Cross-jurisdictional management of a trophy-hunted species

Author’s Accepted Manuscript Cross-Jurisdictional Management of a TrophyHunted Species Jacob Hochard, David Finnoff

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S0022-5193(17)30056-5 http://dx.doi.org/10.1016/j.jtbi.2017.02.001 YJTBI8957

To appear in: Journal of Theoretical Biology Received date: 9 May 2016 Revised date: 24 January 2017 Accepted date: 2 February 2017 Cite this article as: Jacob Hochard and David Finnoff, Cross-Jurisdictional Management of a Trophy-Hunted Species, Journal of Theoretical Biology, http://dx.doi.org/10.1016/j.jtbi.2017.02.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Full title: Cross-Jurisdictional Management of a Trophy-Hunted Species Authors: (1) Jacob Hochard (corresponding author) Department of Economics Institute for Coastal Science and Policy East Carolina University 252.328.6383 | [email protected] Greenville, NC 27858 (2) David Finnoff Department of Economics and Finance University of Wyoming Laramie, WY 82071 Abstract. Gray wolves (Canis lupus) are managed for competing uses in the Greater Yellowstone Ecosystem (GYE). Tourism benefits Yellowstone National Park (YNP) visitors while trophy hunting benefits hunters outside of the park. We investigate the policy scope of gray wolf management across jurisdictional boundaries by incorporating three foundations of the behavioral ecology of wolves: refuge-seeking behavior, optimal foraging group size and territoriality. Tradeoffs between and within consumptive and non-consumptive human benefits and wolf population fitness and life history indicators are quantified as a set of elasticities, providing clear implications to resource managers. Our approach highlights that hunting intensity affects the provision of consumptive and non-consumptive human benefits across jurisdictional boundaries and ought to be managed accordingly. We also show that population levels are an incomplete indicator of species fitness, which may depend on how hunting policies impact underlying group ecology. Our findings suggest traditional optimization approaches to wildlife management may lead to suboptimal policy recommendations when the boundaries on the natural system are oversimplified. Highlighting the human element of wildlife management, we show that understanding tourist and hunter responses to wildlife population abundances is critical to balancing provision of consumptive and non-consumptive human uses. Keywords: behavioral ecology, competing uses, elasticities, gray wolves, wildlife management.

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Cross-Jurisdictional Management of a Trophy-Hunted Species

2

Abstract. Gray wolves (Canis lupus) are managed for competing uses in the Greater Yellowstone Ecosystem (GYE). Tourism benefits Yellowstone National Park (YNP) visitors while trophy hunting benefits

4

hunters outside of the park. We investigate the policy scope of gray wolf management across jurisdictional boundaries by incorporating three foundations of the behavioral ecology of wolves: refuge-seeking

6

behavior, optimal foraging group size and territoriality. Tradeoffs between and within consumptive and non-consumptive human benefits and wolf population fitness and life history indicators are quantified as a

8

set of elasticities, providing clear implications to resource managers. Our approach highlights that hunting intensity affects the provision of consumptive and non-consumptive human benefits across jurisdictional

10

boundaries and ought to be managed accordingly. We also show that population levels are an incomplete indicator of species fitness, which may depend on how hunting policies impact underlying group ecology.

12

Our findings suggest traditional optimization approaches to wildlife management may lead to suboptimal policy recommendations when the boundaries on the natural system are oversimplified. Highlighting the hu-

14

man element of wildlife management, we show that understanding tourist and hunter responses to wildlife population abundances is critical to balancing provision of consumptive and non-consumptive human uses.

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Keywords: behavioral ecology, competing uses, elasticities, gray wolves, wildlife management.

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Gray wolf (Canis lupus) reintroduction into Yellowstone National Park (YNP) may be the Endangered 18

Species Act’s (ESA) most successful recovery program. The now recovered population is managed for competing uses across state and federal boundaries throughout the Greater Yellowstone Ecosystem (GYE).

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Tourism benefits Yellowstone National Park (YNP) visitors while trophy hunting occurs outside of the park. Bioeconomic models often inform such wildlife management decisions by optimizing consumptive and non-

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consumptive human benefits while ensuring the survival of a minimum viable population (Clark and Munro 1978, Wacker 1999, Bulte and van Kooten 2001, Eiswerth and van Kooten 2009). In practice, maintaining

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population stability while enjoying a balanced flow of consumptive and non-consumptive benefits can be challenging. Federal wolf hunting moratoriums have been reinstated in Wyoming to safeguard population

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stability thus terminating a flow of benefits to wolf hunters (Smith et al. 2016). We develop a wolf population model with behavioral foundations that shows population levels are an incomplete indicator of wolf popu-

28

lation fitness and life history indicators when hunting affects underlying group ecology. As an alternative to optimization-based management, a set of management elasticities are presented that capture a portfolio of

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wolf population fitness and life history indicators (e.g. reproductive fitness, recruitment, dispersal, etc.) and human use (e.g. recreation days, tourism views, etc.) tradeoffs.

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With notable exceptions (Sanchirico and Wilen 1999, Naevdal et al. 2012), few bioeconomic models examine wildlife use management at the sub-population level, which is particularly relevant for social grouping

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species such as gray wolves. Such group-level social structure, found in obligate cooperative breeders, creates a high risk of group-level extinction (Courchamp et al. 1999) while population levels, especially of

36

large carnivores in the presence of a protected area, are poor predictors of local extinctions (Woodroffe and Ginsberg 1998). We constructed a bioeconomic model of wolf management in the presence of a protected

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area that serves as a wolf refuge from hunting, such as the Greater Yellowstone Ecosystem (GYE). The impacts of hunting on group ecology are integrated into our model by recognizing three features of gray

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wolf behavior: refuge-seeking behavior, optimal foraging group size and territoriality. Using behavioral detail to model sub-population dynamics increases the precision with which a wildlife

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manager can estimate the fitness of an actively managed gray wolf population under alternative management scenarios. Group size, dispersal rates, the creation of new groups and within-group recruitment are all

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capable indicators of population fitness useful for making informed management decisions. Geographic 2

isolation may reduce the fitness of a group’s average individual, or create an Allee effect, which may lead 46

to group-level extinction (Pardini et al. 2010) while dispersal among subpopulations will generally offset this risk (Taylor 1990, Frank and Brickman 2000, Hill et al. 2002). Similarly, wildlife managers can avoid

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dispersal-induced instability by recognizing hunting policies may generate spatial synchrony, via correlated shocks to the wolf population from hunting (Abbott 2011).

