GRAY WOLF POPULATION PROJECTION WITH INTRASPECIFIC COMPETITION

NATURAL RESOURCE M ODELINGVolum e 27, Number 3, August 2014

GRAY WOLF POPULATION PROJECTION WITHINTRASPECIFIC COMPETITION

J. HOCHARD*Department of Economics and Finance, University of Wyoming, Laramie, WY, 82071

E-mail: [email protected]

D. FINNOFFDepartment of Economics and Finance, University of Wyoming, Laramie, WY, 82071

E-mail: [email protected]

Abstract. Competition effects are incorporated into a model of wolf-population dynamics. A classic single-state model is augmented into a dual-state mapping of the evolution of the size of wolf packs and the number ofwolf packs. This dual-state model, unlike the single-state density dependentmodel, is amenable to analyzing intraspecific competition. The single-state,dual-state and dual-state with competition models are estimated using Yel-lowstone National Park (YNP) data on wolf populations and pack structuresfrom 1996 to 2011. The dynamic properties of each model are examinedunder an array of harvesting policies. Results suggest that intraspecific com-petition matters when projecting wolf populations. Wolf pack removal hascompetition-reducing effects from added territory availability, making popu-lations more sensitive to pack size reduction than reduction in the number ofpacks. This research suggests that wildlife managers may consider monitor-ing the composition of wolf kills throughout a harvesting season, adaptivelyadjusting harvesting quotas and delineating harvesting zones over a few packterritories rather than spreading these effects evenly across all packs.

Key Words: Population projection, intraspecific competition, wildlifemanagement, population dynamics, Canis lupus.

1. Introduction. US federal protection of the gray wolf (Canis lupus), priorensured by the Endangered Species Act (ESA) of 1973, has expired across theNorthern Rocky Mountains and management has been handed over to state con-trol. In April 2009, the United States Fish and Wildlife Service (USFWS) removedthe federal protection of wolves throughout the Rocky Mountain region before reim-plementing the protection in 2010. In May 2011, a final court ruling removed thegray wolf from ESA protection in Idaho, Montana, Oregon, Washington, and Utah.On August 31, 2012 Wyoming’s state-level management plan was approved, re-moving federal protection of the gray wolf in Wyoming as of September 30, 2012completing the removal of federal protections.

A reduction in federal wolf protection begins an era of active state wolf-populationmanagement. The feedback between management decisions and population

∗Corresponding author. Jacob Hochard, Department of Economics and Finance, University ofWyoming, Laramie, WY 82071, e-mail: [email protected]

Received by the editors on 24th June 2013. Accepted 27th May 2014.

Copyright c© 2014 W iley Period ica ls, Inc.

360

GRAY WOLF POPULATION PROJECTION 361

dynamics and the potential for further wolf reintroduction make the understand-ing of wolf-population dynamics and management options increasingly important.1

Success of wolf-population management depends critically on wildlife managers’understanding of wolf-population dynamics. Prior modeling of these dynamics hasnot incorporated crucial aspects of wolf population social structure (Mladenoff andSickley [1998], Jensen and Miller [2001], Miller et al. [2002], Varley and Boyce[2006]). Perhaps the most important of these overlooked structural considerationsis intraspecific competition.

Intraspecific competition of wolves is rarely modeled. This is despite the largestsource of wolf mortality (outside of human interference) being interpack aggres-sion (Mech [1991]). “Interpack warfare” has been attributed as a major cause tothe 70% decline in wolves over just a 2-year period on Isle Royale, Michigan inthe early 1980s (Peterson [1988]). Intraspecific competition has also been observedin the closely monitored Yellowstone National Park (YNP) wolf population. Since1995, wolf packs within YNP have been settling for pack territories with lowerprey catchability rates to avoid interpack conflicts (Kauffman et al. [2007]). De-spite wolves’ aversion to interpack conflict, between pack contests have becomecommon. Conflict has made dispersal beyond territory boundaries increasingly dan-gerous. In turn, pack splitting and pack creation have become more difficult (Smithand Ferguson [2012]). These territory boundaries are delineated by increased scentmarkings (Lewis et al. [1997]) and characterized by having less wolf activity (Lewiset al. [1997], Caro and Stoner [2003]). The danger of wolf dispersal beyond theseboundaries is great enough to deter wolves from leaving their pack to find breedingpartners. This behavior runs counter to their preference not to breed incestuouslyand is a clear demonstration of the importance of intraspecific competition (Smithet al. [1997]).

