A P System Modeling an Ecosystem Related to the Bearded Vulture

A P System Modeling an Ecosystem Related to the Bearded Vulture M´onica Cardona1, M. Angels Colomer1, Mario J. P´erez-Jim´enez2, Delf´ı Sanuy3, Antoni Margalida4 1Dpt. of Mathematics, University of Lleida Av. Alcalde Rovira Roure, 191. 25198 Lleida, Spain E-mail: {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 E-mail: [email protected] 3Dpt. of Animal Production, University of Lleida Av. Alcalde Rovira Roure, 191. 25198 Lleida, Spain E-mail: [email protected] 4Bearded Vulture Study & Proteccion Group Adpo. 43 E-25520 El Pont de Suert, Lleida, Spain E-mail: [email protected] Summary. The Bearded Vulture is one of the rarest raptors in Europe and it is an endangered species. In this paper, we present a model of an ecosystem related with the Bearded Vulture which is located in the Catalan Pyrenees, by using P systems. The population dynamics constituted by the Bearded Vulture (that feeds almost exclusively on bones) and other five subfamilies that provide the bones they feed on, is studied. P systems provide a high level computational modeling framework which integrates the structural and dynamical aspects of ecosystems in a comprehensive and relevant way. P systems explicitly represent the discrete character of the components of an ecosystem by using rewriting rules on multisets of objects which represent individuals of the population and bones. The inherent stochasticity and uncertainty in ecosystems is captured by using probabilistic strategies. In order to give an experimental validation of the P system designed, we have constructed a simulator that allows us to analyze the evolution of the ecosystem with different initial conditions. 1 Introduction An ecosystem is a natural unit consisting of all plants, animals and microorganisms (biotic factors) in an area functioning together with all of the non-living physical (abiotic) factors of the environment. 52 M. Cardona et al. Animal species are interconnected in a food chain in which some species depend on others [6], [16]. The variations of different biomass affect the composition of the population structures [15]. In mountain ecosystems where livestock activities are developed in a traditional way, the relationship among ungulates and their predators has been disrupted by the presence of domestic animals [5]. Animals located at the top of the ecological pyramid are influenced by their presence and number. The abandonment of corpses on mountains is a major source of food for necrophagous animals [8]. The study of population ecology and how the species interact with the environment is one of the aspects of the conservation biology with more and more interest for managers and conservationists [2]. A widespread tool used are the ecological models, based on mathematical representations of ecological processes [13]. Ecosystems are dynamic entities composed of the biological community and the abiotic environment. An ecosystem’s abiotic and biotic composition and structure is determined by the state of a number of interrelated environmental factors. Changes in any of these factors, such as nutrient availability, temperature, light intensity, grazing intensity, and species population density, will result in dynamic changes in the nature of these systems. Scientists have recognized that organisms can be organized according to several different functional levels. The functional level known as species refers to a group of organisms that are similar in morphology and physiology and which have the ability to interbreed. A population is formed by all the different organisms of a single species occupying a specific area on the Earth. A community is defined as all the populations of different species inhabiting a particular region of the Earth. The most complex functional level of organization is the ecosystem. An ecosystem consists of the community and its relationship with the abiotic factors of the environment. Ecosystems are primarily governed by stochastic events, the reactions they provoke on non-living materials and the responses by organisms to the conditions surrounding them. Thus, an ecosystem results from the sum of myriad individual responses of organisms to stimuli from non-living and living elements from the environment. As the number of species in an ecosystem increases, the number of stimuli also does. Ever since life began, organisms have survived continuous change through natural selection of successful feeding, reproductive and dispersal behavior. Ecosystems can be