Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure

Shay Be'er; Michael Assaf; Baruch Meerson

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arXiv · 1503.00350

Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure

Shay Be'er·Michael Assaf·Baruch Meerson

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Abstract

We study the dynamics of colonization of a territory by a stochastic population at low immigration pressure. We assume a sufficiently strong Allee effect that introduces, in deterministic theory, a large critical population size for colonization. At low immigration rates, the average pre-colonization population size is small thus invalidating the WKB approximation to the master equation. We circumvent this difficulty by deriving an exact zero-flux solution of the master equation and matching it with an approximate non-zero-flux solution of the pertinent Fokker-Planck equation in a small region around the critical population size. This procedure provides an accurate evaluation of the quasi-stationary probability distribution of population sizes in the pre-colonization state, and of the mean time to colonization, for a wide range of immigration rates. At sufficiently high immigration rates our results agree with WKB results obtained previously. At low immigration rates the results can be very different.

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Shay Be'er, Michael Assaf, Baruch Meerson. 2015-03-01. Colonization of a territory by a stochastic population under a strong Allee effect and a low immigration pressure. https://arxiv.org/abs/1503.00350

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