  Competition is the main driver of population dynamics, which shapes the
genetic composition of populations and the assembly of ecological communities.
Neutral models assume that all the individuals are equivalent and that the
dynamics is governed by demographic (shot) noise, with a steady state species
abundance distribution (SAD) that reflects a mutation-extinction equilibrium.
Recently, many empirical and theoretical studies emphasized the importance of
environmental variations that affect coherently the relative fitness of entire
populations. Here we consider two generic time-averaged neutral models, in both
the relative fitness of each species fluctuates independently in time but its
mean is zero. The first (model A) describes a system with local competition and
linear fitness-dependence of the birth-death rates, while in the second (model
B) the competition is global and the fitness dependence is nonlinear. Due to
this nonlinearity, model B admits a noise-induced stabilization mechanism that
facilitates the invasion of new mutants. A self-consistent mean-field approach
is used to reduce the multi-species problem to two-species dynamics, and the
large-$N$ asymptotics of the emerging set of Fokker-Planck equations is
presented and solved. Our analytic expressions are shown to fit the SADs
obtained from extensive Monte-Carlo simulations and from numerical solutions of
the corresponding master equations.
