Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Time delays in stochastic models of evolutionary games
Jacek Miekisz

Last modified: 2014-06-09


It is usually assumed that interactions between individuals take place instantaneously and their effects are immediate. In reality,
all social and biological processes take certain amount of time. It is natural therefore to introduce time delays into evolutionary
game models. We will discuss combined effects of stochasticity and time delays in various finite-population, discrete-time
evolutionary games. A state of a population of individuals is stochastically stable if it appears with a high frequency in the
limit of small stochastic perturbations of deterministic dynamics. We show the existence of a stochastically stable cycle in
two-player games with a mixed evolutionarily stable strategy for any discrete time delay. The situation is much more complex
in there-player games with two evolutionarily stable strategies, a mixed and a pure one. In particular, we show that if the basin
of attraction of the mixed equilibrium is bigger than the one of the pure equilibrium, then there exists a critical time delay where
the pure equilibrium becomes stochastically stable.


J Alboszta and J Miekisz, Stability of evolutionarily stable strategies in discrete replicator dynamics with time delay, J. Theor.

Biol. 231: 175-179 (2004).

J Miekisz and S Wesoowski, Stochasticity and time delays in evolutionary games, Dynamic Games and Applications 1: 440-448


J Miekisz, Stochasticity and time delays in gene expression and evolutionary game theory, Probabilistic Engineering Mechanics

26: 3338 (2011).

J Miekisz and M Matuszak, Stochastic stability in three-player games with time delays, submitted to Dynamic Games and

Applications (2013).