On reliability of simulations of complex co-evolutionary processes

Infinite population models of co-evolutionary dynamics are useful mathematical constructs hinting at the possibility of a wide variety of possible dynamical regimes - from simple attractive fixed point behavior, periodic orbits to complex chaotic dynamics. We propose to use the framework of shadowing lemma to link such mathematical constructs to large finite population computer simulations. We also investigate whether the imposition of finite precision computer arithmetic or the requirement that population ratios be rational numbers does not leave the infinite population constructs and theories irrelevant. We argue that if the co-evolutionary system possesses the shadowing property the infinite population constructs can still be relevant. We study two examples of hawk-dove game with Boltzmann and (μ, λ) selection. Whereas for Boltzmann selection there is a strong indication of the shadowing property, there is no shadowing in the case of (μ, λ) selection.