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Our model, calibrated loosely to the GYE, measures the responsiveness of consumptive and non-consumptive wolf population uses within and beyond YNP boundaries using management elasticity indicators. Wildlife

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management elasticities highlight that the provision of consumptive and non-consumptive uses are not simply bound by the fitness of the wildlife population. Rather, these services are traded off with wolf population

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fitness and life history indicators at the group and population levels. Using elasticities instead of optimization techniques to present these management tradeoffs avoids imposing structure on the human outcomes

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that are deemed highest priority to the wildlife manager. Instead, the measures can be used to inform tradeoffs that result from specific hunting strategies across space and time. We also show these hunting strategies

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can be tailored to address natural system uniquities. Changes to these management elasticities are examined in the presence of a highly aggressive pack or prey-switching behavior and when distributing hunting quotas

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across many or few packs. As a display of the interdisciplinary nature of wildlife management, we show that assessing management outcomes requires a clear understanding of the human response to announced

62

hunting policies as well as the wildlife response to human presence. Modeling strategic territory choice

64

A model of fitness optimality is constructed for i wolf packs simultaneously choosing a territory, Ai,t , in time t with i = 1, 2, ..., n (Figure 1). We assume that prey and landscape covariates (e.g. grasslands, steep

66

¯ All packs compete over this terrain, etc.) are distributed uniformly across a homogeneous habitat of area A. limited habitat and do not overlap. The territory a pack controls, Ai,t , measured in units of area, represents

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the resources available to that pack to increase reproductive success. Following carnivore group size theory (Mech and Boitani 2003), wolf packs are assumed to be maximizing net energy of predation by choosing

70

an ideal pack size, s∗ , which is predetermined by the biomass of primary prey. For example, s∗ may be 8 in an area where elk are the primary prey or 12 in an area where bison are the primary prey. Deviations in 3

Timing (within one year)

Management and Wolf Group Ecology packs simultaneously defend a territory (strategically) to maximize fitness

February - April

benefits costs

• •

Habitat quality Optimal foraging group size

• • •

Intraspecific strife Defensibility Maintenance costs

recruitment dispersal

• •

May - September

Locate breeding partner Identify vacant territory

Human Outcomes

Consumptive Uses •

Number of recreation days

•

Hunting season length

•

State revenue

•

Hunters’ demand response

new pack creation

October - December

January

total hunting quota (known to hunters) wolf density in hunted area (known only to managers)

harvest quotas set for each pack

redistribution of packs within and outside of refuge

•

prior period wolf density (known to tourists) current wolf density (known only to managers)

Refugeseeking behavior

•

Hunters’ success rate

•

Non-Consumptive Uses Visitors’ demand response

•

Number of wolf views

Figure 1: Model schematic displaying the timing, progression and features of our wolf population model and the production of consumptive and non-consumptive uses. Arrows capture the model’s progression, bullet points capture the model’s features and braces represent decisions made using incomplete information. 72

pack size, si,t , from the ideal pack size downscale the resources available to a pack from a given territory, due to fitness gains from the security of a larger pack and fitness losses from increased competition over

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available resources (Rodman 1981) where the ideal group size maximizes foraging efficiency. Territory and its resources translate into reproductive ability at rate η. This reproductive ability, without loss of generality,

76

can be interpreted directly as reproduction or in intermediary units of reproductive fitness (Both and Visser, 2003). We adopt the latter interpretation where η is measured in units of reproductive fitness per unit area.

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Intermediary units of fitness offer greater flexibility in defining pack-level reproduction. Total reproductive fitness benefits from territory, T B, are

 s∗2 − [si,t − s∗ ]2 T Bi,t = η Ai,t s∗2 

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for t = 1, 2, ..., T measured in units of reproductive fitness. Territory provides benefits, yet is costly to maintain. Following Both and Visser (2003), there is a fixed territory maintenance cost, M, independent of the

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Table 1: Baseline parameters used to display modeling relationships (Figures 1-4), define recruitment process (Figure 5) and execute elasticity simulations (Figures 6-12).

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Parameter

Notation

Units

Baseline values

Defensibility Habitat quality Initial number of packs Intraspecific penalty Location parameter (recruitment process) Max recruitment paramater (recruitment process) Optimal foraging pack size Pack’s natural mortality rate Scale parameter (recruitment process) Terminal time Habitat area Territory maintenance cost

D η n ε µ ρ s∗ δ σ T A¯ M

Fitness / wolf Fitness / km2 Packs Scalar Scalar Scalar Wolves Scalar Scalar Years km2 Fitness

0.1 100 6 0.05 0 6, 000 10 0.125 0.5 12 100 0

size of the territory demanded that reduces reproductive fitness directly. There are also costs from intraspecific competition in territory defense using the total number of wolves, ∑ni=1 si,t , per unit of vacant territory,

84

A¯ − ∑ni=1 Ai,t . Higher densities per unit of vacant territory correspond to less buffer space between wolf packs and greater intraspecific strife. As buffer space disappears, A¯ − ∑ni=1 Ai,t → 0, the total cost of territory

86

defense approaches infinity, which is consistent with convexly increasing cost curves used commonly in economics. Defensive abilities are differentiated by pack-specific defendability parameter Di , accounting for different levels of aggression or defense capabilities across packs. Per-pack defendability is the inverse ratio of how a pack’s representative wolf pays for, in terms of reproductive fitness, territory that is threatened by higher population levels and smaller buffer areas between packs. The inverse form and assuming Di > 0 ensures that a higher per-pack defendability makes territory defense less costly. The total cost of territory, TC, in terms of reproductive fitness, when combining maintenance costs and intraspecific competition costs, is  1 ∑ni=1 si,t TCi,t = M + ¯ Ai,t . n A − ∑i=1 Ai,t Di 

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Pack i maximizes reproductive fitness in time t

max Fi,t = T Bi,t − TCi,t , Ai,t

which depends strategically on the territory choices of all other n − 1 competing packs. We restrict any pack from reaching twice the ideal pack size, which would drive the marginal benefit of territory negative and be inconsistent with optimal foraging behavior. In the first-order condition, the marginal benefit (MB) of an additional unit of territory to the fitness of pack i,

MBi,t =

 ∗2  s − (si,t − s∗ )2 dT Bi,t , =η dAi,t s∗2

or the fixed benefit to territory size discounted by a pack-size deviation from the ideal pack size, equates with the marginal cost (MC) of an additional unit of territory to the fitness of pack i,   dTCi,t 1 ∑ni=1 si,t ∑ni=1 si,t , = MCi,t = + Ai,t ¯ dAi,t Di A¯ − ∑ni=1 Ai,t (A − ∑ni=1 Ai,t )2

Figure 2: A single pack chooses territory to maximize its reproductive fitness in the absence of competition.

