Complex System Dynamics

LTCC Course Feb-March 2012

Dr Hannah Fry hfry@math.ucl.ac.uk

Course Outline

1. An introduction to complex systems.

2. The Mathematics of urban crime.

3. The spread of epidemic disease on a network.

4. Location theory and spatial interaction in a retail system.

Science and ComplexityProblems of Simplicity

Warren Weaver, Science and Complexity, American Scientist, 36: 546 (1948)

Problems of Disorganised ComplexityScience and Complexity

Warren Weaver, Science and Complexity, American Scientist, 36: 546 (1948)

“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)

Science and Complexity

Problems of Organised ComplexityScience and Complexity

Warren Weaver, Science and Complexity, American Scientist, 36: 546 (1948)

What is a complex system?

M. Newman, Complex Systems: A Survey, American Journal of Physics, 800-810 (2011)

“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”

What is a complex system?

A. Geller, The use of complexity-based models in international relations (2011)

Shoals of fish

http://www.youtube.com/watch?v=cIgHEhziUxU&feature=related

Self Similar Broccoli

Self Similar Broccoli

Ant Colonies

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

The Financial Markets

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

Traffic shockwaves

The Double Pendulum

http://www.youtube.com/watch?v=U39RMUzCjiU&feature=related

What is a complex system?

H. Fry (2012)

“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”

dynamic

?

Complex Systems Theory: Rule Based Simulation

M. Newman, Complex Systems: A Survey, American Journal of Physics, 800-810 (2011)

“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.”

AliveDead

Conway’s Game of Life

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

Forest FiresEmpty

Tree

Burning Tree

Charred Tree

The Schelling ModelEmpty

Red team

Blue team

Residents can move if the neighbouring cells contain more than 3 members of the other team.

Agent Based Modelling

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

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, only if they’ve eaten recently

Die after a certain number of time steps

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

Predator Prey Model

• 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.

• 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.

Predator Prey Model

Complex Systems Theory: Mathematical Abstraction

M. Newman, Complex Systems: A Survey, American Journal of Physics, 800-810 (2011)

“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.”