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Conserving large carnivores amidst human-wildlife conflict: Thescope of ecological theory to guide conservation practice
Sumanta Bagchi
PII: S2352-2496(18)30035-1DOI: https://doi.org/10.1016/j.fooweb.2018.e00108Article Number: e00108Reference: FOOWEB 108
To appear in: Food Webs
Received date: 2 June 2018Revised date: 22 September 2018Accepted date: 13 November 2018
Please cite this article as: Sumanta Bagchi , Conserving large carnivores amidst human-wildlife conflict: The scope of ecological theory to guide conservation practice. Fooweb(2018), https://doi.org/10.1016/j.fooweb.2018.e00108
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[Revision of FOOWEB_2018_35]
Conserving large carnivores amidst human-wildlife conflict: the scope of ecological theory
to guide conservation practice
Sumanta Bagchi*
Centre for Ecological Sciences, Indian Institute of Science, Bangalore 560012, India
EMail: [email protected], Fax: +91 80 23601428, Telephone: +91 80 22933528
Running title: Ecological theory and conservation practice
*Correspondence: Sumanta Bagchi, Centre for Ecological Sciences, Indian Institute of Science,
Bangalore 560012, India. Tel: +91 80 22933528; Fax: +91 80 23601428; E-mail:
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Conserving large carnivores amidst human-wildlife conflict: the scope of ecological theory
to guide conservation practice
ABSTRACT
Predator-prey interactions where livestock are killed by carnivores, are a serious global
challenge. Conservation interventions to address this conflict are inadequately guided by
ecological theory, and instead rely on pragmatic experiential decisions. I review four families of
theoretical models that can accommodate essential features of this human-wildlife conflict,
namely – prey-refuge, specialist/generalist predation, social-ecological, and metapopulation
models. I evaluate their relevance for conservation and arrange each model’s predictions along
two conceptual dimensions: coexistence and stability. These models are described with examples
of pastoralists and snow leopards in the Himalayas, but they can have broader relevance to other
regions. All models suggest that livestock-loss can be better controlled in highly productive
habitats, than under low productivity. But, they differed in the ease with which their predictions
may be translated into real-world conservation interventions. These constraints can be
circumvented through animal movement between patches which is represented only in
metapopulation models. But, metapopulation models do not offer much clarity on size of the
predator population. Instead, they can prescribe rotational-grazing policies for livestock –
another pragmatic management concern. This comparison of models identifies lacunae where
ecological theory could be better integrated within conservation practice. One option is better
integration with emerging knowledge of animal movement. Comparative analyses of models
helps identify future directions where outcomes of alternative management interventions can be
predicted and evaluated.
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Keywords: Prey-predator; carnivore conservation; endangered species; grazing management;
1. Introduction
Large carnivores face grim conservation challenges. They range over large expansive habitats,
occupy apex positions in food-webs, and depend on prey species that are often in-turn also wide-
ranging (Gittleman et al., 2001). A suite of other life-history traits make them inherently prone to
extinction in a human-dominated world (Ripple et al., 2014). Most terrestrial large carnivores
currently face real extinction risks and have experienced severe population declines in the last
century (Ripple et al., 2014), particularly among Felidae, Canidae and Ursidae (Fig. 1).
Carnivore extinctions can trigger far-reaching trophic cascades that impact herbivores,
vegetation and soil, disease epidemiology, hydrology, and biogeochemical cycles (Estes et al.,
2011; Peterson et al., 2014). Their interface with humans, through livestock losses, as well as
occasional mortality and morbidity for humans, is of particular concern. In addition to habitat
loss, fragmentation, degradation, prey depletion, and other threats, carnivores also face
retaliatory persecution from ranchers and pastoralists. There are historical precedents to resolve
such conflict by eradicating carnivores altogether (Paddle, 2002), and/or other controversial
measures such as restricting them within small fenced reserves (Creel et al., 2013; Packer et al.,
2013; Woodroffe et al., 2014). For millennia, across different human societies, the tiger
(Panthera tigris), lion (Panthera leo), wolf (Canis lupus), jaguar (Panthera onca), bear (Ursus
spp.), and other big fierce animals have been revered as cultural icons, have featured in our
mythology and fairy-tales. Yet, their future now depends on whether we can mitigate their
damage to our livestock. Repercussions of losses to livestock and livelihoods are most severely
felt in under-developed economies, and this strains societal tolerance of large carnivores (Aryal
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et al., 2014; Harihar et al., 2015; Pooley et al., 2017; Redpath et al., 2013, 2017; Singh and
Bagchi, 2013; Treves and Bruskotter, 2014).
Despite numerous studies on human-carnivore conflict, it remains uncertain whether
conservation efforts actually reduce livestock losses, or not (Eklund et al., 2017). Large
carnivores enter into other types of conflict as well. For example, beyond ranchers and
pastoralists, they often antagonize other stakeholders such as recreational hunters (Schwartz et
al., 2003). Many forms of such conflicts have roots in the ecological realm, but extend deep into
the human dimension (Pooley et al., 2017; Redpath et al., 2017). But, reducing the impact of
large carnivores on livestock is a pressing need; one that can be a first step toward effective
conservation planning (Eklund et al., 2017).
2. Conservation practice and ecological theory
Response to human-carnivore conflict revolves around seeking a pragmatic solution to the
problem of coexistence between the predator, its natural prey, and livestock. There is a long
history of discussion on this challenge, spanning a variety of dimensions including legislature,
economics, peoples’ perceptions, cultural attributes of tolerance, external incentives, etc.
(Harihar et al., 2015; Mishra et al., 2003; Redpath et al., 2013; Treves and Karanth, 2003). Often
it is advocated that inter-disciplinary approaches may help address conflicts better (Pooley et al.,
2017; Redpath et al., 2017), but it remains unclear how disparate disciplines can be mobilized
simultaneously into effective action (Thébaud et al., 2017). The role of ecological theory and
models rarely features in this discussion. This is surprising because ecology is supposedly awash
with theories and models (Marquet et al., 2014; Scheiner, 2013); many of which are thought
pertinent to conservation practice (Doak and Mills, 1994; Ryall and Fahrig, 2006; Simberloff,
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1988). Yet, empirical researchers and conservation practitioners often ignore theoretical work
altogether (Driscoll and Lindenmayer, 2012; Ryall and Fahrig, 2006). Are empirical researchers
and conservation practitioners unaware of existing theoretical predictions? Are models irrelevant
in the real world (Colyvan et al., 2009; van Grunsven and Liefting, 2015)? Or, are theoretical
guidelines poorly communicated (Jacobson, 2009)? A large and sophisticated body of work by
theoretical ecologists, therefore, remains under-utilized.
