Existence of solutions and numerical approximation of a non-local tumor growth model

In order to model the evolution of a heterogeneous population of cancer stem cells and tumor cells, we analyse a nonlinear system of integro-differential equations. We provide an existence and uniqueness result by exploiting a suitable iterative scheme of functions which converge to the solution of the system. Then, we discretize the model and perform some numerical simulations. Numerical approximations are obtained by applying finite differences for space discretization and an exponential Runge-Kutta scheme for time integration. We exploit the numerical tool in order to investigate the effects that niches have on cancer development. In this respect, the numerical procedure is applied in the case when the function of cell redistribution is assumed to be spatially explicit. It allows for finding an approximate solution which is spatially inhomogeneous as time progresses. In this framework, numerical investigation may help in understanding the process of niche construction, which plays an important role in cancer population biology.