Modeling Ecosys ems Using P Sys ems:
The Bea ded Vul u e, a Case S udy
M´onica Ca dona1, M. Angels Colome 1,Ma ioJ.P´e ez-Jim´enez2,
Del ´ıSanuy
3, and An oni Ma galida4
1Dep . o Ma hema ics, Uni e si y o Lleida
A . Alcalde Ro i a Rou e, 191. 25198 Lleida, Spain
{mca dona,colome }@ma ema ica.udl.es
2Resea ch G oup on Na u al Compu ing
Dep . o Compu e Science and A ificial In elligence, Uni e si y o Se illa
A da. Reina Me cedes s/n, 41012 Se illa, Spain
[email p o ec ed]
3Dep . o Animal P oduc ion, Uni e si y o Lleida
A . Alcalde Ro i a Rou e, 191. 25198 Lleida, Spain
[email p o ec ed]
4Bea ded Vul u e S udy & P o ec ion G oup
Adpo. 43 E-25520 El Pon de Sue (Lleida), Spain
[email p o ec ed]
Abs ac . The Bea ded Vul u e (Gypae us ba ba us) is an endange ed
species in Eu ope ha eeds almos exclusi ely on bone emains o wild
and domes ic ungula es. In his pape , we p esen a model o an ecosys-
em ela ed o he Bea ded Vul u e in he Py enees (NE Spain), by
using P sys ems. The e olu ion o six species is s udied: he Bea ded
Vul u e and fi e sub amilies o domes ic and wild ungula es upon which
he ul u e eeds. P sys ems p o ide a high le el compu a ional model-
ing amewo k which in eg a es he s uc u al and dynamic aspec s o
ecosys ems in a comp ehensi e and ele an way. P sys ems explici ly
ep esen he disc e e cha ac e o he componen s o an ecosys em by
using ew i ing ules on mul ise s o objec s which ep esen indi iduals
o he popula ion and bones. The inhe en s ochas ici y and unce ain y
in ecosys ems is cap u ed by using p obabilis ic s a egies. In o de o
expe imen ally alida e he P sys em designed, we ha e cons uc ed a
simula o ha allows us o analyze he e olu ion o he ecosys em unde
diffe en ini ial condi ions.
1 In oduc ion
Animal species a e in e connec ed in a ne wo k in which some species depend
on o he s in e ms o eeding [10], [26]. Va ia ions in biomass affec he compo-
si ion o he popula ion s uc u es [24]. In moun ain ecosys ems, he p esence
o domes ic animals has dis up ed he adi ional ela ionships be ween wild
ungula es and hei p eda o s[6]. Animals loca ed a he op o he ecological
py amid a e suscep ible o he p esence and numbe o hese domes ic animals.
The abandonmen o dead animals in he moun ains is a majo sou ce o ood o
nec ophagous species [15]. This is he case o he Bea ded Vul u e (Gypae us
ba ba us), a h ea ened species which eeds on bone emains o domes ic and
wild ungula es.
The s udy o popula ion ecology and how species in e ac wi h he en i on-
men [13] is one aspec o conse a ion biology o g ea in e es o manage s and
conse a ionis s [2]. A widesp ead ool used in his a ea is he ecological model,
which uses ma hema ical ep esen a ions o ecological p ocesses [21].
In his s udy, we design a model ha s udies he e olu ion o an ecosys em
loca ed in he Py enees, aking ad an age o he capaci y he P sys ems o wo k
in pa allel. P sys ems p o ide a high le el compu a ional modeling amewo k
which in eg a es he s uc u al and dynamic aspec s o ecosys ems in a comp e-
hensi e and ele an way. P sys ems explici ly ep esen he disc e e cha ac e o
he componen s o an ecosys em by using ew i ing ules on mul ise s o objec s
which ep esen indi iduals o he popula ion and biomass a ailable. The inhe -
en s ochas ici y and unce ain y in ecosys ems is cap u ed by using p obabilis ic
s a egies. The ecosys em included six species: Bea ded Vul u e as a sca enge
(p eda o ) species and he Py enean Chamois (Rupicap a py enaica), Red Dee
(Ce us elaphus), Fallow Dee (Dama dama), Roe Dee (Cap eolus cap eolus)
and Sheep (O is cap a) as ca ion (p ey) species. In o de o expe imen ally
alida e o he P sys em designed we ha e cons uc ed a simula o ha allows
us o analyze he e olu ion o he ecosys em unde diffe en ini ial condi ions.
