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Ecological Modelling 316 (2015) 28–40

Contents lists available at ScienceDirect

Ecological Modelling

j ourna l h omepa ge: www.elsev ier .com/ locate /eco lmodel

ynamical scenarios from a two-patch predator–prey system withuman control – Implications for the conservation of the wolf in thelta Murgia National Park

eborah Lacitignolaa,∗, Fasma Dieleb, Carmela Marangib

Dipartimento di Ingegneria Elettrica e dell’Informazione – Università di Cassino e del Lazio meridionale, Via di Biasio 43, 03043 Cassino, ItalyIstituto per le Applicazioni del Calcolo M. Picone, CNR, Via Amendola 122, 70126 Bari, Italy

r t i c l e i n f o

rticle history:eceived 12 December 2014eceived in revised form 17 July 2015ccepted 20 July 2015

eywords:athematical modeling

a b s t r a c t

We evaluate a mathematical model of the predator–prey population dynamics in a fragmented habi-tat where both migration processes between habitat patches and prey control policies are taken intoaccount. The considered system is examined by applying the aggregation method and different dynam-ical scenarios are generated. The resulting implications are then discussed, their primary aim being theconservation of the wolf population in the Alta Murgia National Park, a protected area situated in theApulian Foreland and also part of the Natura 2000 network. The Italian wolf is an endangered species

onservation ecologyifurcation analysisynamical scenariosumerical simulations

and the challenge for the regional authorities is how to formulate conservation policies which enable themaintenance of the said wolf population while at the same time curbing that of the local wild boars andits negative impact on agriculture. We show that our model provides constructive suggestions in how tocombine wild boar abatement programs awhile maintaining suitable ecological corridors which ensurewolf migration, thus preserving wolves from extinction.

© 2015 Elsevier B.V. All rights reserved.

. Introduction and research motivations

The establishment of management policies for protected areasas to face the issue of crop damages originating from wildlife.hese damages have the negative effect of increasing the intoler-nce toward the wildlife and of diverting a substantial part of thesually scarce resources allocated to nature conservation towardompensation actions (Ficetola et al., 2014). Wild boars, which intaly represent the most widespread species throughout the coun-ry, are deemed responsible for the largest amount of damages byildlife. However, their heavy impact on agriculture practices may

e partially compensated by their positive action on the nativeiotic communities of the ecosystem they are embedded in. For

nstance, it is known that wild boars can favor the regeneration ofoods, at least when the population density is kept low enough. To

educe the size of the wild boar population and its ensuing impact

n agriculture, while at the same time retaining the advantagesf their presence in protected areas, culling activities need to bearefully planned by the managing authorities. A further relevant

∗ Corresponding author. Tel.: +39 0775859705; fax: +39 0775859726.E-mail address: d.lacitignola@unicas.it (D. Lacitignola).

ttp://dx.doi.org/10.1016/j.ecolmodel.2015.07.027304-3800/© 2015 Elsevier B.V. All rights reserved.

issue in such a planning task is that, although wild boar is not atrisk of extinction itself, it plays a significant role in the trophicchain of another species which is currently red listed, i.e. the wolf.Wild boars represent in fact the favorite prey of the Italian wolf,especially in areas where other preys such as wild ungulates, areabsent. Wolves, as well as other large carnivores, e.g. coyotes anddingoes, are considered keystone species, as their impact on theirembedding ecosystems is disproportionally large with respect totheir abundance. In Soulé et al. (2003) the authors describe the eco-logical effectiveness of the role of top predators for communitiesand ecosystems. Their extinction can cause a cascade of secondaryextinctions. For that reason, they are a focus of interest in a numberof large-scale re-wilding projects such as the North America Wild-lands Network1 which is aimed at building large reconnected areasof land at continental scale to ensure the viability and sustaina-bility of big wild animal populations (Soulé and Terborgh, 1999).

In several European countries wolves are protected by nationallaws that try to conserve and increment the number of such preda-tors. The legal status of wolves in the European Union countries

1 North America Wildlands Network, http://www.twp.org/.

ical Modelling 316 (2015) 28–40 29

iiwp(

Psbtpt–i2ifmaadtahspPhtsauwMbctcepcbceG2

labciimdpgi2Add

l

Fig. 1. Representation of the two areas (Alta Murgia and Monti Dauni) and ofthe three possible corridors that represent, from top to bottom, three alternative

D. Lacitignola et al. / Ecolog

s directly specified in the Habitats Directive (92/43/EEC)2 whichs also responsible for the establishment of the Natura 2000 net-

ork of protected areas. The Directive requires strict protection,rohibiting any destruction or damage to the wolf populationKaczensky and Chapron, 2013).

The present paper actually focuses on the Alta Murgia Nationalark in Italy, a node of the Natura 2000 network. In this area amall sample of wolves recently showed up and their presence iseing constantly monitored even if the area extension and vege-ation type appears to be unsuited for hosting a viable and stableopulation of predators. Instead, the wild boar population – ini-ially brought to the site in small bunches for hunting purposes

exploded in recent years, especially because hunting activitiesn the area were strictly banned after the inclusion in the Natura000 network. The specific issues related to the presence of wolves

n the Alta Murgia site have been under study within a Europeanunded project (BIO-SOS) aiming at building an Earth observation

onitoring system for assisting the reporting obligations of man-ging authorities. It is known that wolves reached Alta Murgia from

nearby protected area, Monti Dauni, through ecological corri-ors. Wolves are socially organized in packs, normally with onlywo reproductive animals. Wolf packs are territorial and, in Italy,re small in size but can travel long distances and have very largeome ranges. They expand their range via dispersal and typicallyettle into unoccupied territories or assimilate into establishedacks within 50–100 km of their natal pack (Gese and Mech, 1991).resently, it is not known if the area of the Alta Murgia site canost a whole pack. Nevertheless national legal obligations bindhe managing authorities to set up conservation policies even foruch a limited number of individuals. The challenge for the regionaluthorities is to plan actions capable of maintaining the small pop-lation of wolves while at the same time limiting the presence ofild boars. A recent act by the management authority of the Altaurgia Park (PGC, 2012), addresses the issue of the minimum num-

er of wild boars capable of maintaining favorable conservationonditions for wolves without providing any guideline to quan-ify such an amount. The crucial point is: to what extent could oneontrol wild boars without affecting the wolves’ survival? Math-matical modeling can suggest answers for the solution of thisroblem. Mathematical models are in fact exploratory tools thatan be very useful in conservation ecology since they allow theuilding of hypothetical scenarios and provide management indi-ations (and warnings) in planning control programs as well as inxploring the benefits of different control options (Diele et al., 2014;arvie and Golinski, 2014; Gobattoni et al., 2014; Lacitignola et al.,007, 2010; Schmolke et al., 2010; Vincenot et al., 2011).

For this purpose, we recast the wild boars and wolves prob-em through a predator–prey model in a two-patch environmentnd use the modeling approach to evaluate how the interplayetween wolf migration processes between patches and humanontrol policies on wild boars can affect both populations dynam-cs as well as the wolf survival in the Alta Murgia Park. The modelncludes two time scales, a fast one associated to wolves move-

ents between the two patches and a slow one corresponding toemographic events of the wolf and wild boar populations. Theresence of these two different time scales enable us to use theeneral theoretical approach known as aggregation method whichs quite effective when used in population dynamics (Auger et al.,007; Auger and Bravo de la Parra, 2000; Auger and Poggiale, 1998;

uger and Roussarie, 1995). This allows for the reduction of theimension of a multi-patch model into a reduced system whichescribes the dynamics of a small number of global variables. The

2 Habitats Directive (92/43/EEC), http://ec.europa.eu/environment/nature/egislation/habitatsdirective/.

hypotheses of wolf colonization of the Alta Murgia from Monti Dauni: (top) thehypothesis that they have crossed the river Ofanto; (middle) the hypothesis of theshortest path; (bottom) the hypothesis of the path of natural areas (Pennacchioni,2010). In our modeling framework, the two areas are razionalized as two patches.

reduced model is nevertheless capable of providing useful analyt-ical insights into complex multi-patch systems. We refer to Augeret al. (2007) for an extensive review of the underlying philosophybehind the aggregation method as well as for a wide number ofapplications.

