Persistence and Neutrality in Interacting Replicator Dynamics

We study the large-time behavior of an ensemble of entities obeying replicator-like stochastic dynamics with mean-field interactions as a model for a primordial ecology. We prove the propagation-of-chaos property and establish conditions for the strong persistence of the N-replicator system and the existence of invariant distributions for a class of associated McKean–Vlasov dynamics. In particular, our results show that, unlike typical models of neutral ecology, fitness equivalence does not need to be assumed but emerges as a condition for the persistence of the system. Further, neutrality is associated with a unique Dirichlet invariant probability measure. We illustrate our findings with some simple case studies, provide numerical results, and discuss our conclusions in the light of Neutral Theory in ecology. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.