Title: Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids,
Abstract: Plasmids are autonomously replicating genetic elements in bacteria. At cell
division plasmids are distributed among the two daughter cells. This gene
transfer from one generation to the next is called vertical gene transfer. We
study the dynamics of a bacterial population carrying plasmids and are in
particular interested in the long-time distribution of plasmids. Starting with
a model for a bacterial population structured by the discrete number of
plasmids, we proceed to the continuum limit in order to derive a continuous
model. The model incorporates plasmid reproduction, division and death of
bacteria, and distribution of plasmids at cell division. It is a hyperbolic
integro-differential equation and a so-called growth-fragmentation-death model.
As we are interested in the long-time distribution of plasmids we study the
associated eigenproblem and show existence of eigensolutions. The stability of
this solution is studied by analyzing the spectrum of the integro-differential
operator given by the eigenproblem. By relating the spectrum with the spectrum
of an integral operator we find a simple real dominating eigenvalue with a
non-negative corresponding eigenfunction. Moreover, we describe an iterative
method for the numerical construction of the eigenfunction.