Noise driven unlimited population growth

TL;DR

This paper demonstrates that demographic noise can lead to unlimited population growth in stochastic models, characterized by a metastable distribution with power-law tails and multiple WKB modes.

Contribution

It develops a systematic WKB and van Kampen theory to analyze the metastable population distribution and decay time in stochastic birth-death models with immigration.

Findings

Population distribution exhibits power-law tails.

Decay of metastable state is exponentially slow.

Presence of two distinct WKB modes in the solution.

Abstract

Demographic noise causes unlimited population growth in a broad class of models which, without noise, would predict a stable finite population. We study this effect on the example of a stochastic birth-death model which includes immigration, binary reproduction and death. The unlimited population growth proceeds as an exponentially slow decay of a metastable probability distribution (MPD) of the population. We develop a systematic WKB theory, complemented by the van Kampen system size expansion, for the MPD and for the decay time. Important signatures of the MPD is a power-law tail (such that all the distribution moments, except the zeroth one, diverge) and the presence in the solution of two different WKB modes.