  In this manuscript, we prove the existence of slow and fast traveling wave
solutions in the original Gatenby--Gawlinski model. We prove the existence of a
slow traveling wave solution with an interstitial gap. This interstitial gap
has previously been observed experimentally, and here we derive its origin from
a mathematical perspective. We give a geometric interpretation of the formal
asymptotic analysis of the interstitial gap and show that it is determined by
the distance between a layer transition of the tumor and a dynamical
transcritical bifurcation of two components of the critical manifold. This
distance depends, in a nonlinear fashion, on the destructive influence of the
acid and the rate at which the acid is being pumped.