6

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where a pack’s marginal cost is scaled by its defendability parameter. A pack with a limited ability to defend its territory (Di → 0) will pay a higher marginal cost than a pack with stronger defense capabilities (Di → 1).

90

We only consider those feasible equilibria with a total occupied territory less than the total available territory. In the case of a single pack, n = 1, the pack maximizes fitness by choosing territory in a non-competitive

92

environment (Figure 2). This framework resembles closely the conclusion from Both and Visser (2003) that a species chooses territory optimally to maximize net benefits to fitness. Equilibrium territory size

94

is denoted by A∗ and equilibrium fitness is found by evaluating the objective function at A∗ . Benefits to territory increase linearly while costs are convex as the pack occupies the majority of territory driving buffer

96

space towards zero. Even in a single pack environment the pack does not occupy all territory because it becomes increasingly costly to convert buffer space into territory. The intuition here is that the existing pack

98

must still defend against potential entrant packs or that the pack has a natural range limiting the feasible territory size. The model is extended to include two packs simultaneously choosing a territory optimally (Figure 3). In the multi-pack case, each pack i recognizes the existence and strategic territory demand response of pack q for i 6= q. In the first-order condition, which follows from equating marginal costs and benefits to reproductive fitness of territory choice, pack i chooses territory in best response to pack q’s choice of territory and pack q chooses territory in best response to pack i0 s choice of territory

Ai,t = Ai,t (Aq,t ) ∀ i 6= q.

100

These behavioral rules govern territory choice of each pack and trace out two territory choice reaction functions that solve for the territory choice of pack i in terms of the model parameters and the choice of

102

territory by pack q. This form of competition, known as Cournot competition in economics, is used often to represent firms competing on quantity of production in a strategic market environment. These reaction

104

curves intersect at a stable Nash equilibrium where no individual pack could improve fitness by unilaterally choosing to defend a different territory. The Nash aggregate territory demanded (A∗1,t + A∗2,t ) in the two pack

106

case is lower than the territory demanded in the single pack case (A∗ ). Habitat sharing occurs as each pack chooses territory optimally. These choices balance benefits to reproductive fitness from territory and the cost 7

Figure 3: Two packs simultaneously and strategically choosing a territory to maximize reproductive fitness leading to a stable Nash equilibrium. 108

of potential intraspecific strife by expanding territory and reducing the buffer zone between the two packs. Evaluating the fitness objective function at the equilibrium territory levels yields the per pack equilibrium

110

fitness levels. The symmetric nature of parameters employed in this baseline approach yields two packs with identical territory sizes and fitness levels (Figure 3). This framework can be extended to the n pack

112

case where territory reaction curves intersect in an n-dimensional space with implications for reproductive fitness and territory choice at both the group and population levels.

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Spatial dynamics, harvest and refuge-seeking behavior

114

Packs choose territory within or outside of a refuge, which is a protected area without hunting, through a comparison of relative reproductive fitness of territory acquisition. We assume here that packs know how they would fair in a territory-choice game with other in-refuge or out-of-refuge packs. This relative success changes with each additional entrant in-refuge. Packs continue this comparison and adjust to move into the area with the higher reproductive fitness until the relative advantage of one area versus the other is dissipated. Harvesting policies are designed such that the wildlife manager has perfect control over determining the distribution of harvest across packs. This is consistent with the notion of harvest zone delineations isolating each pack and harvest quotas within those zones determining how intensively to hunt each individual pack. Let this additional risk of harvest be represented by a constant reduction to the harvested pack’s size. The set of conditions governing refuge and out-of-refuge territory choice are # s∗ 2 − (si,t − s∗ )2 − ← − = η ← s∗ 2 ← − ← − # " ntr ntr si,t si,t 1 ∑i=1 ∑i=1 + Ai,t r r ¯ − ∑nt Ai,t ]2 Di A¯ − ∑nt Ai,t [ A i=1 i=1 ← − ← − "

and # s∗ 2 − (s j,t (1 − h j,t ) − s∗ )2 → − → − η = s∗ 2 → − → − # " n−nr n−nr t ∑ j=1t s j,t (1 − h j,t ) 1 ∑ j=1 s j,t (1 − h j,t ) + A j,t r r ¯ − ∑n−nt A j,t ¯ − ∑n−nt A j,t ]2 Dj A [ A j=1 j=1 → − → − "

for i = 1, 2, ..., ntr denotes the i’th in-refuge pack and j = 1, 2, ..., n − ntr denotes the j’th out-of-refuge pack. 116

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¯ Here, left arrow accents, η , s∗ and ← A −, track in-refuge parameters while right arrow accents track out-of− ← − ← A¯ , for habitat quality, optimal foraging group size and habitat area. For refuge parameters, η , s∗ and → − − → − → example, bison may be more plentiful with a national park warranting a larger foraging group. Outside of the park, elk and deer may be more plentiful warranting smaller foraging groups.

9

120

Hunting a portion of an out-of-refuge pack, h j,t , reNet reproductive fitness

Fitness

duces that pack’s size to below its ideal level forcOut-of-refuge

122

ing it to operate less efficiently. This is represented

non-hunted pack

by the hunted pack’s total benefits line rotating to 124

In-refuge nonhunted pack

a more gradual slope (Figure 4). All other out-ofrefuge packs maintain their optimal foraging group

126

size and have total benefits line identical to those

Out-of-refuge hunted pack

packs located within refuge (Figure 4). Addition128

ally, non-hunted out-of-refuge packs recognize the

Territory Area

Fitness

debilitated state of their hunted counterparts and

Total benefit (TB) and

“best respond” by expanding their territories. Si-

reproductive fitness

total cost (TC) to

130

TC TC

multaneously, the hunted pack anticipates this ex132

pansion and best responds by reducing its territory

TB TB

size. These strategic responses enable non-hunted 134

TC

packs to compete for larger stakes of unoccupied territory while hunted packs are forced to establish

136

Territory Area

Fitness

territory in those limited areas they know will be

MC Marginal benefit (MB)

uncontested. These effects operate through the de-

MC

and marginal cost (MC) to reproductive

138

nominator of each pack’s cost curve. As such, the

fitness

MC

total cost curve of a hunted pack rotates to a steeper 140

MB

slope while the total cost curve of a non-hunted

MB

pack rotates to a more gradual slope (Figure 4). 142

This results in out-of-refuge non-hunted packs realizing greater reproductive fitness and controlling

144

146

Territory Area

larger territory shares than in-refuge packs (Figure Figure 4: Harvest, fitness and territory response. Parameters: n = 6, ntr = 3, s∗ = 10, h j = 0 ∀ j 6= 1 and 4). This relative surplus occurs at the expense of the h1 = 0.5. Remaining parameters available in Table 1. hunted pack, which realizes lower fitness and controls less territory (Figure 4).

