Immigration-extinction dynamics of stochastic populations

How high should be the rate of immigration into a stochastic population in order to significantly reduce the probability of observing the population become 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 answer 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.