A bio-inspired computing model as a new tool for modeling ecosystems: The avian scavengers as a case study

A bio-inspi ed compu ing model as a new ool o modeling ecosys ems:The a ian sca enge s as a case s udy M. Àngels Colome a, An oni Ma galida b, Del í Sanuy c, Ma io J. Pé ez-Jiménez d a Depa men o Ma hema ics, Uni e si y o Lleida, A . Alcalde Ro i a Rou e, 191. 25198 Lleida, Spain b Bea ded Vul u e S udy and P o ec ion G oup, Apdo. 43, 25520 El Pon de Sue Lleida, Spain c Depa men o Animal P oduc ion, Uni e si y o Lleida, A . Alcalde Ro i a Rou e, 191. 25198 Lleida, Spain d Resea ch G oup on Na u al Compu ing, Dp . 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 Keywo ds: P sys ems Ecosys em A ian sca enge s Conse a ion abs ac The models used o ecosys ems modeling a e gene ally based on di e en ial equa ions. Howe e , in ecen yea snewcompu a ionalmodelsbasedonbiologicalp ocesses,o bioinspi edmodels,ha ea isen, amongwhicha e Psys ems.These a einspi edby he unc ionso cellsandp esen impo an ad an ages wi h espec o adi ional models, such as a high compu a ional e ficiency, modula i y and hei abili y o wo k in pa allel. They a e simple, indi idual-based models ha use biological pa ame e s ha can be ob ained expe imen ally. In his wo k, we p esen he amewo k o a model based on P sys ems applied o he s udy o an ecosys em in which h ee a ian sca enge s (p eda o s) in e ac wi h 10 wild and domes ic ungula es (p eys). The compu a ion ime o 100 epe i ions, co esponding o 14 simula ion yea s each, wi h an ini ial popula ion composed o 385,422 indi iduals, was 30min. Ou esul s sugges ha he model p esen ed, based on P sys ems, co ec ly simula es he popula ion dynamics in he pe iod o ime analyzed. We discuss he use ulness o his ool in simula ing complex ecosys ems dynamics o aid manage s, conse a ionis s and policy-make s in making app op ia e decisions o he imp o emen o managemen and conse a ion p og ams. 1. In oduc ion Ma hema ical models desc ibing p eda o –p ey ela ionships a e used o s udy dynamics be ween wo popula ions when one o hem depends on he o he o ood and su i al. The ela ionships among species, pa icula ly e eb a es, in asymme ical in aguild p eda o sys ems a e complex, and gene aliza ions emain elusi e (Li ai is and Villa ue e, 1996), bu ha e impo an implica ions o conse a ion biology (Polis and Hol , 1992). Rega ding he use o managemen and conse a ion measu es o endange ed species, i is pa icula ly impo an o ha e an a ailable model ha allows us o eliably p edic he dynamics o such popula ions, especially o he mos endange ed species (e.g., Me e sky e al., 2000; O ega e al., 2009; O o e al., 2008; Chap on e al., 2009; G ande e al., 2009). This ool can be use ul in decision-making and in op imizing managemen o he gi en ecosys em. The e a e many p ocesses in an ecosys em ha un in pa al- lel and a e in e ela ed, hence i is essen ial o be able o o ganize hese in e ela ions in a schema ic and g aphic way. Food webs a e cha ac e ized by many weak in e ac ions and a ew s ong in e ac ions, which appea o p omo e communi y pe sis ence and s abili y (Bascomp e e al., 2005). In his sense, quan ifica ion o he s eng h o in e ac ions be ween species is essen ial o unde - s anding how ecological communi ies a e o ganized and how hey espond o human exploi a ion. Gene ally, he ools used o ecosys ems modeling a e based on di e en ial equa ions. Fo example, he Lo ka-Vol e a model has been one o he mos equen ly used o he modeling o wo species, p eda o and p ey (Mu ay, 2002), and Ve huls ’s model (logis ic di e en ial equa ions) has been u ilized as a undamen al g ow h model in ecological s udies because o i s ma hema ical simplici y and single biological defini ion (Sakanoue, 2007, 2009). Fuzzy se heo y has been used o es ima e he pa ame e s o he models based on di e en ial equa ions s udying he in e ac ion be ween a p ey and i s p eda o (Da Sil a e al., 2008). Despi e he sa is ac o y esul s p esen ed by hese models, some au ho s ques- ion he de e minis ic and con inuous app oach hey imply. Russell e al. (2009), o example, de eloped a s ochas ic model o h ee species by using o dina y logis ic di e en ial equa ions analyzed h ough nume ical analyses echniques. Viabili y models do no conside op imal solu ions bu a he define all possible e olu ions o a dynamic sys em unde gi en cons ain s (Mullon e al., 2004). Viabili y heo y conside s ha he e olu ion o a sys em is non-de e minis ic bu belongs o a se o possibili ies, depending on i s s a e. Due o he ma he- ma ical complexi y, he implemen a ion equi es a sys em o low dimensionali y (<4 in e ac ing componen s), which limi s i s e ec- i eness gi en he complexi y o ecological models. Recen ly, some mo ecomplexmodelsha ea isen(i.e., akingin oaccoun ag ea e numbe o pa ame e s and a iables), ha a e di