Numerical Investigation of Some Reductions for the Gatenby-Gawlinski Model

Two (consecutive) reductions of the complete Gatenby-Gawlinski model for cancer invasion are proposed in order to investigate the mathematical framework, mainly from a computational perspective. After a brief overview of the full model, we proceed by examining the case of a two-equations-based and one-equation-based reduction, both obtained by means of a quasi-steady-state assumption. We focus on invasion fronts, exploiting a numerical strategy based on a finite volume approximation, and perform corresponding computational simulations to study the sharpness/smoothness of the traveling waves. Then, we employ a space-averaged wave speed estimate-referred to as the LeVeque-Yee formula-to quantitatively approach the propagation phenomenon. Concerning the one-equation-based model, we propose a scalar degenerate reaction-diffusion equation, which proves to be effective in order to qualitatively recover the typical trends arising from the Gatenby-Gawlinski model. Finally, we carry out some numerical tests in a specific case where the analytical solution is available.