Evolutionary game theory and population dynamics

We begin these lecture notes by a crash course in game theory. In particular, we introduce a fundamental notion of a Nash equilibrium. To address the problem of the equilibrium selection in games with multiple equilibria, we review basic properties of the deterministic replicator dynamics and stochastic dynamics of finite populations. We show the almost global asymptotic stability of an efficient equilibrium in the replicator dynamics with a migration between subpopulations. We also show that the stability of a mixed equilibrium depends on the time delay introduced in replicator equations. For large time delays, a population oscillates around its equilibrium. We analyze the long-run behaviour of stochastic dynamics in well-mixed populations and in spatial games with local interactions. We review results concerning the effect of the number of players and the noise level on the stochastic stability of Nash equilibria. In particular, we present examples of games in which when the number of players increases or the noise level decreases, a population undergoes a transition between its equilibria. We discuss similarities and differences between systems of interacting players in spatial games maximizing their individual payoffs and particles in lattice-gas models minimizing their interaction energy. In short, there are two main themes of our lecture notes: the selection of efficient equilibria (providing the highest payoffs to all players) in population dynamics and the dependence of the longrun behaviour of a population on various parameters such as the time delay, the noise level, and the size of the population.