

at exactly rMNPL. Rather, human-caused removals resulted in lower equilibrium population sizes that 

were closer to the quasi-extinction threshold, and, therefore, were more likely to cross below it due to 

stochasticity (fig. 7). 
24

Table 5. Range of survival rates that must be maintained, in conjunction with sufficient recruitment, to achieve a 

90-percent probability of persistence over 100 years.

[Rates are natural survival of adult females, referenced to maximum net productivity level (MNPL). Panel (a) is for populations subject to 

human-caused removals at exactly the natural per capita growth rate referenced to MNPL (i.e., rMNPL). Panel (b) is for populations without 

human-caused removals. *Asterisks indicate that persistence was not possible even if survival rates were 1.0]

(a)

Survival

Carrying 

capacity

Equilibrium 

population size from to

4000 2760 0.93 0.95

2000 1380 0.93 0.95

1000 690 0.93 0.95

500 345 0.93 0.96

250 173 0.94 0.99

150 104 * *

(b)

Survival

Carrying 

capacity

Equilibrium 

population size from to

4000 4000 0.93 0.95

2000 2000 0.93 0.95

1000 1000 0.93 0.95

500 500 0.93 0.96

250 250 0.94 0.96

150 150 0.94 0.97
25

Figure 6. Contour plot of the combinations of survival and recruitment necessary to achieve a 90-percent

probability of persistence over 100 years for different levels of carrying capacity. Survival is the natural adult female 

survival rate, averaged over the three adult female stages (4, 5, and 6) in the life cycle graph. Recruitment is the 

number of yearlings per adult female. Vital rates are referenced to population density at maximum net productivity 

level. The three lines correspond to different levels of carrying capacity. 
26

Figure 7. Graphs showing example projections for a small population with and without human-caused removals.

The thick solid line is median population size (N). The thin solid lines are the values of N for 100 example iterations. 

The dotted gray line is the median carrying capacity (K), and the gray shaded area is the 95-percent confidence 

interval of interannual variation in carrying capacity. The solid gray line is the quasi-extinction threshold of 100 

individuals. Sample results are shown for a population with medium resilience and K = 200 individuals. Panel (a) 

includes no human-caused removals. The corresponding probability of persistence was 100 percent. Panel (b) 

includes human-caused removals at a rate exactly equivalent to the per capita population growth rate at maximum 

net productivity level (i.e., rMNPL). The corresponding probability of persistence was 97 percent because N crosses 

below the quasi-extinction threshold for three iterations. 
27

MMPA Criteria

We evaluated the relationships between inputs to the state-dependent management framework 

and the risk of violating a candidate MMPA criterion, defined as the probability of population size 

decreasing to less than NMNPL (i.e., P<MNPL). As expected based on equations 1 and 2, P<NMPL was 

positively correlated with the management factor FO (sample results are shown in fig. 8). Higher values 

of FO lead to higher human-caused removal rates, which decrease equilibrium population size. The 

probability P<MNPL was negatively correlated with data precision (sample results are shown in fig. 8). 

That is, if management decisions are made using more precise information on population status, it is 

possible to implement a higher value of FO, and, therefore, a higher removal rate, without increasing the 

risk of negative outcomes. 

The placeholder degree of risk tolerance with respect to maintaining populations greater than

NMNPL depended on the resilience of the population: upper limits on P<MNPL, above which human-caused 

removals would result in Ppersist <90 percent, were 0.22, 0.36, and 0.62 for populations with high, 

medium, and low resilience, respectively (sample are results shown in fig. 9). Threshold values for FO, 

defined as the values necessary to remain below these upper limits on P<MNPL, were a function of 

population resilience, data precision, and the approach used to select N� from its sampling distribution 

(table 6). 

Human-caused removals that are implemented using the state-dependent management 

framework, with a 10-year management interval and values of FO less than those in table 6, would be 

expected to achieve the candidate MMPA criterion using the placeholder degree of risk tolerance. These 

findings were relatively insensitive to the model used for density dependence. For example, for a 

population with medium resilience, threshold values for FO generated using the linear density-dependent 

functions for vital rates were, on average, only 1.2 percent lower than the values in table 6b, which 

incorporated the more realistic density-dependent functions for polar bears. 

The relationships between FO and P<MNPL demonstrate a common tradeoff in wildlife 

management (sample results are shown in fig. 8). When there is uncertainty in the population data used 

to inform management decisions, increasing removals toward the maximum sustainable level also will 

increase the risk of population size decreasing to less than NMNPL, which can lead to negative population 

outcomes and reduce future yield. The methods presented here can be used to inform this tradeoff, to 

concurrently meet conservation goals for persistence and provide for long-term sustainable use. For 

example, annual yield is shown in figure 10 as a function of time, for biological and management 

conditions under which the upper limit on FO is 0.86 (table 6). In this example, removal levels

calculated using FO >0.86 increase current yield, but result in smaller equilibrium population sizes or 

extirpation, and, therefore, reduce future yield. 
28

Table 6. Threshold values of the management factor (FO) used to calculate human-caused removals under the 

state-dependent management framework, which meet conservation goals based on a placeholder degree of risk 

tolerance.

[Population resilience is defined in terms of the natural per capita population growth rate referenced to maximum net productivity level 

(i.e., rMNPL). Data precision levels are defined in table 4. Population size (N) was selected from its sampling distribution using the lower 

5th, 15th, or 50th percentiles. Values of FO were evaluated within the range of 0.5–1.25]

(a)

Low resilience (rMNPL = 1.5%)

Data precision 

level

N lower 5th

percentile

N lower 15th

percentile N 50th percentile

true 1.18 1.18 1.19

1 >1.25 >1.25 1.15

2 1.13 1.07 0.99

3 0.82 0.72 0.61

4 <0.50 <0.50 <0.50

(b)

Medium resilience (rMNPL = 4.3%)

Data precision 

level

N lower 5th

percentile

N lower 15th

percentile N 50th percentile

true >1.25 >1.25 >1.25

1 >1.25 >1.25 >1.25

2 1.21 1.15 1.07

3 0.96 0.86 0.72

4 0.95 0.76 0.56

(c )

High resilience (rMNPL = 8.5%)