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In marginal space, hunting an out-of-refuge pack causes the hunted pack’s marginal benefit line to shift downward resulting in a smaller territory size and lower reproductive fitness. The strategic response, in

150

marginal space, is given by an outward (inward) rotating marginal cost curve of the non-hunted (hunted) out-of-refuge pack, which results in a more gradual (steeper) slope (Figure 4). The strategic nature of

152

packs’ competing territory choices causes all out-of-refuge total cost and marginal cost curves to adjust, in response to any harvest policy, regardless of the distribution of harvest across packs. Harvested packs

154

avoid harvest by seeking refuge. Those packs migrate from out-of-refuge to in-refuge until the reproductive fitness of the worst-off out-of-refuge pack equilibrates to the representative in-refuge pack for a given pack

156

distribution following

∗ Min(Fj,t )|nt∗r ∀ j = Fi,t∗ |nt∗r .

The intuition of this equilibrium condition is that hunted packs seek refuge to avoid being hunted. This 158

allows refuge-seeking packs to operate at the ideal pack size. Hunting pressure shifts to a new out-of-refuge pack, which may also choose to seek refuge. Every additional refuge entrant drives up the level of in-refuge

160

intraspecific competition and drives down the in-refuge per-pack fitness. In-refuge migration occurs until the marginal entrant would receive higher reproductive fitness by operating at a less-than-ideal pack size

162

than operating at the ideal pack size in-refuge while facing higher intraspecific pressure. This migration condition governs a clear tradeoff between out-of-refuge pack-fitness loss from hunting and in-refuge loss

164

from competition and clears based on an updating distribution of packs across space, ntr and n − ntr .

Temporal dynamics, recruitment and new pack creation

Each pack reproduces once annually. A recruitment function links units of reproductive fitness, which are realized by resource availability, territory choices and hunting regimes, with natural mortality that occurs within each pack. Because packs generally have a single breeding pair, annual recruitment is bounded between zero and some upper threshold. The lognormal cumulative distribution function (CDF), used commonly to specify stochasticity around recruitment processes (Shelton 1992; Higgins et al. 1997), is used here to translate per-pack fitness into per-pack recruitment. Distribution parameters are chosen to yield rea11

sonable rates of annual recruitment following F

Log( ρi,t ) − µ 1 1 √ si,t+1 = si,t [1 − δ ] + [ + Er f ] 2 2 2σ 166

where µ and σ are the mean and standard deviation of the CDF. Biologically, µ determines the CDF’s inflection point corresponding to the fitness level for which recruitment is at 50% of its maximum while

168

σ determines the additional fitness required to reach maximum recruitment. The minimum and maximum recruitment capacity for one pack is determined by a scalar, ρ. For all of our simulations, we choose

170

ρ = 6, 000, µ = 0 and σ = 12 , which bounds maximum recruitment asymptotically at 8 pups and minimum recruitment at zero pups) and δ is a natural mortality rate for each pack set at 0.125 (Table 1 and Figure 5). Here, i notation is used while this pack-size growth process applies also to all j out-of-refuge packs. Recruitment max

0

Fitness Fitness

Figure 5: Lognormal Cumulative Distribution Function (CDF) recruitment process used to determine recruitment based on units of reproductive fitness. 172

Recruitment drives packs above their optimal size (s∗ ). Dispersal occurs until each pack is driven back to its optimal size and dispersing (lone) wolves conjoin to form new packs of size s∗ . The number of new packs follows

12

n −nr

nr

t t t sj si + ∑ j=1 ∑i=1 nt+1 = s∗ [nt ε)]

where the numerator is the total wolf population (sum of in-refuge and out-of-refuge wolves) and the de174

nominator divides the total wolf population into packs of ideal size s∗ . The denominator scaling component, (nt ε), imposes an intraspecific penalty, ε, on pack creation for each additional pack. We set the intraspecific

176

penalty at = 0.05 (Table 1). Effectively, as packs become congested (an increasing nt ) it requires more than the ideal number of per-pack wolves, s∗ , to sustain the optimal pack size because, as observed by Cubaynes

178

et al. (2014), density-dependent intraspecific aggression mediates wolf survival. This is consistent with dispersing wolves failing to create a pack, failing to locate suitable and vacant territory or failing to locate a

180

breeding partner in the face of increasing congestion.

Timing

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The timing of the model is measured in yearly increments (Figure 1). Four stages occur within any given unit of time. First, packs choose their territory in February through April and use that fitness to reproduce. In the

184

summer following reproduction, packs have been pushed above their optimal pack size and the remainder wolves disperse thus returning natal packs to their optimal size. All dispersing wolves conjoin into new

186

packs at the ideal pack size. In October to December, the hunting season begins and managers determine how to allocate harvest quotas across out-of-refuge packs. Packs respond immediately to this hunting by

188

adjusting their territory choices. These territory adjustments occur in several ways. First, hunted packs may relocate for refuge. Second, hunted packs may relinquish territory in their weakened state if seeking refuge

190

is too costly. Third, out-of-refuge packs that are not hunted may cannibalize those areas given up by hunted packs. In January, after the hunting season completes, packs relocate until the in-refuge migration condition

192

is satisfied. After pack relocation in-refuge and out-of-refuge is settled, territory choices are again updated and reproduction in time t + 1 occurs.