At the same time, it is also often lamented that conservation policies remain uninformed
by ecological theory and are heavily dependent on expert opinions and experiential decisions
(Doak and Mills, 1994; Driscoll and Lindenmayer, 2012; Ryall and Fahrig, 2006). For example,
a well-meaning decision to connect fragmented habitats through corridors can hasten extinction
due to destabilizing effects of dispersal (Earn et al., 2000), and progress stalls under the strain of
such conflicting results. Admittedly, managers are frequently required to take decisions under
data-deficient conditions as the focal species are imperiled and preclude detailed study. From a
pedagogical viewpoint, it may also be worrisome that the next generation of conservation
scientists is receiving inadequate training in theory, and may be under-appreciative of any role
theory can potentially play (Bagchi, 2017; Jordan et al., 2009; Kendall, 2015; Knapp and
D’Avanzo, 2010; Marquet et al., 2014; Ryall and Fahrig, 2006). Interestingly, at the same time,
the roots of many familiar ecological models lie in the search for solutions to practical problems.
For example, the history of classical prey-predator models (Volterra, 1926), can be traced to a
need for fisheries management (Kingsland, 1995).
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3. Models as simplifications of the real-world
3.1. Identifying relevant theoretical models
From literature, I identified theoretical models that are relevant to human-wildlife conflict, based
on key characteristics. I included models that attempt to explain dynamics and coexistence of at
least three interacting species, and be broadly applicable to different ecosystems rather than a
specific study area, or particular taxa. Insights can be obtained from these models with linear
algebra and calculus, and no advanced techniques are needed. These model are sufficiently
simple to have tractable analytical solutions. The mathematical interpretation of 3-species
coexistence is already fairly complex. More complex formulations that require numerical
analyses are excluded from this review. Often, researchers use dimensionless variants to aid
mathematical analysis. But, since this offers coefficients that may be difficult to interpret in the
real world (e.g., ratio of growth rate and carrying capacity, Ranjan and Bagchi, 2016), I review
the models in their full form. However, a necessary drawback is that simplified models do not
incorporate many potentially interesting features of real-world interactions (see section 4,
below). For e.g., individual-level variation in predator behavior may be a feature in real
ecosystems. Such features are excluded from further analysis via simplifying assumptions, but
this does not preclude broad insights (see section 3.2 below).
From the literature (Case, 1999; Gotelli, 1995; Hastings and Gross, 2012; Murdoch et al.,
2003; Turchin, 2003; Yodzis, 1989), four families of models were judged suitable for review and
analysis (Table 1). Of these, three models are fairly well-studied, as they were proposed 3-4
decades ago, and one is more recent. I review and compare their predictions in the context of
controlling livestock losses to carnivores (Table 1), and also evaluate the extent to which they
can accommodate real-world conflict scenarios. I use illustrative examples linked with loss of
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livestock to snow leopards (Panthera uncia) in the Himalayas, but these models can have broad
and general relevance for other systems as well. Specifically, I consider model parameters that
can be influenced through conservation interventions, and discuss their implications against the
predicted outcomes. I arrange model interpretations over two conceptual dimensions: the
conditions for coexistence, and conditions for stability. This exercise can afford some clarity on
how to design interventions, distinguish the options which are likely to succeed from the ones
that are difficult to implement, and add value to current discussion on the multi-dimensional
nature of human-carnivore interactions (Pooley et al., 2017; Redpath et al., 2017), and why
conservation interventions do not seem to work (Eklund et al., 2017). It also provides clear
testable predictions that can be the basis for empirical studies, and for design of conservation
interventions. The goal is to compare predictions from different models, and assess their real-
world implications, particularly the nature of their stabilizing features. And, identify anticipated
outcomes that can be potentially implemented through management interventions, at least
qualitatively. These steps can offer predictions, which can become relevant to ask under what
conditions conservation interventions should work (Eklund et al., 2017; Ferraro and Pattanayak,
2006).
3.2. Relevance of theoretical models to the real-world
For decades, researchers have pondered whether or not theory and models have any direct real-
world applicability, and if so, how much (Colyvan et al., 2009; Marquet et al., 2014; Scheiner,
2013). Part of the confusion stems from simplifying assumptions that are made when questions
across diverse disciplines are formulated mathematically. Often, these assumptions create a
distance between equations and reality. In thermodynamics, molecular collisions are assumed to
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be elastic. In astrophysics, black bodies are assumed to exist. Famously, model assumptions have
even led to amusing propositions (e.g., spherical chicken; Stellman, 1973). So, it is justified to
question whether simplifying assumptions render their carrier models useless in the real world.
Another related aspect is the time-lag (years, decades, even a century) between the availability of
model predictions and their empirical falsification/acceptance as well as their utilization toward
pressing problems (e.g., Adler et al., 2018; Wainright et al., 2017) – it is seldom that models
become useful right away.
Mathematical models are caricatures of the natural world, rather than replicas. Models
remove clutter, and create idealized conditions within which mathematical formalism can take
hold. They exclude many familiar characteristics of the real world in order to do so. The four
families of models I evaluate below contain their own sets of assumptions; they can only
partially accommodate the complexities seen in real-world conditions. But, these models can
help our understanding of how a complex natural system might work (i.e., description), or it
ought to work (i.e., prediction). Importantly, they yield falsifiable predictions which can be
targets of thought-experiments, and actual empirical data. In other words, models can tell us
something useful about chicken that are spherical; from this we can learn about the ones which
are more familiar in shape.
4. Prey-refuge model (Sih, 1987; Vance, 1978)
4.1. Model structure
Prey-predator interactions reflecting human-carnivore conflict can be captured by models that
consider refuges for prey (Sih, 1987; Vance, 1978). Here, two alternative prey differ in their
degree of susceptibility to a predator; this makes them a suitable template for understanding
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systems with livestock, wild prey and carnivores. The basic model for density of wild-prey (W),
livestock (L), and predator (P) is of the following form:
PWc fL
K
rW
K
rrW
dt
dW)(
PεcL
K
rβW
K
rrL
dt
dL)(
DLεcbWWbcfPdt
dP )()(
The three equations, for dynamics in W, L, and P are inter-linked such that each influences the
others. Here, r is the maximum per-capita growth rate of wild-prey and K is the carrying capacity
for the habitat. Per-capita growth is considered equal for W and L. Capture rate of prey by
predator is c; β is competitive advantage of one prey over another; ε is predator avoidance
advantage of L through husbandry; b is conversion efficiency of predator and D its density
independent mortality through metabolic demands of the predator. Predation on wild prey
follows a functional response depending on availability of refuges (R) as:
RW,
RW/W,W-Rf(W)
0
)(. In this way, the number of wild prey captured is influenced by their
abundance relative to the availability of refuges.