Data precision 

level

N lower 5th

percentile

N lower 15th

percentile N 50th percentile

true >1.25 >1.25 >1.25

1 >1.25 >1.25 1.23

2 >1.25 1.22 1.12

3 >1.25 1.17 1.00

4 >1.25 1.11 0.82

 
29

Figure 8. Graph showing probability of population size decreasing to less than NMNPL (P<MNPL) as a function of the 

management factor (FO), for different levels of data precision. The equilibrium population size NMNPL corresponds to 

a population density at maximum net productivity level. Sample results are shown for a population with medium 

resilience. The estimate of population size used to calculate human-caused removals (N�) was selected as the 

lower 15th percentile of the sampling distribution for N. Data precision levels are defined in table 4. 
30

Figure 9. Graph showing relationship between the probability of extirpation (1-Ppersist) and the probability of 

population size decreasing to less than NMNPL (P<MNPL). The equilibrium population size NMNPL corresponds to a 

population density at maximum net productivity level. Sample results are shown for a population with medium 

resilience. The solid black line represents the best polynomial fit between values of (1- Ppersist) and P<MNPL estimated 

from the simulation to evaluate inputs to the state-dependent management framework. The vertical dotted line 

corresponds to an upper limit on P<MNPL of 0.36, above which human-caused removals alone would result in a 

probability of persistence (Ppersist) less than 90 percent. 
31

Figure 10. Graph showing annual yield as a function of time, for different levels of the management factor (FO).

Sample results are shown for a population with medium resilience, stable carrying capacity of 1,000 animals, and 

simulated population assessments performed on a 10-year management interval using a data precision level of 3 

(table 4). The estimate of population size used to calculate human-caused removals (N�) was selected as the lower 

15th percentile of the sampling distribution for N. The upper limit on FO for these conditions was 0.86 using the 

placeholder risk tolerance presented here (table 6). At higher values of FO, removals lead to reduced future annual 

yield, as demonstrated by lines with negative slopes. Yield is shown as fitted linear regression lines to demonstrate 

relative trends.
32

A sample application of the state-dependent management framework with FO = 0.75, SR = 2, and 

a data precision level of 3, was robust over a wide range of biological conditions (table 7). Increased 

risk of extirpation compared to no human-caused removals (i.e., ΔPextirpation) was greatest for populations 

with rMNPL in the range of -0.05–0. Populations with rMNPL <0 have zero capacity for positive growth; 

they can do nothing but decline and any human-caused mortality will be additive and serve to accelerate 

population declines. Although the combination of rMNPL <0 and stable K is probably not biologically 

realistic for polar bears under current conditions, it is included to illustrate the potential risks of humancaused removals under severe density-independent limitation. The increased risk of extirpation for 

populations with human-caused removals largely was due to sampling error—the chance of a positively

biased estimate of rMNPL leading to an overestimate of the sustainable removal rate. For populations with 

rMNPL in the range 0 to 0.025, values of ΔPextirpation ranged from 4 to 8 percent, depending on the trend in 

K. These values are consistent with the placeholder degree of risk tolerance (i.e., that removals alone not 

result in Ppersist <90 percent). Compared to populations with stable carrying capacity, values of 

ΔPextirpation were similar for populations that experienced the mean estimated decline of 7 percent per 

decade in the proxy for K developed using sea-ice data for the CS and SB regions. This is because 

human-caused removals conducted at a sustainable rate tend to result in an equilibrium population size 

that declines in parallel to K, but not at an accelerated rate compared to K (figs. 11a–11c). Values of 

ΔPextirpation were higher for populations that experienced a 14-percent decline per decade in K, especially 

if those populations had low or medium resilience (table 7). This is because K declined so quickly that 

small-population effects became important toward the end of the projection. Human-caused removals 

led to smaller equilibrium population sizes and increased relative variance, which together put 

populations at greater risk of declining below the quasi-extinction threshold due to stochasticity. At a 

14-percent decline rate per decade in K, any population would be headed for extirpation in the next 100 

years. Populations with human-caused removals, even if conducted at a sustainable rate, would be 

expected to face extirpation slightly sooner (figs. 11c and 11f). 

The state-dependent management framework can be placed in the context of historical polar bear 

harvests by using a sample value of FO to derive sustainable removal rates for different subpopulations. 

We continue the previous example using FO = 0.75, SR = 2, and assuming that estimates of r derived 

from the vital rates for case studies (table 1) are equivalent to rMNPL. Application of equations 1 and 2 

with these inputs and assumptions would suggest a mean sustainable removal rate of 5.3 percent (95-

percent CI = 1.7–9.7 percent).
33

Table 7. Increased risk of extirpation compared to no human-caused removals (ΔPextirpation), under a sample 

application of the state-dependent management framework, for different rates of declining carrying capacity.

[Population resilience is defined in terms of the natural per capita population growth rate referenced to maximum net productivity level 

(i.e., rMNPL). Increased risk of extirpation is derived as the probability of persistence (Ppersist) for populations with no human-caused

removals, minus Ppersist for populations with removals. Removals were calculated using an example application of the state-dependent 

management framework with a management factor (FO) of 0.75, a male-to-female sex ratio (SR) of 2, and simulated population 

assessments with a data precision level of 3 (table 4). Populations with moderate negative resilience (e.g., -0.05 < rMNPL < -0.025) have the 

greatest increased risk of extirpation, because using population data with low precision can lead to the spurious conclusion that there is a 

sustainable harvest rate, when in fact no removals were sustainable. In this example, populations with strong negative resilience (e.g., 

rMNPL < -0.05) do not have an increased risk of extirpation, because the intrinsic growth rate was low enough that there was little risk of 

reaching the spurious conclusion that there was a sustainable harvest rate]

Percent change in carrying 

capacity per decade

Population resilience 

(rMNPL) 0% -7% -14%

< -0.05 0% 0% 0%

-0.05 to -0.025 11% 10% 10%

-0.025 to 0 6% 6% 7%

0 to 0.025 4% 5% 8%

0.025 to 0.05 1% 1% 4%

> 0.05 0% 0% 2%
34

Figure 11. Graphs showing example population projections with and without human-caused removals for stable 

and declining carrying capacity. The thick solid line is median population size (N). The thin solid lines are the values 

of N for 100 example iterations. The dotted gray line is the median carrying capacity (K), and the gray shaded area 

is the 95-percent confidence interval of interannual variation in carrying capacity. The solid gray line is the quasiextinction threshold. Sample results are shown for a population with medium resilience and K(t=1) = 1,000 

individuals. The left column of panels represent populations with human-caused removals using the statedependent management framework with management factor (FO) = 0.75, a male-to-female sex ratio (SR) of 2, and 

simulated population assessments performed on a 10-year management interval using a data precision level of 3 

(table 4). The estimate of population size used to calculate human-caused removals (N�) was selected as the lower 

15th percentile of the sampling distribution for N. The right column of panels has no human-caused removals. 

