Simple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław

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1 Simple models for complex systems toys or tools? Katarzyna Sznajd-Weron Institute of Theoretical Physics University of Wrocław

2 Agenda: Population dynamics Lessons from simple models Mass Extinction and Bak-Sneppen Model SOC and Sand-pile model Population dynamics and logistic equation Minimum viable population a lattice model Logistic equation and determistic chaos (c) 20 Katarzyna Sznajd-Weron 2

3 Lessons from simple models Computational philosophy: lessons from simple models (October -3, 2007, Niels Bohr Institute) This meeting celebrates the 20 year anniversary of selforganized criticality and the approach to science personified by one of its inventors the late Per Bak. Organizers: Dante Chialvo, Maya Paczuski and Kim Sneppen (c) 20 Katarzyna Sznajd-Weron 3

4 Simple models Easier to deal with Do not attempt many detailed predictions Help to understand universal features Pose fundamental questions related to cooperativity self-organization communication regulation and functionality How do complicated systems keep any organization? Simple rules complex behaviour? Complex system simple behavior? (c) 20 Katarzyna Sznajd-Weron 4

5 Various complex systems Biological evolution Population dynamics Opinion dynamics Culture, Lenguages Traffic, pedestrian trafific, evacuation And many others (c) 20 Katarzyna Sznajd-Weron 5

6 Why Physicits? Temptation to use tools from statistical physics Microscopic models can explain collective behaviour Complex Systems: Nonlinear dynamics: determninistic chaos (c) 20 Katarzyna Sznajd-Weron 6

7 A mass extinction Relatively sudden, global decrease in the diversity of life forms Extinctions occur all over the world A large number of species go extinct The extinctions are clustered in a short amount of geological time (c) 20 Katarzyna Sznajd-Weron 7

8 The five largest mass extinctions About 438 million years ago 00 families extinct The late Devonian (about 360 mya) 30% of animal families extinct The end of the Permian period (about 245 mya) 50% of all animal families, 95% of all marine species, and many trees die out The late Triassic (208 mya) 35% of all animal families die out About 65 mya about half of all life forms died out, including the dinosaurs, many families of fishes and many others (c) 20 Katarzyna Sznajd-Weron 8

9 What Causes Mass Extinctions? Cataclysmic events: collision with asteroids or comets, extraordinarily massive volcanism Result of several different factors working in tandem: volcanic activity, acid rain, global warming, a drop in sea levels Direct reason is often unknown One of the greatest mysteries (c) 20 Katarzyna Sznajd-Weron 9

10 Bak - Sneppen Model 993 Natural selection - a key mechanism of evolution Food web - interactions between species (c) 20 Katarzyna Sznajd-Weron 0

11 Natural Selection There is variation in traits Some beetles are green and some are brown The environment cannot support unlimited population growth Green beetles reproduce less often than brown beetles do The surviving brown beetles have brown baby beetles because this trait has a genetic basis Fitness of brown beetles is higher (c) 20 Katarzyna Sznajd-Weron

12 Bak - Sneppen Model 993 fitness (c) 20 Katarzyna Sznajd-Weron 2

13 Self-organizing criticality in BS (c) 20 Katarzyna Sznajd-Weron 3

14 Self-organizing criticality? Per Bak, Chao Tang and Kurt Wiesenfeld Physical Review Letters (987) A property of dynamical systems which have a critical point as an attractor In standard critical phenomena control parameter which an experimenter can vary SOC a critical state is reached by their intrinsic dynamics The archetype of a self-organized critical system is a sand pile (c) 20 Katarzyna Sznajd-Weron 4

15 A sand pile Sand is slowly dropped onto a surface, forming a pile Pile grows and reaches treshold (critical) slope Avalanches occur which carry sand from the top to the bottom of the pile Critical states of a system are signaled by a power-law distribution in some observable PDF(Avalanches) 34 (c) 20 Katarzyna Sznajd-Weron 5

16 Rice pile V. Frette et al., Nature 996 (c) 20 Katarzyna Sznajd-Weron 6

17 Sand pile model (Per et al. 987) (c) 20 Katarzyna Sznajd-Weron 7

18 avalanche s=4 (c) 20 Katarzyna Sznajd-Weron 8

19 Power-laws and sand-piles (c) 20 Katarzyna Sznajd-Weron 9

20 Rainfall Sources: G. Peters, B. Fischer, T. Andersson, Bor. Env. Res., 2002 O. Peters, C. Hertlein, and K. Christensen, A complexity view of rainfall. Phys. Rev. Lett. 88, 0870, -4 (2002). O. Peters and K. Christensen, Rain: Relaxations in the Sky. Phys. Rev. E. 66, 03620, -9 (2002). (c) 20 Katarzyna Sznajd-Weron 20

