Numerical investigation of some reductions for the Gatenby-Gawlinski model

TL;DR

This paper numerically investigates simplified versions of the Gatenby-Gawlinski cancer invasion model, analyzing traveling wave behavior and validating reductions against the full model through simulations.

Contribution

It introduces and tests reduced models of the Gatenby-Gawlinski framework, demonstrating their effectiveness in capturing key invasion dynamics.

Findings

Reduced models qualitatively replicate full model trends

Finite volume method effectively simulates traveling fronts

Reduction within degenerate reaction-diffusion equations is successful

Abstract

The Gatenby-Gawlinski model for cancer invasion is object of analysis in order to investigate the mathematical framework behind the model working by means of suitable reductions. We perform numerical simulations to study the sharpness/smoothness of the traveling fronts starting from a brief overview about the full model and proceed by examining the case of a two-equations-based and one-equation-based reduction. We exploit a numerical strategy depending on a finite volume approximation and employ a space-averaged wave speed estimate to quantitatively approach the traveling waves phenomenon. Concerning the one equation-based model, we propose a reduction framed within the degenerate reaction-diffusion equations field, 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.