Carrying capacity was stable for panels (a) and (d), declined by 7 percent per decade for panels (b) and (e), or 

declined by 14 percent per decade for panels (c) and (f).
35

Discussion

We developed a demographic model that builds upon existing PVA frameworks for polar bears 

(e.g., Taylor and others, 2005; Hunter and others, 2010). We used the model to estimate conditions that 

maintain polar bear populations with a 90-percent probability of persistence over 100 years. We also 

developed a state-dependent management framework by linking the demographic model to simulated 

population assessments. This linkage is useful because management decisions are informed by 

infrequent estimates of population parameters that include uncertainty and possible bias (Nichols and 

Williams, 2006). Although we focused on human-caused removals, our methods could be applied to 

other management issues, such as quantifying the effects of reproductive failures (e.g., due to den 

disturbance; Amstrup, 1993). Future use of the demographic model could be integrated with decision 

analysis methods (Williams and Johnson, 2013) to help focus on key uncertainties or optimize tradeoffs 

in conservation planning. 

Demographic Model

Density dependence has not been included in previous matrix-based models for polar bears 

(Hunter and others, 2010; Lunn and others, 2014). This is justifiable if projections are for short time 

horizons over which density is constant (Taylor and others, 2002) or if projections are based on 

estimated relationships between vital rates and the environment that reflect the full range of densitydependent behaviors. However, inclusion of density dependence is necessary if a PVA will be used to 

evaluate changes in environmental carrying capacity, human-caused removals, and interactions between 

these two factors (Guthery and Shaw, 2013). 

How density dependence is represented can affect estimates of population persistence (Mills and 

others, 1996) and, therefore, should be based on the biology of the species (Henle and others, 2004). For 

large mammals with long life expectancy and low reproductive rates, density-dependent effects occur 

mostly at population sizes close to K (Fowler, 1987). The density-dependent functions in our 

demographic model led to behaviors consistent with theory and empirical observations. Our estimate of 

MNPL was within the range of 0.5–0.85 suggested for marine mammals (Taylor and DeMaster, 1993), 

and similar to the range of 0.75–0.90 suggested for polar bears (Taylor, 1994). Fowler (1988) derived an 

empirical equation to estimate MNPL based on population data for a wide range of taxonomic groups. 

Using that equation with reasonable inputs for polar bears (generation length from 10 to 15 years, rmax

from 5 to 10 percent) would estimate MNPL = 0.65 (95-percent CI = 0.57–0.74), also similar to our 

range of estimates. The upper limits on the management factor (FO) in the state-dependent management 

framework, necessary to meet conservation goals, were insensitive to the alternative, linear densitydependent functions for the vital rates. We did not evaluate the hypothesis of no density dependence 

because it is not biologically plausible for polar bears. Although the details of density dependence are 

difficult to estimate from field data (Dennis and Taper, 1994), its presence is necessary to explain that 

populations do not naturally increase without limit (Caughley and Sinclair, 1994) and can be exploited 

without being extirpated (McGowan and others, 2011). 
36

For polar bears, the demographic effects of sea-ice loss could occur through both densitydependent mechanisms (e.g., increased competition for limited resources) and density-independent 

mechanisms (e.g., insufficient temporal availability of resources, regardless of competition), although 

the details of such effects and their interactions are poorly understood. Within the demographic model, 

density-dependent processes are represented through variation in K, and we have illustrated how sea-ice 

data could be used to derive a proxy for K. The demographic model allows for density-independent 

limitation in two ways. First, populations can have different maximum vital rates, and thus different 

values of rmax, which represents the capacity for growth at very low densities and in the absence of 

human-caused removals. Conceptually, the demography of populations with low resilience (e.g., rMNPL = 

0.015) is relevant both to current subpopulations in less-productive regions of the Arctic, and to future 

subpopulations that experience density-independent reductions in rmax due to ecological change. Second, 

the nonlinear density-dependent functions for the vital rates resulted in growth curves that were nearly 

horizontal for densities less than 0.5×N/K (fig. 2c), meaning that population responses at low densities 

were effectively density-independent. Because the demographic model includes both density-dependent 

and density-independent processes, it is sufficiently general to include a range of mechanisms by which 

climate change might affect populations and their ability to sustain human-caused removals. For 

example, if the primary effect of sea-ice loss is to reduce K, then the sustainable harvest rate would not 

change, but the harvest level (i.e., the number of animals removed per year) would decline with 

declining N. Additionally, if the primary effect of sea-ice loss is to reduce rmax, then the sustainable 

harvest rate would decline to the point that, if rmax approached 0, any human-caused mortality would be 

additive. The demographic model also allows the dynamic properties of compensation for humancaused mortality (Péron, 2013) to be adjusted through the density-dependent functions of the vital rates. 

As currently implemented, the ratio rMNPL/rmax = 0.82 indicates the potential for fairly strong 

compensation. Note that the mechanism for compensation takes place across years. That is, humancaused removals are not immediately compensated by an increase in survival and reproductive rates in 

the current year, but rather in the following year.

To our knowledge, this is the first time that energetic requirements have been incorporated into a 

demographic model for a species of bear. Our approach considered theoretical relationships between 

population dynamics and individual energetic requirements (Savage and others, 2004). In conjunction, 

the effects of metabolic energetic equivalents, removals at a 2:1 male-to-female sex ratio, and agespecific harvest vulnerabilities led to population sex ratios skewed toward females, and equilibrium 

population sizes greater than the expected value based on asymptotic population dynamics. For 

example, the projection in figure 11a had a starting population size of NMNPL = 690, and subsequently 

experienced a period of transient dynamics (Caswell, 2007) during which the proportion of females 

increased from 0.52 to 0.68, and the median population size increased from 690 to approximately 814. 