of any size. An ecosystem can be as large as a desert or a lake or as small as a tree or a puddle. A terrarium can be described as an artificial ecosystem. Ecosystem models, or ecological models, are mathematical representations of ecosystems. Given the complexity of the problems to be modeled, simplifications need to be made in order to achieve a good numerical approach. Ecosystem models are a development of theoretical ecology that describes the major dynamics of ecosystems. Ecosystem models aim not only to integrate the understanding of these systems but also to be able to predict their behavior (in general terms, or in response to particular changes). Due to the complexity of ecosystems in terms of A P System Modeling an Ecosystem Related to the Bearded Vulture 53 numbers of species/ecological interactions, ecosystem models tend to simplify the systems they are studying down to a limited number of pragmatic components. These may be particular species of interest, or may be broad functional types such as autotrophs, heterotrophs or saprotrophs. In biogeochemistry, ecosystem models usually include representations of non-living “resources” such as nutrients, which are consumed by living components of the model. The process of simplification reduces an ecosystem to a small number of state variables. Depending upon the system under study, these may represent ecological components in terms of numbers of discrete individuals or quantify the component continuously as a measure of the total biomass of all organisms of that type, often using a common model currency. The components are then linked together by mathematical functions that describe the nature of the relationships between them. For instance, in models which include predator-prey relationships, the two components are usually linked by some function that relates total prey captured to the populations of both predators and prey. Deriving these relationships is often extremely difficult given habitat heterogeneity, the details of component behavioral ecology (including issues such as perception, foraging behavior), and the difficulties involved in unobtrusively studying these relationships under field conditions. Typically, relationships are derived statistically or heuristically. For example, some standard functional forms describing these relationships are linear, quadratic, hyperbolic or sigmoid functions. Besides establishing the components to be modeled and the relationships between them, another important factor in ecosystem model structure is the representation of space used. Historically, models have often ignored the confusing issue of space by using zero-dimensional approaches such as ordinary differential equations. As computing power increases in, models which incorporate space are being increasingly used, for example, based on partial differential equations or cellular automata. The inclusion of the factor of space allows dynamics not present in non-spatial frameworks to be considered, and sheds light on processes that lead to the formation of patterns in ecological systems. One of the earliest and best-known ecological models is the predator-prey model described by Alfred J. Lotka (1925) and Vito Volterra (1926). This model is composed of a pair of ordinary differential equations where one represents a prey species and the other its predator. Volterra originally devised the model to explain fluctuations in fish and shark populations observed in the Adriatic Sea following the First World War when fishing had been curtailed. However, the equations have subsequently been applied more generally. Although simple, they illustrate some of the salient features of ecological models: the considered biological populations grow, interact with other populations (either as predators, prey or competitors), and suffer mortality. The objective of this paper is to design a model that studies the evolution of an ecosystem located in the Pyrenees, taking advantage of the capacity the P Systems to work in parallel. The ecosystem includes six species: the Bearded Vulture (Gypaetus barbatus) as scavenger (predator) species and the Pyrenean Chamois (Rupicapra pyrenaica), Red Deer (Cervus elaphus), Fallow Deer (Dama 54 M. Cardona et al. dama), Roe Deer (Capreolus capreolus) and Sheep (Ovis capra) as carrion (prey) species. The paper is structured as follows. In the next section, basic concepts of the ecosystem to be modeled are introduced. The most outstanding aspects of each species are detailed as well as interactions among them. In Section 3, a probabilistic P system is presented in order to describe the ecosystem. A simulator of that probabilistic P system is designed in Section 4, in order to study the dynamics of the ecosystem. Section 5 is devoted to the analysis of the results produced by the simulator. Finally, conclusions are presented in the last section. 