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Consumptive and non-consumptive uses

194

Wildlife viewers, henceforth tourists, are attracted to the prospect of seeing a wolf, which is increasing 196

198

in the park’s wolf population density. This information is available to park visitors with a one-year lag. The number of visitors in time t, τt , is an increasing function with the refuge’s prior-period wolf density,  ∗ r  s nt−1 − τt ← . Here, the numerator of the τt function’s argument is the product of the number of packs within A¯ ← −

the refuge and the optimal pack size within the refuge. This product represents the in-park population 200

because packs return to their optimal pack size after dispersal. The actual probability a visitor sees a wolf, s∗ ntr

− ), which is bounded between 0 and ptv , is an increasing function with the park’s current wolf density, ptv ( ← A¯ ← −

202

1. The total number of wolf views in time t is the product of the number of visitors and their probability of  ∗ r  s nt−1 s∗ ntr v( ← − − ). p seeing a wolf, τt ← t A¯ A¯ ← −

204

← −

Hunters have information about the harvest quota but not about the population distribution of wolves across space. The number of hunters in time t, γt , is an increasing function of the total harvest quota across all

206

n −nr

t t h j,t ), where the number of hunted packs, zt , is a subset, or vector subspace, out-of-refuge packs, γt (∑ j=1

of the total number of out-of-refuge packs, (zt ⊂ nt − ntr ). The initial density of wolves in pack j across 208

its territory is

s∗ → − A j,t

with an end-of-hunting-season density of

s∗ (1−h j,t ) → − . A j,t

The average density of hunted packs,

throughout the season, is weighted across all packs based on how intensively each pack is hunted

zt

∑

j=1

210

h j,t zt ∑ j=1 h j,t

"

s∗ (2 − h j,t ) → − 2A j,t

## .

This density of wolves across hunted areas determines the probability a hunter records a successful hunt, ptk ,    ∗  s (2−h j,t ) h j,t zt k → − ; a measure of hunter success. Hunter success is on a given day in time t, pt ∑ j=1 ∑zt h 2A j,t j=1

212

"

j,t

determined by two competing forces. First, a harvested pack maintains a smaller territory in its weakened state increasing wolf density. Second, as harvest occurs throughout that season, there are fewer wolves

214

spread across that territory reducing wolf density. This specification captures both competing effects. Combining the number of hunters with the probability a hunter records a successful hunt in a given day yields,   ∗     s [2−h j,t ] h j,t nt −ntr zt k → − γt ∑ j=1 h j,t pt ∑ j=1 ∑zt h the number of successful wolf hunts per day in a hunting 2A j,t j,t j=1

14

season where both γt and ptk are functions increasing in their arguments shown in parenthesis. The expected length of that hunting season is zt (1 − h j,t ) s ∗ ∑ j=1 → −     s∗ (2−h j,t ) h j,t n−ntr zt k → − γt (∑ j=1 h j,t )pt ∑ j=1 ∑zt h 2A j,t j,t j=1

or the total season harvest divided by the daily rate of harvest while the total number of recreation days in that season is

zt (1 − h j,t ) s ∗ ∑ j=1 −    ,  ∗→ s (2−h j,t ) h j,t zt k → − pt ∑ j=1 ∑zt h 2A j,t j,t j=1

the product of the number of hunters and the length of the hunting season. To arrive at the number of recre216

ation days in a season, we make the assumption that hunters are not removed after recording a successful hunt. In reality, a license generally expires after a successful hunt is registered. Because gray wolf hunting

218

success rates are so low, between 1%-3% of all hunters reporting a successful hunt, we retain the current form for simplicity. Parameterizing the production process

220

Wildlife benefits across management jurisdictions are determined in part by four implicit functional forms 222

relating (i) refuge visitation and prior-period wolf density, (ii) wolf viewing rates and current period wolf density, (iii) the number of hunters and the current period wolf harvest quota and (iv) wolf kill rates and the

224

density of wolves across hunting areas throughout the hunting season. For illustration, we impose several simplified functional forms to human and ecological data collected from the GYE. These data offer a loose

226

calibration to an area where gray wolves are being hunted actively beyond the boundaries of YNP, which is a known refuge area.

228

A linear trend for 2011, 2012 and 2013 was used to predict YNP visitation in time t based on the wolf population in t − 1 giving a constant visitation of 1, 842, 023 people annually with an additional 16, 201

230

people visiting in time t for every additional wolf in time t − 1. The constant visitation term captures park visitation for reasons other than wolf tourism, which we choose not to focus on in our model. Based on a

232

survey between 2004 - 2006, 11.6% of YNP visitors reported seeing a wolf (Duffield et al. 2006) over which 15

time an average of 142 wolves were present in the park.1 Using a linear trend, assuming no wolf views are 234

recorded when wolves are not present, an additional in-park wolf increases viewership by ≈

8 100

percentage

points. Assuming no hunters purchase licenses in the absence of a wolf population, we run a linear trend 236

between licenses sold in 2012 (4, 492) and 2013 (2, 153) with corresponding harvest quotas of 52 and 39, respectively. For each additional unit increase in the harvest quota and recognizing that the price of wolf

238

hunting licenses was unchanged from 2012 to 2013, we predict ≈ 75 new hunters to take part in the hunting season.

240

Calibration of hunting success rates rely on detailed harvest-based recreation reports and pack territory data available from Wyoming Game and Fish annual hunting reports. Based on maps in the Fish and Wildlife Ser-

242

vice annual wolf reports for 2012 and 2013 (Wyoming Game and Fish 2013/2014), we identify GYE packs within 11 trophy hunting zones as well as the hunter success rates and wolf populations that accompany

244

each of those zones. The Targhee zone, which was closed in 2013, is excluded from our parameterization. Because we do not observe the territory size of each pack, we implicitly assume that packs do not change

246

their home range between 2012 and 2013 and use the difference in wolves per zone and difference in success rates per zone to predict the impact of increasing wolf density on harvest success. We find that a one wolf

248

increase, within the average harvest zone, increases the probability of a successful hunt by 0.026%. Our model territory sizes are not calibrated to GYE harvest zone areas because of limited data. We impose a

250

scalar to calibrate our 0.026% hunting success measure to our model. The average wolf density for a given pack, assuming packs share available territory equally, is

252

1  ¯

A → − nt −nr

A¯

− . A baseline harvest success of = 0.026% where ι = 0.026% nt → −nr

1% is imposed around the density realized without harvest and adjusted based on this scalar following z

ptk 256

A → − nt −nr

around which a one-wolf increase causes a 0.026% change in hunter success. We solve for our wolfhunting success scalar, ι, by ι 

254

s∗ − ,  → ¯

t ∑ j=1

= ι[[zt − 1] ∑zt

h j,t A j,t

j=1 [1−h j,t

− ]

s∗ → −  ] + 2% ¯

A → − nt −nr

where average season densities higher than uniformly distributed

wolf densities increase the natural rate of harvest success at rate ι. The remaining parameters are biophysical and ecological in nature. These are chosen according to our prior assumptions and symmetrically for in-

258

refuge and out-of-refuge packs. 1 In

2004 there were 171 wolves in the park, 118 in 2005 and 136 in 2006 as per the YNP annual wolf reports here http://www.nps.gov/yell/naturescience/wolfrpts.htm.

