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

We present and analyze kinetic models for the size evolution of a huge number of populations subject to interactions which describe both birth and death in a single population, and migration between them. If the mean size of the population is preserved, we prove that the solution to the underlying kinetic equation converges to equilibrium as time goes to infinity, and in various relevant cases we recover its main properties. In addition, by considering a suitable asymptotic procedure (the limit of quasi-invariant interactions) a simpler kinetic description of the model is derived. 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 a process which is closely related to the Lea-Coulson model of mutation processes in bacteria, a variation of the original model proposed by Luria and Delbrück, and a model recently proposed to describe evolution of the cross-genomic family abundance (i.e. the number of genes of a given family found in different genomes).