The classic logistic model with age structure has been used to project wolf pop-ulations on Michigan’s upper peninsula (Miller et al. [2002]). Courchamp et al.[1999] incorporate an Allee effect into the classic model to represent the popula-tion dynamics of obligate cooperative breeders. Both of these models aggregate thespecies to the population level. As a result, these frameworks ignore intraspecificcompetition. For example, consider an area that has 4 packs of 20 wolves and anidentical area that has 16 packs of 5 wolves. A population-level model will pre-dict identical population growth. Symmetric growth is unlikely to occur since eachpack has a breeding pair and is growing within its own territory. The 16 packpopulation is subject to greater intraspecific competition than the 4 pack popula-tion. Competition over territory will yield greater intraspecific mortality. Popula-tion level modeling based on density dependence of the population will omit thismortality.

Mortality resulting from intraspecific competition is likely to affect both the birthand death rates of wolves. Sex-ratio variation is a driving factor of demographicstochasticity in populations (Melbourne and Hastings [2008]). This variation may

362 J. HOCHARD AND D. FINNOFF

be particularly pronounced in wolves since each pack contains a single breedingpair and wolves, as a top predator, typically have relatively small populations. Lowpopulations increase relative vulnerability to demographic effects (Melbourne andHastings [2008]). YNP’s Druid pack, the largest documented wolf pack in history,included 37 wolves at its peak in 2001. In 2004, the alpha female #42 was killedby neighboring Mollie’s pack (Walker [2012]) contributing to the pack’s decline insubsequent years (Smith et al. [1995–2011]).

We investigate the origins of demographic stochasticity in a simple model.Stochasticity can be incorporated into a logistic growth process to determine rela-tive extinction risk (Melbourne and Hastings [2008]). Here we take a deterministicperspective which allows an inspection of the roots of this stochasticity. A deter-ministic perspective allows a clear view of how management choices may affect andbe affected by these demographic processes.

We model the evolution of a wolf population as a dual-state process respectingthe evolution of pack number and pack size. The dual-state approach allows adirect incorporation of intraspecific competition. Unlike other models (Haight et al.[1998], Vucetich and Creel [1999]), we do not model new pack creation as it relatesto territory vacancy and yearling dispersal. Instead, we simplify this process byimplicitly assuming that the number of packs drives intraspecific pressure on newpack creation and pack-size expansion. This approach allows us to analyze better thesystem’s stability properties under various harvesting regimes. Strategic harvesting,which reduces the size of packs or the number of packs, is explored using thisframework. Traditional analyses do not consider strategic harvesting. For example,in 2009, Montana Fish Wildlife and Parks conducted a wolf-harvesting simulationwith harvesting rates ranging from 5% to 70% (Sime [2009]). This simulation didnot examine strategic harvesting (reduction of packs or reduction of the size ofpacks) and did not account for competition.

Measures of competition, which take into consideration the size of all packs andthe number of packs, are employed to estimate the effect of intraspecific competitionon population growth. A logistic growth model with competition is estimated usingannual data on wolf pack structure in YNP from 1996 to 2010. Our model suggestsintraspecific forces are crucial for accurate population projection. We show thatfailure to include intraspecific competition when setting harvest quotas can resultin severe mismanagement, by wrongly predicting local wolf population extinctionand sustainment.

2. Model. In discrete time, the single-state population (Pt) depends on a pos-itive linear component at rate a and a positive quadratic component at rate baccording to

Pt+1 − Pt = aPt − bP 2t .(1)


The single-state logistic model can be dissected into a dual-state model by respect-ing the evolution of the number of wolf packs and the size of those wolf packs.