The Bea ded Vul u e is an endange ed species and so he e a e many p ojec s
ha s udy i s beha io and how i is affec ed by i s en i onmen . Thanks o
hese s udies he e is a la ge amoun o in o ma ion a ailable which is equi ed
o define he P sys em and o alida e he esul s ob ained.
The pape is s uc u ed as ollows. In he nex sec ion, basic concep s o he
ecosys em o be modeled a e in oduced. The mos ou s anding aspec s o each
species a e de ailed as well as he in e ac ions among hem. In Sec ion 3, a
dynamic p obabilis ic P sys em o desc ibe he ecosys em is p esen ed. In o de
o s udy he dynamics o he ecosys em, a simula o o ha p obabilis ic P
sys em is designed in Sec ion 4. The ollowing sec ion is de o ed o he analysis
o he esul s p oduced by he simula o . Finally, conclusions a e p esen ed in
he las sec ion.
2 Modeling he Ecosys em
The ecosys em o be modeled is loca ed in he Ca alan Py enees, in he No h-
eas o Spain. This a ea con ains a o al o 35 b eeding e i o ies ha cons i-
u es 34.3% o he Bea ded Vul u e’s Spanish popula ion in 2007 (n = 102). See
Figu e 1 [15].
The ecosys em o be modeled is composed o six species: he Bea ded Vul u e
(p eda o species) and he Py enean Chamois, Red Dee , Fallow Dee , Roe Dee ,
and Sheep (p ey species). P ey species belong o he bo id amily, hey a e
he bi o es and hei bone emains o m he basic sou ce o nou ishmen o he
Bea ded Vul u e in he Py enees.
Fig. 1. Regional dis ibu ion o he Bea ded Vul u e in he Ca alan Py enees
The Bea ded Vul u e is a cliff-nes ing and e i o ial la ge sca enge dis ib-
u ed in moun ains anges in Eu asia and A ica. This is one o he a es ap o s
in Eu ope (150 b eeding pai s in 2007). This species has a mean li espan in wild
bi ds o 21.4 yea s [4]. The mean age o fi s b eeding is 8.1 yea s, whe eas he
mean age o fi s success ul b eeding was 11.4 yea s [1]. Egg-laying akes place
om Decembe o Feb ua y and a e 52-54 days o incuba ion and a e abou
120 days o chick- ea ing, he chick abandons he nes be ween June and Augus
[19]. Clu ch size in his species is usually wo eggs, bu only one chick su i es
as a consequence o sibling agg ession [18]. The emale’s annual e ili y a e in
Ca alonia du ing he las fi e yea s is es ima ed a ound 38%.
The Bea ded Vul u e is he only e eb a e ha eeds almos exclusi ely on
bone emains. I s main ood sou ce is bone emains o dead small and medium-
sized animals. In he Py enees, he emains o Py enean Chamois, Red Dee ,
Fallow Dee , Roe Dee , and Sheep o m 67% o he ul u e’s ood esou ces,
and he emaining 33% includes he emains o small sized mammals (e.g., dogs,
ca s), la ge mammals (cows, ho ses), medium sized mammals (e.g., wild boa s)
and bi ds [15]. A pai o Bea ded Vul u es needs an a e age o 341 Kg o bones
pe yea [17],[16].
Du ing he dispe sal pe iod ( om fledgling un il he bi ds become e i o ial
a 6 o 7 yea s), non-adul Bea ded Vul u es co e la ge dis ances su eying
diffe en a eas. Fo example, he a e age su ace co e ed by ou young ul u es
moni o ed a e fledging was 4932 km2( ange 950-10294 km2, [23]). B eeding
bi ds a e e i o ial and he app oxima e home anges ob ained o eigh pai s
s udied a ied be ween 250 km2and 650 km2. The a e age annual g ow h in
he popula ion o he Bea ded Vul u es in he Py enees has been es ima ed a
4-5%. The floa ing popula ion p incipally emains in eeding s a ions si ua ed in
he cen al Py enees (A agon).
