
Nonlinear dynamics with applications to biological systems
In the winter term 2007/2008, I teach a course on nonlinear dynamics (Tuesday 2.153.45 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.
Nonlinear 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). Nonlinear dynamics can be studied through nonlinear differential or difference equations and in both cases, graphical methods are very helpful. The range of typical behaviour of nonlinear systems includes bistability, switchlike behaviour and oscillations, which occur in many natural and manmade 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 nonlinear 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, quasispecies equation, payoff matrix, evolutionary games, prisoner's dilemma, titfortat, 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 (cyclindependent protein kinases), circadian rhythms (per genes), HodgkinHuxley 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 EdelsteinKeshet, Mathematical Models in Biology, Random House 1988
 J Keener and J Sneyd, Mathematical Physiology, Springer 1998
 H Haken, Synergetics, Springer 1983
Presentations
 Introduction Oct 16 2007
 Presentation on adhesion clusters Nov 6 2007
 Presentation on muscle Jan 22 2008
 Presentation on circadian rhythms Jan 29 2008
 Presentation on network motifs Feb 5 2007 (by Christian Korn)
Interesting papers
 1978 paper by Bell on modelling adhesion clusters
 1927 paper by Kermack and McKendrick on modelling epidemics
 2003 review on evolutionary game dynamics by Hofbauer and Sigmund
 2007 paper on evolutionary games on graphs by Szabo and Fath
 2003 review on network motifs by Tyson, Chen and Novak
 1952 paper by Hodgkin and Huxley on modelling action potentials
 1961 paper by Fitzhugh introducing the standard model for excitable media
Software
 pplane, a great phase plane analysis tool for Matlab from John Polking at Rice University
 xppaut, a great standalone phase plane analysis and bifurcation tool from Bard Ermentrout at the University of Pittsburgh
 Here you find information on dynamic systems tools in Mathematica
 Matlab program for the glycolysis oscillator
 Matlab program for the van der Pol oscillator
 Matlab program for the FitzhughNagumo model