This document proposes a novel class of population dynamics that are parameterized by a nonnegative scalar ϵ. We show that any rest point of the proposed dynamics corresponds to an ϵ-equilibrium of the underlying population game. In order to derive this class of population dynamics, our approach is twofold. First, we use an extension of the pairwise comparison revision protocol and the classic mean dynamics for well-mixed populations. This approach requires full-information. Second, we employ the same revision protocol and a version of the mean dynamics for non-well-mixed populations that uses only local information. Furthermore, invariance properties of the set of allowed population states are analyzed, and stability of the ϵ-equilibria is formally proven. Finally, two engineering examples based on the ϵ-dynamics are presented: A control scenario in which noisy measurements should be mitigated, and a humanitarian engineering application related to wealth distribution in poor societies.