Propagation fronts in a simplified model of tumor growth with degenerate cross-dependent self-diffusivity

Abstract

Motivated by tumor growth in Cancer Biology, we provide a complete analysis of existence and non-existence of invasive fronts for the reduced Gatenby--Gawlinski model \[ \partial_t U = U\{f(U)-dV\}, \qquad \partial_t V = \partial_x \{f(U)\,\partial_x V\} + r V f(V), \] where $f(u)=1-u$ and the parameters $d,r$ are positive. Denoting by $(\mathcal{U},\mathcal{V})$ the traveling wave profile and by $({\mathcal{U}}_{\pm },{\mathcal{V}}_{\pm })$ its asymptotic states at $\pm \infty$, we investigate existence in the regimes i) $d>1$ (homogeneous invasion) : $({\mathcal{U}}_{-},{\mathcal{V}}_{-})=(0,1)$, $({\mathcal{U}}_{+},{\mathcal{V}}_{+})=(1,0)$; ii) $d<1$ (heterogeneous invasion) : $({\mathcal{U}}_{-},{\mathcal{V}}_{-})=(1-d,1)$, $({\mathcal{U}}_{+},{\mathcal{V}}_{+})=(1,0)$. In both cases, we prove that a propagating front exists whenever the speed parameter $c$ is strictly positive. We also derive an accurate approximation of the front profile in the singular limit $c\to 0$.

arXiv comment33 pages, 4 figures