1 Modeling Nature Modeling Nature 12 October 2011

2 Modeling Nature 1 st LECTURE : Population models 2 nd LECTURE : Network Models

3 Modeling Nature LECTURE : Population models * and predator-prey models …

4 Overview Growth and decay Bounded Growth Volterra’s model of predator-prey systems Why are predator-prey models useful? Examples from nature

5 1. Growth and Decay

6 Growth and Decay Examples of Growth and Decay –Unlimited growth –Limited growth Modelling growth and decay in Nature

7 Growth and decay Growth and decay: two sides of the same coin Growth –At each step: replace each element by n elements Decay –At each step: replace n elements by one element

8 Mathematical description GROWTH: At time “t” seconds the quantity “P” is “n” times the quantity at t-1 seconds : P(t) = n P(t-1)

9 plot for P(t) = nP(t-1) P(t) t

10 Logarithms The rapid growth makes it hard to draw Trick: express quantities in terms of their number of zeros LOG(x) is the number of zeros of x LOG(10) = 1 LOG(1000) = 3 LOG( ) = 6 A logarithmic plot of P(t) = n P(t-1) makes the curves straight…

11 Log(P(t)) t Logarithmic plot for P(t) = nP(t-1)

12 Mathematical description DECAY: At time t seconds the quantity P is 1/n times the quantity at t-1 seconds : P(t) = P(t-1)/n

13 P(t) t plot for P(t) = (1/n)P(t-1)

14 Log(P(t)) t Logarithmic plot for P(t) = (1/n)P(t-1)

15 Unlimited growth As long as there is no limit to the growth Observed in initial growth: –World population growth –Spreading of disease (AIDS) –Internet hype P(t) = nP(t-1)

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17 World wide web

18 Internet Connectivity

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20 Bounded growth Apparently, growth is generally bounded An S-shaped curve is characteristic for bounded growth The logistic curve

21 Bounded growth (Verhulst) P(t+1) = n P(t) ( 1-P(t) ) Logistic model a.k.a. the Verhulst model How do you state this model in a linguistic form? P n is the fraction of the maximum population size 1 n is a growth parameter

22 The Verhulst model exhibits initial growth, with ultimate decay to a assymptote P(t+1) = 1.5 P(t) (1-P(t))

23 2. Predator-Prey Systems

24 Interacting populations The logistic model describes the dynamics of a single population interacting with itself (and available food resources) We now move to models describing two (or more) interacting populations

25 Fish statistics Vito Volterra ( ): a famous Italian mathematician Father of Humberto D'Ancona, a biologist studying the populations of various species of fish in the Adriatic Sea The numbers of species sold on the fish markets of three ports: Fiume, Trieste, and Venice.

26 percentages of predator species (sharks, skates, rays,..)

27 Volterra’s model Two (simplifying) assumptions –The predator species is totally dependent on the prey species as its only food supply –The prey species has an unlimited food supply and no threat to its growth other than the specific predator predatorprey

28 predatorprey Lotka–Volterra equation : The Lotka–Volterra equations are a pair of equations used to describe the dynamics of biological systems in which two species interact, one a predator and one its prey. They were proposed independently by Alfred J. Lotka in 1925 and Vito Volterra in 1926.

29 Lotka–Volterra equation : Two species species #1: population size: x species #2: population size: y

30 Lotka–Volterra equation : Remember Verhulst-equation: Predator ( x ) and prey ( y ) model: x n+1 = x n (α – βy n ): y is the limitation for x y n+1 = y n (γ – δx n ) : x is the limitation for y

31 Behaviour of the Volterra’s model Limit cycleOscillatory behaviour

32 Effect of changing the parameters (1) Behaviour is qualitatively the same. Only the amplitude changes.

33 Effect of changing the parameters (2) Behaviour is qualitatively different. A fixed point instead of a limit cycle.

34 Different modes…

35 Huffaker (1958) reared two species of mites to demonstrate coupled oscillations of predator and prey densities in the laboratory. He used Typhlodromus occidentalis as the predator and the six-spotted mite (Eotetranychus sexmaculatus) as the prey Predator-prey interaction in vivo

36 Why are PP models useful? They model the simplest interaction among two systems and describe natural patterns Repetitive growth-decay patterns, e.g., –World population growth –Diseases –… time Exponential growth Limited growth Exponential decay Oscillation

37 Lynx and hares Very few "pure" predator-prey interactions have been observed in nature, but there is a classical set of data on a pair of interacting populations that come close: the Canadian lynx and snowshoe hare pelt-trading records of the Hudson Bay Company over almost a century.

