Higher-order equivalence of Lotka-Volterra and replicator dynamics

The Lotka-Volterra equations are foundational in ecology, modeling logistic growth in isolated populations and revealing complex dynamics in interacting species. These equations are mathematically equivalent to the replicator dynamics of evolutionary games. This mathematical equivalence, established by Hofbauer and Sigmund, allows dynamic patterns in ecology to be mirrored in evolutionary games and vice versa. More recently, both fields have focussed more on non-linearities and higher-order interactions. So far, it is unclear if the mathematical equivalence can still be exploited. Here, we demonstrate that such an equivalence holds and illustrate this in classical non-linear models from theoretical ecology. This suggests that non-linearities in eco-logical models or evolutionary games do not undermine the foundational connection between these two fields. However, directly using evolutionary games to model ecological dynamics, or vice versa, carries the risk of misinterpretations, as ecological species in Lotka-Volterra dynamics cannot be directly interpreted as strategies in evolutionary games. Our study enhances the understanding of the interplay between ecology and evolutionary game theory, highlighting the robustness of their mathematical connections even in complex scenarios, but also the associated caveats.