P System Based Model of an Ecosystem of the Scavenger Birds

P System Based Model of an Ecosystem of the Scavenger Birds M´onica Cardona1, M. Angels Colomer1, Antoni Margalida4, Ignacio P´erez-Hurtado2, Mario J. P´erez-Jim´enez2, Delf´ı Sanuy3, 1Dpt. of Mathematics, University of Lleida Av. Alcalde Rovira Roure, 191. 25198 Lleida, Spain {mcardona,colomer}@matematica.udl.es 2Research Group on Natural Computing Dpt. of Computer Science and Artificial Intelligence, University of Sevilla Avda. Reina Mercedes s/n, 41012 Sevilla, Spain [email protected] 3Dpt. of Animal Production, University of Lleida Av. Alcalde Rovira Roure, 191. 25198 Lleida, Spain [email protected] 4Bearded Vulture Study & Protection Group Adpo. 43 E-25520 El Pont de Suert (Lleida), Spain [email protected] Summary. The Bearded Vulture (Gypaetus Barbatus) is an endangered species in Europe that feeds almost exclusively on bone remains provided by wild and domestic ungulates. In [1], we presented a P system in order to study the evolution of these species in the Pyrenees (NE Spain). Here, we present a new model that overcomes some limitations of the previous work incorporating other scavenger species (predatory) and additional prey species that provide food for the scavenger intraguild and interact with the Bearded Vulture in the ecosystem. After the validation, the new model can be a useful tool for the study of the evolution and management of the ecosystem. P systems provide a high level computational modelling framework which integrates the structural and dynamical aspects of ecosystems in a compressive and relevant way. The inherent stochasticity and uncertainty in ecosystems is captured by using probabilistic strategies. 1 Introduction Since nature is very complex, the perfect model that explain it will be complex too. A complex model is not practical or good to use, so we should obtain a simple model that keeps the most important natural factors and consequently will be useful. 66 M. Cardona et al. The P system presented in [1] gives good results in order to study the evolution of the ecosystem based on the Bearded Vulture in the Catalan Pyrenees, but it does not take into account important factors such as the population density or the feeding limitations. In the Catalan Pyrenees, in the North-east of Spain, three vulture species inhabits sharing the geographic space and the existent food resources so in this work, we present a P system for modelling an ecosystem based on three vulture species and the prey species present from which scavengers obtain their food. The present model improves the results presented in [1]. Apart from adding two new predator species (the Egyptian Vulture Neophron percnopterus and Eurasian Griffon Vulture Gyps fulvus), we introduce new prey species (making a total of 12 species) in the new model that provide feeding resources for the scavenger community. Besides, new rules are introduced to limit the maximum amount of animals that can be supported by the ecosystem as well as the amount of grass available for the herbivorous species. For a good and efficient management of the ecosystems, it is necessary to know the quantity and the evolution of the biomass that reaches every species. One of the contributions the P system presents is the evolution of the quantity of annual biomass that is left by every species. The paper is organized as follows: First, an ecosystem, which is located in the Catalan Pyrenees and is to be modelled, is described. Section 3 shows a formal framework to model ecosystems by means of probabilistic P systems, and a P system modelling the above mentioned ecosystem is presented. In Section 4, we discuss the results obtained and we compare the model presented in this paper to the one presented in [1], using a P-lingua simulator [4]. 