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Plasmids are autonomously replicating genetic elements in bacteria. At celldivision plasmids are distributed among the two daughter cells. This genetransfer from one generation to the next is called vertical gene transfer. Westudy the dynamics of a bacterial population carrying plasmids and are inparticular interested in the long-time distribution of plasmids. Starting witha model for a bacterial population structured by the discrete number ofplasmids, we proceed to the continuum limit in order to derive a continuousmodel. The model incorporates plasmid reproduction, division and death ofbacteria, and distribution of plasmids at cell division. It is a hyperbolicintegro-differential equation and a so-called growth-fragmentation-death model.As we are interested in the long-time distribution of plasmids we study theassociated eigenproblem and show existence of eigensolutions. The stability ofthis solution is studied by analyzing the spectrum of the integro-differentialoperator given by the eigenproblem. By relating the spectrum with the spectrumof an integral operator we find a simple real dominating eigenvalue with anon-negative corresponding eigenfunction. Moreover, we describe an iterativemethod for the numerical construction of the eigenfunction.

Eigensolutions and spectral analysis of a model for vertical gene transfer of plasmids