Barbara Ikica: Evolutionary dynamics, games and graphs

Povzetek. In this talk we will motivate the use of a mathematical approach to examine how populations evolve through evolutionary processes and how population structure affects the course of these processes. Initially, we will consider infinitely large populations by means of dynamical systems and game theory. Here we will investigate the stability of the species which form the populations. Furthermore, we will devote some time to studying the permanence and persistence of underlying dynamical systems through the prism of index theory. Subsequently, we will tackle finite populations and therefore provide a framework for exploring stochastic processes (Markov chains). In addition to the stability of the species, we will also discuss their fixation probabilities. Finally, using graph theory, we will take into account the spatial distributions of the populations that directly affect the interactions among their members.

The central concept of this talk will be the replicator equation, which will be introduced to deal with infinite populations. Nevertheless, it will emerge in unexpected ways afterwards. It will be observed both in stochastic models of finite populations and in spatial models which place members of the populations on the graphs.