2 Modelling the Ecosystem The ecosystem to be modelled is located in the Catalan Pyrenees (NE Spain). This area contains a total of 36 breeding territories of the Bearded Vulture, 65 of the Egyptian Vulture and 525 of the Eurasian Griffon Vulture. In addition to the three vultures, the ecosystem to be modelled is also composed of 9 prey species: the Pyrenean Chamois (Rupicapra pyrenaica), the Red Deer (Cervus elaphus), the Fallow Deer (Dama dama), the Roe Deer (Capreolus capreolus), sheep (Ovis aries), the cow (Bos taurus), the horse (Equus caballus), the goat (Capra hircus) and the wild boar (Sus scrofa). Prey species remains constitutes the basic feeding source for the vultures in the area under study. The three vultures are cliff-nesting and long-lived species characterized by their low fecundity [3]. Concerning their diet, the Bearded Vulture is the only vertebrate that feeds almost exclusively on bone remains of medium size ungulates (see reviews in [6], [5]), whereas the Egyptian vulture feeds on dead small animals and the Eurasian Griffon vultures on medium and large sized animals [3]. P System Based Model of an Ecosystem of the Scavenger Birds 67 The Bearded Vulture has a mind lifespan in wild birds of 21 years [2]. The mean age of first breeding is 8 years. The number of decedents for breeding is one chick. The female’s annual fertility rate in Catalonia in the last years is estimated to be around 38%. The Egyptian Vulture and the Griffon Vulture has a mind lifespan in wild birds of 25 years. The mean age of first breeding is 5 years. The number of descendants per breeding is one chick. The female’s annual fertility rate in Catalonia in the last years is estimated to be around 59% for the Egyptian Vulture and 75% for the Griffon Vulture ([3],[5]). It is accepted that there is the same proportion of females than males in the three types of vultures. The ecosystem modelled in [1] is characterized by the presence of the domestic species, sheep. We considered that species stay all year in the mountain and thus, the bones they left when they die is available for the vultures. The real situation is that there are some domestic animals which live permanently in the mountain whereas others only spend the summer season there. This fact is assumed in the present paper. Taking all this background information into consideration, the following data are required for each species: •I1: age at which adult size is reached. Age at which the animal eats like an adult animal does. Moreover, at this age it will have surpassed the critical early phase during which the mortality rate is high; •I2: age at which it starts to be fertile; •I3: age at which it stops being fertile; •I4: average life expectancy; •I5: fertility ratio (number of descendants by 100 fertile female); •I6: mortality ratio in the first years, age < I1(per cent); •I7: mortality ratio in adult animals, age ≥I1(per cent); •I8: ratio of females in the population (per cent). •I9: amount of bones from young animals when they die (kg) •I10: amount of bones from adult animals when they die (kg) •I11: amount of meet from young animals when they die (kg) •I12: amount of meet from adult animals when they die (kg) •I13: amount of bones necessary per year and animal (kg) •I14: amount of meet necessary per year and adult animal (kg) •I15: amount of grass necessary per year and adult animal (kg) The required information about each species is shown in Table 4 (see Appendix). 