Due to this behavior, the upper limits on FO that still met the example conservation goals were greater 

than 1.0 under some conditions (table 6). This departs from the theoretical expectation that FO = 1 will 

lead to populations at exactly NMNPL, which in turn would result in a high probability of temporarily

decreasing to less than NMNPL because of stochastic variation. A similar phenomenon of higher 

abundance, relative to a given resource base, has been suggested for female-skewed populations of 

ungulates that result from sexual selection, resource partitioning, and higher energetic requirements of 

males (McCullough, 1999). 
37

We defined population persistence relative to a quasi-extinction threshold (Dennis and others, 

1991) below which small-population effects may become significant. The threshold was based on 

demographic rather than genetic considerations due to the relatively low genetic diversity of polar bears 

(Peacock and others, 2015), their high mobility and potential for exchange among subpopulations 

(Amstrup and others, 2004), and the assumed presence of more than one subpopulation in each 

ecoregion within the timeframes considered. The proportional component of the quasi-extinction 

threshold (i.e., 15 percent of starting population size) is tentative, but is based on the work of Molnár 

and others (2008) suggesting that reductions in population density, along with the depletion of males 

due to sex-selective removals, could lead to Allee effects in the mating system. Future refinements of 

the demographic model could incorporate a mechanistic model for Allee effects (Molnár and others, 

2014), which could have the added benefit of representing the demographic effects of skewed 

population sex ratios at normal densities. For polygynous species, the male segment of the population 

can become quite small before fecundity is affected (Caughley and Sinclair, 1994; Derocher and others, 

1997). Taylor and others (2008c) suggested that, for polar bear populations harvested at maximum 

sustainable yield, a 3:1 male-to-female sex ratio in the harvest can lead to a 0.25 proportion of males in 

the population, with a young male distribution, and most males harvested as 2-year-olds. Our evaluation 

of human-caused removals, which included the effects of imperfect information and time lags, 

suggested that this degree of reduction in the male segment is possible at a 2:1 male-to-female removal 

ratio. For example, for populations with medium resilience subject to removals using the combinations 

of FO and other parameters in table 6b, the mean proportion of females after 50 years was 0.75 (95-

percent CI = 0.70–0.80). Monitoring the proportion of females in a population may be a useful indicator 

of overharvest when removals are sex-selective (Millspaugh and others, 2009). 

The use of matrix-based PVA models (Morris and Doak, 2002) is a preferred approach for 

addressing quantitative questions in wildlife management (Milner-Gulland and Akcakaya, 2001). 

Nonetheless, PVAs come with multiple assumptions and caveats (Pe'er and others, 2013), and may be 

susceptible to negative bias when assessing population trends (Weinbaum and others, 2013). In this 

report, we focused on developing metrics for candidate ESA and MMPA criteria, and demonstrated use 

of the demographic model to estimate threshold values for these metrics. We also focused on estimating 

relative risks rather than absolute risks (Akcakaya and Sjogren-Gulve, 2000). If the demographic model 

is used in the future to prescribe a specific management approach, or to forecast the status of a specific 

subpopulation, additional issues will have to be investigated and justified because such analyses can be 

sensitive to sampling uncertainty and potential bias in estimated vital rates (Hunter and others, 2010), 

and to uncertain or non-stationary relations between vital rates and the environment (Bromaghin and 

others, in press). 

ESA Criteria

We suggest that natural adult female survival, recruitment defined as the number of C1s per 

adult female, and carrying capacity, are useful demographic metrics that collectively serve as a proxy 

for population persistence. We evaluated these metrics at the scale of an individual subpopulation, for 

two reasons. First, in the near- and mid-term, polar bear management and conservation will likely 

continue to focus on subpopulations. Second, variation in the status of subpopulations within an 

ecoregion could make metapopulation dynamics (Brook and others, 2009), which the demographic 

model does not include, an important factor at larger spatial scales. 
38

Natural adult female survival rates of at least 0.93 to 0.96, in conjunction with sufficient 

recruitment, are necessary to achieve a 90-percent probability of persistence over 100 years. These 

values are within the range necessary to maintain bear populations (Eberhardt, 1990) and are similar to 

survival rates associated with stable or positive population growth in case studies (Regehr and others, 

2010; Lunn and others, 2014). Estimates of the recruitment metric, derived from the vital rates in table 

1, were toward the upper end of the plausible range for polar bears, and toward the high end of the range 

necessary for persistence (fig. 5). Direct observations of the recruitment metric from capture samples 

include values of 0.24 for the SB subpopulation (Regehr and others, 2006) and 0.32 for the Davis Strait 

subpopulation (Peacock and others, 2013). If environmental change affects polar bears primarily 

through increased competition for nutritional resources, declining recruitment may serve as a leading 

indicator of declines in adult survival as competition increases (Stirling and others, 1999; Regehr and 

others, 2007). 

In addition to survival and recruitment, which determine population resilience, long-term 

persistence requires a certain level of abundance to buffer against stochastic variation. In our 

simulations, a carrying capacity of approximately 250 animals (for an isolated subpopulation) was 

necessary to reduce the risk of crossing below the quasi-extinction threshold of 100 animals. At more 

than 250 animals, human-caused removals implemented at exactly the sustainable rate rMNPL did not 

have a negative effect on persistence. At fewer than 250 animals, removals decreased the probability of 

persistence because smaller equilibrium sizes, for harvested populations, led to increased vulnerability 

to stochastic events. In practice, increased caution in implementation of human-caused removals would 

be necessary for small and isolated subpopulations because management imperfections could be 

expected to result in increased relative variance in population size, and thus increased risk of 

extirpation, compared to the best-case scenario of removals implemented at exactly rMNPL. 

We referenced the proposed survival and recruitment metrics to a population density at MNPL. 

This was practical for polar bears because many subpopulations are thought to be harvested near 

maximum sustainable yield (Obbard and others, 2010). If that is the case, the equilibrium size of 

harvested populations will be close to NMNPL, and estimates of natural survival from multistate models 

(Lebreton and others, 2009) or live recapture-dead recovery models (Barker, 1997) can be used to 

directly estimate a value of r which, in turn, will be close to rMNPL. The approach taken here is to infer 

population density relative to K (i.e., N/K) based on knowledge of the removal rate, rather than based on 

an independent estimate of carrying capacity, which generally is not available (Gerrodette and 

Demaster, 1990). 

Several assumptions of the demographic model may have underestimated persistence, and are

listed here for future consideration. First, we assumed that density-independent temporal variation in 

vital rates constituted 25 percent of total uncertainty from case studies (Taylor and others, 2002, 2006). 