The na u al beha io o he fi e bo id species is simila because hey a e all
he bi o es and hey all each adul size a one yea o age. In gene al, hey each
sexual ma u i y wi hin wo yea s o bi h. Py enean Chamois and he Red Dee
ha e a longe li e expec ancy han Fallow Dee and Roe Dee ( o a e iew o
popula ion pa ame e s see [7], [8], [3], [9] and [20]). The na u al mo ali y a es
a e simila in all fi e species in he fi s yea o li e i is calcula ed o be 50%
and 6% du ing hei emaining yea s. In spi e o he g ea deg ee o simila i y
be ween hese fi e species, impo an diffe ences exis . Fo example, some a e
na u ally occu ing while o he s ha e been in oduced by human popula ions0.
I is essen ial o bea hese diffe ences in mind while defining a P sys em ha
can simula e he ecosys em in a eliable way.
Red Dee a e app ecia ed by hun e s, no o hei mea bu as a ophy and
so only he males a e hun ed. This causes he na u al e olu ion o he popula ion
o be modified. The hun e only akes he head as a ophy lea ing he animal’s
body in he field. Hence he ca cass is ea en by o he species and he bone
emains may hen be ea en by he Bea ded Vul u e.
Fallow Dee and Roe Dee li e in a eas ha a e difficul o each and o his
eason, he Bea ded Vul u e canno ake ad an age o he emains o all o he
dead animals o hese species.
As sheep [25] a e domes ic animals, humans exe a high le el o con ol o e
hei popula ions. The size and g ow h o he sheep popula ion is limi ed by he
owne s o he flocks. The na u al a e age li e expec ancy o sheep is longe han
hei ac ual li e expec ancy in he field because upon a dec ease in e ili y a e
a he age o eigh , hey a e emo ed om he habi a . Mos o he lambs a e
sold o ma ke and so hey a e emo ed om he habi a in he fi s yea o li e.
Only 20% o 30% o he lambs, mos ly emales, a e le in he field and hese a e
used o eplace sheep ha ha e died na u ally and hose olde sheep ha ha e
been emo ed om he flock. The numbe o animals in he Ca alan Py enees
du ing he yea s 1994 and 2008 is shown in Table 1 (see Appendix).
In his s udy, he eeding o he Bea ded Vul u e is dependen on he e olu ion
o he P sys em. Howe e he P sys em does no conside he ac ha he
a ailabili y o ood limi s he eeding o he he bi o es, and so he g ow h o
ege a ion is no modeled.
Taking all o his backg ound in o ma ion in o conside a ion, he ollowing
da a was equi ed o each species:
–I1: Age a which adul size is eached. Age a which he animal consumes
as much as an adul . A his age he animal will ha e su passed he c i ical
ea ly phase du ing which mo ali y a e is high;
–I2: Age a which i begins o be e ile;
–I3: Age a which i s ops being e ile;
–I4: A e age li e expec ancy;
–I5: Fe ili y a io (numbe o descendan s by 100 e ile emales);
–I6: Mo ali y a io in fi s yea s, age < I1( his quan i y is exp essed in e ms
o pe cen age);
–I7: Mo ali y a io in adul animals, age ≥I1( his quan i y is exp essed in
e ms o pe cen age);
–I8: Ra io o emales in he popula ion ( his quan i y is exp essed in e ms o
pe cen age).
The equi ed in o ma ion abou each species is shown in Table 2 (see he
Appendix).
When an animal dies, he weigh o he bones ha i lea es behind is a ound
20% o i s o al weigh . Table 3 (see Appendix) shows he a e age weigh o each
animal as well as he weigh o bones le behind. In he case o Fallow Dee and
Roe Dee , he alue o he weigh o bones is hen mul iplied by 0,2 (20%) which
is he po ion o bones om which he Bea ded Vul u e may benefi .