38 Lynx and hares

39 The Hudson Bay data give us a reasonable picture of predator-prey interaction over an extended period of time. The dominant feature of this picture is the oscillating behavior of both populations

40 Other populationmodels can also be modeled as Pred/Prey: here two herbivores (e.g. zebra and gnou) that compete (indirectly) for the same food resource (e.g. grass).

41 1.what is the period of oscillation of the lynx population? 2.what is the period of oscillation of the hare population? 3.do the peaks of the predator population match or slightly precede or slightly lag those of the prey population?

42 Adaptations

43 Evolutionary arms race

44 This is the basis for evolution

45 Modeling Nature LECTURE : Network Models * and some applications …

46 Overview Some definitions Basic characteristics of networks Special network topologies Examples from nature and sociology Network synchronization

47 Definition of a Network A network is a system of N similar nodes (a.k.a. vertex), where each node interacts with certain other nodes in the system. This interaction is visualized through a connection (a.k.a. edges). nod e connection

48 Some examples Undirected network Directed network Self- connection and multiple edges

49 A more complex example

50 A large network

51 Characteristic path length (L) Clustering coefficient (C) Degree (k) and Degree distribution Characteristics of Networks

52 Characteristic Path Length (L) –The average number of associative links between a pair of concepts Characteristics of Networks L = 4

53 Characteristics of Networks Clustering Coefficient (C) –The fraction of associated neighbors of a node that are also connected Characteristics of Networks

54 Picture pathlengths and clustercoefficients in these networks

55 Branching Factor a.k.a. Degree (k) –The number of other nodes connected to this node i.e. the number of vertices of a node Characteristics of Networks k = 1 k = 2k = 3k = 4 k = 0

56 Degree distribution –The number of nodes in the network that have a certain degree : i.e. the histogram over the degrees. Characteristics of Networks

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58 Special Network Topologies In many situations networks can have a special structure (topology) or properties. We will consider the following cases.

59 1. Regular network A regular network is a network where each node has an identical connection scheme. ?YES ?NO

60 2. Fully connected network A fully connected network is a network where each node is connected to all other nodes.

61 3. A sparse network A sparse network is a network that exhibits a (very) small amount of connections. (opposite: dense)

62 dense networksparse network

63 4. Random network A random network is a network that is generated by some random process.

64 Small-World (SW) network A SW network is a property of the network rather than a specific topology – though the SW-property has implications for the network architecture.

65 Small-world networks * Many clusters of highly interconnected elements (→ C large) * Small number of connections between clusters (→ L small)

66 Small-world networks "six-degrees-of-separation" concept.

67 Four network types a c b d fully connectedrandom regular “small world

68 Network Evaluation Type of networkkCL Fully-connectedN-1LargeSmall Random<<NSmall Regular<<NLarge Small-world<<NLargeSmall

69 Of course there are many other ways for classifying networks …

70 Scale-Free Networks (Barabasi et al, 1998)

71 In scale-free networks, some nodes act as "highly connected hubs" (high degree, red), although most nodes are of low degree (green). Scale-Free Networks

72 Scale-Free (SF) networks A Scale-Free (SF) network is a network where the degree distribution has a very specific structure More concrete; degree distribution P(k) is the proportion of nodes that have k links (k = )

73 Scale-Free (SF) networks degree distribution P(k) for SF networks: * Few nodes with many connections * Many nodes with few connections More concretely: log P(k) ~ - log k (a power law)

74 Scale-Free Networks P(k) is the proportion of nodes that have k links. (k = ) random graphs :

75 Power law : a log-log plot of P(k) versus k gives a straight line. Scale-Free Networks

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77 Scale-free networks' structure and dynamics are independent of the system's size N, the number of nodes the system has. In other words, a network that is scale-free will have the same properties no matter what the number of its nodes is. Scale-Free Networks

78 Scale-free networks can grow by the process of preferential attachment : new links are made preferably to hubs: the probability of a new link is proportional to the links of a node.