ficul o cali- b a e and alida e. Mo eo e , hei applica ion equi es di ision in o smalle sizes defined a bi a ily and app oxima ing he global model (e.g., Ful on e al., 2003, 2004). In his sense, Law ie and Hea ne (2008) p opose a non-a bi a y algo i hm o he di ision o hese la ge models. Memb ane Compu ing is an eme gen b anch o Na u al Com- pu ing (P˘ aun, 1998; Ciobanu e al., 2006; P˘ aun e al., 2010) ha was in oduced wi h he pu pose o defining compu ing de ices, called P sys ems, which abs ac om he s uc u e and he unc ion o he li ing cells. Ra he han being an al e na i e o mo e classical modeling amewo ks, such as ODEs (O dina y Di e en ial Equa- ions), P sys ems cons i u e a complemen a y app oach o be used when he classical modeling app oaches ail. The mos impo an p ope y o hese models is hei capaci y o wo k in pa allel and o cap u e he andomness o he na u al en i onmen al p ocesses using s ochas ic s a egies based on Gillespie’s heo y o s ochas ic kine ics (Gillespie, 1976, 1977) and he seman ics defined by using p obabilis ic unc ions (Ca dona e al., 2009, 2010b). In p e ious wo ks we es ed he u ili y o his new amewo k o manage sand conse a ionis s byapplying hese modelson a com- muni y o sca enge s (Ca dona e al., 2009, 2010a) and he Zeb a mussel D eissena polymo pha (Ca dona e al., 2010b). He e, aking a sca enge communi y ha depends on ca ion p o ided by wild and domes ic ungula es as a model, we in oduce a new model based on P sys ems gene alizing he p e ious models and enabling hesimul aneousanalysiso somein e ela ed ophicchains.Fi s , in e ac ionsamongspeciesa eshown by means o Ne wo ks. P sys- ems associa e a ule o each in e ac ion obse ed in he ne wo k quan i ying he exis en in e ac ion. In o de o e alua e i s obus - ness, he model is checked and alida ed by using he expe imen al in o ma ion om he ecosys em, which co esponds o a pe iod o 14 yea s, aking h ee a ian sca enge s (as p eda o species) and en ungula e species (as p eys) as case s udies. We discuss he esul s ob ained om a me hodological poin - o - iew and he use ulness o his ool in simula ing complex ecosys emsdynamics o aidmanage s,conse a ionis sandpolicy- make s in making app op ia e decisions o he imp o emen o managemen and conse a ion p og ams. 2. Ma e ial and me hods 2.1. Ecosys em o be modeled The s udy was ca ied ou in he Py enean and P epy enean moun ainso Ca alonia (NESpain,Fig. 1). Theecosys em o be mod- eledis composedo 13species: h eea iansca enge s( he Bea ded ul u e Gypae us ba ba us, he Egyp ian ul u e Neoph on pe c- nop e us and he G i on ul u e Gyps ul us) as p eda o species, and six wild ungula es ( he Py enean chamois Rupicap a py enaica, he Red dee Ce us elaphus, he Fallow dee Dama dama, he Roe dee Cap eolus cap eolus, he Wild boa Sus sc o a and he Mou- flon O is o ien alis) and ou domes ic ungula es ha a e ound in an ex ensi e o semi-ex ensi e egime ( he sheep O is a ies, he goa Cap a hi cus, he cow Bos au us and he ho se Equus caballus) p o iding ca ion o he a ian sca enge s and conside ed as p ey species. P ey species a e he bi o es and hei emains o m he p i- ma y ood esou ce o he a ian sca enge sin he s udy a ea (>80% o he die is based on hese species, see Donáza , 1993; Ma galida e al., 2009). Fig. 1. S udy a ea and subpopula ions conside ed in he ecosys em. Py enees (da k g ey); P epy enees (ligh g ey). The s udy a ea is cha ac e ized by he p esence o wo di e en- ia ed subpopula ions ha a e in e connec ed. The Py enees egion (57,244km2) is cha ac e ized by he p esence o abundan wild ungula es h oughou heyea ,as ongp esenceo domes icungu- la es du ing he summe (as a consequence o anshumance) and low human densi y. In he Py enean egion, he le el o annual ain all anges om 800 o 1200mm and he maximum a e age empe a u es do no exceed 25◦C in he summe and do no all unde −5◦C in hewin e . The o og aphy p esen sal i udinal zones be ween 1000 and 3000m whe e alpine e ain, which is cha - ac e ized by he p esence o meadows abo e 2200m, domina es and subalpine g ound is cha ac e ized by wooded o ma ions o Moun ain pine (Pinus uncina a). Unde 1600m, mon ane e ain is ound whe e wooded o ma ions a e domina ed by Eu opean beech (Fagus syl a ica). The P epy enean egion (77,372km2)is mo e popula ed by humans, wi h a mo e abundan domes ic ungu- la e popula ion ha is egula du ing he win e , wi h low densi ies o wild ungula es. In he P epy enean egion, he le el o annual ain all anges om 800 o 1200mm and maximum a e age em- pe a u es do no exceed 30◦C in he summe o all unde −2◦Cin he win e . The o og aphy p esen s al i udinal zones be ween 600 and 1500m co esponding o submon ane g ound, which is dom- ina ed in i s lowe a ea by Po uguese oak (Que cus aginea) and Holm oak (Q. ilex sp. ballo a) woods, and mon ane g ound which is domina edbyDownyoak(Q. pubescens). Occasionally, some