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We study the large-time behavior of an ensemble of particles obeyingreplicator-like stochastic dynamics with mean-field interactions. We prove thepropagation of chaos property and establish conditions for the strongpersistence of the $N$-replicator system and the existence of invariantdistributions for the associated McKean-Vlasov dynamics. We illustrate ourfindings with some simple case studies, provide numerical results, and discussour conclusions in the light of both, Hubbell's Neutral Theory and thephenomenon of the emergence of ecologies.

Persistence and neutrality in interacting replicator dynamics, and invariant measures for the McKean-Vlasov replicator