Markov branching processes regulated by emigration and large immigration
Here is an informal description of the continuous-time Markov chain on the nonnegative integers discussed in this paper. From any state \(i > 0\) reproduction occurs according to a Markov branching process; in addition catastrophes, or mass emigrations, occur, causing the population to become extinct, at a rate which depends on \(i\). However there is immediate immigration into state 0, in the sense that transitions from there to state \(i\) occur at the rate \(\alpha_i\) where \(\sum \alpha_i = \infty\). Existence and uniqueness are discussed. Further properties, in particular positive recurrence, are derived for the special case where the emigration rate is proportional to \(i\). The methods are analytical. The interest of the work is primarily as study in general Markov chain theory. The fact that 0 is an instantaneous state means that sample path behaviour is rather wild for the model to serve as one for a population.
scientific article; zbMATH DE number 772937
scientific article