4.2. Model characteristics and assumptions
In this model, prey can grow at the rate r in absence of a competitor or predators. But,
growth is influenced by density dependence (i.e., logistic growth), through
K
Wr 1 for wild
prey, and
K
Lr 1 for livestock when they are alone, respectively. This assumes bottom-up
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regulation in absence of predators. The wild prey and livestock are sufficiently similar in their
life history traits (e.g., body size, fecundity, resource consumption rates, etc.) so that their r and
K are not modeled separately, but by common terms (see section 5 when there are separate terms,
below). Search rate of the predator is c, and it can capture wild prey at the rate cf(W)W. The
predator’s functional response f(W) describes how capture rate varies with prey density. If all
wild prey are protected, and hidden in refuges (R), then f(W) = 0. Any surplus wild prey which
are not protected in refuges, W-R, are susceptible to predation. Effectively, the predator can catch
only a fraction of the wild prey population W
RW . This is a reasonable approximation for many
natural systems. For example, in the Himalayas; ibex (Capra sibirica) stay in proximity of steep
cliffs in order to avoid attacks by snow leopards; they are more vulnerable when foraging away
from cliffs (Fox et al., 1992). Similarly, the predator searches the habitat and captures livestock
at the rate (c-ε)L instead of cL, as ε represents the predator avoidance among livestock, since
some encounters (ε) get thwarted by vigilant herders and guard dogs, etc. The two prey compete
with each other; the net competitive effect of one prey over the other is β. The predator converts
prey (food) into offspring with a conversion efficiency b. Net foraging return on wild prey can be
bcf(W)W = bc(W-R), and on livestock it is b(c-ε)L. In absence of any prey, the predator
population declines due to starvation at the rate -DP.
4.3. Model interpretation
Intuitively, this model has two stabilizing elements: the habitat’s carrying capacity K,
which keeps the prey becoming too abundant, and refuges R, which prevent prey from becoming
too rare. One should expect stable coexistence of all three species (Vance, 1978). But, the
important question is whether 3-species coexistence can occur without livestock losses.
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Conservation interventions would aim to establish conditions where L is immune to
attacks, or, c=ε. This is often attempted via increased vigilance while herding, strengthening of
corrals, and guard dogs (Gehring et al., 2010). If successful, then at the nontrivial equilibrium
(where the * symbol denotes equilibrial density) we get
2)1( *
*
*
*
*
WDK
βrc
βWKbc
DR
P
L
W
. At this equilibrium, all three species can have positive densities.
Interestingly, predator density is seen to depend on W, and not on L; and this is consistent with
empirical observations on snow leopards in the Himalayas (Suryawanshi et al., 2017). High
livestock populations can be maintained when competitive interactions (β) are weak. But, high
predator density exists when competition is weak. So, meaningful densities of all species can
exist only within a narrow range of competitive interactions that satisfy these opposing demands:
𝐾
𝑊∗ ≫ 𝛽 and 𝛽 > 1, and this is more likely to be met in high-productivity ecosystems.
Now, the next important question is whether this equilibrium is stable, or not.
Conservation efforts that result in increased predator populations can intensify conflicts (Rigg et
al., 2011). Local stability of such mathematical models can be judged using well-known
principles of linear algebra (e.g., Routh-Hurwitz criteria, Brassil, 2012; May, 1973). Models are
judged stable when a small perturbation at equilibria dissipates over time; they are unstable if the
disturbance grows. It is known (Vance, 1978), that these solutions are stable only when
bc
DRβK and
bc
DR . The key insight from model solutions is that three-species stable
coexistence depends on the size of refuges, R. High predator and wild-prey density can be
achieved only under plentiful refuges; which necessarily also lead to low density of livestock
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(because carrying capacity is split between the two types). Consistent with the coexistence
conditions, these stability conditions are also likely to be met in high-productive habitats.
4.4. Conservation implications
In this model, humans can intervene to alter prey-predator encounters. But, for simplicity,
livestock and wild prey are considered to be sufficiently similar in their life-history traits (e.g.,
sheep/bharal in Himalayas, or zebu cattle/wildebeest in Africa, etc.). It contains a number of
parameters that are ecologically meaningful. But, these parameters, or traits (R, D, b and c), may
be difficult to manipulate in free-ranging animals in order to meet the coexistence and/or stability
criteria. Perhaps, forage competition (β) can be manipulated, and competitive effects of livestock
on wild prey can be reduced via measures such as stall-feeding. But, this can inflate their
populations, and have indirect effects on carrying capacity (K, see section 6 below for further
discussion). Protecting livestock alone can enable coexistence; the model also prescribes
reducing livestock density which will likely compromise human livelihoods, thus undermining
the constituency for conservation (Harihar et al., 2015; Redpath et al., 2013). In summary,
conflict-free scenarios are possible in the prey-refuge model. But, the common practice of
protecting livestock in conflict situations may actually be detrimental for predator’s survival,
unless the carrying capacity of the habitat is high enough to accommodate refuges for the wild
prey. This approach to reducing conflict may lead to favorable outcomes in productive habitats
(high K). It is less likely to succeed in less productive ones (low K), because it is difficult to
manipulate the parameter space determined by β, R, D, b and c. When a habitat is naturally very
productive, it may be possible to prevent all attacks on livestock, and have sufficient refuges for
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wild prey, and have meaningful density of the predator. However, this model offers strong
justification to strive for at least some areas that are inviolate of human use as parks and reserves.