16

We conducted hunting simulations and report the responsiveness of consumptive use outcomes, non-consumptive use outcomes and indicators of population fitness to changes in hunting intensity. We examine hunting policies as shares of the total out-of-refuge wolf population to be consistent with how these quotas are implezt h j,t s j,t ∑ j=1 . For simplicity, we restrict hunting to either one or two packs, mented by wildlife managers, nt −ntr s j,t ∑ j=1 zt ∈ [1, 2]. The former case lends insight into management strategies that expose fewer groups to hunting while the latter case lends insight into the effects of spreading hunting more evenly across groups. The hunting elasticity, φ , of any bioeconomic outcome variable, θ , is the percentage change in that indicator divided by the percentage change in the number of hunted wolves, which we denote as λ , dθ φλ = θ . dλ λ We restrict our hunting simulations to low percentage levels (1% to 5% of all out-of-refuge wolves) because 260

at most only two packs are bearing the brunt of all hunting. The baseline elasticities show the responsiveness of wolf population fitness and life history indicators (Figure 6) and the provision of consumptive and non-

262

consumptive uses (Figure 7) when that hunting is concentrated on one group or spread evenly across two groups. We simulate over twelve periods to allow ample time for all variable to reach their new equilibrium

264

values. The transition paths of these variables are all monotonic. The second set of elasticities displays the impact of distributing hunting across a “high defendability" pack

266

(D1 = 0.2) and a “low defendability" pack, (D2 = 0.1), on wolf population fitness and life history indicators (Figure 8) and the provision of consumptive and non-consumptive uses (Figure 9).

268

The third set of elasticities displays the impact of a distributing hunting across packs with different primary prey, such as elk, bison or deer, and therefore different optimal foraging group sizes (Figures 10 and 11).

270

In this case, the baseline s∗ = s∗ = 10 represents a pack with elk as its primary prey. We increase this to ← − → − s∗ = s∗ = 12 to represent bison as primary prey and decrease it to s∗ = s∗ = 8 to represent deer as primary − ← − → − ← − →

272

prey. Elasticity signs represent the bioeconomic variable’s direction of responsiveness to hunting. For example, indicators with positive elasticities are those that are accentuated by hunting while indicators with

274

negative elasticities are those that are attenuated by hunting whereas elasticity magnitudes are an indicator of how responsive that bioeconomic indicator is to hunting. 17

Management Elasticities

276

Baseline management elasticities

a) Fitness indicators.

b) Population and group distributions.

c) Territory indicators.

d) Recruitment and dispersal.

Figure 6: Baseline management of wolf population fitness and life history indicators’ elasticities. 278

Hunting fewer packs more intensively is disruptive to the bioeconomic system relative to spreading hunting evenly across two packs. The fitness of the hunted pack drops disproportionately (Figure 6a) triggering

280

pack relocation in-refuge, which drives up the in-refuge wolf population and drives down the out-of-refuge wolf population (Figure 6b). Refuge-fleeing packs put downward pressure on in-refuge territory sizes while

282

non-hunted out-of-refuge packs cannibalize vacated territory (Figure 6c). We also find the counterintuitive result that movement of new packs into the refuge increases the area of vacant territory while the area of

284

vacant out-of-refuge territory decreases. The intuition here is that buffer territory between packs mediates intraspecific aggression. As more packs flee for refuge, the aggregate buffer area increases as to insulate

286

packs from each other. In-refuge recruitment and dispersal impacts are driven by several factors. In the presence of hunting, fewer 18

a) Consumptive use indicators.

b) Non-consumptive use indicators.

Figure 7: Baseline management of consumptive and non-consumptive elasticities. 288

out-of-refuge packs are available to recruit and generate dispersing wolves. Further, hunted packs, in a fitness-weakened state, have an inhibited ability to reproduce (Figure 6d). Because a hunted pack’s re-

290

cruitment is unlikely to counterbalance the impacts of hunting, new pups drive the pack toward its optimal foraging group size but hunted wolves do not disperse away from their natal pack. Despite this effect, we see

292

a slight increase in out-of-refuge dispersal, which is driven by the fitness-strengthened state of non-hunted out-of-refuge packs (Figure 6d).

294

An increase in the announced hunting quota elicits a strong demand response from hunters causing the number of hunters, the state revenue and the season kills to be increasing in hunting intensity (Figure 7a).

296

Wolves respond to hunting intensity by fleeing in-refuge, which drives down the average season hunting density. A race occurs where an increasing numbers of hunters are targeting an increasingly reclusive out-

298

of-refuge wolf population. The wolves win this race when hunting is spread evenly across packs but lose this race when all hunters target one pack. In turn, the probability of a successful hunt in a given day decreases

300

in hunting intensity when hunting is spread evenly, though the magnitude is negligible, and increases in hunting intensity when hunting is concentrated on a single pack. The former case causes more hunting to

302

extend the hunting season while the latter case causes more hunting to shorten the hunting season (Figure 7a).

304

Managers are able to exploit their knowledge of wolf density across the landscape to extend the hunting season because hunters are only responding to the total quota when deciding to hunt or not. The number of

19

306

recreation days within the season is decreasing in hunting intensity despite making it more difficult to register a successful hunt in a given day by spreading hunting out evenly. Here, a tradeoff occurs where hunters

308

respond to an increased quota disproportionately to wolves’ ability to endure those hunters. Management techniques that make wolves more difficult to hunt would help generate more recreation days in a season.