In he P sys em only Bea ded Vul u es olde han 8 a e conside ed, because
younge ones a e floa ing bi ds. The e a e se en eeding s a ions in Ca alonia
which p o ide a ound 10500 kg o bone emains annually. These a ificial eeding
si es ha e no been conside ed in he s udy and mos o he floa ing bi ds eed
a hese si es.
3 A P Sys em Based Model o he Ecosys em
Memb ane compu ing is a b anch o Na u al Compu ing ha was ini ia ed
a he end o 1998 by Gh. P˘aun (by a pape ci cula ed a ha ime on he
web and published in 2000 [22]). Since hen i has ecei ed impo an a en-
ion om he scien ific communi y. De ails can be ound on he web page
h p://ppage.psys ems.eu/
In sho , one abs ac s compu ing models om he s uc u e and he unc-
ioning o li ing cells, as well as om he o ganiza ion o cells in issues, o gans,
and o he highe o de s uc u es. The main componen s o such a model a e
a cell-like memb ane s uc u e,in hecompa men s o which one places mul i-
se s o symbol-objec s which e ol e in a synch onous maximally pa allel manne
acco ding o gi en e olu ion ules, also associa ed wi h he memb anes.
The seman ic o he P sys ems is defined as ollows: a configu a ion o a P
sys em consis s o a memb ane s uc u e and a amily o mul ise s o objec s
associa ed wi h each egion o he s uc u e. A he beginning, he e is a config-
u a ion called he ini ial configu a ion o he sys em.
In each ime uni we can ans o m a gi en configu a ion o ano he one
by applying he e olu ion ules o he objec s placed inside he egions o he
configu a ions, in a non–de e minis ic, and maximally pa allel manne ( he ules
a e chosen in a non–de e minis ic way, and in each egion all objec s ha can
e ol e mus do so). In his way, we ob ain ansi ions om one configu a ion o
he sys em o he nex .
Acompu a ion o he sys em is a (fini e o infini e) sequence o configu a ions
such ha each is ob ained om he p e ious by a ansi ion, and shows how he
sys em is e ol ing. A compu a ion ha eaches a configu a ion in which no mo e
ules can be applied o he exis ing objec s is called a hal ing compu a ion.The
esul o a hal ing compu a ion is usually encoded by he mul ise associa ed
wi h a specific ou pu memb ane (o he en i onmen ) in he final configu a ion.
In his sec ion, we p esen a model o he ecosys em desc ibed in Sec ion 2 by
means o p obabilis ic P sys ems. We will s udy he beha io o his ecosys em
unde di e se ini ial condi ions.
Fi s , we define he P sys ems based amewo k (p obabilis ic P sys ems),
whe e addi ional ea u es such as wo elec ical cha ges which desc ibe specific
p ope ies in a be e way, a e used.

Defini ion 1. A p obabilis ic P sys em o deg ee nis a uple
Π=(Γ,μ,w1,...,w
n,R,{c } ∈R),
whe e:
–Γis he alphabe (fini e and nonemp y) o objec s ( he wo king alphabe );
–μis a memb ane s uc u e, consis ing o nmemb anes, labeled 1,2,...,n.
The skin memb ane is labeled by 1. We also associa e elec ical cha ges wi h
memb anes om he se {0,+}, neu al and posi i e;
–w1,...,w
na e s ings o e Γ, desc ibing he mul ise s o objec s ini ially
placed in he n egions o μ;
–Ris a fini e se o e olu ion ules. An e olu ion ule associa ed wi h he
memb ane labeled by iis o he o m :u[ ]i
c
−→ u[ ]i,whe eu, , u, 
a e a mul ise o e Γand c isa ealnumbe be ween0and1associa ed
wi h he ule.
We assume ha a global clock exis s, ma king he ime o he whole sys em
( o all compa men s o he sys em); ha is, all memb anes and he applica ion
o all ules a e synch onized.
The n- uple o mul ise s o objec s p esen a any momen in he n egions
o he sys em cons i u es he configu a ion o he sys em a ha momen . The
uple (w1,...,w
n) is he ini ial configu a ion o he sys em.
