The replicator and the Generalized Lotka-Volterra equations are closely-related, foundational models in evolutionary game theory and community ecology, respectively. The concept of evolutionary stability and its relationship with dynamic stability has received significant attention: in the replicator equation, a mixed evolutionary stable strategy is also dynamically globally stable—i.e., will be reached by any trajectory originating from positive conditions. Intriguingly, the converse is not true: there are replicator equations yielding dynamically stable mixed strategies that are not evolutionary stable. Here we consider two classes of equivalence (i.e., transformations that do not alter the qualitative dynamics) for the replicator equation, to determine whether a globally-stable, but not evolutionary stable strategy maps into an equivalent state that is evolutionary stable—and show that this is the case for the examples that have been put forward so far. We derive the same two classes of equivalence for the Generalized Lotka-Volterra model, obtaining the same conditions for stability as for the replicator equation, and show that in this way we can characterize stability when other methods fail. By unifying the approach to proving stability for the replicator equation and Lotka-Volterra models, we bring these foundational equations even closer together.