moun- ain anges a e ound in he mon ane egion eaching 2000m. In ega ds o he dis ibu ion o he sca enge species, he bea ded ul u e popula ion in he Py enees s. P epy enees is 37.8% s. 62.2% (n=37), he Egyp ian ul u e 10.2% s. 89.8% (n= 59) and he G i on ul u e21.5% s. 78.5%(n=822).Sinceindi iduals can mo e om one a ea o ano he acco ding o he esou ces a ailable, he ecosys em would unc ion as a single se (global ecosys em) com- posedo wosubse s(subpopula ionssepa a edbybiogeog aphical c i e ia). In his way, whene e he e is a lack o ophic esou ces in one o he suba eas, he indi iduals can mo e o he o he one. In bo h, he ecosys em load capaci y has been limi ed o he app o- p ia e a eas and habi a s o he di e en species as well as he maximum densi y ha can be eached (see Appendix A). The h ee sca enge species a e cli -nes ing and only he Egyp- ian ul u e is mig a o y ( hei p esence in he s udy a ea is limi ed Elemen a y memb ane Memb anes Regions Skin En i onmen En i onmen Fig. 2. Rep esen a ion o a memb ane s uc u e. o Ma ch–Augus ). Conce ning hei ophic ecology, he die o he Bea ded ul u e is based p incipally on bone emains o wild and domes ic ungula es (p incipally sheep and Py enean chamois), al hough du ing he chick- ea ing pe iod small animals a e impo - an o he ene ge ic equi emen s o he chicks (Ma galida e al., 2005, 2009). The o aging a eas a e abou 20km a ound he nes (A.M. unpubl. da a). The G i on ul u e eeds mainly on wild and domes ic ungula es and mea emains p o ided by sheep, pigs, cows and ho ses (Donáza , 1993). G azing a eas ha e a adius o app oxima ely 25km a ound he hill. Finally, he Egyp ian ul u e is a mo e oppo unis ic species ha ing a mo e he e ogeneous die which is based mainly on small co pses o mammals and bi ds. I is mo e dependen on ubbish dumps and supplemen a y eeding a eas, and g azing a eas a e limi ed o adii unde 10km om he nes (Donáza , 1993). Conce ning p ey species, domes ic ungula es a e p esen in he Py enean subpopula ion du ing he summe (May–Oc obe ) as a consequence o anshuman mo emen s and become ood esou ces o a ian sca enge s du ing his pe iod (see Olea and Ma eo-Tomás, 2009) whe eas in he P epy enean egion, hey a e p esen h oughou he yea in an ex ensi e o semi-ex ensi e egime. Wild ungula es a e p esen in he a ea all yea wi h he densi ies o Py enenan chamois, Red dee and Mouflon being mo e impo an in he Py enean egion and Wild boa mo e impo an in he P epy enean egion (see Appendix A). 2.2. A P sys em based model o ecosys ems In his sec ion, we will p esen he seman ics and syn ax used o model ecosys ems by means o P sys ems and he specific model p oposed o he ecosys em o he a ian sca enge s. 2.2.1. P sys ems Thes a ingpoin o hisnewmodelo compu a ionis heobse - a ion ha he cell is he smalles li ing hing as well as a iny machine wi h a complex s uc u e, and he assump ion ha he p ocesses aking place in he compa men al s uc u e o a li ing cell can be in e p e ed as compu a ions. The challenge is o ake he cell i sel as a suppo o compu a ions, and o find hose ele- men s use ul o compu a ions in he s uc u e and he unc ioning o he cell asa whole. Thede iceso his modela e called Psys ems, consis ing o a cell-like memb ane s uc u e, in he compa men s o which one places mul ise s o objec s ha e ol e acco ding o gi en ules. The main componen s o P sys ems a e he memb ane s uc u e, mul ise s o objec s, and e olu ion ules (Fig. 2). •Amemb ane s uc u e consis s o se e al memb anes a anged in a hie a chical s uc u e inside a main memb ane ( he skin), anddelimi ing egions( hespace in-be weenamemb aneand he immedia elyinne memb anes,i any).Eachmemb aneiden ifies a egion inside he sys em. •Regions defined by a memb ane s uc u e con ain objec s co e- sponding o chemical subs ances p esen in he compa men s o a cell. The objec s can be desc ibed by symbols o by s ings o symbols, in such a way ha mul ise s o objec s a e placed in egions o he memb ane s uc u e. •The objec s can e ol e acco ding o gi en e olu ion ules, associ- a ed wi h he egions (and hence, wi h he memb anes). The unc ioning o a P sys em is defined as ollows: •Aconfigu a iono a cell-like memb ane sys emconsis so amem- b 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 configu 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 configu a ion 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, maximally pa allel way ( 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 ge ansi ions om one con- figu a ion o he sys em o he nex . •Acompu a ion o he sys em is a sequence o configu a ions such ha each one is ob ained om he p e ious one by a ansi ion, and shows how he sys em is e ol ing. The app oach has a se ies o ea u es ha o e come se e al d awbacks o classical models based on di e en ial equa ions: modula i y (in insic o a