The paper is structured as follows: in Section 2 the two-patchmodel is presented together with the assumptions made and themodel parameters are selected. In Section 3 some preliminarynumerical test about the model behavior are presented, and thereduced model is obtained by applying the aggregation method.The results of the theoretical analysis of different kinds of equilibria,also in terms of their stability properties, are shown. By using fielddata, possible scenarios are discussed for the dynamics of the wolfand wild boar populations in the Alta Murgia National Park. Impli-cations for the wolf conservation are also discussed as a functionof the interplay between human control on the wild boar popula-tion and wolf migration processes between patches. In Section 4,concluding remarks are drawn.

2. Materials and methods

2.1. The two-patch model

The starting point of our analysis was a classical populationdynamics approach redesigned to describe the predator–prey pop-ulation dynamics in a fragmented habitat, where patches areconnected via ecological corridors, as depicted in Fig. 1. Under theassumptions related to the specific case study we derived a two-patch model for the wild boar and wolf population dynamics. Weconsidered the site of Alta Murgia Park as patch1 and the DauniMountains site as patch2. Consequently, we indicated as p1 and p2the density of the wolf population in Alta Murgia Park and in theDauni Mountains site respectively. We also indicated as n the den-sity of the wild boar population in the Alta Murgia Park. In fact,experimental field data indicated that the wild boar populationwas present in the Dauni Mountains at very low densities and nomovements between patches was observed for wild boars, so weconsidered their presence in patch2 as negligible for the aims of thepresent study. In the following, we list our assumptions on (a) themigration processes and (b) the local dynamics within the patches:

(a) Migration between patches1. Wild boars (preys) are sedentary inside their living patch so

that no migration was assumed from one patch to the otherand vice versa.

3 ical M

(

T⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩

0 D. Lacitignola et al. / Ecolog

2. Regarding the wolves (predators), we assumed that theymoved from patch1 (Alta Murgia) to patch2 (Dauni Moun-tains) at a constant rate d1 and from patch2 (DauniMountains) to patch1 (Alta Murgia) at a constant rate d2.Since the ‘attractiveness’ in terms of habitat suitability of thetwo patches was different, we expected the migration ratenot to be the same in both directions, i.e. d1 /= d2.

3. We assumed that migration between patches occurred at afaster time scale than the local predator–prey dynamics.

b) Local dynamics within the patches1. Field data showed that wild boar colonization of the Alta

Murgia Park occurred more slowly in the first stages of arestocking program, applied in the years 2000, 2001 and2002. The animals were released in two different areas of thePark, i.e. Gravina di Puglia (Bosco Difesa grande) and Spinaz-zola (Canale S. Lucia) and the releases were staggered overthree years. The repopulation started in 2000 with 20 indi-viduals, then 47 in 2001 and finally additional 105 animalswere introduced in 2002, for an overall number of 172. In theinitial stages of the reintroduction program the populationgrowth was slower, probably due to stress factors related tothe transportation as well as to the disruption of the socialorganization of animals removed from the wild. Thereafter,wild boars spread faster and the population growth was fos-tered by the hunting ban introduction in 2004 (PGC, 2012).This characteristic appeared to be strongly similar to thespread of wild pigs in California where the occupation at firstsporadic and then permanent of some areas has suggestedthe occurrence of a natural range expansion (Waithman et al.,1999). The same pattern was observed with other invasiveorganisms where range expansion is initially slow and thenrapidly accelerates (Lodge, 1993). For example, althoughhouse finches benefited from human introductions wheninvading North America, they expanded rapidly only afterpopulations reached some threshold size (Veit and Lewis,1996). On the basis of the above considerations, we assumedan Allee growth for prey population in patch1, where r is theintrinsic growth rate, k is the carrying capacity and A is theAllee threshold, with A < k.

2. For the predator population we assumed the same naturalper capita mortality rate � both in patch1 and in patch2.

3. We assumed a linear Lotka–Volterra functional responsewith a1 and b1 as predation coefficients in patch1 for preyand predator populations, respectively. We also supposedb1 = ea1 where e is a positive parameter which indicates theconversion of prey into predator biomass.

4. We assumed a control policy of the wild boar populationwhich corresponds to a harvesting density dependent termin the prey dynamics.

5. We also assumed that such a control policy had the effectof increasing the mortality rate of the wolf population inthe Alta Murgia Park by the rate qE where q is the catcha-bility (vulnerability) coefficient and E is a parameter whichrepresents the effort and hence the strength of the controlpolicy.

aking into account the above assumptions, the full model reads:

dn

d�= �

[rn

(1 − n

k

) (n

A− 1

)− a1np1 − qEn

],

dp1 = d2p2 − d1p1 + � [−(� + qE)p1 + ea1np1] , (1)

d�

dp2

d�= d1p1 − d2p2 + � [−�p2] ,

odelling 316 (2015) 28–40

where all the involved parameters are assumed positive constants;� is a small positive parameter, i.e. � � 1, � is the fast time scaleand t = �� is the slow time scale. Model (1) clearly shows that thefast part of the model only describes the displacement of wolvesbetween the two patches whereas, at slow time scale, the evolu-tion of the wild boar and wolf populations is described by a classicalpredator–prey system including (i) a logistic growth of the preywith Allee effect; (ii) a linear functional response for predation and(iii) a harvesting density dependent term for the human control onthe wild boar population. We stress that the control on wild boarshas also the effect of increasing the mortality rate of the wolf pop-ulation in patch1 by the rate q E. Moreover, it was assumed thatin patch2 – where no wild boars are present – mortality was theprominent process (with respect to other demographic processesthat are here considered as negligible) and also that it occurred ata slow time scale. In this regard, one can obviously ask what inter-est the wolves that inhabit patch1 would have in moving back topatch2. The most probable reason is likely to be the specific habitatfeatures of Dauni Mountains that, in spite of the absence of prey,would justify the presence of wolves in this site. It is well knownthat wolves are habitat generalists and, in addition to prey abun-dance, secluded sites and relatively low levels of human activity aregeneral habitat requirements for wolves (Fish and Service, 1987). Astudy by Mladenoff et al. (1995) in the northern Great Lakes Region(USA) showed for example that low densities of roads and lowfractal dimension of vegetation patches were the most importantpredictors of wolf colonization in logistic regression models. More-over, wolf pack territories were shown to be negatively correlatedwith amounts of agricultural land, road density, human populationdensity and positively correlated with coniferous forest cover andcounty-managed forest lands (Mladenoff et al., 1999).

In the framework of the present model, a further point thatdeserves a deeper explanation is the meaning that we attach tothe word migration. Migration can in fact take a number of formsand has been described by ecologists in many different ways. Westress that the word migration refers here to a form of individ-ual movement. Movements can generally occur either within theanimal home range or can take the individual beyond it (Dingle,1996). The behavior known as ‘station keeping’ (Kennedy, 1985),belongs to the formers kind of movements: it is mainly concernedwith finding and appropriating resources, (e.g. food, mate, etc.)and typically occurs on short timescales and small spatial scales.Its extended form – known as ‘commuting’ – regards longer to-and-fro journeys that are made regularly (often daily) to spatiallyseparated resource patches, roost sites and other localities, withinthe home range, where specific activities occur (Dingle and Drake,2007; Dingle, 1996). In the present model, we denote the latter typeof movements with the term migration. Moreover, given the dis-persal distances of 30–60 km traveled by a wolf at constant rate of6–10 km/h per day (Tardio, 2011), and since the minimum distancebetween the two patches is less than 30km, the wolf migrationbetween Alta Murgia and Dauni Mountains can be reasonably con-sidered as a daily one.

