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Complex System Dynamics LTCC Course Feb-March 2012

Complex System Dynamics LTCC Course Feb-March 2012. Dr Hannah Fry hfry@math.ucl.ac.uk. Course Outline. An introduction to complex systems. The Mathematics of urban crime. The spread of epidemic disease on a network. Location theory and spatial interaction in a retail system.

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Complex System Dynamics LTCC Course Feb-March 2012

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  1. Complex System DynamicsLTCC Course Feb-March 2012 Dr Hannah Fry hfry@math.ucl.ac.uk

  2. Course Outline • An introduction to complex systems. • The Mathematics of urban crime. • The spread of epidemic disease on a network. • Location theory and spatial interaction in a retail system.

  3. Science and Complexity Problems of Simplicity Warren Weaver, Science and Complexity, American Scientist, 36: 546 (1948)

  4. Science and Complexity Problems of Disorganised Complexity Warren Weaver, Science and Complexity, American Scientist, 36: 546 (1948)

  5. Science and Complexity “Scientific methodology went from one extreme to another – from two variables to an astronomical number - and left untouched a great middle region” Warren Weaver, Science and Complexity, American Scientist, 36: 536 (1948)

  6. Science and Complexity Problems of Organised Complexity Warren Weaver, Science and Complexity, American Scientist, 36: 546 (1948)

  7. What is a complex system? “A complex system is a system composed of many interacting parts, which displays collective behaviour that does not follow trivially from the behaviours of the individuals” M. Newman, Complex Systems: A Survey, American Journal of Physics,800-810 (2011)

  8. What is a complex system? A. Geller, The use of complexity-based models in international relations (2011)

  9. Shoals of fish http://www.youtube.com/watch?v=cIgHEhziUxU&feature=related

  10. Self Similar Broccoli

  11. Self Similar Broccoli

  12. Ant Colonies http://www.youtube.com/watch?v=g7VhvoMFn34&feature=related

  13. The Financial Markets http://www.guardian.co.uk/science/2012/feb/12/black-scholes-equation-credit-crunch

  14. Traffic shockwaves

  15. The Double Pendulum http://www.youtube.com/watch?v=U39RMUzCjiU&feature=related

  16. What is a complex system? dynamic “A complex system is a system composed of many interacting parts, which displays collective behaviour that does not follow trivially from the behaviours of the individuals” ? H. Fry (2012)

  17. Complex Systems Theory: Rule Based Simulation “To create comprehensive and realistic models, usually in the form of computer simulations which represent the interacting parts of a complex system, often down to minute details, and then to watch and measure the emergent behaviours that appear.” M. Newman, Complex Systems: A Survey, American Journal of Physics,800-810 (2011)

  18. Conway’s Game of Life Dead Alive Under-populated: Live cell has one neighbour Dies Next generation: Live cell has two or three neighbours Lives Overcrowding: Live cell has four or more neighbours Dies Reproduction: Dead cell has exactly three neighbours Reborn

  19. Empty Forest Fires Tree Burning Tree Charred Tree

  20. The Schelling Model Empty Red team Blue team Residents can move if the neighbouring cells contain more than 3 members of the other team.

  21. Agent Based Modelling

  22. Bird Flocking Separation: Agents avoid collisions with other birds Alignment: Birds direct their path towards the local center of mass Cohesion: Birds align their speed with the local average velocity

  23. Predator Prey Model Die if they don’t meet enough rabbits Breed if they meet a fox, only if they’ve eaten recently Die after a certain number of time steps Die if they meet a fox Reproduce if they meet a rabbit Die after a certain number of time steps

  24. Predator Prey Model Die if they meet a fox Reproduce if they meet a rabbit Die after a certain number of time steps Die if they don’t meet enough rabbits Breed if they meet a fox, if eaten recently Die after a certain number of time steps

  25. Predator Prey Model • In the absence of foxes, the per capita rabbit growth rate is constant. • In the presence of foxes, the per capita rabbit growth rate will fall linearly as a function of the fox population. • In the absence of rabbits, the per capita fox • growth rate is constant and negative. • In the presence of rabbits, the per capita growth rate of foxes increases linearly as a function of • the rabbit population.

  26. Predator Prey Model

  27. Complex Systems Theory: Mathematical Abstraction “The creation and study of simplified mathematical models that, while they may not mimic the behaviour of real systems exactly, try to abstract the most important qualitative elements into a solvable framework from which we can gain scientific insight.” M. Newman, Complex Systems: A Survey, American Journal of Physics,800-810 (2011)

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