The P sys em can pass om one configu a ion o ano he by using he ules
om Ras ollows: a each ansi ion s ep, he ules o be applied a e selec ed
acco ding o he p obabili ies assigned o hem, and all applicable ules a e
simul aneously applied and all occu ences o he le –hand side o he ules a e
consumed, as usual.
3.1 The Model
Ou model consis s o he ollowing p obabilis ic P sys em o deg ee 2 wi h wo
elec ical cha ges (neu al and posi i e):
Π=(Γ,μ, w1,w
2,R,{c } ∈R),
whe e:
–In he alphabe Γ, we ep esen he six species o he ecosys em (index iis
associa ed wi h he species and index jis associa ed wi h hei age, and he
symbols X,Yand Z ep esen he same animal bu in diffe en s a es); i
also con ains he auxilia y symbol B, which ep esen s 0.5kgo bones,and
C, which allows a change in he pola iza ion o he memb ane labeled by 2
a a specific s age.
Γ={Xij,Y
ij,Z
ij :1≤i≤7,0≤j≤ki,4}∪{B, C}
–In he memb ane s uc u e, we conside wo egions, μ=[[]
2]1(neu al
pola iza ion will be omi ed):
• he skin egion whe e he objec s ha ep esen animals e ol e acco ding
o he ules o ep oduc ion and mo ali y.
•an inne memb ane whe e he objec s associa ed wi h animals e ol e
acco ding o he eeding ules.
–In w1and w2, we speci y he ini ial numbe o objec s p esen in each egion
(encoding he ini ial popula ion and he ini ial ood);
•w1={Xqij
ij |1≤i≤7,0≤j≤ki,4}, whe e he mul iplici y qij
indica es he numbe o animals, o species iwhose age is j ha a e
ini ially p esen in he ecosys em;
•w2={CB
α},whe eαis defined as ollows:
α=
21

j=1
q1j·1.10 ·682
Value α ep esen s an ex e nal con ibu ion o ood which is added du -
ing he fi s yea o s udy so ha he Bea ded Vul u e su i es. In he
o mula, q1j ep esen s he numbe o jyea s o age o Bea ded Vul-
u es, he finali y o cons an ac o 1.10 is o gua an ee enough ood o
10% popula ion g ow h. A p esen , he popula ion g ow h is es ima ed
an a e age 4%, bu his alue can each highe alues. Thus, o a oid
p oblems ela ed wi h he unde es ima ion o his alue he fi s yea we
es ima ed he popula ion g ow h (o e es ima ed) a 10%. The cons an
alue 682 ep esen s he amoun o ood needed pe yea o a Bea ded
Vul u e pai o su i e.
–The se Ro e olu ion ules consis s o :
•Rep oduc ion- ules.
Adul males:
∗ 0≡[Xij
(1−ki,13)·(1−ki,15)
−−−→ Yij]1,1≤i≤7,k
i,2≤j≤ki,4.
Adul emales ha ep oduce:
∗ 1≡[Xij
ki,5·ki,13·(1−ki,15)
−−−→ YijYi0]1,1≤i≤7,k
i,2≤j<k
i,3.
Fe ile adul emales ha do no ep oduce:
∗ 2≡[Xij
(1−ki,5)·ki,13·(1−ki,15)
−−−→ Yij]1,1≤i≤7,k
i,2≤j<k
i,3.
No e ile adul emales:
∗ 3≡[Xij
ki,13·(1−ki,15)
−−−→ Yij ]1,1≤i≤7,k
i,3≤j≤ki,4.
Young animals ha do no ep oduce:
∗ 4≡[Xij
1−ki,15
−−−→ Yij ]1,1≤i≤7,0≤j<k
i,2.
•G ow h ules.
∗ 5≡[Xij
(ki,6+ki,10)·ki,15
−−−→ Yiki,2Yij]1,1≤i≤7,k
i,2≤j<k
i,4.
∗ 6≡[Xij
(1−ki,6−ki,10)·ki,15
−−−→ Yij ]1,1≤i≤7,k
i,2≤j<k
i,4.
