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(1000000) = 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
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))
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 (1860-1940): 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.
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.
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?
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
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 = 1..2..3..)
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 = 1..2..3..) random graphs :
75 Power law : a log-log plot of P(k) versus k gives a straight line. Scale-Free Networks
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.
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
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