memb ane sys em), scala- bili y/ex ensibili y ( u he memb anes and/o u he e olu ion ules can be added wi hou essen ially changing he way a sys- em wo ks), unde s andabili y (e olu ion ules di ec ly co espond o chemical eac ions o in e ac ions among species), p og amma- bili y (a ew i ing-based model can be easily ans o med in o a p og am, wi h ce ain p og amming languages, such as JAVA, C++, CLIPS), while p ese ing o he desi able ea u es o di e en ial equa ions models, such as non-linea i y o e olu ion. 2.2.2. The o mal model In his sec ion, we define a P sys em based amewo k whe e addi ional ea u es, such as p obabilis ic unc ions and h ee elec- ical cha ges ha be e desc ibe specific p ope ies, a e used. Askele on o an ex ended P sys em wi h ac i e memb anes o deg ee q≥1, ˘=(,,R), can be iewed as a se o (pola ized) memb anes hie a chized by a s uc u e o memb anes (a oo ed ee) labeled by 0, 1, ...,q−1. All memb anes in a e supposed o be (ini ially) neu al and hey ha e associa ed wi h hem R, a fini e se o e olu ion ules o he o m u[ ]˛ i→u[ ]ˇ i ha can modi y hei pola iza ion bu no hei label. is an alphabe ha ep e- sen s he objec s (i.e., Bea ded ul u e, Py enean chamois, e c., see Fig. 3). A p obabilis ic unc ional ex ended P sys em wi h ac i e mem- b anes o deg ee q≥1 aking T ime uni s, ˘=(,,R,T,{ : ∈R}, M0,...,Mq−1), can be iewed as a skele on (,,R) wi h he mem- b anes hie a chized by he s uc u e labeled by 0, 1, ...,q−1. Tis ana u alnumbe ha ep esen s he simula ion imeo hesys em. Fo each ule ∈Rand a,1≤a≤T, (a) is a whole numbe be ween 0 and 1, which ep esen s a p obabilis ic cons an associa ed wi h ule a momen a. In a gene ic way, we deno e :u[ ]˛ i (a) −→ u[ ]˛ i. The uple o mul ise s o objec s p esen a any momen in he q egions o he sys em cons i u es he configu a ion o he sys em a ha momen . The uple (M0,...,Mq−1) is he ini ial configu a ion o ˘. 0 1 2 3 45 6 a c c b a e b0 0 0 0 00 0Cha ge Memb ane label Objec s 0 213 456 Rules: ][[] ][][ ][ ][ 0 1 0 1 2 6 0 6 2 0 3 0 3 1 aa aa a de k→≡ →≡ →≡ + ... d Memb anes hie a chized Fig. 3. A skele on o an ex end P sys em ˘(,,R) o deg ee 7 whe e =[[]1[[]4]2[[]5[]6]3]0and ={a,b,c,d,e, }. The P sys em can pass om one configu a ion o ano he by using he ules om Ras ollows: •A ule u[ ]˛ i (a) −→ u[ ]˛ iis applicable o a memb ane labeled by i, and wi h ˛as elec ical cha ge i mul ise uis con ained in he memb ane immedia ely ou side o memb ane i,i is osay memb ane a he o memb ane i, and mul ise is con ained in he memb ane labeled by iha ing ˛as elec ical cha ge. When ha ule is applied, mul ise u( espec i ely ) in he a he o memb ane i( espec i ely in memb ane i) is emo ed om ha memb ane, and mul ise u( espec i ely ) is p oduced in ha memb ane, changing i s elec ical cha ge o ˛. •M() is he se o med by he mul ise s o .I u, ∈M(),i∈{0,...,q−1},˛∈{0,+,−} and 1,..., z a e he ules applicable whose le -hand side is u[ ]˛ ia gi en momen a, hen i should be e ified ha 1(a)+...+ z(a)=1, and he ules will be applied acco ding o he co esponding p obabili ies 1(a),..., z(a). Amul ien i onmen p obabilis ic unc ional ex ended P sys em wi h ac i e memb anes o deg ee (m,q) aking T ime uni s (˙, G, RE,,,R,T,{ j : ∈R˘,1≤j≤m},M ij : 0≤i≤q−1,1≤j≤m) can be iewed as a se o men i onmen s e1,...,emlinked by he a cs om he di ec ed g aph G. Each en i onmen ejcon ains a p obabilis ic unc ional ex ended P sys em wi h ac i e memb anes o deg ee q,˘j=(,,R,T,{ j : ∈R˘,1≤j≤m},Mij :0≤i≤q−1, 1≤j≤m) each o hem wi h he same skele on, ˘=(,,R), and such ha M0,j,...,Mq−1,jdesc ibes hei ini ial mul ise s. is an alphabe ha ep esen s he objec s o  ha can be p esen in he di e en en i onmen s (Fig. 4). The communica ion ule be ween en i onmen s in REa e o he o m e:(x)ej px,j,k −→ (y)ek, and o each x∈,1≤j≤m,1≤a≤T, i e - ifies m  k=1 px,j,k(a)=1. When a ule o his ype is applied he objec xmo es om en i onmen ej o en i onmen ekcon e ed in o y, acco ding o he p obabili y pj,k. We assume ha a global clock exis s, ma king he ime o he whole sys em ( o i s compa men s), ha is, all memb anes and he applica ion o all ules a e synch onized. In he P sys ems, a configu a ion consis s o mul ise s o objec s p esen in he men i- onmen s and a each o he egions o he P sys ems loca ed in he en i onmen . The P sys em can pass om one configu a ion o ano he by using he ules om R=RE∪∪ m j=1R˘jas ollows: a each ansi ion s ep, he ules o be applied a e selec ed acco ding o he p obabil- i 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. 