∗ 7≡[Xij
ki,6·ki,15
−−−→ Yiki,2Yij]1,1≤i≤7,j=ki,4.
∗ 8≡[Xij
(1−ki,6)·ki,15
−−−→ Yij ]1,1≤i≤7,j=ki,4.
•Mo ali y ules.
- Young animals
Those which su i e:
∗ 9≡Yij[]
2
1−ki,7−ki,8
−−−→ [Zij]2,1≤i≤7,0≤j<k
i,1.
Those which die:
∗ 10 ≡Yij[]
2
ki,8
−−−→[Bki,11 ]2,1≤i≤7,0≤j<k
i,1.
Those which a e e i ed om he ecosys em:
∗ 11 ≡[Yij
ki,7
−−−→ λ]1,1≤i≤7,0≤j<k
i,1.
- Adul animals ha don’ a i e a an a e age li e expec ancy .
Those which su i e:
∗ 12 ≡Yij[]
2
1−ki,10
−−−→[Zij]2,1≤i≤7,k
i,1≤j<k
i,4.
Those which die:
∗ 13 ≡Yij[]
2
ki,10
−−−→[Bki,12 ]2,1≤i≤7,k
i,1≤j<k
i,4.
- Animals ha a i e a an a e age li e expec ancy:
Those which g ow h popula ion depend on he e ili y a io and die in
he ecosys em:
∗ 14 ≡Yij[]
2
(1−ki,15)·(ki,9+(1−ki,9)·ki,10)
−−−→ [Bki,12 ]2,1≤i≤7,j=ki,4.
Those which g ow h popula ion depend on he e ili y a io and a e
e i ed o he ecosys em:
∗ 15 ≡[Yij
(1−ki,15)·(1−ki,9)·(1−ki,10)
−−−→ λ]1,1≤i≤7,j=ki,4.
Those which g ow h popula ion no depend on he e ili y a io:
∗ 16 ≡Yij[]
2
ki,15
−−−→[Ziki,2]2,1≤i≤7,j=ki,4.
•Feeding ules.
∗ 17 ≡[ZijBki,14 ]2→Xij+1[]
+
2,1≤i≤7,0≤j≤ki,4.
•Balance ules. The p opose o his ules is o make a balance a he end
o he yea . I is o say he le o e ood no se ed o he nex yea , so
i is necessa y elimina e, and i he amoun o ood no is enough some
animals die.
Elimina ion o emaining bones:
∗ 18 ≡[B]+
2→[]
2.
Adul animals ha die because hey ha e no enough ood:
∗ 19 ≡[Zij]+
2→[Bki,12 ]2,1≤i≤7,k
i,1≤j≤ki,4
Young animals ha die because hey ha e no enough ood:
∗ 20 ≡[Zij]+
2→[Bki,11 ]2,1≤i≤7,j<k
i,1
I he ood is equal o he necessa y he objec C allow o change he
pola iza ion.
∗ 21 ≡[C]+
2→[C]2.
The cons an s associa ed wi h he ules ha e he ollowing meaning:
–ki,1: Age a which adul size is eached. This is he age a which he animal
consumes ood as an adul does, and a which, i he animal dies, he amoun
o biomass i lea es behind is simila o he o al le by an adul . Mo eo e ,
a his age i will ha e su passed he c i ical ea ly phase du ing which he
mo ali y a e is high.
–ki,2: Age a which i begins o be e ile.
–ki,3: Age a which i s ops being e ile.
–ki,4: A e age li e expec ancy in he ecosys em.
–ki,5: Fe ili y a io (numbe o descendan s by e ile emales).
–ki,6: Popula ion g ow h ( his quan i y is exp essed in e ms o 1).
–ki,7: Animals e i ed om he ecosys em in he fi s yea s, age < ki,1( his
quan i y is exp essed in e ms o 1).
–ki,8: Na u al mo ali y a io in fi s yea s, age < ki,1( his quan i y is ex-
p essed in e ms o 1).
–ki,9:0 i he li e animals a e e i ed a age ki,4, in o he cases, he alue is 1.
–ki,10: Mo ali y a io in adul animals, age ≥ki,1( his quan i y is exp essed
in e ms o 1).
