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∂tU=Uf(U)−dV,∂tV=∂xf(U)∂xV+rVf(V),where f(u)=1−u and the parameters d,r are positive. Denoting by (U,V) the traveling wave profile and by (U±,V±) its asymptotic states at ±∞, we investigate existence in the regimes d>1:(U−,V−)=(0,1)and(U+,V+)=(1,0),d<1:(U−,V−)=(1−d,1)and(U+,V+)=(1,0),which are called, respectively, homogeneous invasion and heterogeneous invasion. 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→0.