2.2. Parameters selection

The model parameters have been selected as follows:

• A: Allee threshold. From a preliminary analysis of the liter-ature we found that the evaluation of the minimum viableself-sustaining wild boar population size produces somehowinconsistent results. However, some empirical assessments were

done on the specific case. From available data, a reasonable min-imum size was 68, i.e. the minimum of the two animal amountsreleased in Gravina and Spinazzola, respectively. This size wasconsistent with the numbers given in Leaper et al. (1999), where

D. Lacitignola et al. / Ecological M

2000 2002 2004 2006 2008 2010 20120

5

10

15

Years

Den

sity

of w

ild−

boar

pop

ulat

ion

Simplified model (2): simulation curve,introduction of 20 individuals in 2000Simplified model (2): simulation curve,introduction of 47 individuals in 2001Simplified model (2): simulation curve,introduction of 105 individuals in 2002Simplified model (3): simulation curveintroduction of wolf predation in 2004Estimation from census 2010Estimation from census 2011Estimation from census 2012

Fri

•

•

•

ig. 2. Simulated wild boar dynamics corresponding to the optimal value for = 0.0484 with the simplified Eq. (2) in the interval [2000, 2004] and Eq. (3) in thenterval [2004, 2012].

the authors estimated a minimum size in the range 25–50, and theminimum of 75 animals provided by Howells and Edward-Jones(1997) in their model without inbreeding depression. Since thewoodland occupies 110 km2 of the total area of Alta Murgia Park,in terms of density we have A = 68/110 = 0.6182 individuals perkm2.k: carrying capacity. In PGC (2012), the number of individualscounted in each sample area of the park is reported. The max-imum density registered therein was 120 individuals per km2

(Masserie Nuove site) and it was assumed as the carrying capacityfor the site.a1: predation coefficient. By supposing that the intake of biomasswas entirely provided by wild boars, a wolf was estimated to con-sume 13.3 wild boars annually (Nores et al., 2008). By supposingthat this predation corresponds to the prey carrying capacity,from the relation a1k = 13.3 we obtained an estimated value ofa1 = 0.1108 km2/year per individual.r: wild boar growth rate. During the years of the restockingprogram, the wild boar dynamics in the Alta Murgia Park wassupposed to be described by the simplified equation

dn

dt= rn

(1 − n

k

) (n

A− 1

). (2)

The initial wild boar density in early 2000 corresponds tothe first introduction of 20 individuals into the area. We setn(2000) = 20/110 and we simulated the dynamics until early2001 when additional 47 individuals were introduced. Hence,by starting with density set to n(2001) + 47/110 we followedthe dynamics through the years. In order to take into accountthe last introduction of 105 individuals carried out in 2002, themodel (2) was used with an initial wild boar density set ton(2002) + 105/110. Since 2004, when the first tracks of the wolveswere discovered, a different simplified model

dn

dt= rn

(1 − n

k

)(n

A− 1

)− a1np1. (3)

was considered where a constant wolf presence with densityequal to p1 = 4/110 was supposed. In so doing, we were able toestimate r by minimizing the norm of the error among the pre-dicted values in 2010, 2011 and 2012 and the estimated values

deduced by the census made by the Park Authority in the sameyears. The result provided an optimized value of r = 0.0484 year−1.In Fig. 2 we illustrate the simulation of the wild boar populationgrowth starting from the first introduction in 2000 until 2012 in

odelling 316 (2015) 28–40 31

correspondence with this optimized value. All the above data canbe found in PGC (2012).

• �: wolf death rate. We supposed the value of � to be 0.12 year−1

(Knickerbocker, 2013).• e: the conversion coefficient. We assumed e = 0.028

(Knickerbocker, 2013), by supposing that the conversioncoefficient related to wild boars was five times greater than thevalue related to elks. To justify this last assumption, we observethat the conversion of prey into predator biomass is related to anumber of different factors which include the average size of thewolf packs, the size of the prey and its social behavior. When thepredation is done by a large pack, added members of the wolfpack eat portions of prey that would be lost to scavengers if packsize were reduced to a single individual. So the rate of conversioninto predator biomass of one big prey, is different whether thepredator acts in a pack or not. The common situation in AltaMurgia was currently a predation done by a single wolf. Giventhat premise, we remind that wild boars usually live in groupswith one or more adult females and numerous puppies, andthat the average size of a wild boar in such groups is about fivetimes smaller than the average elk size. It was then reasonable toassume that the food intake and the biomass conversion of wildboars, resulting from predation in the case at hand, was about 5times the one for elks (Mech and Boitani, 2006).

All the parameters with their numerical values have been sum-marized in Table 1 where we reported also the related sources.

3. Results and discussion

In the following we illustrate a preliminary numerical investi-gation performed on the full model (1). That in order to set a term ofreference for the behavior of the reduced system obtained by meansof the aggregation method. Thereafter, we describe the aggregatedmodel and qualitative scenarios which were built on it. Potentialimplications for the wolf conservation policies are the subject ofthe next section.

3.1. A preliminary numerical investigation of the full modelbehavior

Numerical investigations on the full model (1) were performedin Matlab by using the built-in function ode15s with absolute andrelative tolerances set to 10−15 and 10−13, respectively. This choicewas motivated by the necessity to tackle stiffness in differentialequations due to the presence of two different time scales.

We started by setting the fast time scale migration dynamicsin day units and the slow evolution in year units, corresponding to� = 1/365. We also assumed that the initial value for wild boar popu-lation in the Alta Murgia patch was the estimated density value pro-vided by the census in 2012 that is n(2012) = 1540/110 = 14 (PGC,2012). The densities of wolves population in the Alta Murgia andthe Monti Dauni sites in 2012 are p1(2012) = 4/110 = 0.0363 (PGC,2012), and p2(2012) = 25/1247 = 0.02 respectively (Pennacchioni,2010), where 25 is the number of wolves (5 packs with anestimated value of 5 wolves per pack) at the Monti Dauni site(Pennacchioni, 2010), whose extension corresponds to 1247 km2

(PPTR, 2013). Therefore, we considered these values as our realisticinitial conditions for the preliminary investigation of the long-termbehaviors of system (1). The model variables, their initial valuesand their related qualitative description have been summarized in

Table 2.

The numerical simulations presented in Figs. 3 and 4 yield thefollowing results. The wolf population survival through coexistencewas attained by a stable equilibrium as shown in Fig. 4(a) or, as in

32 D. Lacitignola et al. / Ecological Modelling 316 (2015) 28–40

Table 1Parameter values used in simulations and the related sources.

Parameter Description Value Units Source

a1 Predation coefficient 0.1108 km2 year−1 N−1 Nores et al. (2008)r Wild boar growth rate 0.0484 year−1 Census datae Conversion coefficient 0.0280 Dimensionless Knickerbocker (2013)� Wolf death rate 0.12 year−1 Knickerbocker (2013)k Carrying capacity 120 N km−2 Leaper et al. (1999)A Allee threshold 0.6182 N km−2 Leaper et al. (1999)

Howells and Edward-Jones (1997)q Catchability coefficient 1 year−1 Assumed

Table 2Variables of the models with initial values and related sources.

Variable Description Initial value Source

p1 Wolf density in the Alta Murgia Park (patch1) 0.0363 PGC (2012)p2 Wolf density in the Dauni Mountains site (patch2) 0.02 Pennacchioni (2010)

Ftas

3

(aa(o

pr⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩vnlfiid⎧⎪⎪⎪⎨⎪⎪⎪⎩w

d

n Wild boar density in the Alta Murgia Parkp Total number of wolves

ig. 4(b) and (c), by small stable periodic oscillations. In the long-erm behavior, extinction for both the species was found in Fig. 3(a)nd in Fig. 4(d), whereas in Fig. 3(b), only wild boar populationurvival was observed.

.2. The aggregated model

As previously stated, the presence of two different time scalesyears/day) gave the chance to benefit from the general theoreticalpproach known as aggregation method (Auger et al., 2007). Thisllowed for the reduction of the dimension of the two-patch model1) through a reduced one-patch system describing the dynamicsf a smaller number of global variables.

