  We present a two-species model with applications in tumour modelling. The
main novelty is the coupling of both species through the so-called Brinkman law
which is typically used in the context of visco-elastic media, where the
velocity field is linked to the total population pressure via an elliptic
equation. The same model for only one species has been studied by Perthame and
Vauchelet in the past. The first part of this paper is dedicated to
establishing existence of solutions to the problem, while the second part deals
with the incompressible limit as the stiffness of the pressure law tends to
infinity. Here we present a novel approach in one spatial dimension that
differs from the kinetic reformulation used in the aforementioned study and,
instead, relies on uniform BV-estimates.