The results were similar to an estimated coefficient of variation of 0.017, obtained by applying variance 

components methods (White, 2000) to 27 annual estimates of adult female survival for the western 

Hudson Bay subpopulation (Lunn and others, 2014). However, our approach of including variation in 

the proxy for K, additive to density-independent temporal variation, may have overstated the amount of 

environmental variation that polar bears generally experience. Second, sampling covariances and the 

structure of capture-recapture models used to estimate vital rates for the SB subpopulation (Regehr and 

others, 2010) could have introduced positive bias into the estimated correlation coefficients among vital 

rates. This would tend to overstate the negative effects of density-independent variation on the longterm population growth rate (Fieberg and Ellner, 2001). Third, the model did not allow individual 

variation in demographic parameters. Although the potential effects of individual variation are not 

understood for polar bears, persistence can be bolstered if animals with lower fitness are preferentially 
39

removed from a population subjected to stress (White, 2000). Other assumptions may have 

overestimated persistence. For example, we did not include potential autocorrelation in the ice-based 

proxy for K (Rockwell and others, 2011), which under some conditions can exacerbate the risk of 

extirpation (Ripa and Heino, 1999). We also did not evaluate the potential effects of catastrophic habitat 

loss or mortality events (Derocher and others, 2013). 

We estimated threshold values for the proposed demographic metrics relative to a specific 

probability of persistence, timeframe, and quasi-extinction threshold. These metrics can serve as initial 

indicators for monitoring but should be interpreted with caution. First, we report true process mean 

values for the metrics. In reality, sampling variation will lead to uncertainty around the process mean, 

and bias could lead to "non-true" estimates (e.g., negative bias in estimates of survival and population 

size from capture-recapture studies, when there is unexplained individual heterogeneity in recapture 

probabilities; Devineau and others, 2006). Uncertainty and bias are particular challenges when 

evaluating adult female survival, which is a strong determinant of the population growth rate and can be 

difficult to estimate with sufficient accuracy to differentiate between biologically significant outcomes 

(McLellan and others, 1999). Second, the demographic model assumed a certain covariance structure 

among vital rates. Departures from this structure (e.g., a large decline in the survival of subadult males) 

could lead to low probabilities of persistence regardless of other vital rates. Third, the magnitude and 

form of temporal variation in vital rates can be expected to vary among subpopulations, and we did not 

evaluate the sensitivity of persistence to such variation. For example, vital rates from case studies for 

polar bears (table 1) suggest that most subpopulations currently have sufficient natural survival rates to 

persist, provided those rates are maintained over the next 100 years. This is not surprising given that 

future negative effects of habitat loss are the primary concern for polar bears (Amstrup and others, 

2008). However, the status of some subpopulations may be different than would be expected on the 

basis of these metrics alone. Time-invariant estimates of adult female survival for the SB subpopulation, 

for 2001–06, were approximately 0.95 (Regehr and others, 2010). Although this rate seems high enough 

to achieve persistence, projected growth rates for the SB population were negative due to high temporal 

variation in vital rates, and correlation between vital rates and declining environmental conditions 

(Hunter and others, 2010). 

MMPA Criteria

We developed a state-dependent management framework that can be used to estimate the 

sustainable rate of human-caused removals for polar bears, including subsistence harvest. Our approach

is state dependent with respect to both N and rMNPL, which is appropriate given the potential for 

ecological change in the Arctic to affect polar bears through both density-dependent and densityindependent mechanisms. Although our simulated population assessments used a single management 

interval of 10 years, actual intervals vary among subpopulations (Obbard and others, 2010), and it can 

be expected that the upper limits on FO, associated with a particular risk tolerance, would increase for 

shorter management intervals. Our simulations demonstrated clearly that the risk associated with 

human-caused removals is due in large part to imprecise data on population status, which emphasizes 

the importance of scientific studies and collection of traditional knowledge (TK; Voorhees and others, 

2014). Our simulations also demonstrate that, as long as K remains positive for the timeframe of 

interest, maintaining current harvest within sustainable limits will serve to safeguard both population 
40

persistence and the opportunity for future sustainable use. Future application of the demographic model 

could be expanded to evaluate tradeoffs between current harvest rates, long-term cumulative yield, and 

various population outcomes (e.g., probability of persistence, expected time to extirpation) under 

different scenarios such as declining K followed by stabilization (e.g., due to a stabilized climate 

system) and declining K to 0. 

We determined that sustainable harvest rates were positively correlated with population 

resilience because more resilient populations have a higher net productivity and the capacity to rebound 

following periods of overexploitation. Our simulations also demonstrate the importance of accounting 

for sex, age, and reproductive status. Young bears, especially males, are most often killed in conflict 

situations (Dyck, 2006). Furthermore, most subsistence harvests select for males (Taylor and others, 

2008c), and national laws and international agreements discourage the removal of adult females that are 

near maternal dens or have dependent young (Obbard and others, 2010). Across the 400 sets of vital 

rates in the parameter space, the mean ratio of the reproductive value (Caswell, 2001) for adult females 

with C1s (stage 6) compared to single adult females (stage 4) was 1.23 (95-percent CI = 1.16–1.30). The 

ratio for stage 6 females compared to 2-year-old females (stage 1) was 2.03 (95-percent CI = 1.83–

2.28). In combination, the increased vulnerability of young bears to being removed by humans, and the 

decreased reproductive value of young bears, lead to a lower risk for a given removal level than would 

be predicted using a naïve model that did not account for such variation. The differences in reproductive 

value also emphasize the importance of protecting adult females, a common management goal for bears 

(Taylor and others, 1987). 

Several assumptions may have overestimated the potential negative effects of human-caused 

removals on population persistence, and are listed here for future consideration. First, changes in 

population density at time step t acted on vital rates at time step t+1. If multiyear time lags exist 

between declines in K and reductions in vital rates, under conditions of rapidly declining K it is possible 

that human-caused mortality would be more compensatory than in our projections (Péron, 2013), 

effectively moving populations toward K while having a limited effect on persistence. Second, during 

simulations to evaluate the candidate MMPA criterion, populations started with a stable stage 

distribution and subsequently underwent a period of transient dynamics due to selective removals. This 

resulted in a trend toward higher values of per capita growth rate and equilibrium population size, 

compared to the starting values of rMNPL and NMNPL, respectively. Future applications of the 

demographic model could include a “burn-in” period prior to t=1, especially if the goal is to evaluate 

human-caused removals over a shorter timeframe. Third, when evaluating harvest in relation to data 

precision, the simulated population assessments assumed a correlation coefficient of 1.0 among 

estimated population parameters. Thus, if sampling variation happened to result in positively biased 

values for the vital rates, it also would require a positively biased value for N. For a given replicate, this 

could have overstated the risk of positive bias in both N� and r̃MMMM, which would have led to excessive 

removals.