21 Earthquakes N N M M b, b a t t, a (,.5 ) (c) 20 Katarzyna Sznajd-Weron 2

22 Toys or Tools? Explains punctuated equilibria concept Stasis is broken up by rare and rapid events Explains universal behavior Power laws in various natural catastrophes Oversimplified for biologists (c) 20 Katarzyna Sznajd-Weron 22

23 Logistic equation is used Verhulst, 845 (c) 20 Katarzyna Sznajd-Weron c n n n r c c c n 23

24 Minimum viable population Limited resources carying capacity A minimum viable population for any given species in any given habitat is the smallest isolated population having a 99% chance of remaining extant for 000 years despite the foreseeable efects of demographic, environmental, and genetic stochasticity, and natural catastrophes. Does the critical population size really exist or is it just a heuristic device? (c) 20 Katarzyna Sznajd-Weron 24

25 Simple microscopic model Environment square lattice Individuals (fitness) survival movement offspring (c) 20 Katarzyna Sznajd-Weron 25

26 Our model The population consists of N(t) individuals No more than one individual can occupy a lattice site Each individual is characterized by a certain survival probability p According to the myopic/blind ant rules the individual is moved to one of its nearest neighboring sites The neighbor for mating is chosen at random I. Mróz, A. Pękalski, K. Sznajd-Weron, Phys. Rev. Lett. 76, 3025 (996) K. Sznajd-Weron, Eur. Phys. J. B 6, 83 (2000) K. Sznajd-Weron, M. Wolaoski, Eur. Phys. J. B 25, 253 (2002) (c) 20 Katarzyna Sznajd-Weron 26

27 An example (c) 20 Katarzyna Sznajd-Weron 27

28 Evolution equation (c) 20 Katarzyna Sznajd-Weron 28 ) ( births - deaths 2 p pr R t N t N t N t N t N ) ( ), ( ) ( 4 3 ), ( t c R t c R t c R t c R ) ( ) ( 4 3 ' / ) ( 2 2 p c pc c c c L p pr R t N t N t N

29 Fixed points * ) ( ) ( ) ( ) ( 4 3 0, 0 ) ( ) ( 4 3 ' 2 2 p p p p p p p p pc pc p c pc c p c pc c c c (c) 20 Katarzyna Sznajd-Weron 29 p p c 3 2 2,2

30 Blind and myopic ant (c) 20 Katarzyna Sznajd-Weron 30

31 Is the nest good or not? (c) 20 Katarzyna Sznajd-Weron 3

32 c(t+) c(t+) c(t+) Feedback from logistic equation Fixed points Limit cycles Deterministic Chaos a= a= a= c(t) c(t) c(t)

33 Fixed points and stability a a a f a f x a f 2 ', (0) ' ), 2 ( ' a a x x x x f x ax f * 0, * * *, a a a a Unstable (repulsive) Stable (attractive) Stable (attractive) Unstable (repulsive)

34 c(t+) c(t) a< c(t) t 0 a a 2 2 a 3 a 3 4 Unstable (repulsive) Stable (attractive) Stable (attractive) Unstable (repulsive)

35 c(t+) c(t) a= c(t) t 0 a a 2 2 a 3 a 3 4 Unstable (repulsive) Stable (attractive) Stable (attractive) Unstable (repulsive)

36 c(t+) c(t) a= c(t) t 0 a a 2 2 a 3 a 3 4 Unstable (repulsive) Stable (attractive) Stable (attractive) Unstable (repulsive)

37 c(t+) c(t) a= c(t) t 0 a a 2 2 a 3 a 3 4 Unstable (repulsive) Stable (attractive) Stable (attractive) Unstable (repulsive)

38 c(t+) c(t) a= c(t) t 0 a a 2 2 a 3 a 3 4 Unstable (repulsive) Stable (attractive) Stable (attractive) Unstable (repulsive)

39 c(t+) c(t) a= c(t) t Deterministic chaos

40 Bifurcation diagram

41 Which one is random? Data: Dr. C. Ting

42 How to check it? Henon system x n+ =.4 - x 2 n y n y n+ = x n White noise

43 Toys or Tools? Feedback from population dynamics Is the World random or just chaotic? Nature - the set of deterministic equations? What about stability? (c) 20 Katarzyna Sznajd-Weron 43