Every day there will be two sessions (three hours each), one in the morning (starting at 9.00) and one in the afternoon (starting at 14.00).

Monday 16, morning

by Sergio Rinaldi

This lecture surveys the main characteristics of continuous-time dynamical systems. Equilibria and their stability are first introduced. Then, more complex attractors, e.g. limit cycles (periodic behavior), tori (quasi-periodic behavior), and strange attractors (chaotic behavior) are presented. Multiplicity of attractors is also discussed with emphasis to the role played by so-called saddles. Examples from population dynamics are illustrated.

Monday 16, afternoon

by Fabio Dercole and Stefano Maggi

Evolutionary change in biological, as well as social and economic systems, results from a mechanism of innovation, altering the characteristic traits of individual agents, and from competition selecting the best performant agents. Different modeling approaches for the description of evolutionary dynamics, focused on different aspects of innovation and competition processes, have been developed and are quickly reviewed in this lecture. Particular emphasis is given to individual-based approaches versus population-based approaches to evolutionary dynamics.

Tuesday 17, morning

by Karl Sigmund

Evolutionary game theory fuses game theory and population dynamics, by describing how frequencies of strategies change in accordance with their success. This lecture gives a brief introduction into the basics — which are closely related with ecological dynamics — and applies this to a study of diverse models describing the emergence of cooperation in societies of selfish agents.

Tuesday 17, afternoon

by Ulf Dieckmann

Whenever the success of a player depends on which other players it has to compete with, adaptation cannot be understood as mere optimization. Adaptive dynamics theory meets this challenge, establishing macroscopic evolutionary laws from the microscopic ecological interactions between individuals. This lecture will introduce three key tools of adaptive dynamics theory — invasion fitness, pairwise invasibility plots, and the canonical equation — before illustrating their application in several examples. Particular emphasis will be given to explaining the endogenous forces responsible for the diversification of complex adaptive systems.

Wednesday 18, morning

by Fabio Dercole and Sergio Rinaldi

The dependence of the behavior of dynamical systems upon one or more parameters is discussed by introducing basic notions of bifurcation theory and related numerical aspects. Then, the dependence of evolutionary dynamics upon parameters is studied through the bifurcation analysis of the Adaptive Dynamics canonical equation. An example concerning the coevolution of resource-consumer systems is presented.

Wednesday 18, afternoon

by Fabio Dercole

In this lecture the dependence of evolutionary dynamics upon parameters is discussed with reference to three specific problems: The emergence of technological diversity due to innovations in markets with competing products; The periodic occurrence of diversification and extinction events in cannibalistic populations; Demographic bistability as a cause for cyclic evolutionary regimes in populations competing for common resources. The analysis is systematically performed through the study of the bifurcations of the Adaptive Dynamics canonical equation.

- Fabio Dercole
^{1}(Director of the course) - Ulf Dieckmann
^{2} - Stefano Maggi
^{1} - Sergio Rinaldi
^{1,2} - Karl Sigmund
^{2,3}

1, Department of Electronics and Information, Politecnico of Milano, Milano, Italy.

2, Adaptive Dynamics Network, International Institute for Applied Systems Analysis, Laxenburg, Austria.

3, Faculty of Mathematics, University of Vienna, Vienna, Austria.

The notes and transparencies used by all lecturers and material for further reading will be electronically available
to all participants.

A certificate of attendance will be given to all participants at the end of the course.