Development of continuous biopharmaceutical manufacturing processes is an area of active research. This study considers the long-term transgene copy number stability of Pichia pastoris in continuous bioreactors. We propose a model of copy number loss that quantifies population heterogeneity. An analytical solution is derived and compared with existing experimental data. The model is then used to provide guidance for stable operating timescales. The model is extended to consider copy number dependent growth such as in the case of Zeocin supplementation. The model is also extended to analyze a continuous seeding strategy. This study is a critical step towards understanding the impact of continuous processing on the stability of Pichia pastoris and the resultant products.