We defined p = p1 + p2 as the total amount of wolves and v1 =1/p and v2 = p2/p as the fraction of wolves in patch1 and patch2,espectively. Model (1) was rewritten as:

dn

d�= �n

[r(

1 − n

k

) (n

A− 1

)− a1v1p − qE

],

dv1

d�= d2 − v1(d1 + d2) + �v1(1 − v1) (e a1 n − q E),

dp

d�= �p[−� + ea1v1n − qEv1].

(4)

The first step was to consider the equilibrium value of the fastariable v1, i.e. the so-called fast equilibrium. This was obtained byeglecting the slow dynamics in system (4). Then the fast equi-

ibrium was given by v∗1 = d2/(d1 + d2). By substituting v∗

1 in therst and the third equation of system (4), we obtained the follow-

ng two-dimensional system which approximately describes theynamics of the total population densities n and p:

dn

dt= n

[r(

1 − n

k

) (n

A− 1

)− a1dp − qE

],

dp

dt= p [−� + ea1dn − qEd] ,

(5)

here the positive parameter d is given by

= d2

d1 + d2, (6)

14 PGC (2012)0.0563 Pennacchioni (2010)

PGC (2012)

and represents the stable fraction of predators in patch1 (Alta Mur-gia Park), attained at the fast time scale. We observe that such aparameter d can be interpreted as the relative migration rate frompatch2 (Dauni Mountain site) to patch1 (i.e. the ratio between themigration rate d2 and the total migration rate d1 + d2) and can hencebe thought as an interesting ecological indicator since it can beindirectly related to the existence of ecological corridors betweenpatches. The description of parameters E, d1, d2 and d has beensummarized and reported in Table 3.

3.3. Qualitative scenarios from the aggregated model

We recall that if system (5) is structurally stable and the param-eter � is small enough, the dynamics of the aggregated model is agood approximation of the dynamics of the global variables in thefull system (1).

Theoretical investigations on the aggregated model (5) havebeen presented in Appendix A. We considered as bifurcationparameters both the parameter d, which is related to the wolfmigration processes between patches, and the parameter E whichrepresents the effort of the human control on the wild boar pop-ulation. It turned out that, according to the specific choice forthe bifurcation parameters, three kinds of equilibria are possible:the predator–prey extinction equilibrium P0 = (0, 0); the predator’sextinction equilibria P1 = (n1, 0) and P2 = (n2, 0); the coexistenceequilibrium P* = (n*, p*).

The predator–prey extinction equilibrium P0 is stable for everyparameter values whereas the predator’s extinction equilibria Piand the coexistence equilibrium P* can be stable or unstable accord-ing to the specific choice of the parameter values. In this regard, aremarkable feature is that – under certain conditions – the coex-istence equilibrium can lose its stability by the means of an Hopfbifurcation when the control parameter E reaches the thresholdvalue Ecr2. The importance of this kind of bifurcation is well knownin the ecological literature since – when supercritical – it producesthe occurrence of self-sustained oscillations in the system dynam-ics (Kot, 2001). With reference to the present case study, thesupercritical nature of the Hopf bifurcation occurring at E = Ecr2 hasbeen established by the means of numerical investigations whichhave revealed the occurrence of stable small amplitude oscillationssurrounding the unstable equilibrium P* in the neighborhood of the

Hopf threshold.

The relevant d and E thresholds, derived in Appendix A andrelated to the existence and stability properties of the equilibria,have been summarized in Table 4.

D. Lacitignola et al. / Ecological Modelling 316 (2015) 28–40 33

2020 2040 2060 2080 21000

5

10

15

2020 2040 2060 2080 21000

0.01

0.02

0.03

0.04

0.05

(a)

2020 2040 2060 2080 21000

50

100

150

2020 2040 2060 2080 21000

0,01

0,02

0,03

0,04

(b)

n(τ)

n(τ)

p2(τ)

p1(τ)p

1(τ)

p2(τ)

Fig. 3. Long-term behaviors of the full model (1) with different parameter sets. (a) E = 2.4 and d1 = 4, d2 = 1. (b) E = 0.2 and d1 = 1.5, d2 = 1. The other parameter values arefixed as in Table 1. For the reasons detailed in the text, the chosen initial conditions are: n(0) = 14, p1(0) = 0.0363 and p2(0) = 0.02. In (a) it is shown that, following a brieftransient, the wild boar population n(�) and the wolf population become extinct in both the patches p1(�) and p2(�). In (b) it is shown that, after a rather long transient, thewolf population become extinct in both the patches p1(�) and p2(�), whereas the wild boar population n(�) tends to its carrying capacity.

2012 2100 2200 2300 2400 25000

100

200

2012 2100 2200 2300 2400 25000

10

20

(a)

2020 2040 2060 2080 21000

50100150

2020 2040 2060 2080 21000

10

20

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(d)

p2(τ)

p1(τ)

n(τ)

p1(τ)

p2(τ)

p2(τ)

p1(τ)

p2(τ)

n(τ)

p1(τ)

n(τ)

n(τ)

Fig. 4. Long-term behaviors of the full model (1) with different parameter sets. (a) E = 0.1 and d1 = 0.667, d2 = 1. (b) and (c) d1 = 0.111, d2 = 1 and E = 0.0537 (d) d1 = 0.111, d2 = 1and E = 0.05. The other parameter values are fixed as in Table 1. For the reasons detailed in the text, the chosen initial conditions are: n(0) = 14, p1(0) = 0.0363 and p2(0) = 0.02.In (a) it is shown that wild boar population n(�) and the wolf population in both the patches, p1(�) and p2(�), tend to a stable coexistence equilibrium. The asymptotic behavioris achieved after a rather long transient. In (b) it is shown that, after a relatively brief transient, coexistence among wild boars and wolves is obtained through stable smallamplitude periodic oscillations. In (c) the simulation, carried out for very long times, shows in details that the coexistence is obtained via a persistent oscillatory dynamics.In (d) it is shown that, after a relatively brief transient, the wild boar population n(�) and the wolf populations p1(�) and p2(�) become extinct.

Table 3Control parameters of the models with the related description.

Control parameter Description Units

d1 Wolf migration rate from patch1 to patch2 year−1

d2 Wolf migration rate from patch2 to patch1 year−1

E Effort of the control year−1

d Relative migration rate from patch2 to patch1 (d = d2/(d1 + d2)) Dimensionless

34 D. Lacitignola et al. / Ecological Modelling 316 (2015) 28–40

Table 4d and E bifurcation thresholds with the related qualitative and mathematicaldescriptions.

d and E thresholds Description Math. description

dlow Existence threshold for P* Formula (A.10)dstab Stability threshold for P* Formula (A.11)Ewf Existence threshold for P1 and P2 Formula (A.1)E1, E2 Existence thresholds for P* Formula (A.8)Ecr2 Hopf threshold for P* Formula (A.11)

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

E=Ewf

(d)E=E

cr2(d)

E=E2(d)

Region 0

Region I

Region IIRegion III

dlow

dstab d

E

Fig. 5. Dynamical scenarios of the aggregated model (5) as function of the wolfmigration processes between patches (i.e. parameter d) and of the effort of thehuman control on the wild boar population (i.e. parameter E). The other param-eter values are fixed as in Table 1. The different dynamical scenarios of the systemare represented through the two-parameter bifurcation diagram in the plane (d, E)with 0 < d ≤ 1. For the chosen parameter values, the d thresholds assume the fol-lowing values: dlow = 0.3223, dstab = 0.6413. The bifurcation curves E = Ewf , E = E2(d),E = Ecr2(d) are transition curves between different scenarios and divide the (d, E) planein four regions: (i) Region 0 in which both the wolf and the wild boar populationwill be extinct; (ii) Region I in which wolf population will be extinct in any case;(ii) Region II in which, according to the initial conditions, either wolf extinctionor wolf survival via a stable coexistence equilibrium is expected (iii) Region III inwtb

daiaebat

(dtf

pRibclwu

Fig. 6. The aggregated model (5). Stability properties of the two wolf- extinctionequilibria P1 and P2 in their region of existence (0 ≤ E ≤ Ewf) as a function of thecombination between wolf migration processes between patches (parameter d) andhuman control on the wild boar population (parameter E). The other parametervalues are fixed as in Table 1. For the chosen parameter values Ewf = 2.3246. Thebifurcation diagrams in the (E, d) plane show that (a) the wolf extinction equilibriumP is always stable in the half plane d > 0, namely where it is ecologically meaningful

hich, according to the initial conditions, (a) wolf extinction or (b) wolf survivalhrough small amplitude periodic oscillations in the strict neighboring of the Hopfifurcation line E = Ecr2(d) are expected.