–ki,11: Amoun o bones om young animals, age < ki,1.
–ki,12: Amoun o bones om adul animals, age ≥ki,1.
–ki,13: P opo ion o emales in he popula ion ( his quan i y is exp essed in
e ms o 1).
–ki,14: Amoun o ood necessa y pe yea and b eeding pai (1 uni is equal
o0.5kgo bones).
–ki,15: Equal o 0 when he species expe ience na u al g ow h (animals ha
emain in he same e i o y h oughou hei li es) and is equal o 1 when
animals a e nomadic ( he Bea ded Vul u e mo es om one place o ano he
un il i is 6–7 yea s old, a which poin i emains in one loca ion).
Values o each species a e shown in Table 4 (see Appendix). Mos alues in
ha able a e equal o hose in Table 2 (see Appendix), bu i is necessa y o
Mo ali y by lack o ood
Elimina ion o le o e ood
Feeding
Na u al / hun e
mo ali y
Rep oduc ion-
G ow h
1
2
0
C
B
X
ij
0
1
2
0
C
B
Y
ij
Y
i0
0
1
2
0
C B
Z
ij
0
1
2
+
C B
Z
1j
0
Rep oduc ion-
G ow h
X
ij
Fig. 2. S uc u e o he P sys em unning
In o de o ob ain a model which allows us o s udy he e olu ion o an
ecosys em in he long– e m, i is necessa y o ake in o accoun ce ain biological
ac o s such as he ollowing:
–Maximum popula ion densi y o each species.
–A ailable eeding in he a ea in which he ungula es may eed.
–Amoun o ood ea en daily by each o he ungula e species conside ing
hei age.
Mo eo e , unde adequa e en i onmen al condi ions, he species exhibi s a
ce ain beha io such ha some alues o he biological pa ame e s can be ac-
cep ed. When essen ial en i onmen al condi ions such as empe a u e and ain-
all a e no ideal, biological cons an s change as a eac ion o he en i onmen .
A model based on Ma ko chains in o de o model empe a u e and ain all
can be accep ed. P sys ems modeling Ma ko chains we e p e iously p esen ed
in [5] and hey should be conside ed in o de o imp o e some esul s.
In u u e wo k, we will also y o model in e ac ions be ween neighbo ing
ecosys ems.
Acknowledgemen s. M.J. P´e ez–Jim´enez acknowledges he suppo o he
p ojec TIN2006-13425 o he Minis e io de Educaci´on y Ciencia o Spain, co–
financed by FEDER unds, and o he P ojec o Excellence TIC 581 o he Jun a
de Andalucia.
Financial suppo o A. Ma galida was ob ained om he Depa amen de
Medi Ambien i Habi a ge o Gene ali a de Ca alunya.
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Appendix
Table 1. Numbe o animals p esen s in he Ca alan Py enees be ween 1994–1998
Species 1994 2008
Bea ded Vul u e pai s 20 37
Py enean Chamois 9000 12000
Red dee 1000 5500
Fallow dee 600 1500
Roe dee 1000 10000
Sheep 15000 200000
Table 2. Na u al cons an s used in he model
Species I1I2I3I4I5I6I7I8
Bea ded Vul u e 1 8 20 21 38 612 50
Py enean Chamois 1 2 18 18 75 60 655
Red Dee 1 2 17 17-20 75 34 650
Fallow Dee 1 2 12 12 55 50 675
Roe Dee 1 1 10 10 100 58 667
Sheep 1 2 8 8 75 15 396
Table 3. Desc ip i e a iables used o model he ecosys em
