Survival for a Galton-Watson tree with cousin mergers

We introduce a generalization of Galton-Watson trees where, individuals have independently a number of Poi(1 + p) offspring and, at each generation, pairs of cousins merge independently with probability q. If q = 0 we recover a usual Galton-Watson tree and the survival threshold for the process has p > 0. Our main thoerem gives sufficient conditions on p and q for extinction and survival of the Galton-Watson trees with cousin mergers. In the setting q > 0, the Markovian property of regular Galton-Watson trees is lost and so the analysis of the model becomes more involved. In particular, the main obstacle are the intergeneration dependencies since the genealogy of the individuals, having possibly more than one parent, is no longer represented by a tree but by a graph. (C) 2021 The Authors. Published by Elsevier B.V.