  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 $c \to 0$.
