Fungi normally do not senesce, but in some species mitochondrial plasmids are known to occur that induce senescence. In this paper models for the dynamics of a senescence plasmid in a fungal population are developed and analysed. In the first model it is assumed that total fungal biomass density is constant, while in the second model the resource dynamics and its effect on fungal growth is modelled explicitly. An additional death rate describes the effect of the plasmid on the senescent subpopulation. Plasmids can be transferred to non-senescent fungus. Criteria for the coexistence of the non-senescent and senescent fungal strains are derived, all of which have a clear biological interpretation. It is shown that coexistence is not possible in the first model, but is possible in the second model for a large range of parameter values. We show that the interplay between resource dynamics, fungal growth and plasmid transmission is crucial for coexistence. We develop a biological interpretation of how these mechanisms have to interact to promote coexistence. A numerical study of the second model further clarifies the relations between the numerical value of several parameters and coexistence of non-senescent and senescent fungal strains.