2.2.3. The model Fi s ly, we g aphically p esen he p oblem o be modeled by means o ne wo ks ha a e desc ip o s o ecological sys ems ha can show he composi ion o nume ous elemen s and he in e - ac ions among hem (Bascomp e, 2009). The ne wo k app oach p o ides a powe ul ep esen a ion o he ecological in e ac- e1 e2 e3 e4 e1e2 e3 e4 Associa e g aph, G Fig. 4. Mul ien i onmen p obabilis ic unc ional ex ended P sys em wi h ac i e memb anes o deg ee (4,7) ( ou en i onmen s and se en memb anes). Fig. 5. Ne wo ks o he ene ge ic needs and con ibu ions: (a) Py enean subpopula ion and (b) P epy enean subpopula ion. Nodes ep esen sca enge s’ ene ge ic needs in he o m o mea and he ene ge ic con ibu ions o each ungula e species. The nodes’ nume ic labels co espond o ungula e species: (1) Py enean chamois, (2) Red dee , (3) Fallow dee , (4) Roe dee , (5) Mouflon, (6) Wild boa , (7) sheep, (8) cow, (9) goa and (10) ho se. A link be ween nodes means ha he ungula e o ms pa o he sca enge eeding. The hickness o he link deno es he ene ge ic con ibu ions and needs o all he species as a whole and is exp essed in pe cen ages. ions among species and highligh s hei global in e dependence (Ulanowicz, 2004; Bascomp e, 2009; Miehls e al., 2009). In his sense, he s eng h o he in e ac ion among he h ee species o a ian sca enge s (p eda o s) on he ungula e communi y (p eys) is quan ified o each subpopula ion, and is measu ed as he biomass p o ided by p ey species and he ene ge ic equi emen s o hese h ee a ian sca enge species ha a e exp essed as megacalo ies pe yea acco ding o he popula ion size. We conside he eg- e able biomass on which wild and domes ic ungula es depend as no being a limi ing ac o bu exceeding he needs (ene ge ic equi emen s) o hese species (Ga cía, 2008). Subsequen ly, he model p esen ed associa es a ule quan i ying he in e ac ion o each o he ela ions shown in he ne wo ks (Fig. 5). In he ne - wo ks, he ene ge ic needs in he o m o mea o he h ee species o a ian sca enge s s udied inhabi ing he Py enees (Fig. 5a) and he P epy enees (Fig. 5b), a e shown by means o nodes as is he p opo ion o biomass made up by wild and domes ic ungula es. Themodelp oposedmus conside :(a) hepopula iondynamics o he nine wild species ( h ee a ian sca enge s and six ungu- la es) and ou domes ic species, (b) he in e ac ions among he 13 species,(c) he p esence o wo zones in hes udya ea,(d) he com- munica ion p o ocol be ween he wo a eas and (e) he ecosys em maximum load capaci y o each o he a eas. In o de o model he ecosys em, we use a mul ien i onmen p obabilis ic unc ional ex ended P sys em wi h ac i e memb anes o deg ee (2,2) ( wo memb anes and wo en i onmen s) aking T ime uni s (simula ion yea s). (˙, G, RE,,,R,T{ j : ∈R˘,1≤j≤2},M ij : 0≤i≤1,1≤j≤2) The skele on consis s o he wo king alphabe , , o med by all objec s ha belong o he ini ial configu a ion and he objec s ha appea in he e olu ion o he P sys em (all o hem appea in Appendix A). The memb ane s uc u e is o med by he skin mem- b ane labeled 0 and an in memb ane labeled 1, bo h ha ing neu al cha ge. The se o ules a e shown in he appendix; he e a e 49 ypes o ules. The p obabilis ic unc ional ex ended P sys em ˘=(,,R,T, { : ∈R˘},M0,M1) is defined as ollows: o each , is a cons an unc ion, he ini ial configu a ion is M0=Xqi,j ,d iand M1={R0, F0}. The objec s Xi,j,1 a e associa ed wi h one animal belonging o he species i,jyea s old in he ins ance 1and qi,jis he amoun o objec s Xi,j,1, we use he objec di o a con ol o he maximum load o he animals o species i.F0is used o gene a ing ex e nal con- ibu ions o di e en kinds o ood, and finally, objec R0allows us o synch onize he P sys em. The alphabe ha can be p esen in he en i onmen is ˙= {Zi,j,s,Z i,j,s}and he ela ionships in g aph G a e o each en i on- men o i sel and o o he en i onmen s. The algo i hmic scheme o he model is s uc u ed ollowing a se ies o modules which a e un sequen ially co esponding o he passing o 1 yea in he ecosys em (Fig. 6). Excep o ep oduc ion, all he p ocesses a e con inuous and annual. In he model, p o- cesses a e disc e ized in Ca dona e al. (2009), which e ified ha he o de in which modules a e applied does no a ec he final esul ob ained om he model. Some ungula es (Red dee and all domes ic ungula es) o he ecosys em ha e been classified in o wo di e en g oups acco d- ing o hei managemen . Specifically, in he case o he Red dee , sexes ha e been sepa a ed because mo ali y in males is highe han in emales due o hun ing ac i i ies. Thus, he o al numbe o g oups o animals conside ed is 18. The spa io- empo al dis i- bu ion o domes ic animals has been de e mined bea ing in mind ha some animals spend he en i e yea in he s udy a ea whe eas o he s spend a iable pe iods o ime he e (e.g., summe anshu- mance). Rega ding anshumance, sheep, cows and ho ses a e in moun ain passes o 6 mon hs o he yea , making use o summe g azing ( om mid May o mid Oc obe depending on he clima e; Roigé, 1995). Da a abou anshumance we e aken om he in o - ma ion p o ided by he Depa amen d’Ag icul u a i Ramade ia o he au onomous go e nmen o Ca alonia. The model used is based on objec s ha e ol e by means o a se ies o ules among which hose associa ed wi h he in e - ac ions shown in he Ne wo ks a e ound (see Fig. 5). Objec s Xi,j,y,Y i,j,y,Z i,j,y,Z i,j,y and Wi,j,y ep esen he same animal a i s di e en s ages h oughou he module se ies. Wi hin he objec s, he fi s index codifies he g oup o which he animal belongs, he second one codifies i s age and he hi d one he simula ion yea . The numbe o yea s (T) o be simula ed is an inpu o he model. 