An age structured cell cycle model with crowding

We study a two compartment, nonlinear, age structured model for the cell cycle. The phases of the cell cycle G1, S, G2 and M are grouped into two phases, which we call Phase 1 and Phase 2, where Phase 1 consists of the phase G1 and Phase 2 consists of the phases S, G2 and M. It is assumed that Phase 1 has a variable duration while the duration of Phase 2 is fixed. The model consists of a system of nonlinear PDEs describing the number densities ni(t,τ), i=1,2, of cells in Phase i of age τ (counted from when the cell entered the phase) and time t, together with initial and boundary conditions for ni. We first prove that this initial and boundary value problem is equivalent to solving a system of integral equations. We then prove existence and uniqueness of this system of integral equations, and hence also of the original PDE system. Qualitative behaviour of solutions with small initial and input data is studied, and an application to quorum sensing is discussed. Finally, some simple numerical examples are computed using the derived integral equations.