Effective population sizes for asymmetrically regulated birth-death processes

In multispecies birth-death processes, how population regulation -- through suppressed replication, elevated mortality, or both -- affects macroscopic stochastic dynamics has escaped detailed analysis.

Here, we show that the distribution of regulation mechanisms can be invisible in deterministic or mean-field dynamics but play a significant role in the diffusive evolution of population frequencies.

By introducing a tunable regulation partitioning parameter $\alpha_i$ and projecting a $d$-species birth-death process onto a $(d{-}1)$-dimensional Moran process, we find a regulation-mechanism-dependent diffusion tensor.

For the simple two-species case, we derive exact fixation times and probabilities to show how different regulation mechanisms stochastically favors a more birth-regulated species, even under complete deterministic neutrality.

Our model also allows us to define an $\alpha$-dependent effective population size $N_{\rm e}(\alpha)$ among neutral species, generalizing its classical interpretation.

For near-neutral populations or populations that are heterogeneous in their regulation mechanism, we used perturbation theory to calculate the spectral gap, identifying it with a diversity loss timescale which can also be interpreted as setting an effective population size.

Our results are particularly applicable to interacting subpopulations of T cells ("clones") which are near-neutral, are regulated through proliferation and apoptosis, and lose diversity with time.