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Scale Invariant Properties of Ecological Species Cecile Caretta Cartozo, Diego Garlaschelli, Luciano Pietronero Carlo Ricotta, Guido Caldarelli University of Rome“La Sapienza” Coevolution and Self-Organization in Dynamical Networks Contents Network Topological properties (degree distribution etc) 1) Give new description of phenomena allowing to detect new universal behaviour. to validate models 2) Can sometime help in explaining the evolution of the system As example of this use of graph I will present 1) Food Webs 2) Linnean Trees Scale-Free Network arise naturally in RANDOM environments •“Food Chain” (ecological network): sequence of predation relations among different living species sharing the same physical space (Elton, 1927): Flow of matter and energy from prey to predator, in more and more complex forms; The species ultimately feed on the abiotic environment (light, water, chemicals); At each predation, almost 10% of the resources are transferred from prey to predator. •“Food Web” (ecological network): Set of interconnected food chains resulting in a much more complex topology: Trophic Species: Set of species sharing the same set of preys and the same set of predators (food web aggregated food web). Trophic Level of a species: Minimum number of predations separating it from the environment. Basal Species: Species with no prey (B) Top Species: Species with no predators (T) Intermediate Species: Species with both prey and predators ( I ) Prey/Predator Ratio = BI IT • Food Web Structure Pamlico Estuary (North Carolina): 14 species Aggregated Food Web of Little Rock Lake (Wisconsin)*: 182 species 93 trophic species How to characterize the topology of Food Webs? Graph Theory * See Neo Martinez Group at http://userwww.sfsu.edu/~webhead/lrl.html Degree Distribution P(k) in real Food Webs Unaggregated versions of real webs: irregular or scalefree? P(k) k- R.V. Solé, J.M. Montoya Proc. Royal Society Series B 268 2039 (2001) J.M. Montoya, R.V. Solé, Journal of Theor. Biology 214 405 (2002) •Spanning Trees of a Directed Graph A spanning tree of a connected directed graph is any of its connected directed subtrees with the same number of vertices. In general, the same graph can have more spanning trees with different topologies. Since the peculiarity of the system (FOOD WEBS),some are more sensible than the others. • Tree Topology (2) 1 1 1 1 5 Out-component size: w AX XY AY 1 Out-component size distribution P(A) : 0,5 3 1 1 5 11 8 Y nn X 0,6 1 2 22 10 1 3 1 Sum of the sizes: CX 1 Y Y X Allometric relations: 33 CX CX A X 35 P(A) A C C A C(A) 30 33 0,5 25 0,4 22 20 0,3 15 0,2 11 10 0,1 0,1 0,1 0,1 0,1 0,1 5 A 0 1 2 3 4 5 6 7 8 9 10 5 A 3 1 0 0 2 4 6 8 10 12 • Optimisation A0: metabolic rate B C0: blood volume ~ M Kleiber’s Law: B(M) M 3 / 4 C( A ) A General Case (tree-like transportation system embedded in a D-dimensional metric space): D1 the most efficient scaling is C( A ) A D West, G. B., Brown, J. H. & Enquist, B. J. Science 284, 1677-1679 (1999) Banavar, J. R., Maritan, A. & Rinaldo, A. Nature 399, 130-132 (1999). | 4 3 •Allometric Relations in River Networks AX: drained area of point X Hack’s Law: C( A ) A L A0.6 3 2 •Area Distribution in Real Food Webs •Allometric Relations in Real Food Webs (D.Garlaschelli, G. Caldarelli, L. Pietronero Nature 423 165 (2003)) • Data and Model Little Rock Webworld Little Rock Webworld S 182 182 S 93 93 L 2494 2338 L 1046 1037 B 0.346 0.30 B 0.13 0.15 I 0.648 0.68 I 0.86 0.84 T 0.005 0.02 T 0.01 0.01 Ratio 1.521 1.4 Ratio 1.14 1.16 lmax 3 3 lmax 3 3 C 0.38 0.40 C 0.54 0.54 D 2.15 2.00 D 1.89 1.89 1.11±0.03 1.12±0.01 1.15±0.02 1.13±0.01 2.05±0.08 2.00±0.01 1.68±0.12 1.80±0.01 Original Webs Aggregated Webs •Spanning trees of Food Webs 1 0 1 0 C( A) A efficient C(A) A 1 2 P(A) A1 stable P(A) A 0 C( A) A 2 inefficient P(A) cost unstable •Ecosystems around the world Lazio Utah Amazonia Iran Peruvian and Atacama Desert Argentina Ecosystem = Set of all living organisms and environmental properties of a restricted geographic area we focus our attention on plants in order to obtain a good universality of the results we have chosen a great variety of climatic environments •From Linnean trees to graph theory Linnean Tree = hierarchical structure organized on different levels, called taxonomic levels, representing: phylum subphylum • classification and identification of different plants class • history of the evolution of different species subclass order family A Linnean tree already has the topological structure of a tree graph genus species • each node in the graph represents a different taxa (specie, genus, family, and so on). All nodes are organized on levels representing the taxonomic one • all link are up-down directed and each one represents the belonging of a taxon to the relative upper level taxon Connected graph without loops or double-linked nodes •Scale-free properties P(k) Degree distribution: k P ( k ) k ~ 2.5 0.2 The best results for the exponent value are given by ecosystems with greater number of species. For smaller networks its value can increase reaching = 2.8 - 2.9. •Geographical flora subsets Tiber Mte Testaccio Aniene Lazio City of Rome Colli Prenestini k k =2.52 0.08 =2.58 0.08 k 2.6 ≤ ≤ 2.8 •What about random subsets? In spite of some slight difference in the exponent value, a subset which represents on its own a geographical unit of living organisms still show a power-law in the connectivity distribution. P(k) P(k) P(k) random extraction of 100, 200 and 400 species between those belonging to the big ecosystems and reconstruction of the phylogenetic tree LAZIO k P(k) • Simulation: k P(k) k ROME P(k)=k -2.6 k k ? Memory? Particular rule to put a species in a genus, a genus in a family….? NO! P(kf, kg) that a genus with degree kg belongs to a family with degree kf kf=1 kf=2 kf=3 kf=4 kg = ∑g kg P(kf,kg) fixed P(kf,kg) kg - fixed ~ 2.2 0.2 kf kg P(ko,kf) that a family with degree kf belongs to an order ko=1 ko=2 ko=3 ko=4 P(ko,kf) kf with degree ko kf = ∑f kf P(ko,kf) fixed - fixed kf ~ 1.8 0.2 ko • A simple Model 1) create N species to build up an ecosystem 2) Group the different species in genus, the genus in families, then families in orders and so on realizing a Linnean tree - Each species is represented by a string with 40 characters representing 40 properties which identify the single species (genes); - Each character is chosen between 94 possibilities: all the characters and symbols that in the ASCII code are associated to numbers from 33 to 126: P g H C ) %o r ? L 8 e s / C c W & I y 4 ! t G j z AB 4 2£ ) k , ! d q 2= m: f V Two species are grouped in the same genus according to the extended Hamming distance dWH: c1i = character of species 1 c2i = character of species 2 ba Z with i=1,……….,40 with i=1,……….,40 dEH = ( ∑i=1,40 |c1i - c2i| )/40 c14 P g H C ) %o r ? L G j |c1i - c2i| = 17 4 2£ ) k , ! d c24 species 1 dEH ≤ C same genus Fixed threshold species 2 genus = average of all species belonging to it c14 c(g)4 P g H C ) %o r ? L ( c1i + c2i )/2 G j : 4 2£ ) k , ! d c24 Same proceedings at all levels with a fixed threshold for each one At the last level (8) same phylum for all species (source node) Two ways of creating N species No correlation: species randomly created with no relationship between them Genetic correlation: species are no more independent but descend from the same ancestor • No correlation: ecosystems of 3000 species each character of each string is chosen at random quite big distance between two different species: P(k) dEH ~ 20 (S . ` U d ~j <@a ~N f K Mg X w´ * : * 4 " j ° z G 9 / F y 2 J ´ R _ x 5 K L ` < G ´ D Q b mV U W ; d L U x o g Z k * 8 y u N v D K Z + { C x 6 I 6 d z (top ~ 1.7 0.2 k bottom ~ 3.0 0.2 ) • Coevolution correlation: single species ancestor of all species in the ecosystem at each time step t a new species appear: - chose (randomly) one of the species already present in the ecosystem - change one of its character 3000 time steps natural selection Environment = average of all species present in the the ecosystem at each time step t. At each time step t we calculate the distance between the environment and each species: dEH < Csel survival dEH > Csel extinction small distance between different species: dEH ~ 0.5 g 5 0 _ " & y = E o [ l R C ( x z G ? g = X %W @ @ / X r ] T K g ? 6 Y G ^ Q z g 5 0 _ " & y = E o [ : R C ( x z G ? 0 = / %W ´ S / X r ] T K g ? 6 K ^ ^ Q z P(k) ~ k - k ~ 2.8 0.2 A comparison Correlated: k k P(k) Not Correlated: • Power-laws out of the Random Graph model Vertices fitnesses are drawn from probability distribution r(x) Edges are drawn with probability f(xi,xj) We investigated the several choices of r(x) and f(xi,xj) SOME OF THEM PRODUCE SCALE-FREE NETWORKS! Analytical derivation successfull for: r(x)= xb (Zipf, Pareto law) and f(xi,xj) xi xj r(x)= ex and f(xi,xj) (xi +xj –z(N)) i.e. a link is drawn when the sum of fitnesses exceeds a threshold value G.C, A. Capocci, P. De Los Rios, M.A. Munoz PRL 89, 258702 (2002). Without introducing growth or preferential attachment we can have power-laws We consider “disorder” in the Random Graph model (i.e. vertices differ one from the other). This mechanism is responsible of self-similarity in Laplacian Fractals •Dielectric Breakdown •In a perfect dielectric •In reality Different realizations of the model a) b) c) have r(x) power law with exponent 2.5 ,3 ,4 respectively. d) has r(x)=exp(-x) and a threshold rule. Degree distribution for cases a) b) c) with r(x) power law with exponent 2.5 ,3 ,4 respectively. Degree distribution for the case d) with r(x)=exp(-x) and a threshold rule. Conclusions Results: networks (SCALE-FREE OR NOT) allow to detect universality (same statistical properties) for FOOD WEBS and TAXONOMY. Regardless the different number of species and environment STATIC AND DYNAMICAL NETWORK PROPERTIES other than the degree distribution allow to validate models. NEITHER RANDOM GRAPH NOR BARABASI-ALBERT WORK Future: models can be improved with particular attention to environment and natural selection FOR FOOD WEBS AND TAXONOMY new data COSIN COevolution and Self-organisation In dynamical Networks RTD Shared Cost Contract IST-2001-33555 http://www.cosin.org • • • • • Nodes Period of Activity: Budget: Persons financed: Human resources: EU countries Non EU countries EU COSIN participant Non EU COSIN participant 6 in 5 countries April 2002-April 2005 1.256 M€ 8-10 researchers 371.5 Persons/months