Mathematical Modelling of Complex Systems

Lecture content:

This lecture introduces the use of mathematical modelling as a tool of studying complex system. Its aim is to present the necessary mathematical/computational techniques and analysis developed in the emerging interdisciplinary field of complex systems science to get insight into the function of the complex environmental systems.

The course is organized along relatively simple models and classical examples of complex systems from selected fields in the natural and earth sciences (e.g. phase transitions and pattern formation, systems biology and ecology, biogeochemistry and oceanography). These models are used as examples to explain the predominantly numerical mathematics that are needed for solving them, and that are also used in large scale complex models such as the earth’s global climate.

Every few lectures we will have a practical exercise where students write / change models that are given as small computer programs in Python or Matlab/Octave and discuss the results.

The lecture consists of the following three blocks:

I Introduction:
-Complex systems in a nutshell
-Fundamentals of modelling

II Systems with a small number of variables
- Static models
- Discrete-time models
- Continuous-time models
- Bifurcations
- Chaos

III Systems with a large number of variables
- Cellular automata
- Continuous-field models
- Network models
- Agent-based models

The course is self-contained, so there is no mandatory reading. The treatment of numerical methods of continuous-time and field models is based strongly on the book 'Numerical Recipes' by William H. Press, Saul Teukolsky, William T. Vetterling und Brian P. Flannery