Other assumptions may have underestimated the negative effects of human-caused removals, 

under certain conditions. First, although we evaluated the state-dependent management framework 

under a wide range of conditions, including different maximum vital rates and temporal trends in K, we 

did not include temporal trends in rmax. If rmax were to decline between population assessments (e.g., due 

to negative density-independent effects of climate change), the sustainable harvest rate estimated at the 

beginning of the management interval could become unsustainable toward the end of the interval. 

Second, in the simulated population assessments, we did not consider uncertainty in how close 

estimated values of r were to the desired parameter rMNPL. For polar bears, the expense and logistical 

challenges of conducting population assessments make it difficult to obtain accurate estimates of vital 
41

rates and population size (Vongraven and others, 2012). Furthermore, knowledge of population density 

is necessary to interpret estimates of vital rates. For example, if population density is less than MNPL

(e.g., due to high harvest or increasing K), direct estimates of r from population assessments will be 

between rMNPL and rmax. In the context of the state-dependent management framework, using direct 

estimates of r in equations 1 and 2 could overestimate the sustainable harvest rate. The potential for 

such overestimation does not seem substantial for polar bears because nonlinear density dependence 

results in rMNPL/rmax ≈ 0.82. Conversely, if population density is greater than MNPL (e.g., due to low 

harvest or declining K), estimates of r will be between rMNPL and 0. Under these conditions, using direct 

estimates of r in equations 1 and 2 could significantly underestimate sustainable harvest rate, especially

if K is declining rapidly. In practice, it may be difficult to obtain valid estimates of rMNPL, unless r can 

be estimated during a period of population stability. In some instances, it may be preferable or necessary

to base inference for rMNPL on other demographic or ecological indicators, rather than on direct estimates 

of r, and to account for this additional uncertainty in management decisions. 

Although the state-dependent management framework is an extension of the PBR method, our 

approach differed from other applications of PBR that viewed take of marine mammals (e.g., due to 

fisheries by-catch) as a negative outcome to be avoided, rather than a positive outcome with value to 

humans as in the case of subsistence harvest. We demonstrated how to apply the state-dependent 

management framework based on a placeholder degree of risk tolerance. The resulting estimates of FO, 

and thus sustainable removal rate, represent the upper limits that still achieved consistency with an ESA 

criterion for persistence. Future applications should be based on clear conservation goals that reflect

statutory and stakeholder values. Although linking fundamental goals to demographic criteria is mostly 

a scientific process, subtle forms of risk tolerance can arise there as well. For example, the threshold 

values of FO in the current example were influenced by the regression methods used to establish upper 

limits on P<MNPL as a function of Ppersist. Furthermore, we evaluated P<MNPL at a single point in time (i.e., 

at t = 50 years), rather than at each year t = 1,2,... 50, which likely underestimated the risk of 

populations decreasing to less than NMNPL in the near term. 

Our results suggest that human-caused removals conducted at a sustainable rate are unlikely to 

accelerate population declines that may result from climate change, provided that climate change affects 

polar bears primarily through density-dependent mechanisms or that negative density-independent 

effects occur at a slow rate relative to the management interval. It follows that a well-managed 

subsistence harvest seems unlikely to have negative effects on persistence beyond those deemed 

acceptable when setting values of FO and other inputs to the state-dependent framework. For this to hold 

true requires periodic assessments of population status, and a management system that provides accurate

and timely harvest reporting and that can adjust removal levels. Under conditions of ecological change, 

information obtained from TK (e.g., Voorhees and others, 2014) and other sources on polar bear 

distribution, human-bear interactions, and patterns of subsistence use, will likely become increasingly 

important to responsive management. 

The state-dependent management framework implements harvest as a fraction of current 

population size, which generally is a robust strategy (Quinn and Deriso, 1999). A consequence of this 

approach is that, if populations are declining due to declining K, the level of human-caused removals 

will decline as well. Thus, implementation of subsistence harvest under this framework requires an 

ability to decrease future use if populations get smaller due to climate change or other factors. We 

suggest that the following considerations also are important to mitigating risk if environmental 

conditions are deteriorating rapidly (i.e., such that large declines in K or rmax occur between population 

assessments): (1) performing more regular population assessments and shortening the management 

interval, (2) planning a stopgap measure to reduce removals if a lower population threshold is crossed, 
42

and (3) evaluating the statistical power to detect whether the lower population threshold has been 

crossed. Threshold harvest strategies, for which there are no human-caused removals below a specified 

population size, are an effective method of minimizing the incremental risk of extirpation due to humancaused removals, while maintaining the opportunity for sustainable use if populations rebound in the 

future (Lande and others, 1997). 

Comparison with Current Polar Bear Harvest Management

The sample demonstration of the state-dependent management framework led to a removal rate 

higher than the standard 4.5 percent, which has a history of application to polar bear populations (Taylor 

and others, 1987). However, the 4.5-percent rate usually has been applied to the best estimate (i.e., mean 

value) of population size, whereas we used the more robust approach of applying removal rates to a 

lower percentile of the sampling distribution for N (Wade, 1998). For the purpose of demonstration, we 

can adjust for this difference by assuming a normal distribution for estimates of N with a coefficient of 

variation equal to 0.15 (this corresponds to data precision level 3 in the simulations; table 4). The lower 

15th percentile of such a distribution is approximately 0.85 of the mean value. Thus, the sustainable 

removal rate from the state-dependent management framework would be adjusted as follows: 5.3 

percent × 0.85 = 4.5 percent. This suggests that application of the state-dependent management 

framework using FO = 0.75, a moderately conservative input under most conditions (table 6), would be 

broadly consistent with historical harvest rates for polar bear subpopulations. That the 4.5-percent rates 

align exactly is an artifact of using FO = 0.75 in this example. In practice, either lower or higher values 

of FO could be appropriate based on population resilience, data precision, and risk tolerance. 

Furthermore, our analyses suggest that under certain conditions, polar bears can support removal rates 

higher than 4.5 percent. This is consistent with (1) the suggestion in Eberhardt (1990) that populations 

of brown bears (Ursus arctos) and polar bears can support higher human-caused mortality than 

recommended by Taylor and others (1987); (2) with observations that polar bear populations harvested 

at a 4.5 percent rate have not experienced long-term declines when environmental conditions are stable 

(Obbard and others, 2010); and (3) with evidence that brown bears, which have a similar life history to 

polar bears, can support removal rates of 6–10 percent under favorable conditions (Bunnell and Tait, 

1981; Hovey and McLellan, 1996; VanDaele, 2007; Krofel and others, 2012). Thus, our analyses 

suggest that a 4.5-percent harvest rate is not a theoretical maximum, but rather a reasonable approach 

for populations with moderate resilience in light of imperfect information on population status and time 

lags in management.