5. Generalist/Specialist predators (Holt, 1984)
5.1. Model structure
Key features of human-carnivore conflict can also be captured by models for generalist versus
specialist predators (Holt, 1984). These models revolve around alternative prey that share a
common predator, and are distributed across distinct patches in space. A generalist predator will
include both prey in its diet. But, a specialist predator includes only the most profitable prey,
depending on model parameters. The predator (P) encounters two non-overlapping patches, with
wild-prey (W) and livestock (L) where its densities are PW and PL respectively, and the model
system is:
WWWWWWW EIDWbaP
dt
dP )(
WPaWWfdt
dWWWW )(
LLLLLLL EIDLbaP
dt
dP )(
LPaLLfdt
dLLLL )(
As before, the equations for L, W and P, are inter-linked, as each influences the others. Here,
fW(W) and fL(L) are per-capita growth rates of wild-prey and livestock, respectively. aW and aL
are rates at which predator’s searches for prey in the two patches. bi is predator’s conversion
efficiency for food to new predators. DW and DL are predator’s metabolic expenditure in these
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patches. Ii and Ei are the rates of predator immigration and emigration for patch type i,
respectively.
5.2. Model characteristics and assumptions
In this model, the wild prey can grow at the rate fW(W) in absence of predation, and
similarly fL(L) for livestock. The predator searches the habitat and encounters wild prey at the
rate aWPW and livestock at the rate aLPL. More complex foraging patterns, e.g., Holling type-II
functional responses can be considered, but are not necessary to understand the broad and
general behavior of this system (see below). The predator converts prey (food) into new
predators with an efficiency of bW and bL, so that its net foraging returns on wild prey is aWbWPW,
and returns on livestock is aLbLPL. The predator’s movement into a W-type patch is given by IW
and its movement away from it is given by EW; similarly it is IL and EL for the L-type patch. For
simplicity, this model assumes that the predator does not starve while moving between patches.
5.3. Model interpretation
The solutions to this model can be examined under conditions of specialist versus
generalist predation. All predators should be found in the patch with greater net foraging returns;
if the net foraging returns are equal between patches, then the predators will follow an ideal-free
distribution (Fretwell, 1972; Holt, 1984). It is known, that if the predator is initially specialized
on wild prey in patch type W, it should expand its diet to kill livestock in patch type L, only if
*Wba
baK
LL
WWL ; where W* is the equilibrial density of wild prey, and KL is the carrying capacity
of L determined by fL(L) (Holt, 1984). This prediction requires optimal decision-making behavior
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on behalf of the predator. This inequality has straightforward conservation implications:
livestock losses can be reduced if this condition remains unfulfilled.
5.4. Conservation implications
Can conservation interventions strive to make livestock perpetually unprofitable for the
predator? In this way, predators remain specialized on wild prey and do not become generalists
and include livestock in their diet. From the left hand side of the inequality condition,
,*Wba
baK
LL
WWL this is possible only if one lowers livestock densities. However, as seen
previously, such measures would compromise human livelihoods. The right hand side of the
inequality contains parameters related to search rate and conversion efficiency, which cannot be
easily manipulated by conservation interventions. Once again, as seen before, predictions from
this model may also lead to a path of antagonizing people (Redpath et al., 2013), unless the
habitat is already very productive. In summary, although the predictions from
generalist/specialist predator models are more encouraging, they may be practically difficult to
implement in real-world scenarios. As in the prey-refuge model, this model also prescribes a
reduction in livestock abundance. The perceptive reader will notice that implementing a
nonlinear functional responses does not fundamentally alter this broad interpretation. Although
equilibrial densities will change, but the model’s solution would continue to remain difficult for
real-world action. This is because the parameters for search rate and metabolic conversion
efficiency are difficult to manipulate for free-ranging mammals. So, while this model is
biologically informative, there is limited scope to mobilize it toward conservation action.
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6. Social-ecological models (Wilman and Wilman, 2016)
6.1. Model structure
Social-ecological models include conditions where the livestock population (L) is determined by
economics, while wild prey (W) and predator (P) populations are determined by species
interactions (competition and predation). In such a scenario, the dynamics of the 3-species can be
represented as (adapted from Wilman and Wilman, 2016):
aWPK
LWrW
dt
dW
1
ePLmPcaLPcaWPdt
dP
LgLfPpdt
dL
As before, the these equations are coupled with one another. Here, r is the intrinsic growth rate, β
is the effect of competition between L and W, a is rate of prey capture, c is the conversion
efficiency of predators, m is metabolic requirements of predators, and e is retaliatory persecution
by humans, p is capital value of livestock, f is resources spent on protecting livestock from
predators and g is their maintenance costs (Wilman and Wilman, 2016).
6.2. Model characteristics and assumptions
In this model the wild prey can grow at a maximum rate or r in absence of competing
livestock and the predator, but this is influenced by density dependence (i.e., logistic growth) as
K
Wr 1 . The effect of livestock competition (β) on wild prey is captured through
K
LWr
1 . A predator searches the habitat at the rate a, and encounters wild prey at the rate
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aW and livestock at the rate aL. The predator converts prey (food) into new predators with an
efficiency c, so that net return on W is caW and on L is caL. In absence of any prey, the predator
population declines due to starvation at the rate –mP, and a second source of mortality in
predators is persecution by people at a rate proportional to their densities, -ePL. The livestock
population can grow at the rate p (their capital value), but this growth is reduced by two factors.
One is the expenditure costs of pastoralists in upkeep of their herds, -gL. This may include
aspects such as veterinary care, supplemental feeding, corral space, and other limitations due to
manpower. Another cost is the additional expenditure for pastoralists to protect their herds from
predators, -fP. This includes investment in vigilance, guarding and protection, and similar limits
to manpower. The size of the livestock population is set by balancing these economic
considerations (Wilman and Wilman, 2016). These constraints may be more severe for
individual pastoralists who raise a few head of livestock, than for ranchers who operate with
large herds (Schiess-Meier et al., 2007). Interaction between livestock and wild prey is
represented by β as a numerator-term
K
LW . This captures scenarios of land-sharing or
forage competition, e.g., pastoralism in the Himalayas. Competition term in the denominator
would represent land-sparing, or interference competition, where livestock have exclusive access
to a fraction of the habitat
LK
W
, e.g., ranching in the Pampas. Readers may notice that both
scenarios are qualitatively similar (they are approximately interchangeable via a Taylor series
expansion).
6.3. Model interpretation
For a model showcasing the conflict scenario, the nontrivial equilibrium densities are
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.
)())((
2
2
2
2
*
*
*
racfefracfrcgKa
fmracfKrcKpa
racfefracfrcgKa
pracepracprgmracgKr
racfefracfrcgKa
agKrfmfkraKpace
L
P
W
Here the predator kills livestock, and people
retaliate by killing the predator. At this equilibrium, all three species can attain positive densities.