79 Some examples…

80 Nodes: -addresses, links: s

81 Nodes: people, links: # of sexual partners

82 Web pages : Inlinks and outlinks (red and blue) Network nodes (green) The World-Wide-Web is scale free

83 Degree distributions in human gene coexpression network. Coexpressed genes are linked for different values of the correlation r, King et al, Molecular Biology and Evolution, 2004

84 Social Networks A social network is a social structure made of nodes (which are generally individuals or organizations) that are tied by one or more specific types of interdependency, such as values, visions, ideas, financial exchange, friendship, kinship, dislike, conflict or trade. The resulting graph-based structures are often very complex.

85 NETWORK SYNCHRONIZATION: Synchronization is the harmonization of the time evolution of various dynamics systems. Of special interest is the synchronization of (semi) periodic processes such as oscillators. Examples are: - synchronization of fire flies, - clapping of audience after concert, - menstrual cycles of women living together, - heart cells in healthy heart

86 SYNCHRONIZATION: 1. phase locking: Kuramoto model coupled oscilators individual oscillator phase θ interaction network

87 Kuramoto: Phase locking

88 phase locking: Kuramoto model Kuramoto found that the degree of synchronization – represented by an order parameter r – depends on the strength of the coupling K between the oscillators. *NO* synchronization for weak coupling synchronization for strong coupling There is a critical value of the coupling, K c, below which *no* synchronization can happen!!!

89 Science of rhythmic applause A nice application of the Kuramoto model is the synchronization of clapping of an audience after a performance, which happens when everybody claps at a slow frequency and in tact. In this case the distribution of ‘natural clapping frequencies’ is quite narrow and Kc is low – so there is synchronization as K > Kc. When an individual wants to express especial satisfaction with the performance he/she increases the clapping frequency by about a factor two, as measured experimentally, in order to increase the noise level, which just depends on the clapping frequency. Measurements have shown, see figure, that the distribution of natural clapping frequencies is broader when the clapping is fast. This leads to an increase in Kc and it happens that now K < Kc. So, no synchronization is possible when the applause is intense. Low frequency: synchronization High frequency: NO synchronization synchronization of rhythmic applause

90 Example 2: (DIS)SYNCHRONIZ|ATION ON THE HEART

91 END of LECTURE

92 APPENDIX: Example of a Small-World Network: The accumulation of knowledge and the growth of the ‘semantic network’ in children

93 Example : Semantic Network apple orange pear lemon Newto n Einstein gravitation

94 Growth of knowledge semantic networks apple orange pear lemon Newto n Einstein gravitation Average separation should be small Local clustering should be large

95 Strongest links of/with APPLE PIE(20) PEAR(17) ORANGE(13) TREE( 8) CORE( 7) FRUIT( 4) NEWTON APPLE(22) ISAAC(15) LAW( 8) ABBOT( 6) PHYSICS( 4) SCIENCE( 3)

96 Semantic net at age 3

97 Semantic net at age 4

98 Semantic net at age 5

99 The growth of semantic networks obeys a logistic law

100 L as a function of age (× 100) = semantic network = random network

101 C as a function of age (× 100) = semantic network = random network

102 Small-worldliness Walsh (1999) Measure of how well small path length is combined with large clustering Small-worldliness = (C/L)/(C rand /L rand )

103 Small-worldliness as a function of age adult

104 Some comparisons Semantic Network Cerebral Cortex Caenorhabditis Elegans Small-Worldliness

105 What causes the small- worldliness in the semantic net? Optimal efficient organization

106 Strongest links in semantic net of adult males [Shields, 2001] TOP 40 of concepts Ranked according to their k-value (number of associations with other concepts)

107 Semantic top 40

108 END of APPENDIX