Definition of Sustainability for Polar Bear Harvest Management

There are conflicting interpretations of “sustainability” in wildlife management (Sutherland, 

2001), yet ensuring that harvests are sustainable is a common goal. We suggest a general definition that 

may be useful for polar bear management: removals conducted at a sustainable rate will (1) result in 

population sizes at or greater than NMNPL with respect to current conditions, and (2) have a limited 

negative effect on population persistence. The degree of risk tolerance for both conditions must be 

determined based on management objectives, the costs and benefits of use, and statutory and 

stakeholder values. This definition is well-suited for 21st-century conservation challenges. It 

acknowledges that carrying capacity can change with time (e.g., due to sea-ice loss) and that whether 

human-caused removals are sustainable depends on current conditions. Because populations produce 

maximum sustainable yield near NMNPL, this definition allows managers to maximize long-term returns 

while seeking to avoid excessive near-term removals that could reduce opportunities for future 
43

generations. Finally, this definition uses objective methods to estimate risk, and acknowledges that other 

factors must be considered when deciding how much risk is acceptable. This is intended to improve 

transparency and does not preclude application of the “precautionary principle” or any other 

conservation standard. It also is consistent with definitions of “sustainable use” according to the 

Convention on Biological Diversity (2004) and International Union for Conservation of Nature (2000), 

and with the growing recognition that balancing preservation and human needs is an integral part of 

long-term conservation (Kareiva and Marvier, 2012). 

Acknowledgments

Primary support for this analysis was provided by U.S. Fish and Wildlife Service (USFWS) and 

U.S. Geological Survey (USGS). Advice on the modeling framework and analytical approach was 

provided by members of the Science and Traditional Ecological Knowledge Working Group of the 

Polar Bear Recovery Team, including T. Atwood, P. Boveng, D. Douglas, K. Johnson, M. Pederson, R. 

Sparks, K. Titus, and A. Von Duyke. We thank H. Stern (University of Washington) for providing seaice data. We thank T. Atwood, P. Conn, A. Von Duyke, K. Oakley, G. Pendleton, and R. Sparks for 

comments that improved the report. The findings and conclusions in this report are those of the authors 

and do not necessarily represent the views of the USFWS. This report was reviewed and approved by 

USGS under their Fundamental Science Practices policy (http://www.usgs.gov/fsp/).

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Appendix A. Methods to Adapt Published Vital Rates to the Polar Bear Life 

Cycle Graph

If a study did not report C1 (polar bear yearling) survival, we assumed it to be equivalent to the 

reported value of subadult (independent polar bear aged 2–4 years) survival. If a study reported survival 

for a combined C1 and subadult age class, we assumed it to be equivalent to subadult survival as 

defined in the life cycle graph (fig. 1). We used estimates of litter production rate for females >6 years 

(Obbard and others, 2010, table 3) as the vital rate β4, and as the vital rate β3 if a separate estimate of β3

was not available. Recruitment parameters in the life cycle graph (σL0, σL1, and f) were calculated from 

C0 (cub-of-the year) litter sex ratio, C0 litter size, C0 survival, and C1 survival as described in Hunter 

and others (2007, appendix B). All projections used a constant value of 0.5 for the proportion of females 

in C0 litters because this is the mean estimate across subpopulations (Obbard and others, 2010, table 3) 

and we did not have a biological expectation for variation in this parameter. All population projections 

used a constant value of 0.10 for β5 (Regehr and others, 2010) because this parameter is relatively 

unimportant to population growth (Hunter and others, 2007) and has not been estimated in most polar 

bear studies. 
53

Appendix B. Methods to Generate Density-Dependent Functions for the Vital 

Rates

We defined relationships between vital rates and density using a form of the logistic equation: 

(B3) 

where y is a vital rate; 

x is a dimensionless measure of density, calculated as the ratio of population size to 

carrying capacity (i.e., N/K);

μ is the median value of the vital rate;

hf is the half range of the vital rate, calculated as twice its standard deviation;

m is a slope coefficient; and

c is a location coefficient denoting the value of x at which the median (i.e., inflection 

point) occurs. 

We calculated the median value (μ) for each vital rate from its maximum value and its half range (hf). 

The maximum of each vital rate (i.e., the value in the absence of density effects) was a specified input 

for any given population projection. We used a half range (hf ) of 0.05 for the vital rates σ4, σ6, and σ6, 

which was calculated as twice the standard deviation of point estimates of adult female survival from 

case studies (table 1). This approach assumed that total variation in survival for adult females, as 

estimated across polar bear populations that exist under different densities and different environmental 

conditions, was a reasonable approximation of the amount of plasticity in this parameter. The true 

amount of plasticity likely was somewhat lower due to the presence of sampling error in estimates of 

survival, which in future analyses could be delineated using variance components. 

Adult female survival is the most important determinant of population growth for many longlived species (Eberhardt, 2002). For polar bears, it is also estimated with higher precision than other 

demographic parameters. We estimated the value of hf for other vital rates from the value for adult 

females, using a linear relationship on the log-log scale between a vital rate’s coefficient of variation 

and elasticity as calculated from the matrix model (Caswell, 2001). This was a quantitative way to 

incorporate the hypothesis of demographic buffering, or negative correlation between the variance of a 

vital rate and its importance to population growth (Pfister, 1998), into the density-dependent functions. 

Elasticity values were not available for male survival because the life cycle graph did not reflect the role 

of males in reproduction. Therefore, we set hf for male survival equal to hf for female survival within 

the same age class. 