With parameters in model (5) chosen as in Table 1, the above thresholds become: dlow = 0.3223; dstab = 0.6413. Recalling (6), itlso follows that 0 < d ≤ 1 and hence we considered d = 1 as the max-mum feasible value for the parameter d. As far as the E thresholdsre concerned, we observe that Ewf does not depend on the param-ter d and is such that Ewf = 2.32. On the contrary, E1 and E2 areoth functions of d and are such that, when d > dlow, E1 = E1(d) < 0nd E2 = E2(d) > 0. The Hopf threshold Ecr2 also depends on d andurns out to be strictly positive if and only if d > dstab.

The bifurcation curves E = Ewf, E = E2(d) and E = Ecr2(d) divide thed, E) plane in four regions. As a consequence, the existence of fourynamical scenarios can be proved, as depicted in Fig. 5 wherehe different dynamics of the aggregated model (5) are shown asunctions of the two bifurcation parameters d and E.

In Fig. 5, the regions in the d − E parameter space where wolfopulation survival can be ensured were labeled as Region II andegion III. Inside these regions, wolf population survival can occur

n two different ways since system dynamics can approach a sta-le coexistence equilibrium (Region II) or a stable coexistence limit

ycle for d and E in the strict neighborhood of the Hopf bifurcationine E = Ecr2(d) (Region III). However both Region II and III also have

olf extinction as a possible output: in fact, whether the wolf pop-lation will survive or will be extinct depends on the chosen initial

1

(b) when d > 0, the wolf extinction equilibrium P2 can change its stability propertiesby crossing the bifurcation curve depicted in the figure.

conditions. On the other hand, in Region 0 or in Region I, there isno chance of survival for the wolf population.

In particular, in Region 0 – which corresponds to very highvalues of the control parameter E – both the wolf and the wildboar population will be extinct independently of the value of theparameter d. Moreover, as showed in Fig. 6, the wolf extinctionequilibrium P1, whenever existent and ecologically meaningful(namely in Regions I, II, III), is always unstable. On the contrary, thewolf extinction equilibrium P2 is stable in Region I and unstable inRegions II and III.

In Table 5 we have summarized the existence and stability prop-erties of the equilibria of the aggregated model (5) in the fourregions of the two-parameter bifurcation diagram in Fig. 5.

In conclusion, the bifurcation diagram in Fig. 5 indicates that(i) too high values of the control parameter E, i.e. E > Ewf, lead boththe species toward extinction independently of the value of the

parameter d (Region 0); (ii) too low values of the parameter d, i.e.d < dlow, can lead either to only wolves extinction (Region I) or toboth wild boar and wolf populations extinction (Region 0), accord-ing to the specific value of the control parameter E; (iii) wolves

D. Lacitignola et al. / Ecological Modelling 316 (2015) 28–40 35

Table 5The aggregated model (5). Existence and stability properties for the equilibria of theaggregated system (5) in the four regions of the two-parameter bifurcation diagramin Fig. 5.

Region 0 Region I Region II Region III

P0 stable P0 stable P0 stable P0 stable

swcbvtRtl

aot(lRoatfd(Iilwst

(–S

atgctitb

RdficlbIbsiepys

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.05

0.1

0.15

0.2

0.25

0.3

Ecr2

(d)

E2(d)

dlow d

stab d

E

Region I

Region III

(Region II)

(Region II)

Risky Region

Security Region

A C

B

D

Fig. 7. A detail of the (d, E) bifurcation diagram in Fig. 5 which specifically focuseson Region II. The bifurcation parameter d is related to the wolf migration processesbetween patches whereas the bifurcation parameter E represents the effort of thehuman control on the wild boar population. In Region II coexistence between wolfand wild boar population can be ensured through a stable equilibrium. Theoreticalresults suggest that, from the point of view of the control strategies, such region canbe divided in two different sub regions: a Security Region and a Risky Region. (For

P1 unstable P1 unstable P1 unstableP2 stable P2 unstable P2 unstable

P* stable P* unstable

urvival is possible only in Region II in which coexistence betweenolf and wild boar populations can be ensured through a stable

oexistence equilibrium or in Region III where, in the strict neigh-oring of the Hopf bifurcation line, coexistence can be obtainedia a stable small amplitude limit cycle. Away from the Hopf line,he amplitude of oscillations tends to increase and this portion ofegion III is hence not a good one. In fact large amplitude oscilla-ions should be prevented since environmental fluctuations couldead to the extinction of the wolf population.

The numerical simulations presented in Figs. 3 and 4 show complete qualitative match between the theoretical scenariosbtained through the aggregated model (5) and the dynamics ofhe full model (1). In fact, recalling the definition of d provided in6), population coexistence has been attained through a stable equi-ibrium in Fig. 4(a) where the parameters E = 0.1 and d = 0.6 lie inegion II. Coexistence has been achieved through small stable peri-dic oscillations in Fig. 4(b) and (c) where the parameters E = 0.0537nd d = 0.9 have been set in Region III, in the strict neighboring ofhe Hopf bifurcation line. Extinction for both the species has beenound in both Fig. 3(a) and Fig. 4(d), where the parameters E and

have been chosen in Region 0 (E = 2.4, d = 0.2) and in Region IIIE = 0.05, d = 0.9) far from the Hopf bifurcation line, respectively.nstead, in Fig. 3(b) only wild boar population survival is expectedn the long-term behavior, since the parameters E = 0.2 and d = 0.4ie in Region I. Because of the qualitative matching outlined above,

e are allowed to use the bifurcation diagram of the aggregatedystem (5) depicted in Fig. 5, to discuss the dynamical scenarios ofhe full model (1).

A closer look at the dynamics of system (5) (and, hence of model1)) suggests that – from the point of view of the control strategies

Region II can be further divided in two different sub regions: aecurity Region and a Risky Region, Fig. 7.

The Security Region is characterized by sufficiently low E valuesnd intermediate d values, i.e. dlow < d < dstab ∧ 0 < E < E2. It representshe perfect region since it allows a monitoring of the control strate-ies. In fact, for a fixed value of d in the Security Region, coexistencean be ensured by keeping the value of E below E2. For E tendingoward E2, the wolf population level progressively decreases untilt approaches zero entering Region I. In this case, a careful moni-oring of the control policy can avoid wolf extinction by lowering Eelow the E2 threshold.

Coexistence can be ensured via a stable equilibrium also in theisky Region. However such a region could be associated to a moreangerous situation in terms of control perspectives. In fact, for axed d value in the Risky Region, specific changes into the E levelan lead to the sudden collapse of both the wild boar and wolf popu-ation. To better elucidate this point, suppose an ecological situationelonging to the Risky Region and labeled with the point A in Fig. 7.f a higher removal level is decided by the authorities for the wildoar population, then system dynamics can be brought progres-ively toward wolf extinction (point D, in Region I). However, alson this case, a careful planning of the control policy can avoid wolf

xtinction by lowering E below the E2 threshold. But what hap-ens if the control level is set too low? Going well into Region III,ou can suddenly pass from coexistence to extinction for both thepecies, (point B, in Region III) and, in this case, wolf survival is not

interpretation of the references to color in text, the reader is referred to the webversion of the article.)

necessarily guaranteed. As a consequence, although the RiskyRegion could ensure coexistence between wild boar and wolf popu-lations, it would be wiser to avoid such a situation, preferring thesafer Security Region. This can be obtained by properly reducing thevalue of the parameter d, for example reaching the point C in Fig. 7.We refer to Section 3.4 for discussions on the practical implicationsof this choice.

