A higher-order equivalence of Lotka-Volterra and replicator dynamics reveals tight connections between ecology and evolution

The Lotka-Volterra equations are foundational in ecology, modelling logistic growth in isolated populations and complex dynamics in interacting species. Hofbauer and Sigmund established that these equations are equivalent to the replicator dynamics of evolutionary games, so dynamic patterns in ecology are mirrored in evolutionary games and vice versa. Both fields have since turned to non-linearities and higher-order interactions, where it is unclear whether this equivalence still holds. Here, we demonstrate the general equivalence and illustrate it in classical non-linear models from theoretical ecology. Non-linearities in either field leave this foundational connection intact. Yet directly modelling ecological dynamics with evolutionary games, or the reverse, risks misinterpretation, since species in Lotka-Volterra dynamics cannot be equated with strategies in evolutionary games. Our study reveals the tight interplay between ecology and evolutionary game theory, and the robustness of their mathematical connection even in complex scenarios, alongside its caveats.