After this course advanced Bachelor students of natural science and life science are able to do basic calculations using the most prominent toy models in biology, chemistry and physics. Moreover, students are able to transform complex problems into the simple mathematical rules that serve as input in these models.
Global course description. ‘Toy models’ are models that use as input (very) simple rules, and as ‘output’ are able to describe a wide variety of (complex) behavior. In this course, some of the most successful toy models will be treated. These models are able to put complex behavior into perspective in terms of generic underlying rules, and have led (and are still leading) to a deeper understanding in biology, chemistry and physics. Besides that, successful toy models have strong predictive power, and often have significant impact beyond the disciplinary boundaries for which they were originally designed.|
Aim of the course: introduce advanced Bachelor students of natural science and life science to some prominent toy models, and provide them with the mathematical and statistical mechanical tools and background that are necessary to ‘play’ with these toys.
Detailed course description
- Introduction to the Ising model and its different macroscopic (stationary) solutions or phases in 1,2,3 dimensions, properties of critical points, scale invariance and renormalization group. Tools: Boltzmann weight, partition function, thermodynamics, macroscopic order parameters, and mean-field theory. (Henk Stoof)
- Simple ODE models in biology. Understanding complex systems with simple models. Bifurcations and bistability. Learn to think the unthinkable. Tools: Formulation of toy models, phase plane analysis, steady state analysis, Lotka Volterra model. (Rob de Boer)
- Pattern formation: Turing models and their application in development and ecology. Tools: elementary analysis of ODEs, linear algebra, steady state stability analysis based on Jacobian of linearized system (Kirsten ten Tusscher)
- The ‘random walk’ and applications in diffusion, polymer statistics and rare events. Tools: basic statistics (Willem Kegel)
- Random adsorption models. Fundamentals, MWC theory of allosteric interactions, simple genetic repression and activation. Tools: grand ensemble theory; undetermined multiplier method of Lagrange. (Willem Kegel)
- Simple Cellular Automata and individual based models in biological evolution. Understanding complex systems and counterintuitive dynamics by local interactions and self-organisation. Tools: Basic theory of Cellular automata, mesoscale patterns. Replicator equations, Hypercycles, RP systems. Computational analysis. (Paulien Hogeweg)