2 Modeling the Ecosystem The ecosystem to be modeled is located in the Catalan Pyrenees, in the Northeast of Spain. This area contains a total of 35 territories that constitutes the 34.3% of the population of the Bearded Vulture Spanish population in 2007 (n = 102). See Figure 1 [8]. CATALONIA Fig. 1. Regional distribution of the Bearded Vulture in the Catalan Pyrenees (NE Spain). The ecosystem is composed of six species: the Bearded Vulture (predator species) and the Pyrenean Chamois, Red Deer, Fallow Deer, Roe Deer, and Sheep (prey species). The last five prey species belong to the bovid family, they are herbivores and their bone remains form the basic source of nourishment for the Bearded Vulture in the Pyrenees. The Bearded Vulture (Gypaetus barbatus) is a cliff-nesting and territorial large scavenger distributed in mountains ranges in Eurasia and Africa. It is one of the rarest raptors in Europe (150 breeding pairs in 2007), it inhabits areas of high altitude (1500-4000 m), though it can be seen in areas of lower altitude (500-800 m) in the winter when high mountains are covered with snow. The mean lifespan of wild Bearded Vultures is 21.4 years [4]. The mean age of first breeding is 8.1 years, A P System Modeling an Ecosystem Related to the Bearded Vulture 55 whereas the mean age of first successful breeding is 11.4 years [1]. Egg-laying take places during December-February and, after 52-54 days of incubation and around 120 days of chick-rearing, the chick abandons the nest between June-August [12]. The clutch size in this species is usually two eggs, but only one chick survives as a consequence of sibling aggression [11]. Bearded Vultures are fertile from the age of eight, when they become adult, and they cease to be fertile at the age of twenty. The female’s annual fertility rate in Catalonia during the last five years is estimated around 38% [8], the female lays two eggs and incubates them for 55-57 days. However, as with most birds of prey, only one of the youngsters fledges. The Bearded Vulture is the only vertebrate animal that feeds almost exclusively on bone remains. Its main food source is bone remains of dead small and mediumsized animals. It searches for food either alone or in pairs. In the Pyrenees the bone remains of Pyrenean Chamois, Red Deer, Fallow Deer, Roe Deer, and Sheep form 67% of the vulture’s food resources of nourishment, and the other 33% includes the bone remains of small size mammals (e.g., dogs, cats), badgers, least weasel, large mammals (cows, horses), medium size mammals (e.g., wild boars) and birds [8]. A pair of Bearded Vultures needs an average 341 Kg of bones per year [10], [9]. During the dispersal period (from fledgling until the bird become territorial at 5-6 years), the non-adult Bearded Vultures birds cover large distances surveying different areas. For example, the averaged surface covered by four young monitored after fledging was 4932 km2(range 950-10294 km2, [14]). They may return to breed in the area where they were born but it is not frequent. The choice of the area where they settle down definitively is based on different parameters such as the availability of food and the vulture population density. Breeding mature birds are territorial and the approximate home ranges obtained for eight pairs studied varied between 250 km2and 650 km2. They almost never leave their territory to settle in neighboring mountainous areas even if these are close by, and this fact makes it more difficult to estimate the growth of its population. Studies by Margalida estimate the average annual growth in the population of Bearded Vultures in the Pyrenees to be 4-53% of the existing population, and that the average floating population is 20 principally remains in feeding stations situated in the central Pyrenees (Aragon) birds. The natural behavior of the five bovid species is similar as they are all herbivores and they all reach the size of the adult animal when they are one year old. In general, they arrive at the sexual maturity within two years from birth. Chamois and the Red Deer have a longer life expectancy than Fallow Deer and Roe Deer. The natural mortality rates are similar in all five species, in the first year of life it is calculated to be 50% and 6% during the remaining years. In spite of the great degree of similarity among these five species, there are differences among them, some are of natural origin and other are induced by human action. It is essential to bear them in mind in order to define a P system that can reliably simulate the ecosystem. 