Mathematical analysis of a model for proliferative-to-invasive transition of hypoxic glioma cells

Abstract Our aim in this paper is to study a mathematical model for the proliferative-to-invasive transition of hypoxic glioma cells. We prove the existence and uniqueness of nonnegative solutions and then address the important question of whether the positive solutions undergo extinction or permanence. More precisely, we prove that this depends on the boundary conditions: there is no extinction when considering Neumann boundary conditions, while we prove extinction when considering Dirichlet boundary conditions.