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We consider a population of hosts infected by a pathogen that exists in two strains. We use a two-parameter family of Markov kernels on $[0,1]$ to describe the discrete-time evolution of the pathogen type-composition within and across individuals. First, we assume that there is no interaction between pathogen populations across host individuals. For a particular class of parameters, we establish moment duality between the type-frequency process and a process reminiscent of the \emph{Ancestral Selection Graph}. We also show convergence, under appropriate scaling of parameters and time, to a Wright-Fisher diffusion with drift. Next, we assume that pathogen-type compositions are correlated across hosts by their empirical measure. We show a propagation of chaos result comparing the pathogen type-composition of a given host with the evolution of a non-linear chain. Furthermore, we show that under appropriate scaling, the non-linear chain converges to a McKean-Vlasov diffusion. To illustrate our results, we consider a population affected by mutation rates that depend on the instantaneous distribution across multiple hosts. For this example, we study the uniform-in-time propagation of chaos and the ergodicity of the limiting McKean-Vlasov SDE., Comment: 43 pages, 2 figures

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.