56 M. Cardona et al. Red Deer are very much appreciated by the hunters, not for their meat but as trophies and so only males are hunted, and this causes the natural evolution of the population to be modified. The hunter only takes the head as a trophy leaving the animal’s body on the field, and so the carcass is eaten by other species and the bones may then be eaten by the Bearded Vulture. Fallow Deer and Roe Deer live in areas that are difficult to reach and for this reason the Bearded Vulture cannot take advantage of the bones of all of the dead animals. Studies estimate that the Bearded Vulture only uses 20% of the bones available on the field. As sheep are domestic animals, humans exert a high level of control over the sheep populations. The size and growth of the sheep population is limited by the owners of the flocks. The natural average life expectancy of sheep is longer that their actual life expectancy in the field because when its fertility rate decreases at the age of eight, they are taken out of the habitat. Most of the lambs are sold to market and so they are taken out of the habitat too in the first year of life. Only 20% of the lambs, mostly females, are left on the field and they are used to replace sheep that have died naturally and the old ones that have been removed from the flock. As the Bearded Vulture is an endangered species, there are many projects that study its behavior and the way it is affected by its environment. Thanks to these studies there is available a large amount of information which is required to define the P System and to validate the results obtained. In this study, the feeding of the Bearded Vulture is dependent on the evolution of the P System. However, the P System does not consider that the availability of food limits the feeding of the herbivores, and so the growth of the vegetation is not modeled. Taking all of this background information into consideration, the following data was required for each species: •ki,1: age at which adult size is reached. This is the age at which the animal eats like the adult does, and at which if the animal dies, the amount of biomass it leaves is similar to the ole left by an adult. Moreover, at this age it will have surpassed the critical early phase during which the mortality rate is high. •ki,2: age at which it starts to be fertile. •ki,3: age at which it stops being fertile. •ki,4: average life expectancy. •ki,5: fertile ratio (number of descendants by fertile female). •ki,6: population growth. •ki,7: mortality ratio in first years (age < ki,1) in which biomass in the form of bones is not left on the field. •ki,8: mortality ratio in first years (age < ki,1) in which biomass in the form of bones is left on the field. •ki,9: mortality ratio in adult animals (age ≥ki,11) in which biomass in the form of bones is not left on the field. A P System Modeling an Ecosystem Related to the Bearded Vulture 57 •ki,10: mortality ratio in adults animals (age ≥ki,1) in which biomass in the form of bones is left on the field. •ki,11 is equal to 1 if the animal dies at the age of ki,4leaving biomass, and it is equal to 0 if the animal dies at the age of ki,4without leaving bones. •ki,12: amount of bones from young animals (age < ki,1). •ki,13: amount of bones from adult animals (age ≥ki,1). •ki,14: percentage of females in the population. •ki,15: type of food. •ki,16: amount of food necessary per year and animal (1 unit is equal 0.5 kg of bones). When an animal dies, the weight of bones which it leaves is around 20% of its total weight. Table 1 shows the average weight of each animal as well as the weight of bones they leave. In the case of Fallow Deer and Roe Deer, the value of the weight of bones must then be multiplied by 0,2 (20%) which is the proportion of bones that are accessible for the Bearded Vulture. Table 1. Bones Species Weigh Weigh Percentage Average Biomass: Biomass: Kg accessible Male Female Female weigh bons adult bons young by B.Vulture (adult/young) B.Vulture 5 6.5 60 2 - - - Chamois 28 32 50 30 6 3 6/3 Red Deer Female - 75 - 75 15 7.5 15/7.5 Red Deer Male 120 - - 120 24 12 24/12 Fallow Deer 63 42 80 46 9 4.5 2/1 Roe Deer 27 23 66 24 5 2.5 1/0.5 Sheep 42 35 97 35.2 7 3.5 7/3.5 33% of