This dual-state logistic model follows

Xi,t+1 − Xi,t = aiXi,t − biX2i,t i = S,N,(2)

where the growth in the number of packs (N) and the size of those packs (S)depends linearly and quadratically on its level in the subsequent period. Consistentwith the single-state logistic model, leading coefficients a and b would take a positivesign.2 The dual-state logistic model is amenable to the deterministic incorporationof intraspecific competition. Two competition indices are employed to determinecompetition effects for robustness. The Herfindahl-Hirschman Index (HHI),

HHIt =Nt∑1

[Sj,t

Pt

]2

j = 1, 2, ..., Nt ,(3)

is used commonly in economics to measure the extent of competition in a market.It measures increases in competition based on the number of firms and size of thosefirms. A market with many smaller firms is said to be competitive relative to amarket with a single firm or few firms of different sizes. The Shannon-Wiener Index(SWI),

SWIt = −Nt∑1

[Sj,t

Pt

]log

[Sj,t

Pt

]j = 1, 2, ..., Nt ,(4)

takes a logged form and is used commonly in biology and ecology as an indicatorof species diversity. This measure takes into account the evenness of species in asystem and can also proxy for competition levels.

The competition measure derived from the HHI (cHHIt ) is decreasing and convex

in the number of packs and ranges from 0 to 10, 000 with higher values correspond-ing to less competition. The competition measure derived from the SWI (cSWI

t )is increasing and concave in the number of packs and typically ranges from 0 to2 (although larger values are possible) with larger values corresponding to morecompetition.

Following our assumption of intraspecific competition resulting from territorydemands by packs, the existence of a pack is the primary driver of competitivepressure, rather than the size of the pack. Under this assumption, pack sizes areaveraged, which eliminates the effect of pack-size asymmetries as a competition-driving factor.


The competition indices at time t (Ct) can be rewritten as

CHHIt =

ρ

Nt,(5)

where ρ = 10, 000 is a scalar consistent with the common calculation of the HHIand

CSWIt = −log

1Nt

.(6)

The ranges of these competition indices are strictly positive, requiring us to in-clude an intercept in our empirical specification to correctly determine competitioneffects on population growth. Incorporating a vertical intercept defines the appro-priate horizontal intercept levels of HHI and SWI that correspond to no effect ofcompetition on growth.

The dual-state model with each competition index,

Xi,t+1 − Xi,t = aiXi,t − biX2i,t +

Competition effect︷︸︸︷cHHIi

[ρ

Nt

]+ βHHI

i i = S, N(7)

and

Xi,t+1 − Xi,t = aiXi,t − biX2i,t +

Competition effect︷︸︸︷cSWIi Log

[1Nt

]+ βSWI

i i = S, N,(8)

include competition in an additively separable fashion. Additively separable compe-tition effects deliberately differentiate the influence of competition from the effectsof density-dependence. We expect the leading competition coefficients, cHHI

i andcSWIi , to take positive signs. This sign would suggest that increased competition,

driven by a higher pack density, creates a drag on both the evolution of pack sizeand the creation of new packs. This effect is expected with worsened predator–preyratios, increased intraspecific strife and higher fitness expenditures for territory de-fense. This specification allows for a simple and clear exposition of how interpackcompetition influences the dynamics of the system.

3. Estimation. Data on YNP wolf populations was collected from the NationalPark Service’s (NPS) annual “Yellowstone Wolf Project Annual Reports” from 1996


to 2011 (Smith et al. [1995–2011]). Data from 1995 was excluded because wolfreintroduction continued into 1996. The reports also provided data on the numberof packs within the park and the size of each pack. “Groups” of two wolves werealso defined as packs. Lone wolves that were not associated with a pack were notincluded. Pack sizes were averaged annually and competition indices were computedwithout averaging pack sizes. Competition estimates associated with these dataare appropriate for actual pack number and pack size variation while our modelsuppresses pack size variation. Since we are attempting to explain the variation inwolf population growth with the variation in the level of competition, our model’svariation-suppressing assumption about the level of competition will provide uswith conservative estimates of competition’s effect on population growth.

The single-state logistic model was estimated using ordinary least squares (OLS)while both dual-state models were estimated using the Seemingly Unrelated Re-gression Equations (SURE) model (Zellner [1962]). The SURE model,

Xi,t+1 − Xi,t = Xi,tβi + εi,t i = S,N,(9)

assumes that fluctuations in the error term of each equation are uncorrelated overtime, E[εS,tS εN ,tN = 0], but may move together contemporaneously, E[εS,tS εN ,tN =σSN ]. This assumption is appropriate for our system of equations since exogenousfactors such as seasonality, weather, disease, and behavioral changes in prey arelikely to affect both growth equations within any given year. Contemporaneouscorrelations in our system of equations’ errors would render OLS no longer themost efficient estimator. The SURE approach ensures efficient estimates.

