Immigration-induced phase transition in a regulated multispecies birth-death process

Power-law-distributed species counts or clone counts arise in many biological settings such as multispecies cell populations, population genetics, and ecology. This empirical observation that the number of species $c_{k}$ represented by $k$ individuals scales as negative powers of $k$ is also supported by a series of theoretical birth-death-immigration (BDI) models that consistently predict many low-population species, a few intermediate-population species, and very high-population species. However, we show how a simple global population-dependent regulation in a neutral BDI model destroys the power law distributions. Simulation of the regulated BDI model shows a high probability of observing a high-population species that dominates the total population. Further analysis reveals that the origin of this breakdown is associated with the failure of a mean-field approximation for the expected species abundance distribution. We find an accurate estimate for the expected distribution $\langle c_k \rangle$ by mapping the problem to a lower-dimensional Moran process, allowing us to also straightforwardly calculate the covariances $\langle c_k c_\ell \rangle$. Finally, we exploit the concepts associated with energy landscapes to explain the failure of the mean-field assumption by identifying a phase transition in the quasi-steady-state species counts triggered by a decreasing immigration rate.