Ruprecht Karls Universität Heidelberg

Non-linear dynamics with applications to biological systems

In the winter term 2007/2008, I teach a course on non-linear dynamics (Tuesday 2.15-3.45 pm, BIOQUANT, INF 267, lecture room on ground floor, 2 credit points). This course is given in German and addresses students after the Vordiplom from physics and related disciplines.

Non-linear dynamics is the study of dynamical processes in nature which do not obey linear laws, which implies that the superposition principle does not hold. On the one hand, this means that small perturbations can decay again, which is an important ingredient to get stable limit cycles (oscillations). On the other hand, small perturbations need not to stay small, thus small variations in initial conditions can lead to very different results (chaos). Non-linear dynamics can be studied through non-linear differential or difference equations and in both cases, graphical methods are very helpful. The range of typical behaviour of non-linear systems includes bistability, switch-like behaviour and oscillations, which occur in many natural and man-made systems. The course offers an introduction to the basic tools to understand these responses as well as to different applications in biology, including molecular processes like enzyme kinetics, cellular processes like hearing and evolutionary processes like coexistence of competing species.

The course is organized in the following parts:

  • Basic tools of non-linear dynamics with simple examples: 1D flow, fixed points, bifurcations, 2D flow, phase plane analysis, oscillations, Hopf and global bifurcations (finished until Xmas)
  • Evolutionary game theory: mutation and selection, fitness landscape, replicator dynamics, quasi-species equation, payoff matrix, evolutionary games, prisoner's dilemma, tit-for-tat, cellular automata (together with Christian Korn)
  • Modelling molecular biological systems: oscillators and switches, network motifs, stability of adhesion clusters, Huxley model for muscle, relaxation oscillations in muscle, cell cycle control (cyclin-dependent protein kinases), circadian rhythms (per genes), Hodgkin-Huxley model for neural excitation (lecture on network motifs by Christian Korn)

The course does not cover stochastic dynamics (compare my lecture two years ago), fractals, structure formation, graph theory, networks or epidemiology.

Recommended literature

  • SH Strogatz, Nonlinear dynamics and chaos, Westview 1994
  • M Nowak, Evolutionary Dynamics, Harvard University Press 2006
  • C Fall et al, eds, Computational Cell Biology, Springer 2002

Additional literature

  • JD Murray, Mathematical biology, 3rd edition (now in volumes I and II), Springer 2002
  • L Edelstein-Keshet, Mathematical Models in Biology, Random House 1988
  • J Keener and J Sneyd, Mathematical Physiology, Springer 1998
  • H Haken, Synergetics, Springer 1983


Interesting papers


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