All coefficient estimate signs, for both the single-state and dual-state models, areconsistent with density dependent growth (Table 1). Competition coefficients alsotake positive signs suggesting that lower levels of competition create conditionsfavoring larger pack sizes and the creation of new packs (Table 1). Although eachof these models has joint significance, our small dataset causes the significance ofour individual estimates to be sporadic. For illustrative purposes, we simulate thesemodels and compare them against the actual wolf population in YNP from 1996 to2011 (Figure 1). The purpose of this estimation is not to compare rigorously theperformance of various models of population evolution. Rather, our intention hereis to get unbiased estimates of key ecological parameters that have not prior beenexplored. These parameter estimates are used in the subsequent section to maketwo notable contributions to the literature on wildlife population dynamics underactive management. First, we explore the dynamic implications of moving from amodel of classic density dependence to a dual-state model of density dependence.Second, we use the changing wolf pack composition, an inherent feature of the dual-state model, to examine the implications of intraspecific competition on harvestedpopulation levels and stability.

366 J. HOCHARD AND D. FINNOFFT

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FIGURE 1. Yellowstone National Park simulated wolf-population dynamics (1996–2011):projection comparison of all models.

FIGURE 2. Phase plane analysis of dual-state logistic model.

4. Dynamical implications. The single and dual-state models without com-petition, consistent with classic density dependence, yield a single and stable equi-librium. The introduction of competition creates a bifurcation to the dual-statesystem, and reveals a second equilibrium (Figure 2). This figure is a discrete-timeequivalent of a continuous-time phase diagram. The vertical line is the discrete-timeisocline of the number of packs and the nonlinear line the discrete-time isocline ofthe size of packs. Equilibrium points of the system are found at the intersectionof these lines and arrows indicate the directions of movement, for any combination


of pack number and pack size, at a point in time. The upper and lower equilibriumpoints each support the same number of packs, though of different sizes, for eachpopulation level. As indicated on the figure, the “upper” equilibrium supportingpacks of larger size is a stable node while the “lower” equilibrium point supportingpacks of smaller size is an unstable saddle point.

The stable node is characterized by a monotonic approach of both the size of packsand the number of packs to the equilibrium level (Holling [1973]). Perturbations thatpush packs out of the equilibrium size or number will be met with countervailingforces that return pack size and pack number to their equilibrium level over time.The unstable saddle will attract off-equilibrium pack numbers or pack sizes, butonly from very specific directions as indicated by the two approach arrows. Anysmall perturbation, or shock, to either the average pack size or the number of packswill put the system on a path away from the equilibrium point, eventually beingrepelled toward the stable node (Cushing et al. [1998]), which is characterized bya higher population level.

The location of the steady states and their stability properties depend on thespecifics of the management strategy employed. Wildlife managers can choose toharvest wolves strategically, either by reducing the number of packs in a habitat,reducing the size of packs in the habitat, or by reducing both the average packsize and pack number. Pack-size reduction is likely to occur under a traditionalharvesting regime where the harvest quotas, although divided into harvest zones,tend to be spread evenly across large geographic regions (Sime [2009], Wydevenet al. [2012]) while the intentional removal of packs is typically reserved to combata particular threat, such as livestock depredation (Bjorge and Gunson [1985]).

Here we focus on the stability properties of the upper equilibrium of the competi-tion model. For management purposes, this is the relevant equilibrium because it isinherently stable in the absence of harvest, and the lower equilibrium is never sta-ble. Unlike the equilibria in the models without competition, which remain stableregardless of the harvesting strategies employed, the equilibrium with intraspecificcompetition is susceptible to stability shifts (Figure 3), which depend on the compo-sition of wolf harvest. Annual pack-size reductions of 20%, without an accompanyingreduction in the number of packs, result in population extinction. Reducing the sizeof each pack hinders each pack’s fitness, while not affecting the overall level of in-traspecific competition. This is consistent with each pack having fewer members, butfacing unaltered costs to fitness, in the form of territory defense against intraspecificstrife.