Species Weigh Weigh Pe cen age A e age Biomass: Biomass: Kg accessible
Male Female Female weigh bone bone by B. Vul u e
adul young
kg kg kg kg kg (adul /young)
Bea ded
Vul u e 56.5 50 5.75 - - -
Py enean
Chamois 28 32 50 30 6 3 6/3
Red Dee
Female -75 -75 15 7.5 15/7.5
Red Dee
Male 120 - - 120 24 12 24/12
Fallow Dee 63 42 80 46 94.5 2/1
Roe Dee 27 23 66 24 52.5 1/0.5
Sheep 42 35 97 35.2 73.5 7/3.5
Table 4. Cons an s used in he P sys em based model
Species i ki,1ki,2ki,3ki,4ki,5ki,6ki,7ki,8ki,9ki,10 ki,11 ki,12 ki,13 ki,14 ki,15
Bea ded
Vul u e 1 1 8 20 21 0.38 0.04 00.06 10.12 0 0 0.50 460 1
Py enean
Chamois 2 1 2 18 18 0.75 - 0 0.60 10.06 612 0.55 - 0
Red Dee
Female 3 1 2 17 17 0.75 - 0 0.34 10.06 15 30 1.00 - 0
Red Dee
Male 4 1 2 - 20 - - 0 0.34 10.36 24 48 0 - 0
Fallow Dee 5 1 2 12 12 0.55 - 0 0.50 10.06 2 4 0.75 - 0
Roe Dee 6 1 1 10 10 1.00 - 0 0.58 10.06 1 2 0.67 - 0
Sheep 7 1 2 8 8 0.75 -0.57 0.15 00.03 714 0.96 - 0
Table 5. Es ima ion o numbe o animals pe age in 1994
Age Bea ded V. Chamois Red d. emale Red d. male Fallow dee Roe dee Sheep
1 0 741 167 58 83 121 20832
2 0 740 133 44 73 121 20208
3 0 668 107 35 69 121 19601
4 0 667 85 28 63 121 19014
5 0 667 68 23 59 109 18443
6 0 596 41 14 55 108 17890
7 0 594 33 11 51 108 17353
8 2 518 26 947 96 16659
9 2 517 21 735 96 0
10 2444 17 533 0 0
11 2444 13 530 0 0
12 2444 11 4 0 0 0
13 2373 9 3 0 0 0
14 1373 7 2 0 0 0
15 1372 5 2 0 0 0
16 1296 4 1 0 0 0
17 1296 3 1 0 0 0
18 1252 0 0 0 0 0
19 1 0 0 0 0 0 0
20 1 0 0 0 0 0 0
21 1 0 0 0 0 0 0
Table 6. Es ima ion o numbe o animals pe age in 2008
Age Bea ded V. Chamois Red d. emale Red d. male Fallow dee Roe dee Sheep
1 0 988 978 254 125 1210 27776
2 0 987 780 192 110 1207 26944
3 0 890 625 154 103 1207 26135
4 0 889 500 124 95 1207 25352
5 0 889 400 99 89 1085 24591
6 0 795 240 60 83 1083 23854
7 0 792 195 48 77 1083 23137
8 6 690 155 38 71 959 22212
9 6 689 123 30 52 959 0
10 6592 97 24 50 0 0
11 6592 78 20 45 0 0
12 5592 62 16 0 0 0
13 5497 50 12 0 0 0
14 5497 40 10 0 0 0
15 5496 32 8 0 0 0
16 5395 25 6 0 0 0
17 5394 20 5 0 0 0
18 5336 0 0 0 0 0
19 5 0 0 0 0 0 0
20 5 0 0 0 0 0 0
21 5 0 0 0 0 0 0
Table 7. Numbe o animals p oduced by he simula o
Yea Bea ded Py enean Py enean Red Fallow Roe Sheep
Vul u e Chamois Chamois Dee Dee Dee
1994 20 9000 1000 600 1000 150000
1995 21 9541 1115 667 1213 152074
1996 21 10023 1263 710 1371 153951
1997 22 10590 1432 758 1568 156183
1998 23 11121 1617 808 1812 158571
1999 24 11718 1834 859 2106 161318
2000 25 12366 2087 908 2469 164391
2001 27 13032 2368 967 2906 167914
2002 28 13767 2705 1032 3459 171940
2003 29 14597 3067 1111 4132 174713
2004 31 15488 10000 3470 1202 4969 177973
2005 33 16468 10594 3917 1297 5883 181300
2006 35 17508 11133 4437 1399 6974 184790
2007 36 18647 11709 5004 1495 8272 188357
2008 38 19866 12297 5631 1602 9774 192097