A general multitype branching process with age, memory and population dependence
2010
Jacob , Christine (INRA , Jouy-En-Josas (France). UR 0341 Unité de recherche Mathématiques et Informatique Appliquées)
We present a general class of multitype branching processes in discrete time with age, memory and population dependent individual transitions. Except in simple particular cases, the asymptotic behavior of this general process, as the time tends to infinity, is an open problem. So we instead study the behavior of limit models, as the initial population size tends to infinity, assuming that, at the initial time, either the types of interest are nonrare, or are rare. In the first case the limit model is a deterministic system on probabilities, and in the second case, the limit model is a multitype Bienaym´-Galton-Watson process on the rare types, with a Poissonian transition. Theelimit model in the first case allows to approximate the asymptotic behavior of the normalized process, and the one in the second case allows to calculate quantities such as the extinction time distribution and, in the subcritical case, the tree size distribution.
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