There is a large and growing body of research applying formal game models to the study of evolutionary competition within species and co-evolutionary (ecological) competition between species. In this paper we discuss ergodic properties and limiting behavior in a class of dynamical systems of which the replicator dynamics of (biological) evolutionary game theory are a special case. We study a class of quadratic dynamical systems on a simplex of arbitrary dimension. They arise in biology at both ends of the evolutionary scale, in models of animal behavior and molecular kinetics. The game theoretical aspects also suggest possible applications in the social sciences. The simplest description of such an evolution is based on the replicator equation. The basic properties of replicator dynamics are analyzed, and some low-dimensional examples such as the Rock-Scissors-Paper game are discussed. Finally we discuss a construction of quadratic dynamical systems which defined on infinite-dimensional simplex.