2.2.4. Rep oduc ion module A he ini ial ins ance, an objec o ype Xis associa ed wi h each animal. When ules om he ep oduc ion module a e applied o objec s X, hey e ol e o objec s o ype Y. Objec s Xi,j,y associa ed wi h emales ha ep oduce when hey each e ili y also gene a e objec s Yi,0,yassocia ed o newbo n animals. In his module, objec s associa ed wi h he amoun o ood p oduced by he ecosys em i sel (g azing o ex e nal con ibu ions o No Yes (no change en i onmen ) REPRODUCTION MORTALITY FEEDING + DENSITY REGULATION (1) CHANGE ENVIRONMENT UPDATING FEEDING + DENSITY REGULATION (2) 1 s ep 8 s eps (synch onizing) 5 s eps 1 s ep 1 s ep 1 s ep 2 s eps Fig. 6. Modules ha o m he model. mea and bones by man) a e also gene a ed. This module akes one simula ion s ep. B eeding pa ame e s o each species we e ob ained om he li e a u e and unpublished da a (see Donáza , 1993; G ande, 2006; Ma galida e al., 2003; O o e al., 2008; Le Goua e al., 2008, see Table 1,Appendix A). 2.2.5. Mo ali y module Hun ing, in he case o wild animals, and mo ali y due o bo h na u al causes and human managemen , in he case o domes ic animals, p o ide he animal biomass on which a ian sca enge s eed.Thei su i alhasbeenes ima ed by using bibliog aphical e - e ences (Mon se a and Villa , 2007; Blasco e al., 1992; Casasús e al., 1999; Ma galida e al., 2009) as well as by means o pe sonal su eys(au ho sunpubl.da a).Mo eo e ,ape cen ageo deadani- mals ha may be accessible o sca enge s was es ima ed. The inpu o his module is o med by objec s o he ype Yi,j,y, which do no e ol e bu a he mo e o ano he memb ane and gene a e a new objec D o each animal ha does no die. Objec s associa ed wi h animals ha die e ol e o objec s associa ed wi h mea (C,M) and bones (H,B). This module akes one simula ion s ep. 2.2.6. Feeding and densi y egula ion module (1) Annual ene ge ic equi emen s (exp essed as calo ies o megacalo ies) as well as he maximum load capaci y in he a ea unde s udy ha e been es ima ed o all he species. In his sense, he e is no o e exploi a ion o moun ain g azing and ca le ais- ing loads ha e dec eased in ecen yea s such ha he ege able biomass a ailable is no a limi ing ac o (Ga cía, 2008). Rega ding he a ian sca enge communi y, i s popula ion has inc eased in he las 20 yea s and maximum alues ha e been es ima ed acco d- ing o he unoccupied space as well as o he maximum densi y ha each o hese species can achie e (see Table 1,Appendix A). Whe he o no he maximum load capaci y o he ecosys em o eachspecieshasbeen eachedisde e minedbyusingobjec sDp e- iously gene a ed o each su i ing animal. Fu he mo e, objec s Ye ol e o objec s Z o begin he eeding p ocess. In he second s ep o his module, objec s Ze ol e o objec s Wi he e is enough physical space and ood. 2.2.7. Change en i onmen module When one o he suba eas eaches i s maximum load capaci y, i has been conside ed ha any o he species modeled can mo e o ano he suba ea acco ding o i s ecological equi emen s. This module will apply i he e is some objec Z ha has no e ol ed in he p e ious s ep; ha is, i he esou ces ha e been insu ficien o all o he animals. In his case, he model simula es animals’ mo e- men s o find he necessa y esou ces o hei su i al. Objec s Zgo ou o he en i onmen by aking wo simula ion s eps and subsequen ly mo e o ano he en i onmen by e ol ing o objec s Z. Nex , in wo mo e s eps, hey en e he inne memb ane o he P sys em whe e he objec s associa ed wi h he possible unused esou ces a e ound. This module akes fi e simula ion s eps. 2.2.8. Feeding and densi y egula ion module (2) This module will apply i he e a e enough esou ces o animals coming om o he a eas. In such a case, objec s Ze ol e o objec s o ype Win one simula ion s ep. 2.2.9. Upda ing module This module will apply a e eigh simula ion s eps; ha is, a e he unning o he hi d module. A e unning one cycle wi hin he modulese ies, he ini ial configu a ion mus be e-es ablished such ha a new yea (cycle) begins. Objec s associa ed wi h he su i - ing animals, Wi,j,y, e ol e o objec s Xi,j+1,y+1. The emaining ood is emo ed and he objec s associa ed wi h hose animals which ha e no su i ed e ol e o objec s ep esen ing ood. This module akes one simula ion s ep. In summa y, he unning o one cycle in he module succession akes 11 simula ion s eps and ep esen s he passing o a 1-yea pe iodin he ecosys em.A e he unning o he cycle, he Psys em e u ns he numbe