Stability of patterns and of constant steady states for a cross-diffusion system

This paper considers mathematical and numerical models of normal and abnormal tissue growth. To this aim, we analyse a nonlinear system of partial differential equations, called as cross-diffusion system, in a rather general form. We investigate the dynamics of the system in dependence on the system parameters. We show analytically the existence of a periodic stationary solution for the set of certain positive initial data. The other choices of the parameters are investigated by construction of corresponding numerical models. We also analyse the stability of the stationary solutions. Numerical examples demonstrate our results.