Global stability of ecological and evolutionary dynamics via equivalence

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, an evolutionarily 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 strategies that are not evolutionarily 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 evolutionarily stable strategy maps into an equivalent state that is evolutionarily 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.