Plasmid invasion in biofilms is often surprisingly limited in spite of the close contact of cells in a biofilm. We hypothesized that this poor plasmid spread into deeper biofilm layers is caused by a dependence of conjugation on the growth rate (relative to the maximum growth rate) of the donor. By extending an individual-based model of microbial growth and interactions to include the dynamics of plasmid carriage and transfer by individual cells, we were able to conduct in silico tests of this and other hypotheses on the dynamics of conjugal plasmid transfer in biofilms. For a generic model plasmid, we find that invasion of a resident biofilm is indeed limited when plasmid transfer depends on growth, but not so in the absence of growth dependence. Using sensitivity analysis we also find that parameters related to timing (i.e. a lag before the transconjugant can transfer, transfer proficiency and scan speed) and spatial reach (EPS yield, conjugal pilus length) are more important for successful plasmid invasion than the recipients' growth rate or the probability of segregational loss. While this study identifies one factor that can limit plasmid invasion in biofilms, the new individual-based framework introduced in this work is a powerful tool that enables one to test additional hypotheses on the spread and role of plasmids in microbial biofilms.