But, the important question is whether this equilibrium is stable, or not. Stability of this 3-species
equilibrium depends on a series of exceedingly stringent conditions (i.e., a large number of terms
in Routh-Hurwitz criteria obtained from the Jacobian matrix). Thus, even if there is coexistence,
it is unlikely to be stable across large regions of the parameter space. Now, conservation
interventions can ameliorate the conflict by protecting livestock (caLP=0; fP=0) and
simultaneously preventing retaliatory persecution (ePL=0). Such attempts often involve a mix of
different interventions such as improved herding, and insurance for livestock (Gehring et al.,
2010; Mishra et al., 2003). When successful, livestock would be protected, and people would not
persecute predators. Under these conditions, the new nontrivial equilibrium densities are
.)(
2
*
*
*
gp
cgKa
pacgmacgKr
acm
L
P
W
Now, meaningful and stable equilibrium can be achieved if
ac
m
g
pK
. This condition is less stringent than those mentioned above.
6.4. Conservation implications
In this model, conflict-free 3-species coexistence can be achieved only if carrying
capacity of the habitat is greater than a threshold level. Readers will notice that this result is
qualitatively similar to the previous two models. A solution may exist in habitats that are
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sufficiently high in productivity (high K). The left hand side of the inequality concerns overall
habitat quality. Since K is often determined by biophysical constraints, it may not always be
possible to design interventions at sufficiently large spatial scales that can increase a habitat’s
carrying capacity. Also, increasing K (e.g., fertilization) may be destabilizing (i.e., paradox of
enrichment, Holyoak, 2000; Rip and McCann, 2011; Rosenzweig, 1971), and ultimately
counterproductive. However, preventing declines in habitat quality due to negative impacts of
livestock on ecosystem function is an option (Bagchi et al., 2012; Bagchi and Ritchie, 2010;
Mishra et al., 2010). Otherwise, one has to evaluate the implications of the right-hand side of the
above condition. But, many parameters here are related to life-history traits of the participant
species (e.g., m, a, and c), and therefore, cannot be effectively altered through conservation
interventions. It is hard to imagine that metabolic rate, search rate and conversion efficiency to
be affected by management action. To design interventions over the economic parameters (p and
g) requires lowering the capital value of livestock, and/or increasing their maintenance costs,
both of which would likely undermine human livelihoods. The only remaining option is
designing interventions around β, in order to reduce competition between livestock and wild
prey. How does one alter the per-capita competitive effects of one species on another? One
option is lease of designated livestock-free patches which are for exclusive use by wild prey
(Mishra et al. 2003), and would require considerable investment to offset loss of grazing rights
for people. Or, previous work suggests that when two species are matched in their body-sizes,
they may compete less strongly than when they differ widely in their body-size (Bagchi and
Ritchie, 2012; Ranjan and Bagchi, 2016). But, with greater difference in body-size, the two prey
populations also begin to differ in their sensitivity to predation, and this could make stable
coexistence more difficult (Holt, 1984). One can explore options where the composition of
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livestock is such that they are similar in body-size to the wild prey, rather than being much
smaller/larger than the wild prey. However, the scope to implement any changes in livestock
herd composition may be very limited in real-world scenarios (e.g., one cannot expect to
substitute sheep with cattle, or vice-versa). In summary, while the predictions of this model are
encouraging, similar to the previous two cases (section 4 and 5), implementing the predicted
outcome from socio-ecological models, and translating them into effective policy interventions,
may be practically challenging. Perceptive readers will notice that, as before, more complex
models which implement nonlinear functional responses would likely change the equilibrial
densities and also introduce new terms in the stability conditions. This additional complexity
makes the model analytically intractable. But the difficulty in converting the model’s prediction
into practical management interventions would continue to persist, without fundamentally
altering the broad inference.
7. Metapopulation models (Hanski, 1997; Nee et al., 1997)
7.1. Model structure
Key aspects of human-carnivore conflict can also be captured by metapopulation models.
These concern occupancy of patches by a single species, or by communities of interacting
species. These models allow inclusion of spatial structure into ecological dynamics, and date
back to Levins (1969), although the underlying philosophy appeared earlier in epidemiology
(Kermack and McKendrick, 1927). The basic model for single species accounts for the fraction
of occupied sites (P) and number of empty sites (1-P), using local colonization (c) and extinction
(e) parameters (Hanski, 1997) as: ePPcPdt
dP )1( . At equilibrium, the fraction of occupied
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sites are c
eP 1* , and the metapopulation persists regionally as long as .1
c
eThis basic model
can be extended to include prey-predator interactions (Bascompte and Sole, 1998; Nee et al.,
1997; Swihart et al., 2001), and we explore this scenario for wild prey, livestock and predators.
First, in pristine conditions with wild prey and predators, and without livestock, let the fraction
of empty patches be o; fraction occupied by wild prey alone be w, and the fraction occupied by
both wild prey and predator be p. Dynamics of this system are given by:
owcpewedt
dowpw
pewpcowcdt
dwwpw
pewpcdt
dppp
Here, ew is wild prey extinction rate in absence of predators, ep is extinction rate of both wild
prey and the predator. Likewise, cw is colonization rate of wild prey, and cp is colonization rate
by both wild prey and predator.
7.2. Model characteristics and assumptions
In this model, empty patches increase when wild prey go extinct at the rate eww, as well
as when wild prey and predator simultaneously go extinct at the rate epp. Colonization of a
previously empty patch occurs at the rate cwow by wild prey, and the model assumes that
predators do not colonize previously empty patches since there are no prey in these. Patches
occupied by wild prey can decrease at the rate cpwp due to effect of predator, and at the rate ewp
even without the predator. Patches used by predator increase when it colonizes a patch featuring
wild prey at the rate cpwp, and is reduced due to extinctions at the rate epp.
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7.3. Model interpretation
This system reaches the following nontrivial equilibrium, which is known to be stable
(Nee et al. 1997):
w
w
p
p
pw
w
p
p
c
e
c
e
cc
c
c
epw
p
w
o
1
1 **
*
*
*
. At this equilibrium, there are some empty
patches, some that are occupied by wild prey, and others that are occupied by both wild prey and
the predator. The predator and wild prey can persist, if 1
w
w
p
p
c
e
c
e, which can be compared
with the previous result for single species (i.e., i i
i
c
e1 ).
