Mean field dynamics of collisional processes with duplication, loss and copy

In this paper we introduce and discuss kinetic equations for the evolution of the probability distribution of the number of particles in a population subject to binary interactions. The microscopic binary law of interaction is assumed to be dependent on fixed-in-time random parameters which describe both birth and death of particles, and the migration rule. These assumptions lead to a Boltzmann-type equation that in the case in which the mean number of the population is preserved, can be fully studied, by obtaining in some case the analytic description of the steady profile. In all cases, however, a simpler kinetic description can be derived, by considering the limit of quasi-invariant interactions. This procedure allows to describe the evolution process in terms of a linear kinetic transport-type equation. Among the various processes that can be described in this way, one recognizes the Lea-Coulson model of mutation processes in bacteria, a variation of the original model proposed by Luria and Delbrück.

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