68 M. Cardona et al. 3 A formal framework: A P System Based Model of the Ecosystem In this section, we present a model of the ecosystem described in Section 2 by means of probabilistic P systems. We will study the behaviour of this ecosystem under different initial conditions. First, we define the P systems based framework (probabilistic P systems), where additional features such as electrical charges which describe specific properties in a better way, are used. Definition 1. A probabilistic P system of degree qis a tuple Π= (Γ, µ, M1,...,Mq, R, {cr}r∈R) where: •Γis the alphabet (finite and nonempty) of objects (the working alphabet); •µis a membrane structure (a rooted tree), consisting of qmembranes, labelled 1,2,...,q. The skin membrane is labelled by 1. We also associate electrical charges with membranes from the set {0,+,−}, neutral and positive; • M1,...,Mqare strings over Γ, describing the multisets of objects initially placed in the nregions of µ; •Ris a finite set of evolution rules. An evolution rule associated with the membrane labelled by iis of the form r:u[v]α i cr −→ u′[v′]α′ i where u, v, u′, v′are multisets over Γ,α, α′∈ {0,+,−}, and cris a real number between 0 and 1. Besides, for each u, v ∈M(Γ),i∈ {1,2, . . . , q}and α∈ {0,+,−}, it must verify Pt j=1 crj= 1, being r1,...,rtthe rules whose lefthand side is u[v]α i. We denote by [ v→v′]α ithe rule u[v]α i→u′[v′]α′ iin the case u=u′=λ, and α=α′. We assume that a global clock exists, marking the time for the whole system (for all compartments of the system); that is, all membranes and the application of all the rules are synchronized. The multisets of objects present at any moment in the nregions of the system constitutes the configuration of the system at that moment. Particularly, tuple (M1,...,Mq) is the initial configuration of the system. The P system can pass from one configuration to another one by using the rules from Ras follows: •A rule u[v]α i cr −→ u′[v′]α′ iis applicable (with a probability cr) to a membrane labelled by i, and with αas electrical charge, when the multiset uis contained P System Based Model of an Ecosystem of the Scavenger Birds 69 in the father of membrane i, and the multiset vis contained in membrane i. When rule u[v]α i cr −→ u′[v′]α′ iis applied, multiset u(resp. v) in the father of membrane i(resp. membrane i) is removed of that membrane and multiset u′ (resp. v’) is produced in that membrane. •The rules are applied in a maximal consistent parallelism, that is, all those rules of type u1[v1]α i→u′ 1[v′ 1]α′ iand u2[v2]α i→u′ 2[v′ 2]α′ imight be applied simultaneously in a maximal way. •The constants crassociated with the rules indicate the affinity of the above mentioned rule for its application. 