Markov Branching Processes with Immigration and Resurrection

Anyue Chen, Junping Li

2006, v.12, №1, 139-168

ABSTRACT

We consider a modified Markov branching process incorporating both state-independent immigration and resurrection. The effect of state-independent immigration is firstly investigated in detail. Explicit expressions for the extinction probabilities and mean extinction times are presented. It is revealed that if the death rate is great than the mean birth rate for the underlying branching structure, then a considerably large immigration is necessary to rescue a species from extinction while if the death rate is equal to the mean birth rate, then only a mild immigration will do. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are also established.

Keywords: Markov branching process,immigration,resurrection,regularity,extinction,recurrence,ergodicity,exponential ergodicity

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