The study of tumor growth dynamics is an important topic in mathematical biology, and a focal point of research in this area is the development of differential equation models of tumor populations. We consider a partial differential equation (PDE) model of growth dynamics in a tumor population comprised of two cell types: cancer stem cells (CSCs) and nonstem tumor cells (TCs). Interactions between these cell types influence tumor behavior and responses to treatment, and hence an understanding of CSC/TC interactions can help explain various phenomena in tumor biology. One such problem is the tumor growth paradox, which is the unexpected effect wherein the rate of tumor growth increases with the rate of cell death. 

In this presentation, we will examine an analytic approach to understanding the behavior of the CSC/TC model. We will show how the reduction of the PDE model to an ordinary differential equation (ODE) model permits the application of standard techniques used to study systems of differential equations, such as Jacobian linearization and eigenvalue stability analysis. Additionally, we will introduce geometric singular perturbation theory and use its methods to resolve the tumor growth paradox in our model. More generally, we will describe how geometric singular perturbation theory is
a powerful tool for studying dynamical systems that arise naturally in the sciences. The clinical implications of the CSC/TC model’s predictions will also be discussed—the human immunological response to oncogenesis will be presented as an instructive example.