310

The key is that those techniques must be privy only to the wildlife manager as not to deter potential hunters. The wolves’ elusiveness is used to the manager’s advantage to elicit a high hunter turnout and keep those

312

hunters in the field as long as possible before the season closure. Non-consumptive uses too are more sensitive to the targeted hunting of one pack (Figure 7b). These uses

314

are derived from in-refuge activities and refuge-seeking behavior, which is more intense when hunting only one pack. A larger in-refuge wolf population stemming from intensive out-of-refuge hunting increases

316

the likelihood of viewing a wolf, which attracts new tourists to the refuge. The number of wolf views increases disproportionately because more tourists are traveling to the refuge and those tourists are more

318

likely to see a wolf. This double dividend effect is represented by the larger hunting elasticity of wolf views magnitude relative to the elasticities on wolf viewing and park visitors. A tension exists in the provisioning

320

of consumptive and non-consumptive uses. State game agencies setting wolf hunting quotas are concerned with consumptive-use outcomes. Targeting one pack intensively increases hunter turnout and revenue but

322

reduces the season length and recreation days experienced by those hunters. To avoid early hunting season closures, a game agency may make it as difficult as possible to hunt wolves

324

by spreading those quotas across hunting zones. This form of management limits refuge-seeking behavior of wolves at the expense of tourism-related non-consumptive uses of the wolf population. A game agency

326

concerned with earning state revenue and encouraging a high hunter turnout may choose to hunt fewer packs more intensively. This form of management increases refuge-seeking behavior of wolves and accentuates

328

tourism-related consumptive uses of the wolf population. This management strategy is generally more disruptive to group fitness, territory choice and population level both in- and out-of-refuge.

20

Management elasticities for low-

330

and high-defendability packs

a) Fitness indicators.

b) Population and group distributions.

c) Territory indicators.

d) Recruitment and dispersal.

Figure 8: High-defensibility and low-defensibility pack management of ecological elasticities. 332

Managers must also decide how intensively to target packs with varying abilities to defend themselves against rival packs. For example, relative pack size and age composition of members are important determi-

334

nants of interpack conflict success (Cassidy et al. 2015). We present three cases where the manager targets (i) a low-defendability pack, (ii) a high-defendability pack and (iii) both packs evenly. Consistent with

336

our baseline analysis, targeting a single pack is generally more disruptive to the bioeconomic system than spreading hunting evenly across packs (Figure 8a-8d). This shock to the system is amplified when the high-

338

defendability pack is targeted. Here, the high defendability pack out-competes the low-defendability pack in equilibrium leaving the low-defendability pack with little territory and low-fitness. The system impact

340

of hunting is greater on the high-defendability pack because it has more fitness and territory to lose. This farther-to-fall effect results because hunting the high-defendability pack evens the playing field allowing

342

the low-defendability pack to recapture territory and increase its fitness in the presence of its superior rival. 21

a) Consumptive use indicators.

b) Non-consumptive use indicators.

Figure 9: High-defensibility and low-defensibility pack management of consumptive and non-consumptive elasticities. Contrarily, the high-defendability pack has little fitness to be gained when its weak rival is hunted. 344

The presence of a high-defendability out-of-refuge pack influences too the distribution of recruitment inand out-of-refuge (Figure 8d). Because the highest positive recruitment returns from increased fitness occur

346

at intermediate levels of fitness (Figure 5). A high-defendability pack captures a large share of all habitable territory generating high fitness where marginal returns to recruitment are low. Similarly, weaker rival

348

packs are driven to low fitness levels where marginal returns to recruitment are low. Following Jensen’s Inequality, hunting serves as an equality-increasing redistribution of territory that boosts aggregate levels

350

of fitness yielding higher out-of-refuge recruitment creating an opportunity-enhancing effect to non-hunted packs (Figure 8d).

352

Hunting the high-defendability pack and the low-defendability pack both allow for vacant territory to be cannibalized (Figure 8c) allowing those non-hunted out-of-refuge packs to increase fitness along the inter-

354

mediate portion of the lognormal CDF (Figure 5) where returns to recruitment are highest. In the former case, these fitness increases come at the expense of the high-defendability pack losing fitness along the right

356

tail where recruitment returns are low. In the latter case, these fitness increases come at the expense of the low-defendability pack losing fitness along the left tail where recruitment returns are low. The presence of

358

a high-defendability pack, via its ability to increase out-of-refuge recruitment when hunted, also makes the wolf population more enduring of hunting. Here, the presence of hunting corrects an inefficiency in the

360

ecological system arising from one pack having an innate ability to out-compete its rival packs. This cor-

22

a) Fitness indicators.

b) Population and group distributions.

c) Territory indicators.

d) Recruitment and dispersal.

Figure 10: Management of ecological elasticities with prey-switching behavior. rection is reflected by higher elasticity magnitudes on season kills, state revenue and the number of hunters 362

as compared to the case with equally-defendable packs (Figure 9a).

Management elasticities under prey-switching behavior

364

Consistent with our baseline and low-high defendability analyses, targeting a single pack is generally more disruptive to the bioeconomic system than spreading hunting evenly across packs (Figure 10). This shock

366

to a pack’s fitness is amplified when hunting smaller packs - i.e., deer as primary prey - relative to when hunting larger packs -i.e., bison as primary prey (Figure 10a). Here, hunting is more crippling to the small

368

pack because that pack’s resulting size is disproportionately distant from its optimal foraging group size relative to the larger pack. The foraging efficiency of a larger pack is partially insulated against losing

370

members. The relative vulnerability of small packs results in higher refuge-seeking behavior (Figure 10b), in-refuge 23

a) Consumptive use indicators.

b) Non-consumptive use indicators.

Figure 11: Management of consumptive and non-consumptive elasticities with prey-switching behavior. 372

population (Figure 10b) and non-consumptive benefits (Figure 11b). The double dividend to wolf views remains but is amplified when hunting a pack whose primary prey is deer as compared to bison. The

374

intuition governing impacts on territory, recruitment and dispersal indicators (Figure 10c and Figure 10d) as well as consumptive use outcomes (Figure 11a) follows closely with those simulations already discussed.