In contrast to pack-size reduction, wolf harvest in the form of pack removal cansustain 40% harvest before shifting the stability of the equilibrium. In excess of40% pack-number harvest, the equilibrium is unstable, which increases populationvolatility and makes active management difficult. This result is consistent withhaving a few large packs, excess territory and high rates of dispersal. The formation


FIGURE 3. Wolf-population stability of upper equilibrium under various harvesting policies.

of new breeding partners and territory establishment would be highly uncertain, andwould make the appropriate assignment of harvest quotas, across space, extremelydifficult.

A counterintuitive result is that increased population harvest can be sustained ina stable equilibrium where pack-size reduction and pack-number reduction are usedas complementary management strategies. This result is consistent with each indi-vidual pack having a weakened state of fitness but also facing less territory defensecosts as intraspecific pressures are eased with the removal of rivalrous packs. Theexistence of this complementary relationship reveals the importance that strategicharvesting (i.e., taking into account the composition of wolf harvest across space)has on defining appropriate harvest quotas and harvest zones that do not cross theboundaries of stability. In addition to stability effects, harvest composition also in-fluences steady state population levels. The effect of pack-size reductions and packremoval, when moving from a single-state density dependent model to a dual-statemodel of density dependence, on population levels is comparable (Figure 4). Bothsurfaces are relatively symmetric with population level in the dual-state model be-ing more sensitive to pack-size reductions. This is apparent as the extinction zonein the dual-state model, which we define as a population falling below two wolves,is broader and occurs at a lower level of pack-size reduction (60%) than the single-state model (90%).


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FIGURE 5. Harvesting combinations and steady state populations for models with competition.

The impact of incorporating competition is striking (Figure 5). Population levelsare again more sensitive to pack-size reductions than pack removals. ComparingFigures 4 and 5 shows that harvests can be used strategically to manipulate thecompetitive forces. In Figure 5, the populations in the stable zone (Figure 3) arehigher with competition than without. Also, areas of extinction that were predictedin the models without competition and depended entirely on pack-size harvest nowdepend on both pack-size and pack-number harvest. These results are consistentwith the complementary relationship between these two forms of harvest that wefound in the stability analysis. Mismatched extinction zones imply that basingharvest decisions on a model that does not incorporate intraspecific competitionmay result in severe mismanagement.

5. Discussion. Wildlife managers’ ability to predict wolf population dynamics,in a system disturbed by harvesting, is becoming increasingly important as federalprotection decreases and reintroduction initiatives increase. Traditional wolf pop-ulation dynamic models may be inadequate since these models do not incorporatethe structural characteristics of wolf populations. Perhaps the most important ofthese considerations is interpack competition.

This paper contributes to the existing literature in several respects. Wepresent a wolf population dynamic model that incorporates interpack competition.


FIGURE 6. False wolf stability projections when competition is not included.

Competition is modeled by separating a standard, single-state population modelinto a dual-state problem where the size of packs and the number of packs aretreated as interdependent processes. We use YNP data to estimate the parametersof our models. The dynamic properties of the models vary significantly.

The majority of wolf population modeling considers symmetric harvesting of packsize and pack number on total population. Our model predicts an asymmetric effectwith total population and population stability being more sensitive to systematicreductions in pack size as opposed to pack number. We also find that pack-sizereductions can be increased when they are accompanied by pack-number reductionsbecause of the competition-reducing effect.

Our dynamic analysis indicates that including competition effects reveals an addi-tional (unstable) equilibrium. Furthermore, the properties of the stable equilibriumdepend critically on the composition of harvests. The composition of harvests alsoinfluences steady-state population predictions. Situations are revealed where thewildlife manager who does not consider competition effects may wrongfully predicteither wolf population survival or wolf population extinction (Figure 6).

Our results suggest that wildlife managers should consider competition effectswhen prescribing hunting policies. Three policy recommendations from this modelinclude (1) considering local pack distributions when determining harvesting zones,(2) monitoring the composition of wolf tags through a harvesting season, and (3)


adaptively adjusting harvesting quotas through a hunting season to avoid equallydistributed pack-size reductions. These three recommendations will benefit the sta-bility, population potential, and harvest value of wolf populations. All recommen-dations follow from recognizing that intraspecific strife is a valuable resource con-suming itself.

