A Two-Type Bellman–Harris Process Initiated by a Large Number of Particles
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We investigate a two-type critical Bellman–Harris branching process with the following properties: the tail of the life-length distribution of the first type particles is of order o(t −2); the tail of the life-length distribution of the second type particles is regularly varying at infinity with index −β, β∈(0,1]; at time t=0 the process starts with a large number N of the second type particles and no particles of the first type. It is shown that the time axis 0≤t<∞ splits into several regions whose ranges depend on β and the ratio N/t within each of which the process at time t exhibits asymptotics (as N,t→∞) which is different from those in the other regions.
KeywordsEvolutionary stages Two-type critical Bellman–Harris process Regular variation Renewal theory
The research was supported in part by the Programs of RAS “Dynamical Systems and Control Theory” (V. Vatutin) and “Development of the methods for investigating some stochastic models aimed towards population and biomedical applications” (V. Topchii). The authors thank two anonymous referees for making a number of valuable remarks that helped improving the presentation of this work.
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