Second Annual French Complex Systems Summer School

Continuous Space, Lattices and Swarms:
Pattern Formation in Complex Systems

Course Materials

Copyright 2008 by Hiroki Sayama (,

*** Note: These materials are continuously updated, so download the most recent version ***

Course Description

One of the hallmarks of complex systems is that they self-organize and spontaneously form nontrivial spatio-temporal patterns from initially homogeneous states without any blueprint. This course will introduce you to various pattern formation phenomena seen in complex systems, and will help you learn key concepts of, and necessary skills for, describing and examining their dynamics in mathematical ways. A particular emphasis will be laid on spatial models written in partial differential equations. We will also discuss lattice models (cellular automata) and agent-based models so as to illustrate the variety of the modeling frameworks ranging from continuous-time/space/state to discrete-time/space/state systems.

It is strongly recommended, though not strictly required, to arrange and bring to classes your own laptop computer, as we will work on several hands-on exercises in every class. We will use Mathematica 6 as a primary programming language for modeling, simulation and mathematical analysis. We expect to provide several free installers of a term-limited version of that software.

A tentative course schedule:

Lectures 1~2: PDE-based models [slides] [Mathematica notebook]
  • Course introduction / Mathematical tutorial
  • Continuous-field models
  • Developing PDE-based models
  • Numerical simulations of PDE-based models
  • Analytical treatments of PDE-based models
  • Reaction-diffusion equations
Lectures 3~4: Lattice models [slides] [Mathematica notebook]
  • Cellular automata: A simplified discrete-state model
  • Properties of cellular automata
  • Discretizing PDE-based models into lattice models
  • Applications to the modeling of biological systems
  • Evolutionary patterns in CA-based artificial life [website]
Lecture 5: Swarm models [slides] [Mathematica notebook]
  • General formulation of agent-based modeling
  • Models with fixed number of agents
  • Models with agent-environment interaction
  • Models with agent replacement
  • Pattern formation in heterogeneous particle swarms: Swarm Chemistry [website]
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