o li ing animals o each species as well as he esou ces in he o m o mea ha each has con ibu ed. 3. Resul s Fo execu ion o he model, MeCoSim so wa e ( ee so wa e unde licence), de eloped by membe s o he Na u al Compu a ion G oup a he Uni e si y o Se illa (GNU GPL; h p://www.p- lingua.o g), has been used. The popula ion end o he h ee sca enge species and he six wildungula esob ainedby hemodelwi h espec oda aob ained by di ec censuses om 1994 o 2008 is shown in Figs. 7 and 8. In o de o check and alida e he model, only ini ial (1994) and final (2008) da a we e a ailable o he six wild ungula es whe eas in e -annual censuses we e a ailable o a ian sca enge s. The popula ion end o he species p esen in he ecosys em h ough- ou he pe iod unde s udy has been ob ained by unning he simula o 100 imes o 14 yea s wi h he same inpu da a. The simula o execu ions ha e allowed us o es ima e he popula ion confidence in e als o he di e en species. The ecosys em mod- M.À. Colome e al. / Ecological Modelling 222 (2011) 33–47 39 Table 1 Values o pa ame e s used in he model o each species (F= emale, M= male, A= spend he en i e yea in he moun ain, P= spend pa o he yea in he moun ain). g1g2g3g4g5g6g7k1k2k3m1m2m3m4 1 2 3 4 5 6 7 8 9 Gypae us ba ba us 1 1 1 6 20 21 0 0.65 0.35 1 0.06 0.08 0 1 00001350450 0 Neoph on pe cnop e us 1 0.5 1 5 24 25 1 0.80 0.57 1 0.28 0.08 0 1 0000001000 0 Gyps ul us 1 1 1 5 24 25 0 0.75 0.56 1 0.06 0.07 0 1 0000002300 0 Rupicap a py enaica 1 1 1 2 18 18 0 0.55 0.75 1 0.6 0.06 0 1 346240000.50.5 Ce us elaphus (Female) 1 1 1 2 17 17 0 1 0.75 1 0.34 0.06 0 1 7 13 15 60 0 0 0 0.6 0.6 Ce us elaphus (Male) 1 1 1 2 20 20 0 0 0 0 0.34 0.06 0 1 12 15 24 96 0 0 0 0.6 0.6 Dama dama 1 1 1 2 12 12 0 0.75 0.55 1 0.5 0.06 0 1 1 14 2 37 0 0 0 0.25 0.25 Cap eolus cap eolus 1 1 1 1 10 10 0 0.67 1 1 0.58 0.06 0 1 141190000.25 0.25 O is o ien alis 1 1 1 2 12 12 0 0.5 0.9 2 0.6 0.06 0 1 346220000.60.6 Sus sc o a 11114600.50.55 4 0.14 0.1 0 1 4 6 12 60 0 0 0 0.25 0.25 O is a ies (Adul ) 01128800.96 0.75 1 0.15 0.03 0 0 347380000.70.7 O is a ies (Young) 00.5128800.96 0.75 1 0.15 0.03 0 0 347380000.70.7 Bos au us (Adul ) 012291400.90.910.057 0.045 0 0 10 60 6 518 0 0 0 0.6 0.6 Bos au us (Young) 00.42291400.90.910.057 0.045 0 0 10 60 6 518 0 0 0 0.6 0.6 Cap a hi cus (Adul ) 01128800.97 0.9 1 0.12 0.015 0 0 349370000.60.6 Cap a hi cus (Young) 00.5128800.97 0.9 1 0.12 0.015 0 0 349370000.60.6 Equus caballus (Adul ) 013392000.97 0.9 1 0.034 0.0142 0 0 10 60 9 891 0 0 0 0.8 0.8 Equus caballus (Young) 0 0.55 3392000.97 0.9 1 0.034 0.0142 0 0 10 60 9 891 0 0 0 0.8 0.8 g1: 1 wild animal and 0 domes ic animals. g2: p opo ion o ime ha animals emain in he moun ains du ing he yea . g3: age a which adul size is eached. This is he age a which he animal consumes an adul die , and a which i he animal dies, he amoun o biomass i lea es 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. g4: age a which e ili y begins. g5: age a which e ili y ends. g6: a e age li e expec ancy in he ecosys em. g7: 1 i an impo an p opo ion o he die o he species can be based on o he small species and 0 o he emainde . k1: p opo ion o emales in he popula ion (pe one). k2: e ili y a io (p opo ion o e ile emales ha ep oduce). k3: numbe o descendan s o e ile emales ha ep oduce. m1: na u al mo ali y a io in fi s yea s, age <g4(pe one). m2: mo ali y a io in adul animals, age ≥g4(pe one). m3: pe cen age o domes ic animals emo ed om non-s abilized popula ions a ea ly ages. m4: is equal o 1 i he animal dies a he age o g6and is no emo ed, and is equal o 0 i he animal does no die a he age o g6bu is emo ed om he ecosys em. 1: amoun o bones om young animals, age <g4. 2: amoun o mea om young animals, age <g4. 3: amoun o bones om adul animals, age <g4. 4: amoun o mea om adul animals, age <g4. 5: amoun o bones necessa y pe yea and animal (kg). 6: amoun o g ass necessa y pe yea and animal (kg). 7: amoun o mea necessa y pe yea and animal (kg). 8: Pe cen age o use ul bones. 9: Pe cen age o use ul mea . eled in his wo k is made up o 13 species o aling 18 animal ypes. Themodelshows hepopula ion endob ainedby equallydi iding he e olu ions o each animal. The ini ial popula ion is composed o 385.422 indi iduals and he compu a ion ime (on a pe sonal com- pu e ) o 100 epe i ions, co esponding o 14 simula ion yea s each, was 30min. The compa ison be ween he eal popula ion endency, es i- ma ed by means o censuses ca ied ou , and ha ob ained by he simula o isshown in Fig. 7. Theadjus men o hepopula ion end shown o he h eea iansca enge specieswi h espec o heda a ob ainedby hesimula o shows ha hemodel unc ionsp ope ly. Wi h espec o he esul s ob ained o wild ungula es (Fig. 8), i is obse ed ha in simula ion yea 10, Roe dee eaches i s max- imum load capaci y in he P epy enean a ea (zone 2) and some o he animals mo e o he