Limit models for a new general class of multitype branching processes with memory and population dependence

We present here a new general class of multitype branching processes in discrete time with memory and population dependent individual transitions. Except in simple particular cases (essentially BGW (Bienaym´e-Galton-Watson) or asymptotically BGW processes), the asymptotic behaviour of this kind of processes, as time tends to infinity, is an open problem. So in order to be able to study this behaviour, we derive the limit models, as the initial size of the population tends to infinity, of either the process suitably normalized, when the types of interest are nonrare at the initial time, or of the process itself, when the types of interest are rare. In the first case, the normalized process has the same asymptotic behaviour as that of a deterministic dynamical system on densities or on probabilities. In the second case, the limit process, as the initial population size tends to #, is reduced to a multitype BGW process on the rare types with Poissonian transitions. We derive its asymptotic behaviour, and in the subcritical case, the distribution of its extinction time and of the size of its tree until extinction. We give some examples, and an error bound between the process and its limit