3.1 The model The model proposed consists in the following probabilistic P system of degree 2 with two electrical charges (neutral and positive): Π= (Γ, µ, M1,M2, R, {cr}r∈R) where: •Γ={Xij, Yij, Vij , Zij : 1 ≤i≤n, 0≤j≤gi,7} ∪ {b0i, bi: 1 ≤i≤n} ∪ {B, G, M, B′, G′, M′, C, C′} ∪ {Hi, H′ i, Fi, F ′ i, Ti, di, ai: 1 ≤i≤n} is the working alphabet. In our model, n= 17 represents the different types of animals (according to the management) of the 12 species which compose the ecosystem under study. Symbols X,Y,Vand Zrepresent the same animal but in different states. Index iis associated with the species and index jis associated with their age. It also contains the auxiliary symbols B, B′, which represent 0.5 kg of bones, M, M′, which represent 0.5 kg of meet and G, G′, which represent 0.5 kg of grass. Objects Hi, H′ irepresent 0.5 kg of biomass of bones, and objects Fi, F ′ irepresent 0.5 kg of biomass meet left by specie iin different states. Tiis an object that is used for counting the existing animals of species i. If a species overcomes the maximum density, values will be regulated. At the moment when a regulation takes place, object aiallows us to eliminate the number of animals of species ithat exceed the maximum density. Object di is used to put under control domestic animals that are withdrawn from the ecosystem for their marketing. •µ= [ [ ]2]1is the membrane structure. We represent two regions, the skin and an inner membrane. The first is important to check the densities of every species do not overcome the threshold of the ecosystem. Animals reproduce, feed and die in the inner membrane. For the sake of simplicity, neutral polarization will be omitted. • M1and M2are strings over Γ, describing the multisets of objects initially placed in the regions of µ(encoding the initial population and the initial food); 70 M. Cardona et al. –M1={b0i, Xqij ij : 1 ≤i≤n, 0≤j≤gi,7}, where the multiplicity qij indicates the number of animals of species iwhose age is jthat are initially present in the ecosystem; –M2={C}, •The set Rof evolution rules consists of: r0≡[C→G′βC′]0 2. r1≡[b0,i →bi]0 1. – Reproduction-rules. ·Adult males: r2≡[Xij (1−ki,1)·(1−ki,4) −−−→ Yij]0 1,1≤i≤n, gi,5≤j≤gi,7. ·Adult females that reproduce: r3≡[Xij ki,2·ki,1·(1−ki,4) −−−→ YijYki,3 i0]0 1,1≤i≤n, i 6= 5, gi,5≤j < gi,6. r4≡[X5j 0.5·k5,1 −−−→ Y5jYki,3 50 ]0 1, g5,5≤j < g5,6. r5≡[X5j 0.5·k5,1 −−−→ Y5jYki,3 60 ]0 1, g5,5≤j < g5,6. ·Adult females that do not reproduce: r6≡[Xij (1−ki,2)·ki,1·(1−ki,4) −−−→ Yij]0 1,1≤i≤n, gi,5≤j < gi,6. r7≡[Xij ki,1·(1−ki,4) −−−→ Yij]0 1,1≤i≤n, gi,6≤j≤gi,7. ·Young animals that do not reproduce: r8≡[Xij 1−ki,4 −−−→ Yij]0 1,1≤i≤n, 1≤j < gi,5. – Growth rules. r9≡[Xij ki,5·ki,4 −−−→ Yi(gi,5−1)Yij]0 1,1≤i≤n, gi,5≤j≤gi,7. r10 ≡[Xij (1−ki,5)·ki,4 −−−→ Yij]0 1,1≤i≤n, gi,5≤j≤gi,7. – Mortality rules. r11 ≡bi[ ]0 2→[biagi,8 i]+ 2: 1 ≤i≤n. ·Young animals that survive: r12 ≡Yij[ ]0 2 1−mi,1−mi,3 −−−→ [VijTi]+ 2: 1 ≤i≤n, 0≤j < gi,4. ·Young animals that die: r13 ≡Yij[ ]0 2 mi,1 −−−→[H′fi,1·gi,3 iF′fi,2·gi,3 iB′fi,1·gi,3M′fi,2·gi,3]+ 2: 1 ≤i≤ n, 0≤j < gi,4. P System Based Model of an Ecosystem of the Scavenger Birds 71 ·Young animals that are retired from the ecosystem: r14 ≡[Yij mi,3 −−−→ λ]0 1: 1 ≤i≤n, 0≤j < gi,4. ·Adult animals that do not reach an average life expectancy and survive: r15 ≡Yij[ ]0 2 1−mi,2 −−−→[VijTi]+ 2: 1 ≤i≤n, gi,4≤j < gi,7. ·Adult animals that do not reach an average life expectancy and die: r16 ≡Yij[ ]0 2 mi,2 −−−→[H′fi,3·gi,3 iF′fi,4·gi,3 iB′fi,3·gi,3M′fi,4·gi,3Vki,4 i,gi,5−1Tki,4 i]+ 2: 1≤i≤n, gi,4≤j < gi,7. ·Animals that reach an average life expectancy and die in the ecosystem: r17 ≡Yigi,7[ ]0 2 c17 −−−→[H′fi,3·gi,3 iF′fi,4·gi,3 