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

TL;DR

This paper develops kinetic equations to model population dynamics with birth, death, and migration, deriving simplified descriptions via quasi-invariant interactions, and applies the framework to bacterial mutation processes.

Contribution

It introduces a kinetic framework for collisional processes with duplication, loss, and copying, including a linear transport approximation and application to bacterial mutation models.

Findings

Derived Boltzmann-type equations for population evolution.

Obtained analytic steady-state profiles in certain cases.

Connected the model to the Lea-Coulson bacterial mutation model.

Abstract

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 Delbr\"uck.