  In this paper we consider the Shigesada-Kawasaki-Teramoto (SKT) model to
account for stable inhomogeneous steady states exhibiting spatial segregation,
which describe a situation of coexistence of two competing species. We provide
a deeper understanding on the conditions required on both the cross-diffusion
and the reaction coefficients for non-homogeneous steady states to exist, by
combining a detailed linearized analysis with advanced numerical bifurcation
methods via the continuation software pde2path. We report some numerical
experiments suggesting that, when cross-diffusion is taken into account, there
exist positive and stable non-homogeneous steady states outside of the range of
parameters for which the coexistence homogeneous steady state is positive.
Furthermore, we also analyze the case in which self-diffusion terms are
considered.
