Persistence and neutrality in interacting replicator dynamics

TL;DR

This paper analyzes the long-term behavior of interacting replicator systems in ecology, showing conditions for persistence and neutrality emergence without assuming fitness equivalence, supported by theoretical and numerical results.

Contribution

It establishes the propagation-of-chaos property and conditions for persistence and neutrality in mean-field replicator dynamics, revealing neutrality as an emergent property.

Findings

Persistence conditions for replicator systems.

Neutrality emerges without fitness equivalence.

Existence of invariant distributions and unique Dirichlet measure.

Abstract

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.