the Bearded Vulture’s nutrition is formed by bone remains of other species that belong to the ecosystem but which are not studied in this model, and so it was necessary for the P system to include an annual contribution from the ecosystem. This contribution was calculated to be 12500 kg bones by considering the size of the Bearded Vulture population and its weight. By subtracting the 3500 kg of food in form of bones that is consumed by the floating population of Bearded Vultures, the annual external contribution is calculated to be 9000 kg of bones. There are seven feeding stations in Catalonia which provide around 10500 kg of bone remains annually. These artificial feeding sites have not been considered in the study and most of the floating birds feed at these sites. The required information about each species is shown in Table 2. 58 M. Cardona et al. Table 2. Constants Specie i ki,1ki,2ki,3ki,4ki,5ki,6ki,7ki,8ki,9ki,10 ki,11 ki,12 ki,13 ki,14 ki,15 ki,16 B.Vulture 1 1 8 20 21 0 3 6 0 7 0 0 0 0 50 bones 341 Chamois 2 1 2 18 18 75 0 0 55 0 6 1 6 12 55 grass 0 Red Deer Female 3 1 2 13 17 50 0 0 50 0 6 1 15 30 100 grass 0 Red Deer Male 4 1 2 18 20 0 0 0 50 0 36 1 24 48 0 grass 0 Fallow Deer 5 1 2 11 12 50 0 0 50 0 7 1 2 4 75 grass 0 Roe Deer 6 1 2 9 10 90 0 0 55 0 5 1 1 2 67 grass 0 Sheep 7 1 2 7 7 75 0 59 15 0 4 0 7 14 96 grass 0 3 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 behavior of this ecosystem in diverse initial conditions. First, we define the P systems based framework (probabilistic P systems), where additional features such as two electrical charges which describe specific properties in a better way, are used. Definition 1. A probabilistic P system of degree nis a tuple Π= (Γ, µ, M0,...,Mn−1, R) where: •Γis the alphabet (finite and nonempty) of objects (the working alphabet). •µis a membrane structure, consisting of nmembranes, labeled 0,1, . . . , n −1. The skin membrane is labeled by 0. We also associate electrical charges with membranes from the set {0,+}, neutral and positive. • M0, . . . , Mn−1are 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 is of the form r:uk −→ v, where u, v are a multiset over Γand k∈[0,1] is a real number between 0 and 1 associated with the rule. 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 rules are synchronized. The n-tuple of multisets of objects present at any moment in the nregions of the system constitutes the configuration of the system at that moment. The tuple (M0, . . . , Mn−1) is the initial configuration of the system. We can pass from one configuration to another one by using the rules from R as follows: at each transition step the rules to be applied are selected according A P System Modeling an Ecosystem Related to the Bearded Vulture 59 to the probabilities assigned to them, and all applicable rules are simultaneously applied and all occurrences of the left–hand side of the rules are consumed, as usual. 3.1 The model Our model consists in the following probabilistic P system of degree 2 with two electrical charges: Π= (Γ, µ, M0,M1, R) where: •In the alphabet Γwe represent the six species of the ecosystem (index iis associated with the species and index jis associated with their age, and the symbols X,Yand Zrepresent the same animal but in different state); it also contains the auxiliary symbols Band C. Γ={Xij, Yij, Zij : 1 ≤i≤7,0≤j≤ki,5} ∪ { B, C} •In the membrane structure we represent two regions, the skin (where animals reproduce) and an inner membrane (where animals feed and die): µ= [ [ ]1]0 (neutral polarization will be omitted) •In M0and M1we specify the initial number of objects present in each regions (encoding the initial population and the initial food). –M0={Xqij ij : 1 ≤i≤7,0≤j≤ki,}, where the multiplicity qij indicates the number of animals, of species iwhose age is jthat are initially present in the ecosystem. –M1={C B18000}, where the object Brepresent 0.5 kg of bones, and 9000 kg is the external contribution of bones to the P system corresponding to the 33% of feeding that come from animals do not modeled in the P system. •The set Rof evolution rules consists of: – Reproduction-rules Adult males: r0≡[Xij 1−ki,14 −−−→ Yij]0,1≤i≤7,0≤j≤ki,4 Adult females that reproduce: r1≡[Xij ki,5ki,14 −−−→ YijYi0]0,1≤i≤7, ki,2≤j < ki,3 Adult females that do not reproduce: r2≡[Xij (1−ki,5)ki,14 −−−→ Yij]0,1≤i≤7, ki,2≤j < ki,3 Young animals that do not reproduce: r3≡[Xij →Yij]0,1≤i≤7, ki,3≤j < ki,2 – Young animals mortality rules: Those which survive: 66 M. 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