Complex Coevolutionary Dynamics - Structural Stability and Finite Population Effects

Unlike evolutionary dynamics, co-evolutionary dynamics can exhibit a wide variety of complex regimes. This has been confirmed by numerical studies e.g. in the context of Evolutionary Game Theory (EGT) and population dynamics of simple two-strategy games with various types of replication and selection mechanisms. Using the framework of shadowing lemma we study to what degree can such infinite population dynamics (1) be reliably simulated on finite precision computers and (2) be trusted to represent co-evolutionary dynamics of possibly very large, but finite populations. In a simple EGT setting of two-player symmetric games with two pure strategies and a polymorphic equilibrium we prove that for (μ, λ), truncation, sequential tournament, best-of-group tournament and linear ranking selections, the co-evolutionary dynamics do not possess the shadowing property. In other words, infinite population simulations cannot be guaranteed to represent real trajectories or to be representative of co-evolutionary dynamics of potentially very large, but finite populations.