Such reasoning was insightfully expressed in terms of the basinsof attraction of the involved system attractors. We recall that thebasin of attraction of a given attractor (i.e. a stable equilibrium, astable limit cycle) is the set of initial conditions which leads to thelong-time behavior that approaches that attractor (Guckenheimerand Holmes, 1997). This means that the qualitative long-timebehavior of a given system can be significantly different accord-ing to which domain of attraction the considered initial conditionlies.

Fig. 8 shows that if we are in the Risky zone of Region II (pointA in Fig. 7) we clearly have a bistability situation with two dif-ferent basins of attraction. The smaller one (the green region) isrelated to the coexistence equilibrium P* and the other one (thered region) is related to the two species extinction equilibrium P0.By suitably reducing the value of the parameter d, we can moveinto the Security zone of Region II (point C in Fig. 7) and – althoughthe two system’s attractors remain the same – we can note a differ-ence in terms of the related basins of attraction since the basin ofattraction of the coexistence equilibrium P* considerably enlargesin the Security Region, thus suggesting a situation of higher eco-logical resilience (van Nes and Scheffer, 2007). Interestingly, thisalso implies that a stochastic event could move the system in analternative basin of attraction less likely in the Security zone thanin the Risky Region.

Fig. 8 also clarifies that if we are in the Risky Region (point A inFig. 7) and reduce the effort of human control on the wild boar pop-ulation (i.e. decreasing the E values), we enter Region III (point B in

Fig. 7) with two possible outputs: (i) if E is slightly reduced, coexis-tence is still possible by the means of an oscillatory dynamics. PanelRegion III (NH) in Fig. 8 shows the basins of attraction related to this

36 D. Lacitignola et al. / Ecological Modelling 316 (2015) 28–40

Fig. 8. Basins of attraction related to the different cases shown in Fig. 7. The red region (dark grey in the printed version) represents the basin of attraction of the two speciesextinction equilibrium P0; the green region (grey in the printed version) represents the basin of attraction of the coexistence equilibrium P*; the light red region (light greyin the printed version) represents the basin of attraction of equilibrium P2 corresponding to the extinction of the wolf only; the light green region (white in the printedversion) represents the basin of attraction of the coexistence limit cycle surrounding the equilibrium P*. In each panel, stable equilibria are marked through filled circleswhereas unstable equilibria are marked through crosses. The stable limit cycle is marked through a blue empty circle. The different panels are distinguished by the differentnumerical values assigned to the two bifurcation parameters E and d. The numerical values of the other parameters are fixed as in Table 1. The panel related to the RiskyRegion refers to the generic point A in Fig. 7, E = 0.1, d = 0.9. In this case, the system attractors are the equilibria P0 and P*. The panel related to the Security region refers to thegeneric point C in Fig. 7, E = 0.1, d = 0.6. The attractors are the same as in the Risky Region even if the basin of attraction of the coexistence equilibrium P* here considerablyenlarges. The panel related to Region III (NH) refers to the generic point B in Fig. 7, E = 0.0537, d = 0.9. In this case, the system attractors are the equilibrium P0 and the limitcycle surrounding the equilibrium P*. The panel related to Region III (AH) refers to the generic point B in Fig. 7, E = 0.05, d = 0.9. In this case, the only attractor is the equilibriumP0. The panel related to Region I refers to the generic point D in Fig. 7, E = 0.2, d = 0.6. In this case, the system attractors are the equilibria P0 and P2. In the panels related tot 1 < n <l ions oe color i

cbntrttltfs

icwobe(

3M

rmwwotpsw

s

he Security/Risky Region and Region III, initial conditions of the kind (n, 0) with night red (light grey in the printed version). This means that when in the above Regquilibrium where only wild boars survive. (For interpretation of the references to

ase: the smaller (light green region) refers to the stable limit cycleorn by supercritical Hopf bifurcation in the neighboring of theow unstable equilibrium P*; the other one (red region) representshe two species extinction equilibrium P0 (ii) whenever E is greatlyeduced, the bistability situation ceases and system dynamics tendsoward the unique system attractor which drives both the specieso extinction, Panel Region III (AH). Both these situations should beooked upon with some caution: the former because of the poten-ial for oscillatory behavior to trigger extinction when stochasticorces are involved (May, 1972), the latter because it represents aharp transition toward extinction.

Finally, Fig. 8 outlines that if we are in the Risky Region (point An Fig. 7) and – acting unwisely – we increase the amount of humanontrol on the wild boar population (i.e. increasing the E values),e enter Region I (point D in Fig. 7) where a bistability situation

ccurs between the two species extinction equilibrium P0, whoseasin of attraction is the red region, and the only wolf extinctionquilibrium P2, whose basin of attraction is the light red regionPanel Region I in Fig. 8).

.4. Implications for the conservations of the wolf in the Altaurgia Park

The achieved results strongly suggest that Region II may rep-esent the most desirable situation in terms of a sustainableanagement of the wolves in the Alta Murgia Park. In this regard,e can identify those factors crucial for wolf conservation: humanild boar control and wolf migration. Indeed, a careful combination

f processes such as human control of the wild boar population –hrough proper planning programs – and wolf migration betweenatches – through the existence of suitable ecological corridors –

hould be used in order to properly limit the wild boar populationhile preserving the wolf population from extinction.

Our findings give some qualitative warnings for the specific casetudy. In primis, either too low d values and high E values have to be

n2 belong to the stable manifold of the saddle point P2 so that they are marked inne starts with these initial conditions, the system asymptotically tends toward ann this figure legend, the reader is referred to the web version of the article.)

avoided. Moreover sufficiently low values of the control parameterE have to be combined with intermediate values of the migrationparameter d in order to ensure coexistence between the wolf andwild boar populations and hence wolf survival in the Alta MurgiaPark.

Now, the obvious question is: which Region in the two-parameters bifurcation diagrams of Figs. 5 and 7 can currentlyrepresent the Alta Murgia case? As far as the parameter d is con-cerned, we observe that it can be reasonable to suppose that thedensities of wolf population in Alta Murgia and Monti Dauni sitesin 2012 correspond respectively to the fast equilibria d2 and d1 ofthe full model (1). It follows that d2 = 0.0363 and d1 = 0.02 so that –according to (6) – one gets d = 0.6446. Recalling the formula (A.8),it follows that the threshold E2 = 0.1788.

Moreover, as explained above, the parameter E is related tothe abatement level of wild boars decided by the managementauthority of the Alta Murgia Park. In the official act of 2012, anabatement proposal was launched to reduce the estimated pop-ulation of 1540 individuals to a value in the range 550–839.By choosing the maximum value for the huntability coefficientq (i.e. q = 1), such abatement can be estimated with values of Ein the range [Emin, Emax] where Emin = log(1540/839) = 0.673 andEmax = log(1540/550) = 1.0296. For this chosen value of the param-eter d, the considered range [Emin, Emax] is such that Emin > E2 andour analysis predicts the extinction of the wolf population as a long-term system behavior. We are in fact in Region I of Fig. 5. Our resultssuggest that, to ensure wolf population survival, we have to enterthe coexistence Region II. To this aim, the value of the effort Emax

should hence be lowered below the threshold value E2. Such find-ings are completely confirmed by numerical investigations on thefull model (1). Fig. 9 shows how a situation of wolf extinction, corre-

sponding to the effort value E = Emin = 0.673, can be avoided simplyby lowering the effort E to the level E = 0.16. However, even in thislast situation, we would be in the Risky zone of Region II whereas, fora sustainable management of the wolf and wild boar populations, it

D. Lacitignola et al. / Ecological Modelling 316 (2015) 28–40 37

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1

2

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4

5

(b)

n(τ)

p2(τ)

p1(τ)p

1(τ)

p2(τ)

n(τ)

Fig. 9. Long-term behaviors of the full model (1) when the parameters d1, d2 and E are chosen so that the parameters d and E are (a) E = Emin = 0.673 and d1 = 0.02 d2 = 0.0363so that d = 0.6446; (b) E = 0.16 and d1 = 0.02 d2 = 0.0363 so that d = 0.6446. The other parameter values are fixed as in Table 1. For the reasons detailed in the text, the choseni at, aftt ng trav

wkwSdlc

tAamSrt(ihrtthMiwtltwtspK

otiib

nitial conditions are: n(0) = 14, p1(0) = 0.0363 and p2(0) = 0.02. In (a) it is shown thhe patches p1(�) and p2(�) become extinct. In (b) it is shown that, after a rather loia a stable equilibrium.

ould be preferable to move into the Security zone of Region II. Stilleeping E = 0.16, this can be obtained by acting on the parameter dhich has hence to be lowered below the threshold dstab = 0.6413.

ince the parameter d is related to the existence of ecological corri-ors between the Alta Murgia and the Dauni Mountains sites, this

ast observation could throw new light on the role of the ecologicalorridors for species survival in multi-patch environments.