The model’s competition effects are muted by our assumption that all packswithin a habitat are of equal size. This conservative assumption causes our modelto likely underestimate the effects of competition on population growth. Withinthis conservative approach, our competition model provides insight into popula-tion stability, which appears to be more complex than stable density-dependentmodels suggest. Wolf pack size is determined not only by wolf population densitybut also by prey species availability (Schmidt and Mech [1997], Boitani and Mech[2003]). For example, wolves that are preying on primarily American bison (Bisonbison) typically have larger packs than those preying on elk (Cervus canadensis).This model could be expanded to accommodate this predator-prey effect. Furtherresearch may also include stochastic effects included in the pack-size process or cali-brated pack-size asymmetries to provide more realistic competition effects and morepowerful population projections. This model may also be augmented to include theprobabilistic destruction of packs based on the harvesting of breeding wolves. Weleave these extensions to further work.

Acknowledgments. We gratefully acknowledge two anonymous reviewerswhose suggestions have improved the quality of this manuscript as well as usefulcomments from Jacob Goheen, Klaas van’t Veld, Katie Lee, David Aadland, andAlexandre Skiba. All estimations and simulations were conducted using softwarelicensed to the University of Wyoming’s Department of Economics and Finance.Model codes are available from the authors upon request. The authors are solelyresponsible for any remaining errors.

ENDNOTES

1. In addition to the gray wolf’s reduction in federal protection, suitable locations for furtherreintroduction have been identified by Larsen and Ripple [2006] in the Pacific Northwest, Carrollet al. [2003] in the Southern Rocky Mountains, Mladenoff and Sickley [1998] in the NortheasternUnited States, Glenz et al. [2001] in Valais, Switzerland and for the timber wolf in the AdirondackPark, Paquet et al. [2001].

2. Population of the dual-state model is defined as the product of the two-state variablesPt = Xs , t Xn , t .

REFERENCES

R.R. Bjorge and J.R. Gunson [1985], Evaluation of Wolf Control to Reduce Cattle Predation inAlberta, J. Range Manage. 38(6), 483–487.

L. Boitani and L.D. Mech [2003], Wolves: Behavior, Ecology and Conservation, University ofChicago Press, Chicago, IL.


T. Caro and C. Stoner [2003], The Potential for Interspecific Competition Among AfricanCarnivores. Biol. Conserv. 110(1), 67–75.

C. Carroll, M.K. Phillips, N.H. Schumaker, and D.W. Smith [2003], Impacts of LandscapeChange on Wolf Restoration Success: Planning A Reintroduction Program Based on Static andDynamic Spatial Models. Conserv. Biol. 17(2), 536–548.

F. Courchamp, B. Grenfell, and T. Clutton-Brock [1999], Population Dynamics of ObligateCooperators. Proc. R. Soc. Biol. Sci. 266(1419), 557–563.

C. Glenz, R. Schlaepfer, A. Massolo, and D. Kuonen [2001], A Wolf Habitat Suitability Predic-tion Study in Valais (Switzerland). Landscape Urban Plan. 55(1), 55–65.

J.M. Cushing, B. Dennis, R.A. Desharanais, and R.F. Costantin0 [1998], Moving TowardAn Unstable Equilibrium: Saddle Nodes in Population Systems. J. Anim. Ecol. 67(2), 298–306.

A. Jensen and D. Miller [2001], Age Structured Matrix Predation Model for the Dynamics ofWolf and Deer Populations. Ecol. Model. 141(1–3), 299–305.

R. Haight, D. Mladenoff, and A. Wydeven [1998], Modeling Disjunct Gray Wolf Populations inSemi-Wild Landscapes. Conserv. Biol. 12(4), 879–888.

C.S. Holling [1973], Resilience and Stability of Ecological Systems. Ann. Rev. Ecol. Syst. 4,1–23.

M. Kauffman, N. Varley, D.W. Smith, D.R. Stahler, D.R. MacNulty, and M.S. Boyce [2007],Landscape Heterogeneity Shapes Predation in a Newly Restored Predator Prey system. Ecol. Lett.10, 690–700.