Sensitivity analysis

376

As hunter and visitor demand responses in the model are parameterized using few data points, the sensitivity 378

of the model to alternative specifications was examined. Our baseline model assumes that each additional wolf hunted attracts 75 new hunters. We also assume that YNP will receive 1,842,023 visitors annually

380

with an additional 16,201 visitors per YNP wolf. Ecological elasticities are unaffected by these demand responses because the production of consumptive and non-consumptive uses occurs at the end of the model’s

382

progression. Indeed, a more complex model that recognizes the mere presence of hunters or tourists may impact wolf reproductive fitness (Gill et al. 2001) would be more sensitive to these specifications. We leave

384

this modeling extension to future work. To allow us to isolate the importance of the demand response on the results, we alter only one demand

386

response at a time. Consider the impact of less responsive hunters and tourists to increased quotas and YNP wolf populations. In this case, an additional YNP wolf attracts 12,960.8 new tourists, instead of 16,201 new

388

tourists, to YNP, which is a 20% decrease in visitor responsiveness. We find that less responsive visitation

24

a) Consumptive use indicators.

b) Non-consumptive use indicators.

Figure 12: Sensitivity analysis examining impact of demand response specifications on consumptive and non-consumptive use elasticities. to gray wolf presence in the park offsets partially the impact of out-of-refuge management on the provision 390

of in-refuge consumptive uses. Here, hunted wolves seeking refuge in the park induce YNP visitation at a relatively lower rate. Unlike wolf-related tourism, which explains only a small portion of total YNP

392

visitation, wolf hunting quotas are the only attractor for new hunters. Therefore, reducing the rate of hunter responsiveness in this way has no impact on elasticities, which are computed as relative percentage changes.

394

In contrast, elasticities are more sensitive to specification when the reasonable assumption is made that

396

hunter and tourist demand responses are increasing concavely in hunting quotas and YNP wolf populations. q nt −ntr The demand response for hunters is given by 75 ∑ j=1 h j,t and the demand response for tourists follows q r nt si . Under the logarithmic specification, which is parameterized similarly to our 1, 842, 023 + 16, 201 ∑i=1

398

baseline case, wolf tourism and hunting remain an attractor but the tourism and hunting generated from additional YNP wolves and higher hunting quotas increases visitation and hunter turnout at a decreasing

400

rate. Tourist visitation increasing concavely in YNP drives elasticities on the number of wolf views and the number of park visitors to negligible magnitudes near zero. This finding highlights that clear understanding

402

of human responses to environmental fluctuations is necessary to predict the impacts of management within and across jurisdictional boundaries. Diminishing effects in hunter responses reduces the number of hunters

404

drawn to a higher wolf hunting quota, which drives down state revenues. Season kills remain unchanged with the harvest quota being met over a longer hunting season. Unlike our baseline case, where increased

406

hunting decreased the season length as eager hunter turnout outpaced the ability of wolves to seek refuge,

25

we find higher hunting quotas increase hunting season length. In this case, the ability of wolves to seek 408

refuge outpaces the sluggish response of hunters to increased quotas.

Discussion and Conclusion

410

We construct a bioeconomic model to examine the production of tourism-related and hunting-related benefits from an actively managed wolf population. Although we calibrate our model loosely to the Greater

412

Yellowstone Ecosystem (GYE), the purpose of the model is not to make pointed policy recommendations. Rather, we focus on integrating detail from the group ecology of wolves with the incentives facing and in-

414

formation sets available to tourists and hunters, to highlight the specific challenges of managing a keystone predator for multiple uses and across multiple jurisdictions.

416

Unlike wildlife models used often in resource economics (Conrad 2010, Fenichel et al. 2010, Hussain and Tschirhart 2010; Polasky et al. 2011), our methodological approach does not attempt to recommend an “op-

418

timal" rate of harvest while balancing tradeoffs inherent to consumptive and non-consumptive uses. Rather, we seek to identify new tradeoffs that arise when accounting for the detailed behavioral tendencies of the

420

species being managed. To support wildlife management decisions, elasticity indicators are calculated that capture how responsive wolf population fitness and life history indicators and the provision of consumptive

422

uses and non-consumptive uses are to various hunting strategies. Using elasticities to support wildlife management avoids imposing structure on the human outcomes that

424

are deemed highest priority to the wildlife manager while enabling managers to balance consumptive uses, non-consumptive uses and population fitness flexibly. As such, elasticity measures may be a preferred

426

management tool when the value of wildlife benefits or relative importance of wildlife fitness indicators are uncertain or changing. Elasticity measures may also be a preferred and less computationally-intensive

428

management tool for a species with a complex population ecology. Our analysis generates several insights into the policy scope of gray wolf management for hunting and

430

tourism benefits. Hunting outside of a refuge area impacts the provision of non-consumptive uses within a refuge area. Among other factors, the degree that management spills over across jurisdictional boundaries

432

is shown to depend on the distribution of hunting across packs and the demand responses of tourists and 26

hunters to environmental conditions. Unlike Borg et al. (2016), we find that active wildlife management 434

outside of a protected area can, under certain conditions, improve the provision of non-consumptive uses. We highlight a double dividend effect where wolf viewing increases disproportionately because of both

436

ecological and human factors. Here, refuge-seeking wolves drive up park wolf density, which increases wolf visibility triggering greater YNP visitation. Both of these predictions contribute to a large literature

438

recognizing natural resource management as an inherently interdisciplinary pursuit (Pearce and Turner 1990; Tschirhart 2000; Carter et al. 2014; Fenichel et al. 2016).

440

We also show that managers may consider targeting packs with a high level of defendability. These packs, which tend to be larger and have older group members (Cassidy et al. 2015), weaken the state of rival packs

442

decreasing total wolf population fitness and recruitment. Targeting these highly capable packs for harvest offsets the superior pack’s relative advantage, which has a net positive impact on population-wide fitness

444

and recruitment. Lastly, we predict that pack switching from low biomass to high biomass prey stabilizes an actively managed wolf population. Packs with a larger optimal foraging group size are better able to endure

446

harvest whereas losing a member has a detrimental effect on the reproductive fitness of a smaller pack. Our model has several shortcomings. First, we assume hunting is the only impact of humans on wolf popu-

448

lations. We do not model the impact of mere human presence on wolf population ecology or habitat change. Modeling wolves as skittish from refuge-visiting tourists may offset the refuge’s desirability creating pre-

450

dictions more similar to Cassidy et al.’s (2015) observation. Although we assume wolf packs are strategic in their choice of territory, we do not recognize that hunters may be strategic in the zones they target, too.

452

Lastly, our model is only loosely parameterized to the GYE, so we emphasize that our model is used primarily for illustrative purposes and to provide a roadmap for evaluating wildlife management decisions of a

454

behaviorally complex species using elasticity indicators. We leave these addressing these shortcomings as future work.

27

456

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