Into this pristine condition, one can now introduce livestock and investigate their effects
(Fig. 2). In absence of any human-wildlife conflict, livestock would occupy a fraction of the total
patches, l, such that [1-l] patches now remain available for the wild prey and the predator.
Conservation action would attempt to keep these patch-types distinct in order to prevent conflict.
In other words, reduce the interface between the predator and livestock, similar to the prey-
refuge model which now has spatial structure as a metapopulation. Previously, we saw
(o+w+p=1), and after introduction of livestock, the available habitat for wild prey and the
predator shrinks to (o+w+p)=1-l (Fig. 2). Substituting this into the previous equations, we get
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new equilibrium densities as:
w
w
p
p
pw
w
p
p
c
e
c
el
cc
c
c
e
pwl
p
w
o
)1(
)1( **
*
*
*
. Under these restrictions
imposed by habitat lost to accommodate the livestock, the wild prey and the predator can coexist
if lc
e
c
e
w
w
p
p
1 . There exists a threshold level of livestock patches, up to which it is possible
to achieve conflict-free coexistence of wildlife and humans; given by
w
w
p
p
c
e
c
el 1ˆ . The
model predicts a threshold fraction on patches that can be used to raise livestock, while the
predator subsists on wild prey in the remaining ones.
7.4. Conservation implications
This approach to prey-predator metapopulation effectively considers that a fraction of the
habitat becomes unavailable to wild prey and predators, much like habitat degradation/loss
(Bascompte and Sole, 1998; Nee et al., 1997; Swihart et al., 2001). More elaborate models can
consider colonization-extinction of all the 3-species across all possible patch-types (Dos Santos
and Costa, 2010; Holt, 1997), but they reach a similar qualitative conclusion. In a scenario where
predators kill livestock, the conflict can be ameliorated by maintaining a prescribed fraction of
the habitat available to wild prey, and allocating the rest for livestock. This effect may be most
pronounced in situations where livestock outnumber wild prey by orders of magnitude (Mishra et
al., 2010, 2003). However, this model does not directly provide information on size of the
predator population, which is often of high relevance to managers (but see section 8, below). In
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fact, predator populations may respond positively to conservation efforts, and this may intensify
the nature and severity of conflicts (Rigg et al., 2011).
Importantly, this model may not preclude humans from raising meaningful densities of
livestock within their designated patches (Fig. 2). In a livestock-patch, or across a collection of
designated patches, people can strive for sustainable livestock densities, determined by the
habitat’s carrying capacity. It can be challenging to ascertain livestock stocking rates accurately,
as overstocking may cause habitat degradation and loss of ecosystem functions (Bagchi et al.,
2012; Bagchi and Ritchie, 2010). But conservation interventions can strive to implement
rotational grazing practices that foster ecosystem stewardship (Briske et al., 2011). In fact,
management concerns about rotational grazing may also be effectively addressed by
metapopulation models that involve population asynchrony across patches brought about through
animal movement between habitat patches (Abrams and Ruokolainen, 2011; Wang et al., 2015).
Dynamics of local livestock populations can be determined by resource-dependent vegetation
growth and movement between patches (Abrams and Ruokolainen, 2011; Wang et al., 2015). In
the simplest form, one can consider a 2-patch habitat used for livestock grazing (Abrams and
Ruokolainen, 2011), where the population dynamics of vegetation (Vi) and livestock (Li) in patch
i are given by:
),( iiii LVfkVrV
dt
dV
)()(),( jijijiiiii WgmLWgmLdcLVfL
dt
dL ,
i, j = 1, 2, where )( jiij WWW from dcLVfW iii ),(
Here, r is the maximum growth rate of vegetation, and k is density dependent reduction in r, such
that vegetation reaches a carrying capacity r/k in absence of livestock. For simplicity, the two
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patches are considered sufficiently similar to not warrant separate analyses of their r and k, but
this restriction can be removed in more elaborate assessments (Ruokolainen et al., 2011). Forage
consumption by livestock is given by the functional response f(Vi, Li). Vegetation biomass is
maintained below carrying capacity in presence of herbivory by an amount governed by f(Vi, Li).
Livestock convert vegetation (food) into new livestock with an efficiency c. Due to metabolic
expenditure, the livestock population declines at the rate d in absence of forage. Movement
between patches is given by m, and net movement depends on the difference in foraging returns
between the two patches given by a function g(ΔWij). Here, foraging return is the net gain given
by f(Vi, Li)c-d.
Several interesting interpretations emerge when one considers a density-dependent
saturating functional response for livestock along a gradient of vegetation biomass (i.e., Holling
type-II). First, livestock and vegetation enter limit cycles in any patch (Abrams and Ruokolainen,
2011). With time it becomes unsuitable to continue foraging there and seek the alternative patch.
This is simply because animals can potentially consume plants are a rate faster than the rate at
which plants can grow. One can readily notice that the same reasoning also applies to the
predator; when it travels between patches it encounters livestock and can be persecuted in
retaliation – this is the origin of the conflict (Abrams et al., 2012). Indeed, many pastoral systems
show periodic movement of livestock between patches (Kuiper et al., 2015), and this affords
further discussion on a rotational grazing policy (Briske et al., 2011). Second, with movement
between patches, out-of-phase cycles in either patch reduces the variability in total livestock
population size (Abrams and Ruokolainen, 2011), which aligns favorably with sustaining human
livelihoods. Similar conclusions are reached even when the patches differ and have non-identical
r and k (Ruokolainen et al., 2011). This branch of theory on metapopulation regulation through
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local movement is still developing (Wang et al., 2015); this can afford management options to
avoid the destabilizing elements, and encourage livestock husbandry to stay within the
boundaries of the parameter space that offers stable livestock populations (Abrams and
Ruokolainen, 2011; Wang et al., 2015). In summary, predictions from metapopulation models
may allow the goals of human livelihoods to remain broadly aligned with those of wildlife
conservation. They also offer opportunities to implement accurate grazing practices (Abrams and
Ruokolainen, 2011; Briske et al., 2011; Wang et al., 2015), as there are clear parallels between
adaptive movement between habitat patches and livestock husbandry. This demonstrates that
pragmatic management concerns about human-wildlife conflict and rotational grazing, may both
be addressed using theoretical models (Fig. 2).