We used a slope coefficient (m) of -7.25 in equation B1. This produced convex vital rate versus 

density curves for which most density-related changes occurred in the range 0.5 <N/K<1.5. It also led to 

functional responses that were consistent with population dynamics theory (see section, “Results”). We 

adjusted the location coefficient c for each vital rate, relative to the others, so that density dependence 

would begin to affect vital rates in the following order: subadult (independent polar bear aged 2–4 

years) survival, C0 (cub-of-the year) survival, breeding probability, and finally C1 (polar bear yearling) 

survival and adult (polar bear aged 5 or more years) survival (Eberhardt, 2002). With the location 

coefficient set to c for adult survival, we set the location coefficient to 0.8c for the vital rates σ1, σ2, σ3, 

σ7, σ8, and σ9. This reflected the hypothesis that, as N/K increases, density effects will first show up in 

subadult survival because subadults are inexperienced hunters that have high energetic demands and 

generally are in poorer nutritional condition than other bears (Noyce and Garshelis, 1994; Eberhardt, 

2002; Schwartz and others, 2006). We set the location coefficient to 0.85c for σL0, reflecting the 

yy = (μμ − hff) + (2hff)⁄{1 + exp[−mm(xx − cc)]}
54

hypothesis that C0 litter survival is the next vital rate affected. Body size and survival for C0s are 

related to maternal nutritional condition, which depends on food availability (Derocher and Stirling, 

1996) and declines in C0 survivorship have been suggested as one of the first responses to density 

effects (Miller and others, 2003). Finally, we set the location coefficient to 0.975c for β3 and β4, 

reflecting the hypothesis that changes in breeding probability may be delayed (Noyce and Garshelis, 

1994) because studies of several subpopulations thought to be experiencing density effects suggest that 

the threshold for maternal nutritional condition, below which females fail to give birth, is low (Robbins 

and others, 2012). This potentially can lead to successful C0 litter production (i.e., relatively high β4) 

followed by C0 litter loss (i.e., relatively low σL0).

The density-dependent functions generated by equation B1 were constrained to the range [0,1]. 

Equation B1 was applied to all vital rates in the life cycle graph (fig. 1) except for the parameter f, the 

expected number of 2-year-olds arriving in stages 1 and 7. This is because f was not independent but 

rather was calculated as a function of C0 litter sex ratio, C0 litter size, C0 survival, and C1 survival 

(Hunter and others, 2010, appendix B). We used a time-constant value of C0 litter size; therefore, the 

demographic model did not allow for potential density-dependent effects on C0 litter size. 
55

Appendix C. Methods to Generate a Sea Ice-Based Proxy for Carrying Capacity

We obtained passive microwave satellite imagery of daily sea-ice concentration for 1979–2013 

(data source, National Snow and Ice Data Center, Boulder, Colorado; http://nsidc.org; data provided by 

H. Stern, University of Washington) within the Chukchi Sea and Southern Beaufort Sea polar bear 

subpopulation boundaries (Obbard and others, 2010). The sea ice concentrations were generated using 

the National Aeronautics and Space Administration Team algorithm and were provided in a polar 

stereographic projection with a nominal grid cell size of 25 × 25 km. Within each subpopulation 

boundary, we first determined the mean area covered by sea ice during March of each year (i.e., the sea 

ice maximum), for 1979–2013. Area covered by sea ice was calculated as the sum of grid cell area 

multiplied by grid cell concentration, over all grid cells with greater than 15-percent sea ice 

concentration. Next, we determined the number of days each year when the area of ice cover was 

greater than 50 percent of the March area, hereafter referred to as “ice-covered days.” We then used 

linear regression to estimate the trend in the number of ice-covered days (trend) and the standard 

deviation of the trend (sd.trend). The standard deviation of the residuals (sd.annual) provided a measure 

of interannual variability. Finally, we calculated the mean number of ice-covered days for the period 

1994–2013 (ndays94-13) to serve as a baseline for projections. Based on these results (table C1), we 

projected the number of ice-covered days forward in time using a gamma distribution to ensure that 

values were greater than zero:

nnn nt~Gamma � h aa = �nn nn94−13+tt� ×t�

2

. a 2 , r = �nn nn94−13+t� ×t�

. a 2 � (C1)

where ndayst is the number of ice-covered days in year t; 

t is the number of years from the beginning of the projection; and 

t� is a random draw, at the beginning of each projection, from a normal distribution 

with mean trend and standard deviation sd.trend. 

We then calculated the proportional change in ndays over time as κ = ndayst / ndays94-13. Thus, the 

dimensionless metric κ reflects proportional changes in the duration of the ice-covered period. 

Table C1. Parameters for linear regressions fit to the number of ice-covered days for 1979–2013.

[Regression parameters were used to project the number of ice-free days forward in time using equation C1]

Subpopulation trend

(days / year)

sd.trend

(days / year)

sd.annual

(days)

ndays94-13

(days)

Chukchi Sea -0.9 0.3 15.3 188

Southern Beaufort Sea -2.1 0.4 24.4 252
56

Appendix D. Methods to Estimate Parameters Using Simulated Population 

Assessments

For the purpose of evaluating the state-dependent management framework, human-caused 

removals were calculated using equations 1 and 2 and were specified at a constant level for 10-year 

management intervals. At the beginning of each management interval, we simulated a population 

assessment to generate the input parameters for equations 1 and 2. Per capita population growth rate at 

maximum net productivity level (i.e., rMNPL) was estimated from correlated random vital rates, selected 

from a multivariate beta distribution using methods described in section, “Density-Independent 

Variation.” The mean values in the multivariate beta distribution were the true vital rates at a population 

density equal to MNPL, as determined from the density-dependent functions for the vital rates. A 

random value for N was selected from a normal distribution. The mean value in the normal distribution 

was the average true value of N for the projected population over the preceding five time steps. Random 

draws were performed using a correlation coefficient of 1 across the vital rates and N. The amount of 

sampling error in the simulated population assessment (i.e., the variances in the beta and normal 

distributions) was based on 75 percent of total uncertainty in parameter estimates from case studies 

(table 4). This was complementary to the previous assumption that process variation constituted 25

percent of total uncertainty in estimates of vital rates. Uncertainty in estimates of N was obtained from 

the most recent case studies for each polar bear subpopulation as summarized in the Status Table 

published by the Polar Bear Specialist Group of the International Union for Conservation of Nature

(http://pbsg.npolar.no/en/status/status-table.html, accessed July 10, 2014). The four levels of data 

precision considered in the simulated population assessments (table 4) were based on the 1st, 10th, 50th, 

and 99th percentiles of estimated sampling uncertainty. Thus, our simulated population assessments 

broadly reflected the observed range of data precision in case studies for polar bears.
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Regehr and others—Resilience and Risk—A Demographic Model to Inform Conservation Planning for Polar Bears—Open-File Report 2015–1029

ISSN 2331-1258 (online)

http://dx.doi.org/10.3133/ofr20151029