T. Larsen and W.J. Ripple [2006], Modeling Gray Wolf (Canis Lupus) Habitat in the PacificNorthwest, U.S.A. J. Conserv. Plan. 2, 17–33.

M. Lewis, K.J. White, and J. Murray [1997], Analysis of a Model for Wolf Territories. J. Math.Biol. 35, 749–774.

L. Mech [1991], The Way of the Wolf, Voyageur Press, Stillwater, MN.B.A. Melbourne and A. Hastings [2008], Extinction Risk Depends Strongly on Factors Con-

tributing to Stochasticity. Nature 454(3), 100–103.D. Miller, A. Jensen, and J. Hammill [2002], Density Dependent Matrix Model for Gray Wolf

Population Projection. Ecol. Model. 151, 271–278.D.J. Mladenoff and T.A. Sickley [1998], Assessing Potential Gray Wolf Restoration in the

Northeastern United States: A Spatial Prediction of Favorable Habitat and Potential PopulationLevels. J. Wildlife Manage. 62(1), 1–10.

P.C. Paquet, J.R. Strittholt, N.L. Staus, P. Wilson, S. Grewal and B.N. White [2001], LargeMammal Restoration: Ecological and Sociological Challenges in the 21st Century, Chapter 2,Island Press, Washington, DC.

R.O. Peterson [1988], The Pit or the Pendulum: Issues in Large Carnivore Management inNatural Ecosystems, The University of Washington Press, Seattle, Washington, DC, Chapter 6,pp 105–117.

P.A. Schmidt and L.D. Mech [1997], Wolf Pack Size and Food Acquisition. Am. Nat. 150(4),513–517.

C. Sime [2009], The 2009 Montana Wolf Hunting Season Summary. Tech. rep., Montana Fish,Wildlife and Parks. Available at: http://fwp.mt.gov/fwpDoc.html?id=41454.

C. Sime [2009], Wolf Harvest Model Simulation Informational Supplement. Tech. rep., MontanaFish, Wildlife and Parks. Available at: http://fwp.mt.gov/fwpDoc.html?id=47299.

D. Smith, D. Stahler, E. Albers, R. McIntyre, M. Metz, J. Irving, R. Raymond, C.Anton, K. Cassidy-Quimby, and N. Bowersock [1995–2011], Yellowstone Wolf Project An-nual Reports. Tech. rep., National Park Service—The Yellowstone Foundation. Available at:http://www.nps.gov/yell/naturescience/wolves.htm.


D. Smith, T. Meier, E. Geffen, L.D. Mech, J.W. Burch, L.G. Adams, and R.K. Wayne [1997],Is Incest Common in Gray Wolf Packs? Behav. Ecol. 8(4), 384–391.

D.W. Smith and G. Ferguson [2012], Decade of the Wolf: Returning the Wild to Yellowstone,First Lyons Press, Guilford, CT.

N. Varley and M.S. Boyce [2006], Adaptive Management for Reintroductions: Updating A WolfRecovery Model for Yellowstone National Park. Ecol. Model. 193, 315–339.

J. Vucetich and S. Creel [1999], Ecological Interactions, Social Organization and ExtinctionRisk in African Wild Dogs. Conserv. Biol. 13, 1172–1182.

Carter G. Walker, Moon Spotlight Wyoming. Avalon Travel, Berkeley, CA, 2012.A. Wydeven, J. Wiedenhoeft, R. Schultz, J. Bruner, and S. Boles [2012], Wisconsin Endangered

Resources Report 143 Status of the Timber Wolf in Wisconsin Performance Report 1 July 2011Through 30 June 2012 (also Progress Reports for April 2011–14 April 2012, and 2011 summaries).Tech. rep., Bureau of Endangered Resources - Wisconsin Department of Natural Resources. Avail-able at: http://dnr.wi.gov/topic/wildlifehabitat/wolf/documents/erreport143.pdf.

A. Zellner [1962], An Efficient Method of Estimating Seemingly Unrelated Regressions and Testsof Aggregation Bias. J. Am. Stat. Assoc. 57, 500–509.

Documents

GRAY WOLF POPULATION PROJECTION WITH INTRASPECIFIC COMPETITION