RECURRENCE OF ANCESTRAL LINES AND OFFSPRING TREES IN TIME STATIONARY BRANCHING POPULATIONS

For time stationary Galton-Watson-branching populations on a general t ype space, the structure of the ''individually positive recurrent part '' of the system is described: its building blocks consist of finitely many ''clans'' with positive recurrent trunks. Conditions are given w hen this subsystem is void, and when it equals the full system. In add ition, positive recurrence on the dan level is characterized. Whereas individual positive recurrence turns out to be a symmetric concept wit h respect to forward and backward time direction (i.e., with respect t o ancestral Lines and offspring trees), with individual null recurrenc e this symmetry can fail even in the absence of branching, i.e., for i ndependently migrating particle systems (Example 13.1). For discrete t ype spaces a classification of types as to the various individual recu rrence concepts (positive, null, forward and backward in time) is prop osed and illustrated by a couple of results and examples. For finite t ype spaces conditions on the branching dynamics and its mean matrix fo r the existence of nontrivial equilibria are given.