Immigration-extinction dynamics of stochastic populations

TL;DR

This paper analytically investigates how immigration influences the extinction probability and population distribution in stochastic models, providing insights into patch occupancy dynamics and conditions for simplified modeling.

Contribution

It offers an exact solution for a stochastic population model with immigration, clarifying the relationship between population distributions and occupancy models.

Findings

Higher immigration rates reduce extinction probability.

Population distributions with immigration differ from those without.

Conditions for the validity of patch occupancy models are identified.

Abstract

How high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population extinct? Is there any relation between the population size distributions with and without immigration? Under what conditions can one justify the simple patch occupancy models which ignore the population distribution and its dynamics in a patch, and treat a patch simply as either occupied or empty? We address these questions by exactly solving a simple stochastic model obtained by adding a steady immigration to a variant of the Verhulst model: a prototypical model of an isolated stochastic population.