8. Meaningful densities in metapopulations
Coexistence without livestock-loss, of the predator, livestock and wild prey in their respective
patches (above), is a necessary condition for conservation. But this alone is not sufficient. The
problem of whether or not these patches can hold meaningful densities of the predator, still
remains. For this, I evaluate food-web models with adaptive dispersal between patches
(Cressman and Křivan, 2013). This helps address the necessary and sufficient conditions for
conservation, through the connection between well-known models of population dynamics with
dispersal (Gadgil, 1971; Hamilton and May, 1977; McPeek and Holt, 1992) and models of
metapopulation dynamics between patches (seen above).
The basic model for population dynamics under dispersal considers the following
coupled equation for one species with population sizes xi and xj in two patches i and j (Cressman
and Křivan, 2013):
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),()( jjiiijiiii xdxdxfx
dt
dx for i=1 and j=2;
where fi is per-capita growth rate in the patch i; δ denotes dispersal speed between patches, and
dij is the probability of dispersing from patch i to patch j. The internal equilibrium of this model
is known to be stable (Cressman and Křivan, 2013), and it can be extended into food-web models
with a prey and its predator. One we include a predator (yi, i = 1, 2) into this model, the
predator’s fitness gi (i = 1, 2) is determined the patch’s prey density (xi), as follows:
)],,(),()[,(),( jjjiiixiiiii yxfyxfyxDyxfx
dt
dx
)],()()[,()( jjiiyiiii xgxgyxDxgy
dt
dy for i=1 and j=2;
where xi and yi are wild prey and predator density in patch i, respectively. Their dispersal rates
are Dx(x, y) and Dy(x, y). These depend on their respective dispersal speeds and population sizes,
ji
ji
xxxx
xxyxD
),( and
ji
ji
yyyy
yyyxD
),( . The dynamics of this model are known to
provide a meaningful, non-trivial, internal equilibrium **** ,,, jiji yyxx , which is independent of
dispersal rates (Cressman and Křivan, 2013). The important implication from this is that
differences between dispersal rates of the prey and the predator do not lead to instability in such
two-patch models. Hence, meaningful densities of the predator and wild prey can exist in both
patches, once they are identified by the metapopulation approach (as seen above).
For simplicity, we can consider linear functional response for the predator (Cressman and
Křivan, 2013), ,)1(),( ii
i
iiiii y
K
xayxf and .)( iiiiii mxcxg Now, the equilibrium
densities are ,*
ii
ii
c
mx
and ,
)(2
*
iii
iiiiii
Kc
mKcay
for i = 1, 2. One can readily infer that for a
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conservation objective of meaningful predator densities, it is necessary thatii
ii
c
mK
. Or, habitat
quality (i.e., carrying capacity) for each patch should be above a minimal threshold level, and the
patches cannot be too degraded. This inference matches the conditions we have seen previously
in the other models (above). A solution may exist for habitats that are already very productive
(high K). Otherwise, this re-iterates that habitat management, reversal and/or prevention of
degradation, rotational grazing, and other interventions must be an integral part of conservation
efforts. This basic interpretation of two-patch dispersal models remains unchanged even if we
consider non-linear saturating functional response for the predator (Cressman and Křivan, 2013).
In summary, metapopulation models prescribe the necessary conditions for livestock,
wild prey and predator to coexist. Next, dispersal between a subset of patches available for wild
prey and the predator can potentially provide the sufficient condition for conflict-free
coexistence. Likewise, rotational grazing in the livestock-patches also allows for raising them at
meaningful densities. Management of habitat quality and controlling degradation appear as key
conservation targets under this framework. These insights are consistent with other recent
theoretical results (Dannemann et al., 2018) which describe how predator movement allows
coexistence through a balance between patch exploitation and regeneration over a wide range of
demographic parameter values of the prey. Emerging knowledge of movement ecology promises
to be applicable across diverse settings (Bagchi et al., 2017; Nathan et al., 2008), perhaps it can
also inform the linkages between prey-predator theory and conservation of carnivores.
9. Ecological theory and conservation practice
The results of four families of models about prey-predator interactions show the implications of
ecological theory for guiding conservation practice. The four families of models were evaluated
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against equilibrium densities, and the stability of the equilibrium. This two-step approach is
necessary because knowledge of population densities is insufficient to evaluate conservation
success. For example, increase in wild prey abundance can coincide with strengthening of
conflict with predators (Stahl et al., 2002; Suryawanshi et al., 2017). So, it is important to know
whether the equilibrial solutions from each model are stable, or not. The insights from each
model could be arranged along two conceptual axes, one for species coexistence, and another for
stability. Encouragingly, all four models show that conservation targets can be achieved via their
respective stabilizing elements. The models also show that reducing livestock-loss is more likely
in habitats that are already highly productive, compared to unproductive ones. Solutions to the
conflict by protecting livestock alone (i.e., prey-refuge), or through generalist/specialist
predation, or socio-ecological models are theoretically plausible, but are difficult to translate into
management actions. Conditions for coexistence and stability are more likely to be
simultaneously satisfied in productive habitats. Metapopulation models appear to be better
equipped of the four families of models, at least qualitatively, to circumvent the constraints
attached with habitat productivity. But, they need to be coupled with knowledge of movement
and dispersal (Cressman and Křivan, 2013; Dannemann et al., 2018; Wang et al., 2015). This
enables the meta-population approach to circumvent the constraints imposed by habitat
productivity, and become more broadly applicable than the other three approaches reviewed
here. They can be applied towards two pragmatic concerns: human-wildlife conflict and grazing-
management. Better integration of knowledge about animal movement into the meta-population
approach can offer insights into the necessary as well as sufficient conditions for conflict-free
coexistence between livestock, wild prey, and predators. Further, metapopulation models
concern parameters that most land managers are already likely familiar with – habitat patchiness,
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carrying capacity, animal movement, etc. (Fig. 2), This familiarity may be advantageous for
communicating theoretical principles to managers, and visualizing concepts and predictions
(Jacobson, 2009).
One can see that impacts of large carnivores on livestock can be managed in the realm of
calculus and algebra. But, how realistic are these scenarios for a practitioner? The final, and
crucial, step is to assess whether real-world efforts can implement strategies that are informed by
these models – without this, the existing gap between theory and practice is not bridged.