A suitable control policy could be designed by acting on eitherhe corridors or the abatement rate or on both. In the case of thelta Murgia Park where the managing authorities have little – ifny – control on the corridors, our findings may recommend theonitoring of corridors to provide estimation of migration rates.

ince we are dealing with ‘natural’ corridors, their control is mainlyelated to the impact of human activities and to the alterations ofhe landscape generated by changes in the land use and land covernew roads, different agricultural practices, reforestation, poach-ng), which might be managed to some extent. Our study alsoighlights qualitative estimates on the strength of the abatementate in order to favor the onset of a coexistence equilibrium of thewo species, advising that the wild boar abatement proposal byhe Apulian regional authorities should not be too severe if oneas to ensure a long-term survival of wolf population in the Altaurgia Park. Moreover, by looking at the structure of the coex-

stence equilibrium (A.6) that represents the asymptotic level tohich the abundances of the predator and prey populations will

end for extremely long times, it is possible to observe that theong-term effect of increasing the control parameter E is to increasehe average wild boar population level and to decrease the averageolf population level. Even if for certain aspects counterintuitive,

his result can be interpreted as a clear admonition against non-pecific control measures that affect indistinctly both preys andredators with possible catastrophic consequences (De Bach, 1974;ot, 2001).

A further interesting result is the evidence of an ambivalent rolef the corridors for the wolf and wild boar dynamics. We point out

hat the spread and impact of antagonistic species, i.e. predators,s generally considered as a negative effect of corridors connect-ng otherwise separated habitat patches. This effect turns out toe highly wanted, of course, if the antagonistic species are the

er a brief transient, the wild boar population n(�) and the wolf population in bothnsient, the wolf populations p1(�), p2(�) and the wild boar population n(�) coexist

endangered and protected ones, whereas the preys are of no inter-est with respect to conservation policies, as in our case. Recentstudies (e.g. Haddad et al., 2014), have shown that the effect ofcorridors on the predator–prey dynamics and the coexistence con-ditions does not have a definite sign: negative, neutral and positiveeffects are all likely to occur and there is currently no theoreticalapproach capable to explain that. This appears to be a point in favorof our model which stresses this double-role of ecological corridorsand suggests the existence of different dynamical scenarios whichqualitatively match behaviors actually observed in nature.

However, for the model to provide quantitative estimationsuseful for management purposes, some key processes have tobe explicitly included. First of all, stochasticity. In fact, althoughdeterministic models are sufficient to predict average behaviors atthe population level, stochasticity is critical for small populationsand could intervene in several processes such as control, popula-tion growth, predation strength and, more importantly, migrationthat is here considered as individual movements between patches.Besides, wolves are reported to have a very strong social structureand a more careful description of the development stages should beconsidered in the model. Enriching the system with compartmentsreferring to an age structure (young/adults) and to a sex structure(male/female) may in fact significantly influence the populationdynamics involved. This feature can in particular affect migrationsince differences have been observed in migration patterns of olderand younger individuals or of individuals of the two sexes. Last butnot least, a more realistic description of the control process has tobe developed. In fact in the model the control of wild boars is con-sidered as a continuous process but it should be more realisticallymodeled as a discrete event since the actual control procedures arebased on the calculation of quotas on annual basis.

4. Concluding remarks

We have used a modeling approach to gain qualitative insights

for the management of the wild boar and wolf population dynam-ics in the Alta Murgia Park. For this purpose, a two-patchpredator–prey model has been considered where both migrationprocesses between patches and prey control policies have been

3 ical M

ttpgcfacdort

mtpqTtdloactoOtuotedrt

A

VftWfuchwtAiC

Ap

tern

q

E

8 D. Lacitignola et al. / Ecolog

aken into account. The full model has been simplified by applyinghe aggregation method to get a reduced dynamical system whoseroperties have been theoretically analyzed. Numerical investi-ations on the two-patch model have provided evidence of theapability of the reduced system to mimic the dynamics of theull model with respect to equilibria and stability properties. Inddition, we have showed the existence of four different dynami-al scenarios which suggest that a suitable control policy could beesigned by acting on either the corridors or the abatement rater on both. In the case of the Alta Murgia Park our findings mayecommend the monitoring of corridors to provide estimations ofhe migration rates.

We stress that the proposed model is meant to be used as aeans for qualitative scenario building more than as a quantita-

ive tool to support environmental decision making on protectionolicies. This is because of the many assumptions that make modeluantitative estimations not fully reliable for practical purposes.his is perfectly consistent with the modeling framework used inhis paper, the top-down approach, where one starts with a simple,eterministic, macroscopic description and then adds successive

evels of details. With reference to the present case study, enrichingur model with elements such as stochasticity, sex/age structuresnd more refined control procedures, corresponds to increasing theomplexity of its mathematical representation and of the relatedheoretical analysis to the point that some results could only bebtained with the help of case-dependent numerical simulations.n the one hand this case-oriented implementation would seem

o limit the overall theoretical power of the model as a means tonderstand general ecological principles and patterns but, on thether hand, it could significantly refine the capability of quantita-ive estimates on the specific case under examination. This is a clearxample of how developing realistic ecological models for practicalecision support is indeed a complex and challenging task, oftenequiring a very careful and calibrated compromise between theop-down and the bottom-up modeling approaches.

cknowledgments

The authors wish to acknowledge the handling Editor, Prof.olker Grimm, and the two anonymous Reviewers for their care-

ul and valuable analysis of the manuscript. Their comments andheir constructive criticism helped us to greatly improve this paper.

e warmly thank Nicholas Tavalion for numerous suggestionsor test improvement. D.L. research work has been performednder the auspices of the Italian National Group for Mathemati-al Physics (GNFM-Indam). The research activities of F.D. and C.M.ave been partially funded by the European Union Seventh Frame-ork Programme FP7/2007-2013, SPA.2010.1.1-04: ‘Stimulating

he development of GMES services in specific area’, under Grantgreement 263435, project title: ‘BIOdiversity Multi-Source Mon-

toring System: from Space To Species (BIO SOS)’, coordinated byNR-ISSIA, Bari-Italy (http://www.biosos.eu).

ppendix A. The aggregated model: equilibria and stabilityroperties

In this mathematical appendix we analytically derive thehresholds, expressed in terms of the system parameters, for thexistence and stability properties of the equilibria. The followingesults hence provide the theoretical bases for the dynamical sce-arios described in Section 3.3 for the specific case study.

Preliminary remarks – Let us introduce the following threshold

uantity for the control parameter E,

wf = r

q

(k − A)2

4Ak, (A.1)

odelling 316 (2015) 28–40

and further consider the threshold quantities for the wild boarintrinsic growth rate parameter r,

r1/2 = kAea1 (k − A)−2 [(k + A) ∓ 2

√Ak

],

rtr = ea1kA (k − A)−1.(A.2)

It is easy to recognize that r1 < rtr < r2. As a preliminary step letus introduce the quantities

C1 = 4rq2�e2a21kA,

C2 = q2e2a21[r2(k − A)2 − 2rea1Ak(k + A) + e2a2

1k2A2],(A.3)

and observe that C1 is always a positive quantity whereas C2 > 0 ifand only if r < r1 ∨ r > r2.