iB′fi,3·gi,3M′fi,4·gi,3Vki,4 i,gi,5−1Tki,4 i]+ 2: 1≤i≤n, being c17 =ki,4+ (1 −ki,4)·(mi,4+ (1 −mi,4)·mi,2). ·Animals that reach an average life expectancy and retire from the ecosystem: r18 ≡[Yigi,7 (1−ki,4)·(1−mi,4)·(1−mi,2) −−−→ λ]1: 1 ≤i≤n. – Regulation rules. r19 ≡[G′→G]+ 2. r20 ≡[B′→B]+ 2. r21 ≡[M′→M]+ 2. r22 ≡[C′→C]+ 2. r23 ≡[H′ i→Hi]+ 2: 1 ≤i≤n. r24 ≡[F′ i→Fi]+ 2: 1 ≤i≤n. – Evaluation of the density of the different species in the ecosystem r25 ≡[Tgi,8 iagi,8−gi,9 i→λ]+ 2: 1 ≤i≤n. r26 ≡[Vij →Zij]+ 2: 1 ≤i≤n, 0≤j < gi,7. – Feeding rules. r27 ≡[ZijaiBfi,5·gi,3Gfi,6·gi,3Mfi,7·gi,3]+ 2→Xi(j+1)[ ]0 2: 1 ≤i≤n, 0≤ j≤gi,7. – Balance rules. The purpose of these rules is to make a balance at the end of the year. That is, the leftover food is not useful for the next year, so it is necessary to eliminate it. But if the amount of food not is enough, some animals die. ·Elimination of the remaining bones, meet and grass: r28 ≡[B→λ]0 2. 72 M. Cardona et al. r29 ≡[G→λ]0 2. r30 ≡[M→λ]0 2. r31 ≡[Ti→λ]0 2: 1 ≤i≤n. r32 ≡[ai→λ]0 2: 1 ≤i≤n. r33 ≡[bi]0 2→bi[ ]0 2: 1 ≤i≤n. r34 ≡[Hi]0 2→Hi[ ]0 2: 1 ≤i≤n. r35 ≡[Fi]0 2→Fi[ ]0 2: 1 ≤i≤n. ·Young animals mortality: r36 ≡[Zij gi,1 −−−→ H′fi,1 iF′fi,2 iB′fi,1M′fi,2]0 2: 1 ≤i≤n, 0≤j < gi,4. r37 ≡[Zij]0 2 1−gi,1 −−−→ di[ ]0 2: 1 ≤i≤n, 0≤j < gi,4. ·Adult animals mortality: r38 ≡[Zij gi,1 −−−→ H′fi,3 iF′fi,4 iB′fi,3M′fi,4]0 2: 1 ≤i≤n, gi,4≤j≤gi,7. r39 ≡[Zij 1−gi,1 −−−→ λ]0 2: 1 ≤i≤n, gi,4≤j≤gi,7. r40 ≡[Hi→λ]0 1: 1 ≤i≤n. r41 ≡[Fi→λ]0 1: 1 ≤i≤n. The constants associated with the rules have the following meaning: •gi,1: 1 for wild animals and 0 for domestic animals. •gi,2: 1 for carnivorous animals and 0 otherwise. •gi,3: proportion of time that they remain in the mountain during the year. •gi,4: age at which adult size is reached. This is the age at which the animal eats like and adult does, and at which if the animal dies, the amount of biomass it leaves is similar to the total one left by an adult. Moreover, at this age it will have surpassed the critical early phase during which the mortality rate is high. •gi,5: age at which it starts to be fertile. •gi,6: age at which it stops being fertile. •gi,7: average life expectancy in the ecosystem. •gi,8: maximum density of the ecosystem. •gi,9: number of animals that survive after reaching maximum density of the ecosystem. •ki,1: proportion of females in the population (per one). •ki,2: fertility ratio (proportion of fertile female that reproduce). •ki,3: number of descendants per each fertile female that reproduce. P System Based Model of an Ecosystem of the Scavenger Birds 73 •ki,4: it is equal to 0 when the species go through a natural growth (animals which remain in the same territory throughout their lives) and it is equal to 1 when animals are nomadic (the Bearded Vulture moves from one place to another until it is 6–7 years old, when it settles down). •ki,5: population growth (per one). •mi,1: natural mortality ratio in first years, age < gi,4(per one). •mi,2: mortality ratio in adult animals, age ≥gi,4(per one). •mi,3: percentage of domestic animals withdrawn in the first ages of the not stabilized populations. •mi,4: is equal to 1 if the animal dies at the age of gi,7and it is not retired, and it is equal to 0 if the animal not dies at the age of gi,7but it is retired from the ecosystem. •fi,1: amount of bones from young animals when they die, age < gi,4. •fi,2: amount of meet from young animals when they die, age < gi,4. •fi,3: amount of bones from adult animals when they die, age ≥gi,4. •fi,4: amount of meet from adult animals when they die, age ≥gi,4. •fi,5: amount of bones necessary per year and animal (1 unit is equal 0.5 kg of bones). •fi,6: amount of grass necessary per year and animal. •fi,7: amount of meet necessary per year and animal. The constants used in this work are the same than those of [1], except for those referring to the maximum density of the population, the meet or the grass. However, they have been renamed in this work in order to be able to group them according to their characteristics. Thus, general characteristics are now named with g, those of reproduction with k, those corresponding to mortality with mand those of feeding with f. See Appendix in table 5. 3.2 Structure of the P system running The model presented in [1], shows some restrictions which produce undesired effects for the study of the ecosystem evolution. Thus, in the first version of the model, it was not born in mind the resources for the feeding of herbivorous species, that is, it was assumed that there was enough grass at the ecosystem for all the population. Similarly, the maximum density of certain species in some areas of the ecosystem was not taken into consideration. It has been experimentally proved that, when the number of animals of a species exceeds a threshold, a phenomenon of autoregulation of the population takes place. In this paper, we present a new model of the ecosystem that includes new ingredients with the aim to overcome the limitations previously described. More specifically, the modifications made are the following: 80 M. Cardona et al. Species i gi,1gi,2gi,3gi,4gi,5gi,6gi,7gi,8gi,9ki,1ki,2ki,3ki,4ki,5mi,1mi,2mi,3mi,4fi,1fi,2fi,3fi,4fi,5fi,6fi,7 Bearded Vulture 1 1 0 1 1 8 20 21 120 120 0.5 0.08 1 1 0.04 0.06 0.12 0 1 0 0 0 0 460 0 0 Egyptian Vulture 2 1 0 0.5 1 5 24 25 160 160 0.5 0.593 1 0 0 0.17 0.07 0 1 0 0 0 0 0 0 332 Griffon Vulture 3 1 0 1 1 5 24 25 1400 1400 0.5 0.75 1 0 0 0.03 0.01 0 1 0 0 0 0 0 0 800 P. chamois 4 1 0 1 1 2 18 18 15000 7500 0.55 0.75 1 0 0 0.6 0.06 0 1 6 8 12 48 0 550 0 Red deer female 5 1 0 1 1 2 17 17 4615 3230 1 0.75 1 0 0 0.34 0.06 0 1 15 26 30 120 0 2540 0 Red deer male 6 1 0 1 1 2 20 20 2885 2020 0 0 0 0 0 0.34 0.36 0 1 24 30 48 192 0 2540 0 Fallow deer 7 1 0 1 1 2 12 12 3000 2400 0.75 0.55 1 0 0 0.5 0.06 0 1 2 28 4 74 0 1100 0 Roe deer 8 1 0 1 1 1 10 10 15000 7500 0.67 1 1 0 0 0.58 0.06 0 1 1 8 2 38 0 600 0 Wild Board 9 1 0 1 1 1 4 11 200000 200000 0.5 0.035 3 0 0 0.705 0.035 0 0 8 12 24 120 0 730 0 Sheep A 10 0 0 1 1 2 8 8 200000 200000 0.96 0.75 1 0 0 0.15 0.030 0.59 0 7 8 14 56 0 1320 0 Sheep P 11 0 0 0.5 1 2 8 8 50000 50000 0.96 0.75 1 0 0 0.15 0.030 0.59 0 7 8 14 56 0 1320 0 Bovine A 12 0 0 1 2 2 9 14 168500 168500 0.9 0.9 1 0 0 0.057 0.045 0 0 21 119 12 1038 0 11000 0 Bovine P 13 0 0 0.5 2 2 9 14 168500 168500 0.9 0.9 1 0 0 0.057 0.045 0 0 21 119 12 1038 0 11000 0 Goat A 14 0 0 1 1 2 8 6 17000 17000 0.97 0.9 1 0 0 0.12 0.015 0.59 0 7 8 19 75 0 1400 0 Goat P 15 0 0 0.5 1 2 8 6 17000 17000 0.97 0.9 1 0 0 0.12 0.015 0.59 0 7 8 19 75 0 1400 0 Horse A 16 0 0 1 3 3 9 20 6600 6600 0.97 0.9 1 0 0 0.034 0.014 0 0 21 119 18 1782 0 12000 0 Horses P 17 0 0 0.5 3 3 9 20 6600 6600 0.97 0.9 1 0 0 0.034 0.014 0 0 21 119 18 1782 0 12000 0 Fig. 5. Constants used in the P system based model. Acronyms (A) and (P) mean annual and periodical respectively