Quantitative tests of these model predictions will be difficult to achieve, but one can use
qualitative guidelines. I am not aware of any existing real-world conservation efforts that can,
sensu stricto, serve as case studies or field-tests of these models (reviewed by Eklund et al.,
2017). However, one attempt to reduce livestock damage by snow leopards in the Himalayas
(Bagchi and Mishra, 2006; Mishra et al., 2010, 2003), does seem to capture many essential
features of the metapopulation model. Here, livestock outnumber wild prey and the snow leopard
is heavily dependent on livestock. From the late 1990s and early 2000s, conservation
interventions created livestock-free patches (strategically identified valleys, and watersheds in
the mountainous landscape) which were expected to help recovery of the wild prey, and reduce
impact of snow leopard on livestock (basic mathematical analyses provided in Mishra et al.,
2010). This essentially creates designated patches for the wild prey and livestock – a key feature
of the metapopulation models. Over the next decade, population-level changes do seem to
qualitatively match the expected outcomes and general reduction of livestock losses (Mishra et
al., 2016), without causing declines in the predator population (Sharma et al., 2015). Better
integration of theory with conservation practice (Doak and Mills, 1994; Driscoll and
Lindenmayer, 2012) can help refine management options and anticipate their outcomes.
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Acknowledgements
This work was supported by grants from MHRD-India, DST, MoEFCC, and DBT-IISc.
Discussions with Mark E. Ritchie, Lasse Ruokolainen, and Navinder J. Singh helped in
preparing the manuscript. Onam Bisht helped in preparing Fig. 1, and many colleagues helped in
preparing Fig. 2. I am grateful to the editors, and the referees.
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Table 1. Description of four families of ecological models, their characteristics, and the
conservation implications of their predictions.
Model type Origins Model
characteristics
Example Conservation
implication
Prey-refuge Sih (1987),
Vance
(1978)
These models depict
dynamics of one
predator and two
alternative prey
(wild-prey and
livestock).
Conservation
implications revolve
around strategies
that protect livestock
from predators, i.e.,
immunity or refugia.
Vigilant
herders
and/or guard
dogs can
thwart
predator
attacks, and
create
conditions to
evaluate this
model.
Protecting
livestock from
predators is not a
very suitable
solution to the
conflict. This will
not support high
density of
predators and of
livestock,
simultaneously.
Generalist/Specialist
predation
Holt
(1984)
These models also
consider one
predator and two
alternative prey. An
otherwise specialist
predator that hunts
only wild-prey will
include livestock in
its diet, based on
relative abundance
of the two prey
types.
Difference
in
profitability
between
wild prey
and
livestock,
can create
conditions to
evaluate this
model.
Conservation
interventions can
strive to avoid the
threshold condition
where it becomes
profitable for the
predator to become
a generalist. But,
this will not
support high
density of
predators and of
livestock,
simultaneously.
Social-ecological
models
Wilman
and
Wilman
(2016):
These models
consider coupled
human-and-natural
ecosystems. Here,
economic factors
influence the size of
livestock population.
Species interactions
determine the size of
the wild-prey and
predator.
Conservation
Retaliatory
persecution
of predators
by herders,
in response
to attacks on
livestock
offer
opportunities
to evaluate
this model.
Conservation
interventions can
create conditions
where the predator
can subsist on wild
prey, without
causing conflict.
Further, it possible
to support
meaningful
densities of all
three species. But
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scenarios concern
three-species
coexistence without
any livestock loss.
the scope for
designing
interventions on
the target
parameters may be
fairly limited in
real-world
scenarios.
Metapopulations Hanski
(1997),
Nee et al.
(1997)
These models
concern patch
dynamics for one
predator and two
prey types.
Conservation
scenarios relate to
situations where
predators can be
sustained on patches
occupied by wild-
prey alone, without
visiting livestock
patches.
When wild
prey and
livestock
occur in
spatially
segregated
patches, then
it creates
conditions to
evaluate this
model.
Predicts a fraction
of patches can be
occupied by
livestock. Predator
and wild prey exist
in the remaining
patches. It is
possible to support
meaningful
densities of all
three species.
Stocking densities
in the livestock
patches can be
calculated using a
rotational grazing
policy.
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Fig. 1. Summary of conservation status of terrestrial species in order Carnivora (data from IUCN
Redlist, www.iucnredlist.org, accessed 1st Jan 2016). Conservation status across families of
terrestrial carnivores in (a), and population trends in (b). Conservation status across body size (at
20-kg intervals) in (c), and population trends in (d). Abbreviations: CR, critically endangered,
DD; data deficient; EN, endangered; EX, extinct; LC, least concern; NT, near threatened; VU,
vulnerable. Carnivore families: AIL, Ailuridae (red panda); CAN, Canidae (dogs); EUP,
Eupleridae (fossas); FEL, Felidae (cats); HER, Herpestidae (mongooses); HYA, Hyaenidae
(hyenas); MEP, Mephitidae (skunks); MUS, Mustelidae (weasels); NAN, Nandiniidae (African
palm civet); PRI, Prionodontidae (linsangs); PRO, Procyonidae (raccoons); URS, Ursidae
(bears); VIV, Viverridae (civets).
a)
0
20
40
60
AIL
CA
N
EU
P
FE
L
HE
R
HY
A
ME
P
MU
S
NA
N
PR
I
PR
O
UR
S
VIV
No.
of
Sp
ecie
s
CRDDENEXLCNTVU
b)
0
20
40
60
AIL
CA
N
EU
P
FE
L
HE
R
HY
A
ME
P
MU
S
NA
N
PR
I
PR
O
UR
S
VIV
No.
of
Sp
ecie
s
decreasingincreasingnastableunknown
c)
0
5
10
15
20
0
20
40
60
80
100
120
160
240
400
Body weight
No.
of
Sp
ecie
s
CRDDENEXLCNTVU
d)
0
5
10
15
20
0
20
40
60
80
100
120
160
240
400
Body weight
No.
of
Sp
ecie
s
decreasingincreasingnastableunknown
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Fig. 2. Schematic representation of the metapopulation model with occupied and empty patches
for livestock (ovals), and wild prey and predator (squares). The model prescribes the maximum
number of patches that can be allocated for livestock, which are kept distinct from the other
patches (dashed line). Occupancy of wild prey and the predator of their respective patches
(dotted arrows) is governed by inherent extinction-colonization dynamics (ei and ci). But,
corresponding movement and stocking rates of livestock in their allocated patches (mi) is
governed by a rotational grazing policy (solid arrows).
mi mimi
ei, ci
ei, ciei, ci
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CONFLICT OF INTEREST:
I declare no conflict of interest.
-Sumanta Bagchi
ORCID ID 0000-0002-4841-6748
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