Existence results – The following result allows us to get aninsight into the different kinds of equilibria of the aggregatedmodel:

Theorem Appendix A.1. The aggregated model

(i) admits the extinction equilibrium P0 = (0, 0) for every value ofthe positive parameter E and the predator’s extinction equilibriaP1 = (n1, 0) and P2 = (n2, 0) for 0 < E < Ewf.

(ii) admits a feasible coexistence equilibrium P* = (n*, p*) if one of thefollowing cases (ii.a)–(ii.d) holds:(ii.a) Let r > r2 and d > �/(ea1A). Then two real positive values of

the control parameter exist, E1 and E2, such that P* is feasiblefor E1 < E < E2.

(ii.b) Let r > r2 and �/(ea1k) < d < �/(ea1A). Then one real posi-tive value E2 of the control parameter exists, such that P*

is feasible for 0 < E < E2.(ii.c) Let rtr < r < r2 and �/(ea1A) < d < − C1/C2. Then two real pos-

itive values of the control parameter exist, E1 and E2, suchthat P* is feasible for E1 < E < E2.

(ii.d) Let rtr < r < r2 and �/(ea1k) < d < min{

�/(ea1A),

−C1/C2}

. Then one real positive value E2 of the control

parameter exists, such that P* is feasible for 0 < E < E2.

Proof. System’s equilibria are the real positive roots of the system⎧⎨⎩

n[

r(

1 − n

k

) (n

A− 1

)− a1dp − qE

]= 0,

p[−� + ea1dn − qEd] = 0.

(A.4)

(i) A straightforward analysis of system (A.4) shows that P0 = (0, 0)is always an equilibrium. Moreover it is easy to check that the preycomponents ni, i = 1, 2 of the predator’s extinction equilibria Pi = (ni,0) are given by the real positive solutions of the following algebraicequation:

n2 − (A + k)n + Ak(r + qE)r

= 0. (A.5)

Observe that � = (k − A)2 − 4AkqEr , so that � > 0 ⇔ E < Ewf. Holding

this, the Descartes’ rule of signs ensures that both the roots ofEq. (A.5) are positive and hence feasible. As a consequence, for0 < E < Ewf, Pi = (ni, 0) exist with ni given by n1/2 = (1/2)[(A + k) ∓√

�].(ii) The coexistence equilibrium P* = (n*, p*) can be easily

deduced by system (A.4) which gives:

n∗ = � + qEd, p∗ = 1

[r(

1 − n∗ ) (n∗

− 1)

− qE]

. (A.6)

ea1d a1d k A

This is feasible when p* is a positive quantity. Namely:

p∗ > 0 ⇔ B2E2 + B1E + B0 < 0, (A.7)

ical M

w

Bif

E

tos�i⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩(

r

tbri(0fm

d

wdftt

Teg

d

fv

D. Lacitignola et al. / Ecolog

here

B2 = rq2d2, B1 = qd[2r� + e2a21dkA − rea1d(k + A)],

B0 = r(ea1dk − �)(ea1dA − �).

Inequality (A.7) can be verified if and only if the quantity �1 =21 − 4B2B0 = d3(C2d + C1) > 0. It follows that if (a) r < r1 ∨ r > r2 or

f (b) r1 < r < r2 and 0 < d < − C1/C2, then inequality (A.7) is verifiedor E1 < E < E2, where

1/2 = −B1 ±√

�1

2B2. (A.8)

Next step for the feasibility of the coexistence equilibrium P* iso search for conditions in order that at least one of the thresh-lds (A.8) is a positive quantity. Applying the Descartes’ rule ofign it follows that Eq. (A.7) has exactly one positive root, E2, if/(ea1k) < d < �/(ea1A) whereas it admits exactly two positive roots

f the following system is verified:

d <�

ea1k∨ d >

�

ea1A,

d >−2r�

ea1[ea1kA − r(k + A)],

r >ea1kA

k + A.

(A.9)

In this regard, we observe that if r > rtr = (ea1kA)/k − A) > (ea1kA)/(k + A), then the following inequalities hold:

�

ea1k<

−2r�

ea1 [ea1kA − r(k + A)]<

�

ea1A.

Recalling system (A.9), one gets that if

> rtr ∧ d >�

ea1A,

hen Eq. (A.7) admits two positive roots E1 and E2. By com-ining the above results on positivity with conditions on theeality of the roots E1 and E2, we can conclude that: (ii.a)f r > r2 and d > �/(ea1A), then P* is feasible for 0 < E1 < E < E2;ii.b) if r > r2 and �/(ea1k) < d < �/(ea1A), then P* is feasible for

< E < E2; (ii.c) if rtr < r < r2 and �/(ea1A) < d < − C1/C2, then P* iseasible for 0 < E1 < E < E2; (ii.d) if rtr < r < r2 and �/(ea1k) < d <

in{

�/(ea1A), −C1/C2}

, then P* is feasible for 0 < E < E2. �

Stability properties of P* – We set:

low = �

ea1k; dup = �

ea1A. (A.10)

In view of the application to the specific case of study, weill focus on the cases (ii.a) r > r2 and d > dup and (ii.b) r > r2 and

low < d < dup of Theorem Appendix A.1. We hence restrict all theollowing investigations to the case: r > r2 and d > dlow. In addition,o be more precise on the expected kind of coexistence, we look athe stability properties of the coexistence equilibrium P*.

heorem Appendix A.2. Let r > r2 and d > dlow. Then the coexistencequilibrium P* is feasible for max {0, E1} < E < E2, where E1 and E2 areiven by the formula (A.8). Define:

stab = 2�

ea (k + A); Ecr2 = [−2� + ea1d(k + A)]

2qd. (A.11)

1

(i) If dlow < d < dstab, then the coexistence equilibrium P* is stableor every value of max {0, E1} < E < E2; (ii) If d > dstab, then a positivealue Ecr2 of the control parameter E exists such that the coexistence

odelling 316 (2015) 28–40 39

equilibrium P* is unstable for max {0, E1} < E < Ecr2 and locally asymp-totically stable for Ecr2 < E < E2. The bifurcation value E = Ecr2 is a Hopfbifurcation point.

Proof. The Jacobian matrix J(n, p) of the aggregated system (5) isgiven by:

J(n, p) =[

G −na1d

pea1d −� − qEd + ea1dn

], (A.12)

with

G = 2rn

A− 3

rn2

Ak− r + 2

rn

k− a1dp − qE.

By evaluating the Jacobian matrix (A.12) in P* one obtains that

det(J(n∗, p∗)) = (� + qEd)p∗a1d,

which is always positive in the range of the parameters, so that P*

is feasible. Moreover,

tr(J(n∗, p∗)) ≥ 0 ⇔ F2E2 + F1E + F0 ≥ 0,

with:

F2 = −2rq2d2, F1 = rqd[−4� + dea1(k + A)],

F0 = �[−2� + dea1(k + A)].

It straightly follows that if d > dstab, then tr(J(n*,p*)) > 0 ⇔ Ecr1 < E < Ecr2 and tr(J(n*, p*)) < 0 ⇔ E < Ecr1 ∨ E > Ecr2,where Ecr1 = − �/(qd) and Ecr2 given by equation (A.11). Moreovertr(J(n*, p*)) = 0 ⇔ E = Ecr2. Hence a couple of complex imaginaryeigenvalues of J(n*, p*) crosses the imaginary axis and a Hopfbifurcation occurs. In the strict neighboring of Ecr2, P* is a stablefocus for E > ≈ Ecr2 and an unstable focus for E < ≈ Ecr2. In this latterrange and close enough to the bifurcation value, an asymptoticallystable limit cycle appears for the system (5) that surrounds theunstable equilibrium P*. Instead, if dlow < d < dstab, then tr(J(n*,p*)) < 0 for every E > 0. By combining the above results with therequirements about the feasibility of the coexistence equilibriumP*, theses (i) and (ii) of Theorem Appendix A.2 easily follow. �

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