Py enean a ea, causing in he la e an impo an popula ion inc ease in simula ion yea 11. Thus, he model is able o show he dispe si e capaci y o some species. A simila si ua ion is obse ed o Red dee in simula ion yea 10 and Py enean chamois in simula ion yea 9. No e ha as simula- ions con inue beyond he ini ial yea o simula ion he confidence in e al inc eases. I is a andom model such ha esul s a e sp ead ho izon ally. 4. Discussion The model p esen ed based on P Sys ems co ec ly simula es he popula ion dynamics in he pe iod o ime analyzed. Ou model conside s he popula ion dynamics and he simul aneous in e - ac ion among he 18 animal ypes. In Ca dona e al. (2009) we desc ibed a model based on P sys ems ( he ecosys em modeled is composed by one a ian sca enge species and fi e p ey species) ha conside s nei he he ecosys em maximum load capaci y no he appea ance o densi y-dependen egula o y phenomena in he species. In Ca dona e al. (2010a), a new model was p e- sen ed ha o e came some limi a ions o he p e ious model by widening he numbe o species ( h ee a ian sca enge species and en p ey species) including some specific cha ac e is ics o he species o ming heecosys em. None heless, his model conside ed he ecosys em o be a closed en i y, ha is, when he necessa y esou ces o a species o su i e (space, eeding,...) a e insu fi- cien , he animal dies wi hou conside ing he possibili y ha he animals mo e o some o he ecosys em. In his s udy, we imp o e he model o Ca dona e al. (2010a) conside ing he he e ogene- i y o he landscape and he possibili y o spa ial mo emen s o he species when ood esou ces a e enough o co e he ene ge ic equi emen s o he species. Models based on indi iduals such as ha p esen ed in his s udy a e gene ally mo e flexible and enable he conside a ion o he he - e ogenei y o he popula ion and he en i onmen . Ou model is composed o modules ha a e applied sequen ially. In ou model he ad an ages wi h espec o o he models a e (1) i is no neces- sa y o di ide he p oblem o be analyzed (Ful on e al., 2003, 2004; Law ieandHea ne,2008);(2) henumbe o in e -andin aspecific in e ac ions among indi iduals is no limi ed (Mullon e al., 2008); (3) i is able o cap u e he andomness inhe en o he p ocesses; and (4) in he unc ion o ecosys em dynamics, i is able o upda e he pa ame e s in a unc ional way, o in o he wo ds, o eadjus (Ca dona e al., 2010a). Ano he ad an age in he applica ion o ou model is ha i is easily p og ammable and is as compa ed o he ime o compu a- ion o a de e minis icmodelwi hsimila cha ac e is ics in ol ing 9h in one simula ion yea wi h 240,000 indi iduals (Mo ales e al., 2006). Among hespecializedp og ams ha enable hes udyo species iabili y (e.g., Gapps, Inma , Ramas, Vo ex) he mos widely used Gypae us ba ba us 0 5 10 15 20 25 30 35 40 14131211109876543210 Yea Te i o ies Py enees P epy enees To al Expe imen al da a Neoph on pe cnop e us 0 10 20 30 40 50 60 70 14131211109876543210 Yea Te i o ies Py enees P epy enees To al Expe imen al da a Gyps ul us 0 200 400 600 800 1000 1200 1400 1514131211109876543210 Yea Pai s (n) Py enees P epy enees To al Expe imen al da a Fig. 7. Compa ison o he esul s ob ained by he simula o (lines) wi h he da a ob ained expe imen ally (do s). Unb oken line ep esen s he whole popula ion whe eas b oken lines ep esen he esul s ob ained in bo h subpopula ions, Py e- nees: ci cles; P epy enees: squa es. is Vo ex, which conside s a highe numbe o ac o s. A com- pa a i e s udy showed ha he esul s ob ained om he di e se p og ams we e di e en i he models we e no no malized (B ook e al., 1999). Like he model p esen ed in his pape , Vo ex enables he modeling o di e en popula ions and conside s e ili y a ios, male- emale ela ionships, numbe o descendan s and in e ac- ions among di e en popula ions. None heless, Vo ex conside s he possibili y ha na u al disas e s may ake place whe eas he model p esen ed he e can only conside such a possibili y by including a new module. Finally, Vo ex can simul aneously model di e en popula ions ha in e ac wi h each o he al hough hey mus co espond o he same species. The model used in his wo k enables he s udy o he dynamics o di e en popula ions whe he o no hey a e composed o he same species, e en allowing in e - ac ion and compe i ion among hem. In addi ion, he a ailabili y o ene ge ic esou ces ha a e essen ial o he dynamics o popula- ions ha a e in e ela ed as well as spa io- empo al egula ion a e conside ed. The flexibili y o ou model enables he inc ease in he num- be o species wi hou ha ing o modi y he model bu simply by adding he new in o ma ion on he biological pa ame e s o he new species o be included o he simula o o modi ying he in o - Fig. 8. Resul s p o ided by he simula o . The s a ing poin is he popula ion o e e y species in he yea 1994. Zone 1: Py enees, Zone 2: P epy enees.