Complex coevolutionary dynamics - Structural stability and finite population effects

Unlike evolutionary dynamics, coevolutionary 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 coevolutionary 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 coevolutionary 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 coevolutionary dynamics of potentially very large, but finite, populations.