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 < c(k)> by mapping the problem to a lower-dimensional Moran process, allowing us to also straightforwardly calculate the covariances < c(k)c(l)>. 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 quasisteady-state species counts triggered by a decreasing immigration rate.